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abstract: 'The problem of identifying intersections between two sets of $d$-dimensional axis-parallel rectangles appears frequently in the context of agent-based simulation studies. For this reason, the High Level Architecture (HLA) specification – a standard framework for interoperability among simulators – includes a Data Distribution Management (DDM) service whose responsibility is to report all intersections between a set of subscription and update regions. The algorithms at the core of the DDM service are CPU-intensive, and could greatly benefit from the large computing power of modern multi-core processors. In this paper we propose two parallel solutions to the DDM problem that can operate effectively on shared-memory multiprocessors. The first solution is based on a data structure (the Interval Tree) that allows concurrent computation of intersections between subscription and update regions. The second solution is based on a novel parallel extension of the Sort Based Matching algorithm, whose sequential version is considered among the most efficient solutions to the DDM problem. Extensive experimental evaluation of the proposed algorithms confirm their effectiveness on taking advantage of multiple execution units in a shared-memory architecture.'
author:
- Moreno Marzolla
- 'Gabriele D’Angelo'
bibliography:
- 'parallel-ddm.bib'
title: 'Parallel Data Distribution Management on Shared-Memory Multiprocessors'
---
=1
<ccs2012> <concept> <concept\_id>10010147.10010341.10010349.10010362</concept\_id> <concept\_desc>Computing methodologies Massively parallel and high-performance simulations</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010169.10010170.10010171</concept\_id> <concept\_desc>Computing methodologies Shared memory algorithms</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
[^1]
Introduction {#sec:introduction}
============
Large agent-based simulations are used in many different areas such human mobility modeling [@Klugl:2007], transportation and logistics [@Cetin:2002], or complex biological systems [@Abraham:2015; @Jacob:2004]. While there exist recommendations and best practices for designing credible simulation studies [@Law:2009], taming the complexity of large models remains challenging, due to the potentially huge number of virtual entities that need to be orchestrated and the correspondingly large amount of computational resources required to execute the model.
The has been introduced to partially address the problems above. The is a general architecture for the interoperability of simulators [@HLA] that allow users to build large models through composition of specialized simulators, called *federates* according to the terminology. The federates interact using a standard interface provided by a component called [@HLA-interf]. The structure and semantics of the data exchanged among the federates are formalized in the specification [@HLA-omt].
Federates can notify events to other federates, for example to signal a local status update that might impact other simulators. Since notifications might produce a significant overhead in terms of network traffic and processor usage at the receiving end, the provides a service whose purpose is to allow federates to specify which notifications they are interested in. This is achieved through a spatial public-subscribe system, where events are associated with an axis-parallel, rectangular region in $d$-dimensional space, and federates can signal the to only receive notifications that overlap one or more subscription regions of interest.
More specifically, allows the simulation model to define a set of *dimensions*, each dimension being a range of integer values from $0$ to a user-defined upper bound. Dimensions may be used as Cartesian coordinates for mapping the position of agents in 2-D or 3-D space, although the specification does not mandate this. A *range* is a half-open interval $[\textit{lower bound},
\textit{upper bound})$ of values on one dimension. A *region specification* is a set of ranges, and can be used by federates to denote the “area of influence” of status update notifications. Federates can signal the the regions from which update notifications are to be received (*subscription regions*). Each update notification is associated with a *update region*: the service identifies the set of overlapping subscription regions, so that the update message are sent only to federates owning the relevant subscription.
As an example, let us consider the simple road traffic simulation model shown in Figure \[fig:road-traffic-ex\] consisting of vehicles managing intersections. Vehicles need to react both to changes of the color of traffic lights and also to other vehicles in front of them. This can be achieved using suitably defined update and subscription regions. Specifically, each vehicle is enclosed within an update region (thick box) and a subscription region (dashed box), while each traffic light is enclosed within an update region only. A traffic light is a pure generator of update notifications, while vehicles are both producers and consumers of events. Each vehicle generates notifications to signal a change in its position, and consumes events generated by nearby vehicles and traffic lights. We assume that a vehicle can safely ignore what happens behind it; therefore, subscription regions are skewed towards the direction of motion.
If the scenario above is realized through an -compliant simulator, entities need to be assigned to federates. We suppose that there are four federates, $F_1$ to $F_4$. $F_1$, $F_2$ and $F_3$ handle cars, scooters and trucks respectively, while $F_4$ manages the traffic lights. Each simulated entity registers subscription and update regions with the ; the service can then match subscription and update regions, so that update notifications can be sent to interested entities. In our scenario, vehicles $2$, $3$ and $4$ receive notifications from the traffic light $8$; vehicles $5$ and $6$ send notifications to each other, since their subscription and update regions overlap. The communication pattern between federates is shown in the bottom part of Figure \[fig:road-traffic-ex\].
As can be seen, at the core of the service there is an algorithm that solves an instance of the general problem of reporting all pairs of intersecting axis-parallel rectangles in a $d$-dimensional space. In the context of , *reporting* means that each overlapping subscription-update pair must be reported exactly once, without any implied ordering. This problem is well known in computational geometry, and can be solved either using appropriate algorithms and/or ad-hoc spatial data structures as will be discussed in Section \[sec:region-matching\]. However, it turns out that implementations tend to rely on less efficient but simpler solutions. The reason is that spatial data structures can be quite complicated, and therefore their manipulation may require a significant overhead that might be not evident from their asymptotic complexity.
The increasingly large size of agent-based simulations is posing a challenge to the existing implementations of the service. As the number of regions increases, so does the execution time of the intersection-finding algorithms. A possible solution comes from the computer architectures domain. The current trend in microprocessor design is to put more execution units (cores) in the same processor; the result is that multi-core processors are now ubiquitous, so it makes sense to try to exploit the increased computational power to speed up the service [@gda-simpat]. Therefore, an obvious parallelization strategy for the intersection-finding problem is to distribute the rectangles across the processor cores, so that each core can work on a smaller problem.
Shared-memory multiprocessors are a family of parallel systems where multiple processors share one or more blocks of Random Access Memory (RAM) through an interconnection network (Figure \[fig:shared-memory\] (a)). A modern multi-core CPU contains multiple independent execution units (cores), that are essentially stand-alone processors. Cache hierarchies within each core, and shared by all cores of the same CPU, are used to mitigate the memory access bottleneck, known as the *memory wall* [@Wulf:1995].
General-Purpose GPU computing is another realization of the multi-core paradigm. were originally intended as specialized devices for producing graphical output, but have now evolved into general-purpose parallel co-processors. A high-level overview of a is shown in Figure \[fig:shared-memory\] (b). A includes a large number of cores that share a common memory, called *device memory*. The device memory is physically separate from the main system RAM, so that programs and data must be transferred from system RAM to device memory before the can start execution. At the end of the computation, results need to be transferred back to system RAM. While a single core is less powerful than a CPU core, a has more cores than a typical CPU, and therefore provides a higher aggregate throughput. However, this comes at a cost: programming is in general more complex than programming a multicore CPU; additionally, CPUs support more memory than , meaning that CPU-GPU communication might be a bottleneck when processing large datasets. Moreover, are based on the paradigm, where a single instruction stream is executed on multiple data items. This paradigm is well suited for applications with regular data access pattern (e.g., linear algebra). Applications with conditional branches or irregular data access patterns may require considerable more effort to be implemented efficiently, but are nevertheless possible: for example, non-trivial but efficient implementations of the Time Warp optimistic synchronization protocol [@Liu:2017] and of the Bellman-Ford shortest path algorithm [@Busato:2016] have been realized, despite the fact that both fall outside the these application exhibits regular data access patterns. Finally, it must be observed that general-purpose are currently not as ubiquitous as multicore CPUs, since they are add-on cards that must be purchased separately.
Shared-memory multiprocessors are interesting for several reasons. first, they are ubiquitous since they power virtually everything from smartphones and single-board computers up to high performance computing systems. Moreover, shared-memory architectures are in general simpler to program than distributed-memory architectures, since the latter require explicit message exchanges to share data between processors. Support for shared-memory programming has been added to traditional programming languages such as C, C++ and FORTRAN [@OpenMP], further reducing the effort needed to write parallel applications.
Unfortunately, writing *efficient* parallel programs is not easy. Many serial algorithms can not be made parallel by means of simple transformations. Instead, new parallel algorithms must be designed from scratch around the features of the underlying execution platform. The lack of abstraction of parallel programming is due to the fact that the user must leverage the strengths (and avoid the weaknesses) of the underlying execution platform in order to get the maximum performance. The result is that, while parallel architectures are ubiquitous – especially shared-memory ones – parallel programs are not, depriving users from a potential performance boost on some classes of applications.
In this paper we present two solutions to the problem that are suitable for shared-memory multiprocessors. The first solution, called , is based on the *interval tree* data structure, that represents subscription or update regions in such a way that intersections can be computed in parallel. The second solution, called *Parallel* , is a parallel version of [@Raczy2005], a state-of-the-art implementation of the service.
This paper is organized as follows. In Section \[sec:region-matching\], we review the scientific literature related to the service and describe in detail some of the existing algorithms that will be later used in the paper. In Section \[sec:interval-tree\] we describe the interval tree data structure and the parallel matching algorithm. In Section \[sec:parallel-sort-matching\], we present the main contribution of this work, i.e., a parallel version of the algorithm. In Section \[sec:experimental-evaluation\] we experimentally evaluate the performance of parallel on two multi-core processors. Finally, the conclusions will be discussed in Section \[sec:conclusions\].
Problem statement and existing solutions {#sec:region-matching}
========================================
In this paper we address the region matching problem defined as follows:
**Region Matching Problem.** Given two sets $\mathbf{S}
= \{S_1, \ldots, S_n\}$ and $\mathbf{U}=\{U_1, \ldots, U_m\}$ of $d$-dimensional, axis-parallel rectangles (also called $d$-rectangles), enumerate all pairs $(S_i, U_j) \subseteq \mathbf{S} \times
\mathbf{U}$ such that $S_i \cap U_j \neq \emptyset$; each pair must be reported exactly once, in no particular order.
Although the specification only allows integer coordinates, we address the more general case in which the coordinates of the edges of the $d$-rectangles are arbitrary real numbers.
Figure \[fig:ddm\_example\] shows an instance of the region matching problem in $d=2$ dimensions with three subscription regions $\mathbf{S} = \{S_1, S_2, S_3\}$ and two update regions $\mathbf{U} =
\{U_1, U_2\}$. In this example there are four overlapping subscription-update pairs $\{ (S_1, U_1),\ (S_2, U_2),\ (S_3,
U_1),\ (S_3, U_2) \}$.
The time complexity of the region matching problem is *output-sensitive*, since it depends on the size of the output in addition to the size of the input. Therefore, if there are $K$ overlapping regions, any region matching algorithm requires time $\Omega(K)$. Since there can be at most $n \times m$ overlaps, the worst-case complexity of the region matching problem is $\Omega(n
\times m)$. In practice, however, the number of intersections is much smaller than $n \times m$.
**return** $x.\emph{low} \leq y.\emph{high} {\ensuremath{\wedge}}\ y.\emph{low} \leq x.\emph{high}$
One of the key steps of any matching algorithm is testing whether two $d$-rectangles overlap. The case $d=1$ is very simple, as it reduces to testing whether two half-open intervals $x=[x.\textit{low},x.\textit{high})$, $y=[y.\textit{low},
y.\textit{high})$ intersect; this happens if and only if $$x.\emph{low} < y.\emph{high} {\ensuremath{\wedge}}\ y.\emph{low} < x.\emph{high}$$ (see Algorithm \[alg:intersect1d\]).
The case $d>1$ can be reduced to the case $d=1$ by observing that two $d$-rectangles overlap if and only if their projections along each dimension overlap. For example, looking again at Figure \[fig:ddm\_example\] we see that the projections of $U_1$ and $S_1$ overlap on both dimensions, so we can conclude that the regions $U_1$ and $S_1$ intersect. Therefore, any algorithm that solves the region matching problem for two sets of $n$ and $m$ segments in time $O\left(f(n,m)\right)$ can be extended to an $O\left(
d \times f(n,m) \right)$ algorithm for the $d$-dimensional case, by executing the 1-D algorithm on each dimension and computing the intersection of the partial results[^2]. Since the parameter $d$ is fixed for every problem instance, and much smaller than $n$ or $m$, it can be treated as a constant so we get $O\left( d \times f(n,m)
\right) \subset O\left( f(n,m) \right)$. Therefore, solving the general case is, under reasonable circumstances, asymptotically not harder than solving the special case $d=1$.
In the rest of this section we provide some background on the region matching problem. We report some known lower bounds for specific formulations of this problem; we then review some algorithms and data structures that have been developed in the context of computational geometry research. Finally, we describe in details some relevant solutions developed within the research community: , , and . Since we will frequently refer to these algorithms in this paper, we provide all the necessary details below. A comprehensive review of the $d$-dimensional region matching problem is outside the scope of this work, and has already been carried out by Liu and Theodoropoulos [@Liu:2014] to which the interested reader is referred.
#### Lower Bounds
Over time, several slightly different formulations of the region matching problem have been considered. The most common formulation is to find all overlapping pairs among a set of $N$ rectangles, without any distinction between subscription and update regions. One of the first efficient solutions for this problem is due to Bentley and Wood [@Bentley:1980] who proposed an algorithm for the two-dimensional case requiring time $\Theta(N \lg N + K)$, where $K$ is the cardinality of the result. They proved that the result is optimal by showing a lower bound $\Omega(N \lg N + K)$ for this type of problem.
Petty and Morse [@Petty:2004] studied the computational complexity of the following formulation of the region matching problem: given an initially empty set $R$ of $d$-dimensional regions, apply a sequence of $N$ operations of any of the following types: (*i*) insert a new region in $R$; (*ii*) delete an existing region from $R$; (*iii*) enumerate all regions in $R$ overlapping with a given test region. They showed that a lower bound on the computational complexity of this problem is $\Omega(N \lg N)$ by reduction to binary search, and an upper bound is $O(N^2)$; the upper bound can be achieved with the Brute Force algorithm that we will describe shortly.
#### Computational Geometry approaches
The problem of counting or enumerating intersections among $d$-dimensional rectangles is of great importance in computational geometry, and as such received considerable attention. Existing approaches rely on a combination of algorithmic techniques such as sorting, searching and partitioning [@Six:1980; @Six:1982; @Bentley:1980; @Edelsbrunner:1983], and data structures used to speed up various types of spatial queries (containment, overlap, counting, filtering).
The aforementioned algorithm by Bentley and Wood [@Bentley:1980] can report all intersections within a set of $N$ rectangles in time $\Theta(N \lg N + K)$. The algorithm is quite complex and operates in two phases: the first phase takes care of rectangles whose sides intersect using the algorithm described in [@Bentley:1979]; the second phase takes care of those rectangles which are contained into another one, using a data structure called *segment tree*. While the algorithm is optimal, it cannot be generalized for the case $d>2$ and its implementation is nontrivial.
A simpler algorithm for the case $d=2$ has been discovered by Six and Wood [@Six:1980]. The algorithm works by sorting the endpoints of the rectangles along one of the dimensions, and then scanning the endpoints updating an auxiliary data structure called *interval tree* (which, despite the similar name, is not related to the interval tree that will be described in Section \[sec:interval-tree\]). The algorithm by Six and Wood runs in time $\Theta(N \lg N + K)$, but is much easier to implement than Bentley and Wood’s. A generalization for the case $d>2$ has been described in [@Six:1982] and requires time $O\left(2^{d-1} n
\lg^{d-1} n + K\right)$ and space $O\left(2^{d-1} n \lg^{d-1}
n\right)$.
Edelsbrunner [@Edelsbrunner:1983] proposed an improved algorithm based on a *rectangle tree* data structure that can report all intersections among a set of $N$ $d$-rectangles in time $O\left(N
\lg^{2d-3} N + K\right)$ and space $O\left(N \lg^{d-2} N\right)$, thus improving the previous results.
Spatial data structures for the rectangle intersection problem include the $k$-$d$ tree [@Rosenberg1985], the quad-tree [@Finkel:1974], the R-tree [@Guttman1984] and the BSP tree [@Naylor:1990]. These data structures use various types of hierarchical spatial decomposition techniques to store volumetric objects. Spatial data structures are widely used in the context of geographical information systems, since they allow efficient execution of range queries, e.g., reporting all objects inside a user-specified range. However, some of these data structures have been adapted to solve the problem. For example, Eroglu et al. [@Eroglu:2008] use a quad-tree to improve the grid-based matching algorithm used in implementations.
In [@Ahn:2012] the authors propose a binary partition-based matching algorithm whose aim is to reduce the number of overlap tests that need to be performed between subscription and update regions. Experimental evaluation shows that the algorithm works well in some settings, but suffers from a worst case cost of $O(N^2 \lg N)$ where $N$ is the total number of subscription and update regions.
Geometric algorithms and data structures have the drawback of being quite complex to implement and, in many real-world situations, slower than less efficient but simpler solutions. For example, Petty and Mukherjee [@Petty:1997] showed that a simple grid-based matching algorithm performs faster than the $d$-dimensional quad-tree variant from [@VanHook:1996]. Similarly, Devai and Neumann [@Devai2010] propose a rectangle-intersection algorithm that is implemented using only arrays and that can enumerate all $K$ intersections among $N$ rectangles in time $O(N \lg N + K)$ time and $O(N)$ space.
#### Brute-Force Matching
\[alg:bfm:for\] \[alg:bfm:body\] \[alg:bfm:endfor\]
The simplest solution to the segment intersection problem is the approach, also called Region-Based matching (Algorithm \[alg:bfm\]). The algorithm, as the name suggests, checks all $n \times m$ subscription-update pairs $(s, u)$ and reports every intersection by calling a model-specific function whose details are not shown.
The algorithm requires time $\Theta(n m)$; therefore, it is optimal only in the worst case, but very inefficient in general. However, exhibits an embarrassingly parallel structure since the loop iterations (lines \[alg:bfm:for\]–\[alg:bfm:endfor\]) are independent. Therefore, on a shared-memory architecture with $P$ processors it is possible to distribute the iterations across the processors; each processor will then operate on a subset of $nm /P$ intervals without the need to interact with other processors. The parallel version of requires time $\Theta(nm/P)$.
#### Grid-Based Matching
![Grid-based matching in $d=2$ dimensions.[]{data-label="fig:ddm_example_grid"}](ddm_example_grid)
$G \gets \textrm{array}[0..\mathit{ncells}-1]\ \textrm{of empty lists}$ $\mathit{lb} \gets \textrm{minimum of the lower bounds of all intervals in $\mathbf{S} \cup \mathbf{U}$}$ $\mathit{ub} \gets \textrm{maximum of the upper bounds of all intervals in $\mathbf{S} \cup \mathbf{U}$}$ $\mathit{width} \gets (\mathit{ub} - \mathit{lb}) / \mathit{ncells}$ \[alg:gbm:first-begin\] $i \gets \lfloor (u.\mathit{lower} - \mathit{lb}) / \mathit{width} \rfloor$ Add $u$ to the list $G[i]$\[alg:gbm:list-add\] $i \gets i + 1$ \[alg:gbm:first-end\] $\mathit{res} \gets \emptyset$ \[alg:gbm:second-begin\] $i \gets \lfloor (s.\mathit{lower} - \mathit{lb}) / \mathit{width} \rfloor$ \[alg:gbm:check-overlap\] $\mathit{res} \gets \mathit{res} \cup (s, u)$ $i \gets i + 1$ \[alg:gbm:second-end\]
The algorithm [@VanHook:1996; @bou2005] improves over by trying to reduce the number of pairs that are checked for overlap. works by partitioning the domain into a regular mesh of $d$-dimensional cells. Each subscription or update region is mapped to the grid cells it overlaps with. Events generated by an update region $U_j$ are sent to all subscription regions that share at least one cell with $U_j$. However, this could generate spurious notifications: for example, the subscription region $S_2$ in Figure \[fig:ddm\_example\_grid\] shares the hatched grid cells with $U_1$, but does not overlap with $U_1$. Spurious notifications can be eliminated by testing for overlap all subscription and update regions sharing the same grid cell. Essentially, this is equivalent to applying the brute-force (or any other region matching) algorithm to the regions sharing the same grid cell [@Tan:2000-2].
Algorithm \[alg:gbm\] shows an implementation of for the case $d=1$. The algorithm consists of two phases. During the first phase (lines \[alg:gbm:first-begin\]–\[alg:gbm:first-end\]) the algorithm builds an array $G$ of lists, where $G[i]$ contains the update regions that overlap with the $i$-th grid cell. The grid cells are determined by first computing the bounding interval $[\mathit{lb},
\mathit{ub})$ of all regions in $\mathbf{S}$ and $\mathbf{U}$. Then, the bounding interval is evenly split into $\mathit{ncells}$ segments of width $(\mathit{ub} - \mathit{lb})/\mathit{ncells}$ so that the $i$-th grid cell corresponds to the interval $[\mathit{lb}
+ i\times\mathit{width}, \mathit{lb} +
(i+1)\times\mathit{width})$. The parameter $\mathit{ncells}$ must be provided by the user.
During the second phase (lines \[alg:gbm:second-begin\]–\[alg:gbm:second-end\]), the algorithm scans the list of subscription regions. Each subscription region is compared with the update regions on the lists of the cells it overlaps with (line \[alg:gbm:check-overlap\]). Since subscription and update regions may span more than one grid cell, the algorithm keeps a set $\mathit{res}$ of all intersections found so far in order to avoid reporting the same intersection more than once.
If the regions are evenly distributed, each grid cell will contain $n/\mathit{ncells}$ subscription and $m/\mathit{ncells}$ update intervals on average. Therefore, function will be called $O(\mathit{ncells} \times n \times m / \mathit{ncells}^2) =
O(n \times m / \mathit{ncells})$ times on average. Initialization of the array $G$ requires time $O(\mathit{ncells})$. The upper and lower bounds $\mathit{lb}$ and $\mathit{ub}$ can be computed in time $O(n +
m)$. If the set $\mathit{res}$ is implemented using bit vectors, insertions and membership tests can be done in constant time. We can therefore conclude that the average running time of Algorithm \[alg:gbm\] is $O(\mathit{ncells} + n \times m /
\mathit{ncells})$.
The average-case analysis above only holds if subscription and update regions are uniformly distributed over the grid, which might or might not be the case depending on the simulation model. For example, in the presence of a localized cluster of interacting agents, it might happen that the grid cells around the cluster have a significantly larger number of intervals than other cells. Additionally, the number of cells $\mathit{ncells}$ is a critical parameter. Tan et al. [@Tan:2000] showed that the optimal cell size depends on the simulation model and on the execution environment, and is therefore difficult to predict *a priori*.
Observe that the iterations of the loop on lines \[alg:gbm:second-begin\]–\[alg:gbm:second-end\] are independent and can therefore be executed in parallel. If we do the same for the loop on lines \[alg:gbm:first-begin\]–\[alg:gbm:first-end\], however, a data race arises since multiple processors might concurrently update the list $G[i]$. This problem can be addressed by ensuring that line \[alg:gbm:list-add\] is executed atomically, e.g., by enclosing it inside a critical section or employing a suitable data structure for the lists $G[i]$ that supports concurrent appends. We will discuss this in more details in Section \[sec:experimental-evaluation\]. Finally, it is worth noticing that some variants of have been proposed to combine the grid-based method with a region-based strategy [@bou2002].
$T \gets \emptyset$ Insert $x.\textit{lower}$ and $x.\textit{upper}$ in $T$ Sort $T$ in non-decreasing order $\texttt{SubSet} \gets \emptyset$, $\texttt{UpdSet} \gets \emptyset$ \[alg:sbm:loop-start\] $\texttt{SubSet} \gets \texttt{SubSet} \cup \{s\}$ $\texttt{SubSet} \gets \texttt{SubSet} \setminus \{s\}$ $\texttt{UpdSet} \gets \texttt{UpdSet} \cup \{u\}$ $\texttt{UpdSet} \gets \texttt{UpdSet} \setminus \{u\}$ \[alg:sbm:loop-end\]
#### Sort-Based Matching
[@Jun2002; @Raczy2005] is an efficient solution to the region matching problem in $d=1$ dimensions. scans the sorted list of endpoints of the subscription and update intervals, keeping track of which regions are active at any point; a region is active if the left endpoint has been scanned, but the right endpoint has not. When the right endpoint of a subscription (resp., update) region $x$ is encountered, all currently active update (resp., subscription) regions are known to overlap with $x$.
The algorithm is illustrated in Algorithm \[alg:sbm\]. Given $n$ subscription intervals $\mathbf{S}$ and $m$ update intervals $\mathbf{U}$, considers each of the $2 \times (n+m)$ endpoints in non-decreasing order; two sets `SubSet` and `UpdSet` are used to keep track of the active subscription and update regions, respectively, at every point $t$. When the upper bound of an interval $x$ is encountered, it is removed from the corresponding set of active regions, and the intersections between $x$ and every active region of the opposite kind are reported. Observe that Algorithm \[alg:sbm\] never calls the function to check whether two regions overlap. Figure \[fig:sbm-example\] illustrates how the `SubSet` variable is updated while sweeps through a set of subscription intervals (update intervals are handled in the same way).
Let $N=n+m$ be the total number of intervals; then, the number of endpoints is $2N$. The algorithm uses simple data structures and requires $O\left( N \lg N \right)$ time to sort the vector of endpoints, plus $O(N)$ time to scan the sorted vector. During the scan phase, $O(K)$ time is spent to report all $K$ intersections. The overall computational cost of is therefore $O\left( N \lg N +
K \right)$.
Li et al. [@Li:2018] improved the algorithm by reducing the size of the vectors to be sorted and employing the binary search algorithm on the (smaller) sorted vectors of endpoints. The execution time is still dominated by the $O\left(N \lg
N\right)$ time required to sort the smaller vectors of endpoints, but the improved algorithm is faster in practice than due to lower constants hidden in the asymptotic notation.
Pan et al. [@Pan2011] extended to deal with a dynamic setting where regions are allowed to change their position or size without the need to execute the algorithm from scratch after each update.
So far, no parallel version of exists. cannot be easily parallelized due to the presence of a sequential scan phase that is intrinsically serial. This problem will be addressed in Section \[sec:parallel-sort-matching\], where a parallel version of will be described.
#### Parallel algorithms for the region matching problem {#parallel-algorithms-for-the-region-matching-problem .unnumbered}
So far, only a few parallel algorithms for the region matching problem have been proposed. Liu and Theodoropoulos [@Liu:2009; @Liu:2013] propose a parallel region matching algorithm that partitions the routing space into blocks, and assigns partitions to processors using a master-worker paradigm. Each processor then computes the intersections among the regions that overlap the received partitions. In essence, this solution resembles a parallel version of the algorithm.
In [@rao2013] the performance of parallel versions of and grid-based matching (fixed, dynamic and hierarchical) are compared. In this case, the preliminary results presented show that the parallel has a limited scalability and that, in this specific case, the hierarchical grid-based matching has the best performance.
In [@Yanbing2015], a parallel ordered-relation-based matching algorithm is proposed. The algorithm is composed of five phases: projection, sorting, task decomposition, internal matching and external matching. In the experimental evaluation, a MATLAB implementation is compared with the sequential . The results show that, with a high number of regions the proposed algorithm is faster than .
Parallel Interval Tree Matching {#sec:interval-tree}
===============================
In this section we describe the parallel algorithm for solving the region matching problem in one dimension. [@gda-dsrt-2013] is based on the *interval tree* data structure. An interval tree is a balanced search tree that stores a dynamic set of intervals; it supports insertions, deletions, and queries to get the list of segments that intersect a given interval $q$.
Figure \[fig:interval\_tree\] shows a set of intervals and their tree representation. The tree is realized using an augmented AVL tree [@avl] as described in [@Cormen2009 Chapter 14.3]. Each node $x$ contains three fields: (*i*) an interval $x.\textit{in}$, represented by its lower and upper bounds; (*ii*) the minimum lower bound $x.\textit{minlower}$ among all intervals stored at the subtree rooted at $x$; (*iii*) the maximum upper bound $x.\textit{maxupper}$ among all intervals stored at the subtree rooted at $x$.
Insertions and deletions are handled according to the normal rules for AVL trees, with the additional requirement that any update of the values of $\textit{maxupper}$ and $\textit{minlower}$ must be propagated up to the root. During tree operations, nodes are kept sorted according to the lower bounds. Since the height of an AVL tree is $\Theta(\lg n)$, insertions and deletions in the augmented data structure require $O(\lg n)$ time in the worst case. Creating a new tree with $n$ nodes requires total time $O(n \lg n)$ and space $O(n)$.
**return**
$T \gets \textrm{create interval tree for $\mathbf{S}$}$\[alg:int-tree-create\] \[alg:int-tree-loop-begin\] \[alg:int-tree-loop-end\]
Algorithm \[alg:itm\] illustrates how parallel works. The first step is to build an interval tree $T$ containing the subscription intervals $\mathbf{S}$ (the details are omitted for the sake of conciseness; Section \[sec:experimental-evaluation\] provides directions to the source code). Function is then used to report all intersections among an update region $u$ and all subscription intervals stored in $T$. The procedure is similar to a conventional binary search tree lookup, using the $x.\textit{minlower}$ and $x.\textit{maxupper}$ fields of every node $x$ to steer the visit away from irrelevant subtrees. Since each node represents one interval, function reports each intersection only once.
#### Asymptotic execution time {#asymptotic-execution-time .unnumbered}
An interval tree can be created in time $O(n \lg n)$ like an ordinary AVL tree. To determine the cost of one invocation of function we first observe that each node can be visited at most once per call, so $O(n)$ is an upper bound on the asymptotic running time of . If region $u$ overlaps with $K_u$ subscription intervals, then the execution time of is also bound by $O(K_u
\lg n)$; combining the two bounds we get that one call costs $O\left(\min\{n, K_u \lg n\}\right)$. Since function is called $m$ times (one for each update region $u$), the total query time is $O\left( \min\{m \times n,
K \lg n\}\right)$, $K$ being the number of intersections.
Once the tree is built, its structure is never modified. Therefore, the iterations of the loop on lines \[alg:int-tree-loop-begin\]–\[alg:int-tree-loop-end\] can be evenly split across $P$ processors on a shared-memory architecture. The parallel query time is then reduced by a factor $P$ and becomes $O\left( \min\{m \times n, K \lg n\} / P \right)$.
Observe that Algorithm \[alg:itm\] allows the roles of $\mathbf{S}$ and $\mathbf{U}$ to be swapped. This can be helpful if the number of update regions $m$ is much lower than the number of subscription regions $m$. If $m \ll n$ it is more convenient to build a tree on $\mathbf{U}$ instead than on $\mathbf{S}$, since the height will be lower. With this optimization we can replace $n$ with $m$ in the asymptotic query time above.
Different implementations of the interval tree are possible. Priority search trees [@McCreight85] support insertion and deletion in time $O(\lg n)$, and can report all $K$ intersections with a given query interval in time $O(K + \lg n)$. While the implementation above based on AVL trees is asymptotically less efficient, it has the advantage of being easier to implement since it relies on a familiar data structure. We have chosen AVL trees over other balanced search trees, such as red-black trees [@rbtree], because AVL trees are more rigidly balanced and therefore allow faster queries. It should be observed that is not tied to any specific interval tree implementation; therefore, any data structure can be used as a drop-in replacement.
#### Dynamic interval management {#dynamic-interval-management .unnumbered}
An interesting feature of is that it can easily cope with *dynamic* intervals. The specification allows federates to modify (i.e., move or grow/shrink) subscription and update regions; the ability to do so is indeed essential in almost every agent-based simulation model. A subscription region $s \in \mathbf{S}$ (resp. $u
\in \mathbf{U}$) changing its position or size will trigger at most $O(m)$ (resp. $O(n)$) new overlaps, so it makes sense to exploit the data structures already built instead of running the matching phase from scratch each time.
The interval tree data structure can be used to implement a dynamic data distribution management scenario as follows. We can use two interval trees $T_U$ and $T_S$ holding the set of update and subscription regions, respectively. If an update region $u \in
\mathbf{U}$ is modified, we can identify the $K_u$ subscriptions overlapping with $u$ in time $O\left( \min \{n, K_u \lg n \}\right)$ ($O\left( \min \{n, K_u \lg n \} / P\right)$ if $P$ processors are used) by repeatedly calling the function on $T_U$. Similarly, if a subscription region $s \in \mathbf{S}$ changes, the $K_s$ overlaps with $s$ can be computed in time $O\left(
\min\{m, K_s \lg m\}\right)$ using $T_U$. When a subscription or update region is modified, the appropriate tree must be updated by deleting and re-inserting the node representing the region that changed. These operations require time $O(\lg n)$ for the subscription regions, and $O(\lg n)$ for the update regions.
Parallel Sort-based Matching {#sec:parallel-sort-matching}
============================
The parallel algorithm from the previous section relies on a data structure (the interval tree) to efficiently enumerate all intersections among a set of intervals with a given query interval $q$. Once built, the interval tree allows a high degree of parallelism since all update regions can be compared concurrently with the subscription regions stored in the tree. can be easily extended to support dynamic management of subscription and update regions.
We now propose a parallel solution to the region matching problem based on a novel parallel algorithm derived from . The parallel version of will be derived incrementally, starting from the serial version that has been described in Section \[sec:region-matching\] (Algorithm \[alg:sbm\]). As we may recall, operates in two phases: first, a list $T$ of endpoints of all regions is built and sorted; then, the sorted list is traversed to compute the values of the `SubSet` and `UpdSet` variables, from which the list of overlaps is derived. Let us see if and how each step can be parallelized.
On a shared-memory architecture with $P$ processors, building the list of endpoints can be trivially parallelized, especially if this data structure is realized with an array of $2 \times (n+m)$ elements rather than a linked list. Sorting the endpoints can be realized using a parallel sorting algorithm such as parallel mergesort [@Cole88] or parallel quicksort [@Wheat92; @Tsigas:2003], both of which are optimal. The traversal of the sorted list of endpoints (Algorithm \[alg:sbm\] lines \[alg:sbm:loop-start\]–\[alg:sbm:loop-end\]) is, however, more problematic. Ideally, we would like to evenly split the list $T$ into $P$ segments $T_0, \ldots, T_{P-1}$, and assign each segment to a processor so that all segments can be processed concurrently. Unfortunately, this is not possible due to the presence of *loop-carried dependencies*. A loop carried dependency is a data dependence that causes the result of a loop iteration to depend on previous iterations. In the algorithm the loop-carried dependencies are caused by the variables `SubSet` and `UpdSet`, whose values depend on those computed on the previous iteration.
$T \gets \emptyset$ Insert $x.\textit{lower}$ and $x.\textit{upper}$ in $T$ Sort $T$ **in parallel**, in non-decreasing order Split $T$ into $P$ segments $T_0, \ldots, T_{P-1}$ $\langle$Initialize $\texttt{SubSet}[0..P-1]$ and $\texttt{UpdSet}[0..P-1] \rangle$\[alg:par-sbm-init\] $\texttt{SubSet}[p] \gets \texttt{SubSet}[p] \cup \{s\}$ $\texttt{SubSet}[p] \gets \texttt{SubSet}[p] \setminus \{s\}$ $\texttt{UpdSet}[p] \gets \texttt{UpdSet}[p] \cup \{u\}$ $\texttt{UpdSet}[p] \gets \texttt{UpdSet}[p] \setminus \{u\}$
Let us pretend that the scan phase *could* be parallelized somehow. Then, a parallel version of would look like Algorithm \[alg:par-sbm\] (line \[alg:par-sbm-init\] will be explained shortly). The major difference between Algorithm \[alg:par-sbm\] and its sequential counterpart is that the former uses two arrays $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$ instead of the scalar variables `SubSet` and `UpdSet`. This allows each processor to operate on its private copy of the subscription and update sets, achieving the maximum level of parallelism.
It is not difficult to see that Algorithm \[alg:par-sbm\] is equivalent to sequential (i.e., they produce the same result) if and only if $\texttt{SubSet}[0..P-1]$ and $\texttt{UpdSet}[0..P-1]$ are properly initialized. Specifically, $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$ must be initialized with the values that the sequential algorithm assigns to `SubSet` and `UpdSet` right after the last endpoint of $T_{p-1}$ is processed, for every $p=1, \ldots, P-1$; $\texttt{SubSet}[0]$ and $\texttt{UpdSet}[0]$ must be initialized to the empty set.
It turns out that $\texttt{SubSet}[0..P-1]$ and $\texttt{UpdSet}[0..P-1]$ can be computed efficiently using a *parallel prefix computation* (also called *parallel scan* or *parallel prefix-sum*). To make this paper self-contained, we introduce the concept of prefix computation before illustrating the missing part of the parallel algorithm.
#### Prefix computations {#prefix-computations .unnumbered}
A prefix computation consists of a sequence of $N > 0$ data items $x_0, \ldots, x_{N-1}$ and a binary associative operator $\oplus$. There are two types of prefix computations: an *inclusive scan* produces a new sequence of $N$ data items $y_0,
\ldots, y_{N-1}$ defined as:
$$\begin{aligned}
{2}
y_0 &= x_0 \\
y_1 &= y_0 \oplus x_1 &&= x_0 \oplus x_1 \\
y_2 &= y_1 \oplus x_2 &&= x_0 \oplus x_1 \oplus x_2 \\
&\vdots \\
y_{N-1} &= y_{N-2} \oplus x_{N-1} &&= x_0 \oplus x_1 \oplus \ldots \oplus x_{N-1}\end{aligned}$$
while an *exclusive scan* produces the sequence $z_0,
z_1, \ldots z_{N-1}$ defined as:
$$\begin{aligned}
{2}
z_0 &= 0 \\
z_1 &= z_0 \oplus x_0 &&= x_0 \\
z_2 &= z_1 \oplus x_1 &&= x_0 \oplus x_1 \\
&\vdots \\
z_{N-1} &= z_{N-2} \oplus x_{N-2} &&= x_0 \oplus x_1 \oplus \ldots \oplus x_{N-2}\end{aligned}$$
where $0$ is the neutral element of operator $\oplus$, i.e., $0 \oplus x = x$.
Hillis and Steele [@Hillis:1986] proposed a parallel algorithm for computing the prefix sum of $N$ items with $N$ processors in $O(\lg
N)$ parallel steps and total work $O(N \lg N)$. This result was improved by Blelloch [@Blelloch89] who described a parallel implementation of the scan primitive of $N$ items on $P$ processors requiring time $O(N/P + \lg P)$. Blelloch’s algorithm is optimal when $N > P \lg P$, meaning that the total amount of work it performs over all processors is the same as the (optimal) serial algorithm for computing prefix sums, i.e., $O(N)$.
A somewhat simpler algorithm for computing prefix sums of $N$ items with $P$ processors in time $O(N/P + P)$ is illustrated in Figure \[fig:scan\]; in the figure we consider the addition operator, although the idea applies to any associative operator $\oplus$. The computation involves two parallel steps (steps and in the figure), and one serial step (step ). In step the input sequence is split across the processors, and each processor computes the prefix sum of the elements in its portion. In step the master computes the prefix sum of the $P$ last local sums. Finally, in step the master scatters the first $(P-1)$ computed values (prefix sums of the last local sums) to the last $(P-1)$ processors. Each processor, except the first one, adds (more precisely, applies the $\oplus$ operator) the received value to the prefix sums from step , producing a portion of the output sequence.
Steps and require time $O(N/P)$ each, while step is executed by the master in time $O(P)$, yielding a total time $O(N/P + P)$. Therefore, the algorithm is optimal when $N > P^2$. Since the current generation of CPUs have a small number of cores (e.g., $P \leq 72$ for the Intel Xeon Phi) and the number of regions $N$ is usually very large, the algorithm above can be considered optimal for any practical purpose. We remark that the parallel algorithm can be readily implemented with the tree-structured reduction operation, and therefore will still be competitive on future generations of processors with a higher number of cores.
\[alg:par-sbm-loop-begin\] $\texttt{Sadd}[p] \gets \emptyset$, $\texttt{Sdel}[p] \gets \emptyset$, $\texttt{Uadd}[p] \gets \emptyset$, $\texttt{Udel}[p] \gets \emptyset$ $\texttt{Sadd}[p] \gets \texttt{Sadd}[p] \cup \{s\}$ $\texttt{Sadd}[p] \gets \texttt{Sadd}[p] \setminus \{s\}$ $\texttt{Sdel}[p] \gets \texttt{Sdel}[p] \cup \{s\}$ $\texttt{Uadd}[p] \gets \texttt{Uadd}[p] \cup \{u\}$ $\texttt{Uadd}[p] \gets \texttt{Uadd}[p] \setminus \{u\}$ $\texttt{Udel}[p] \gets \texttt{Udel}[p] \cup \{u\}$ \[alg:par-sbm-loop-end\] $\texttt{SubSet}[0] \gets \emptyset$, $\texttt{UpdSet}[0] \gets \emptyset$\[alg:par-sbm-step-2-begin\] $\texttt{SubSet}[p] \gets \texttt{SubSet}[p-1] \cup \texttt{Sadd}[p-1] \setminus \texttt{Sdel}[p-1]$ $\texttt{UpdSet}[p] \gets \texttt{UpdSet}[p-1] \cup \texttt{Uadd}[p-1] \setminus \texttt{Udel}[p-1]$ \[alg:par-sbm-step-2-end\]
#### Initialization with prefix computation {#initialization-with-prefix-computation .unnumbered}
We can now complete the description of the parallel algorithm by showing how the arrays $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$ can be initialized in parallel. To better illustrate the steps involved, we refer to the example in Figure \[fig:par-sbm\]; in the figure we consider subscription regions only, since the procedure for update regions is the same.
The sorted array of endpoints $T$ is evenly split into $P$ segments $T_0, \ldots, T_{P-1}$ of $2\times(n+m)/P$ elements each. Processor $p$ scans the endpoints $t \in T_p$ in non-decreasing order, updating four auxiliary variables $\texttt{Sadd}[p]$, $\texttt{Sdel}[p]$, $\texttt{Uadd}[p]$, and $\texttt{Udel}[p]$. Informally, $\texttt{Sadd}[p]$ and $\texttt{Sdel}[p]$ (resp. $\texttt{Uadd}[p]$ and $\texttt{Udel}[p]$) contain the endpoints that the sequential algorithm would add/remove from `SubSet` (resp. `UpdSet`) while scanning the endpoints belonging to segment $T_p$. More formally, at the end of each local scan the following invariants hold:
1. $\texttt{Sadd}[p]$ (resp. $\texttt{Uadd}[p]$) contains the subscription (resp. update) intervals whose lower endpoint belongs to $T_p$, and whose upper endpoint does not belong to $T_p$;
2. $\texttt{Sdel}[p]$ (resp. $\texttt{Udel}[p]$) contains the subscription (resp. update) intervals whose upper endpoint belongs to $T_p$, and whose lower endpoint does not belong to $T_p$.
This step is realized by lines \[alg:par-sbm-loop-begin\]–\[alg:par-sbm-loop-end\] of Algorithm \[alg:par-sbm-cont\], and its effects are shown in Figure \[fig:par-sbm\] . The figure reports the values of $\texttt{Sadd}[p]$ and $\texttt{Sdel}[p]$ after each endpoint has been processed; the algorithm does not store every intermediate value, since only the last ones (within thick boxes) will be needed by the next step.
Once all $\texttt{Sadd}[p]$ and $\texttt{Sdel}[p]$ are available, the next step is executed by the master and consists of computing the values of $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$, $p=0, \ldots,
P-1$. Recall from the discussion above that $\texttt{SubSet}[p]$ (resp. $\texttt{UpdSet}[p]$) is the set of active subscription (resp. update) intervals that would be identified by the sequential algorithm right after the end of segment $T_0 \cup
\ldots \cup T_{p-1}$. The values of $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$ are related to $\texttt{Sadd}[p]$, $\texttt{Sdel}[p]$, $\texttt{Uadd}[p]$ and $\texttt{Udel}[p]$ as follows: $$\begin{aligned}
\texttt{SubSet}[p] &= \begin{cases}
\emptyset & \mbox{if $p=0$} \\
\texttt{SubSet}[p-1] \cup \texttt{Sadd}[p-1] \setminus \texttt{Sdel}[p-1] & \mbox{if $p>0$}
\end{cases} \\
\texttt{UpdSet}[p] &= \begin{cases}
\emptyset & \mbox{if $p=0$} \\
\texttt{UpdSet}[p-1] \cup \texttt{Uadd}[p-1] \setminus \texttt{Udel}[p-1] & \mbox{if $p>0$}
\end{cases}\end{aligned}$$ Intuitively, the set of active intervals at the end of $T_p$ can be computed from those active at the end of $T_{p-1}$, plus the intervals that became active in $T_p$, minus those that ceased to be active in $T_p$.
Lines \[alg:par-sbm-step-2-begin\]–\[alg:par-sbm-step-2-end\] of Algorithm \[alg:par-sbm-cont\] take care of this computation; see also Figure \[fig:par-sbm\] for an example. Once the initial values of $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$ have been computed, Algorithm \[alg:par-sbm\] can be resumed to identify the list of overlaps.
#### Asymptotic execution time {#asymptotic-execution-time-1 .unnumbered}
We now analyze the asymptotic execution time of parallel . Let $N$ denote the total number of subscription and update regions, and $P$ the number of processors. Algorithm \[alg:par-sbm\] consists of three phases:
1. Sorting $T$ in non-decreasing order requires total time $O\left(
(N \lg N) / P\right)$ using a parallel sorting algorithm such as parallel mergesort [@Cole88].
2. Computing the initial values of $\texttt{SubSet}[p]$ and $\texttt{UpdSet}[p]$ for each $p=0, \ldots, P-1$ requires $O\left(N/P + P\right)$ steps using the two-level scan shown on Algorithm \[alg:par-sbm-cont\]; the time could be reduced to $O\left(N/P + \lg P\right)$ steps using the tree-structured scan by Blelloch [@Blelloch89].
3. Each of the final local scans require $O(N/P)$ steps.
Note, however, that phases 2 and 3 require the manipulation of data structures to store sets of endpoints, supporting insertions and removals of single elements and whole sets. Therefore, a single step of the algorithm has a non-constant time complexity that depends on the actual implementation of sets and the number of elements they contain; this issue will be discussed in more detail in Section \[sec:experimental-evaluation\]. During phase 3 total time $O(K)$ is spent cumulatively by all processors to report all $K$ intersections.
#### Some remarks on distributed-memory and GPU implementations {#some-remarks-on-distributed-memory-and-gpu-implementations .unnumbered}
Although the focus of this paper is on shared-memory architectures, we provide here some remarks on possible distributed-memory and implementations of Algorithm \[alg:par-sbm\].
In a distributed-memory system, computing nodes exchange information through some type of high-performance network connection. It turns out that a distributed-memory implementation of Algorithm \[alg:par-sbm\] can be realized with minimal modifications. First, a suitable distributed-memory sorting algorithm (e.g., [@parallel-sorting]) can be used to sort the list of endpoints. Then, the parallel prefix computation shown in Figure \[fig:par-sbm\] can be realized efficiently since it is based on the *Scatter/Gather* communication pattern [@Mattson:2004]. A *Scatter* operation allows a single process to send portions of a local array to multiple destinations, and is executed between steps and in Figure \[fig:par-sbm\]. The symmetric *Gather* allows multiple processes to send portions of an array to a single destination where they are concatenated; this is required between steps and . Since Scatter/Gather primitives are very useful in many contexts, they are efficiently supported by software middlewares (e.g., the `MPI_Scatter()` `MPI_Gather()` functions of the Message Passing Interface specification), or directly at the hardware level [@Almasi:2005].
An efficient implementation of Algorithm \[alg:par-sbm\], however, poses several challenges. Although GPU-based efficient algorithms for sorting and doing prefix computations are available [@Thrust], the data structure used to represent sets of endpoints must be designed carefully. As described earlier in this Section, Algorithm \[alg:par-sbm\] performs $\Theta(N)$ set operations (unions and differences) during the prefix computation, and therefore the data structure used for representing sets must be chosen wisely (we will return to this issue in Section \[sec:experimental-evaluation\]). Data structures based on hash tables or trees are problematic on the , although not impossible [@Awad:2019; @GPU-hash-tables]. A simpler implementation of sets using bit vectors appears to be better suited: a bit vector is a sequence of $N$ bits, where item $i$ is in the set if and only if the $i$-th bit is one. Bit vectors allow union, intersection and set difference to be realized efficiently using nothing more than Boolean operators; however, bit vectors require considerable amounts of memory if the number $N$ of items that could be in the set is large. This issue requires further investigation, and is subject of ongoing research.
Experimental Evaluation {#sec:experimental-evaluation}
=======================
In this section we evaluate the performance and scalability of the parallel and parallel algorithms, and compare them to parallel implementations of and . and have been implemented in C, while and have been implemented in C++. To foster the reproducibility of our experiments, all the source code used in this performance evaluation is freely available on the research group website [@pads] with a Free Software license.
We used the GNU C Compiler (GCC) version 4.8.4 with the `-O3 -fopenmp -D_GLIBCXX_PARALLEL` flags to turn on optimization and to enable parallel constructs at the compiler and library levels. Specifically, the `-fopenmp` flag instructs the compiler to handle OpenMP directives in the source code [@OpenMP], while the flag `-D_GLIBCXX_PARALLEL` enables parallel implementations of some algorithms from the C++ Standard Template Library (STL) to be used instead of their sequential counterparts (more details below).
OpenMP is an open interface supporting shared memory parallelism in the C, C++ and FORTRAN programming languages. OpenMP allows the programmer to label specific sections of the source code as parallel regions; the compiler takes care of dispatching these regions to different threads that can be executed by the available processors or cores. In the C/C++ languages, OpenMP directives are specified using `#pragma` directives. The OpenMP standard also defines some library functions that can be called by the developer to query and control the execution environment programmatically.
The and algorithms are *embarrassingly parallel*, meaning that the iterations on their main loop (line \[alg:bfm:for\] on Algorithm \[alg:bfm\] and line \[alg:int-tree-loop-begin\] on Algorithm \[alg:itm\]) are free of data races. Therefore, a single `#pragma omp parallel for` directive is sufficient to distribute the iterations of the loops across the processors.
requires more care, as we have already observed in Section \[sec:region-matching\], since the first loop (lines \[alg:gbm:first-begin\]–\[alg:gbm:first-end\], Algorithm \[alg:gbm\]) has a data race due to possible concurrent updates to the list $G[i]$ in line \[alg:gbm:list-add\]. A simple solution consists on protecting the statement with a `#pragma omp critical` directive that ensures that only one thread at a time can modify a list. However, this might limit the degree of parallelism since two OpenMP threads would be prevented from updating two different lists concurrently. To investigate this issue we developed an ad-hoc, lock-free linked list data structure that supports concurrent append operations without the need to declare a critical section. Measurements showed that in our experiments the ad-hoc linked list did not perform significantly better, so we decided to use the standard `std::list` container provided by the C++ library [@Stroustrup13], and protect concurrent updates with the OpenMP `critical` directive.
Our implementation of parallel relies on some of the data structures and algorithms provided by the C++ . Specifically, to sort the endpoints we use the parallel `std::sort` function provided by the extensions for parallelism [@parallel-stl]. Indeed, the GNU provides several parallel sort algorithms (multiway mergesort and quicksort with various splitting heuristics) that are automatically selected at compile time when the `-D_GLIBCXX_PARALLEL` compiler flag is used. The rest of the algorithm has been parallelized using explicit OpenMP `parallel` directives.
The algorithm relies on a suitable data structure to store the sets of endpoints `SubSet` and `UpdSet` (see Algorithms \[alg:par-sbm\] and \[alg:par-sbm-cont\]). Parallel puts a higher strain on this data structure than its sequential counterpart, since it also requires efficient support for unions and differences between sets, in addition to insertions and deletions of single elements. We have experimented with several implementations for sets: (*i*) bit vectors based on the `std::vector<bool>` container; (*ii*) an ad-hoc implementation of bit vectors based on raw memory manipulation; (*iii*) the `std::set` container, that in the case of the GNU is based on Red-Black trees [@Bayer1972]; (*iv*) the `std::unordered_set` container from the 2011 ISO C++ standard, that is usually implemented using hash tables; (*v*) the `boost::dynamic_bitset` container provided by the Boost C++ library [@boost]. The most efficient turned out to be the `std::set` container, and it has been used in the experiments described below.
------------------ -------------------- --
CPU Intel Xeon E5-2640
Clock frequency 2.00 GHz
Processors 2
Cores/proc 8
Total cores 16
Threads/core 2
HyperThreading Yes
RAM 128 GB
L3 cache size 20480 KB
Operating System Ubuntu 16.04.3 LTS
------------------ -------------------- --
: Hardware used for the experimental evaluation.[]{data-label="tab:spec"}
#### Experimental setup {#experimental-setup .unnumbered}
The experiments below have been carried out on a dual-socket, octa-core server whose hardware specifications are shown in Table \[tab:spec\]. The processors employ the technology [@HT]: in -enabled CPUs some functional components are duplicated, but there is a single main execution unit for physical core. From the point of view of the , provides two logical processors for each physical core. Studies from Intel and others have shown that in typical applications contributes a performance boost in the range $16$–$28\%$ [@HT]. When two processes are executed on the same core, they compete for the shared hardware resources resulting is lower efficiency than the same two processes executed on two different physical cores.
The number $P$ of OpenMP threads to use can be chosen either programmatically through the appropriate OpenMP functions, or setting the `OMP_NUM_THREADS` environment variable. In our experiments, $P$ never exceeds the total number of (logical) cores, so that over-provisioning never happens. By default, the Linux scheduler spreads processes to different physical cores as long as possible; only when there are more runnable processes than physical cores does come into effect. All tests have been executed with this default behavior.
For better comparability of our results with those in the literature, we consider $d=1$ dimensions and use the methodology and parameters described in [@Raczy2005] (a performance evaluation based on real dataset will be described at the end of this section). The first parameter is the total number of regions $N$, that includes $n=N/2$ subscription and $m=N/2$ update regions. All regions have the same length $l$ and are randomly placed on a segment of total length $L=10^6$. $l$ is defined in such a way that a given overlapping degree $\alpha$ is obtained, where
$$\alpha=\frac{\sum \text{area of regions}}{\text{area of the routing space}} = \frac{N \times l}{L}$$
Therefore, given $\alpha$ and $N$, the value of $l$ is set to $l =
\alpha L / N$. The overlapping degree is an indirect measure of the total number of intersections among subscription and update regions. While the performance of and is not affected by the number of intersections, this is not the case for , as will be shown below. We considered the same values for $\alpha$ as in [@Raczy2005], namely $\alpha \in \{0.01, 1,
100\}$. Finally, each measure is the average of $50$ independent runs to get statistically valid results. Our implementations do not explicitly store the list of intersections, but only count them. We did so to ensure that the execution times are not affected by the choice of the data structure used to store the list of intersections.
#### Wall clock time and Speedup {#wall-clock-time-and-speedup .unnumbered}
The first metric we analyze is the of the parallel programs. The includes the time needed to initialize all ancillary data structures used by each algorithm (e.g., the time needed to build the interval tree, or to fill the grid cells), but does not include the time required to randomly initialize the input regions.
Figure \[fig:wct-1M\] shows the for the parallel versions of , , and as a function of the number $P$ of OpenMP threads, given $N=10^6$ regions and overlapping degree $\alpha = 100$. The algorithm requires the user to define the number of grid blocks ($\mathit{nblocks}$) to use. This parameter should be carefully chosen since it affects the algorithm’s performance [@Tan:2000]. We have empirically determined that the best running time with $P=32$ OpenMP threads with the parameters above is $3000$ regions. Dashed lines indicate when $P$ exceeds the number of physical cores.
We observe that the parallel algorithm is about three orders of magnitude slower than parallel . This is unsurprising, since the computational cost of grows quadratically with the number of regions (see Section \[sec:region-matching\]), while that of and grows only polylogarithmically. performs better than by almost two orders of magnitude. is faster than (remember that Figure \[fig:wct-1M\] uses a logarithmic scale), and provides the additional advantage of requiring no tuning of parameters.
A drawback of is that it requires the number of grid cells to be defined. The optimal value depends both on the simulation model and also on the number of OpenMP threads $P$; our chosen value ($3000$ regions) is optimal for $P=32$, but not necessarily for the other values of $P$; indeed, we observe that the execution time of parallel increases around $P=24$; this shows up as a prominent feature in the speedup graph as explained below.
The *relative speedup* measures the increase in speed that a parallel program achieves when more processors are employed to solve a problem of the same size. This metric can be computed from the as follows. Let $T(N, P)$ be the required to process an input of size $N$ using $P$ processes (OpenMP threads). Then, for a given $N$, the relative speedup $S_N(P)$ is defined as $S_N(P) = T(N, 1) / T(N, P)$. Ideally, the maximum value of $S_N(P)$ is $P$, which means that solving a problem with $P$ processors requires $1/P$ the time needed by a single processor. In practice, however, several factors limit the speedup, such as the presence of serial regions in the parallel program, uneven load distribution, scheduling overhead, and heterogeneity in the execution hardware.
Figure \[fig:speedup-1M\] shows the speedups of the parallel versions of , , and as a function of the number of OpenMP threads $P$; the speedup has been computed using the wall clock times of Figure \[fig:wct-1M\]. Again, dashed lines indicate data points where $P$ exceeds the number of physical processor cores. The algorithm, despite being the less efficient, is the most scalable due to its embarrassingly parallel structure and lack of any serial part. , on the other hand, is the most efficient but less scalable. achieves a $2.6\times$ speedup with $16$ OpenMP threads. When all “virtual” cores are used, the speedup grows to $3.6\times$. The limited scalability of is somewhat to be expected, since its running time is very small and therefore the overhead introduced by OpenMP becomes non-negligible.
The effect of (dashed lines) is clearly visible in Figure \[fig:speedup-1M\]. The speedup degrades when $P$ exceeds the number of cores, as can be seen from the different slopes for on `titan`. When kicks in, load unbalance arises due to contention of the shared control units of the processor cores, and this limits the scalability. The curious “bulge” that appears on the curve is due to some OpenMP scheduling issues on the machine used for the test, that is based on a architecture [@Broquedis2008]. Indeed, the bulge appears even if we replace the body of the inner loop of the algorithm (line \[alg:bfm:body\] of algorithm \[alg:bfm\]) with a dummy statement; moreover, the bulge does *not* appear if we run the algorithm on a non- machine.
The speedup of improves if we increase the work performed by the algorithm. Figure \[fig:speedup-100M\] shows the speedup of parallel and with $N=10^8$ regions and overlapping degree $\alpha = 100$; in this scenario both and take so long that they have been omitted. The algorithm behaves better, achieving a $7\times$ speedup with $P=32$ threads. The reason of this improvement is that increasing the amount of work executed by each processor reduces the synchronization overhead, which is particularly beneficial on multi-socket machines.
![Wall clock time of parallel with $N=10^6$ regions and overlapping degree $\alpha=100$. For each value of the number of OpenMP threads $P$, a red dot indicates the number of grid cells minimizing the . Each is the average of $50$ measurements.[]{data-label="fig:wct-grid"}](solaris-grid-tomacs-colored)
We have said above that the optimum number of grid cells in the parallel algorithm depends on the number of OpenMP threads $P$. Figure \[fig:wct-grid\] shows this dependency for a scenario with $N=10^6$ regions and overlapping degree $\alpha=100$. In the figure we report the as a function of $P$ and of the number of grid cells; for each value of $P$ we put a red dot on the combination of parameters that provides the minimum . As we can see, the optimum number of grid cells changes somewhat erratically as $P$ increases, although it shows a clear trend suggesting that a larger number of cells is better for low values of $P$, while a small number of cells is better for high values of $P$. A more precise characterization of the of the parallel algorithm would be an interesting research topic, that however falls outside the scope of the present paper.
We now turn our attention on how the changes as a function of the number of regions $N$, and as a function of the overlapping degree $\alpha$. We set the number of OpenMP threads to $P=32$, the number of logical cores provided by the test machine. Figure \[fig:wct-n\] shows the of parallel and for $\alpha = 100$, by varying the number of regions $N$ in the range $[10^7, 10^8]$. In this range both and require a huge amount of time, that is orders of magnitude higher than those of and , and will therefore be omitted from the comparison. From the figure we observe that the execution times of both and grow polylogarithmically with $N$, supporting the asymptotic analysis in Section \[sec:parallel-sort-matching\]; however, parallel is faster than , suggesting that its asymptotic cost has smaller constant factors.
In Figure \[fig:wct-alpha\] we report the as a function of $\alpha$, for a fixed $N=10^8$. We observe that, unlike , the execution time of is essentially independent from the overlapping degree.
#### Memory Usage {#memory-usage .unnumbered}
We conclude our experimental evaluation with an assessment of the memory usage of the parallel , , and algorithms. Figure \[fig:memory\] shows the peak of the four algorithms as a function of the number of regions $N$ and OpenMP threads $P$, respectively. The is the portion of a process memory that is kept in RAM. Care has been taken to ensure that all experiments reported in this section fit comfortably in the main memory of the available machines, so that the represents an actual upper bound of the amount of memory required by the algorithms. Note that the data reported in Figure \[fig:memory\] includes the code for the test driver and the input arrays of intervals.
Figure \[fig:memory-extents\] shows that the resident set size grows linearly with the number of regions $N$ for all algorithms. has the smaller memory footprint, which is expected since it requires a small bounded amount of additional memory for a few local variables. On the other hand, requires the highest amount of memory of the four algorithms, since it allocates larger data structures, namely the list of endpoints to be sorted, and a few arrays of sets that are used during the scan phase. In our tests, requires approximately $7$ GB of memory to process $N=10^8$ intervals, about three times the amount of memory required by .
In Figure \[fig:memory-threads\] we see that the does not change as the number of OpenMP threads $P$ increases, with overlapping degree $\alpha = 100$ and $N=10^6$ regions. The anomaly shown by going from $P=1$ to $P=2$ is due to the OpenMP runtime system.
#### Performance Evaluation with the Koln Dataset {#performance-evaluation-with-the-koln-dataset .unnumbered}
So far we have evaluated the implementations using a synthetic workload. We now complement the analysis by considering a more realistic workload taken from the vehicular mobility research domain. Specifically, we use the Cologne dataset [@koln], a realistic (although synthetic) trace of car traffic in the city of Cologne, Germany. The complete dataset[^3] contains the timestamped positions of more than $700.000$ vehicles moving on the greater urban area of Cologne ($400$ square kilometers) over a period of $24$ hours.
We consider a portion of the dataset that includes $541,222$ positions. The $x$ coordinate of each position is used as the center of one subscription and one update region; therefore, there are about $N=10^6$ regions overall. The width of each region is set to $100$ meters, resulting in approximately $3.9 \times 10^9$ intersections.
Figure \[fig:wct-speedup-koln\] shows the and speedup of the parallel versions of the , and algorithms; for we used $3000$ grid cells. As usual, each data point is the average of $50$ independent runs. We observe that the parallel algorithm is the slowest of the three, while parallel is the fastest by a wide margin (three orders of magnitude faster than , two orders of magnitude faster than ). Since is very fast on this benchmark, its poor scalability is caused by the parallelization overhead of OpenMP that has a higher impact on low wall-clock times.
Conclusions {#sec:conclusions}
===========
In this paper we described and analyzed two parallel algorithms for the region matching problem on shared-memory architectures. The region matching problem consists of enumerating all intersections among two sets of subscription and update regions; a region is a $d$-dimensional, iso-oriented rectangle. The region matching problem is at the core of the service which is part of the .
The region matching problem in $d$ dimensions can be reduced to the simpler problem of computing intersections among one-dimensional segments. The first parallel solution to the 1D matching problem, called , is based on an interval tree data structure. An interval tree is a binary balanced search tree that can store a set of segments, and can be used to efficiently enumerate all intersections with a given query segment. Once built, an interval tree can be efficiently queried in parallel. The second solution is based on a parallel extension of , a state-of-the-art solution to the problem.
We have implemented the parallel versions of and using the C/C++ programming languages with OpenMP extensions. These algorithms have been compared with parallel implementations of the Brute-Force and Grid-Based matching algorithms. The results show that the parallel versions of and are orders of magnitude faster than (the parallel versions of) Brute-Force and Grid-Based matching. Among the four algorithms considered, parallel is the fastest in all scenarios we have examined. The algorithm, while slower than , can be easily extendable to cope with dynamic regions since the interval tree allows efficient insertion and deletion of regions. In fact, a version of that can efficiently handle region updates has already been proposed [@Pan2011], but it can not be readily adapted to the parallel version of discussed in this paper. Developing a parallel and dynamic version of is the subject of ongoing research.
In this paper we focused on shared-memory architectures, i.e., multicore processors, since they are virtually ubiquitous and well supported by open, standard programming frameworks (OpenMP). Modern can be faster than contemporary CPUs, and are increasingly being exploited for compute-intensive applications. Unfortunately, the parallel and algorithms presented in this paper are ill-suited for implementations on , since they rely on data structures with irregular memory access patterns and/or frequent branching within the code. Reworking the implementation details of and to better fit the *Symmetric Multithreading* model of modern is still and open problem that requires further investigation.
Notation
========
[lX]{} $\mathbf{S}$ & Subscription set $\mathbf{S} = \{S_1, \ldots, S_n\}$\
$\mathbf{U}$ & Update set $\mathbf{U} = \{U_1, \ldots, U_m\}$\
$n$ & Number of subscription regions\
$m$ & Number of update regions\
$N$ & Number of subscription and update regions ($N = n + m$)\
$K_s$ & N. of intersections of subscription region $s$ with all update regions ($0 \leq K_s \leq m$)\
$K_u$ & N. of intersections of update region $u$ with all subscription regions ($0 \leq K_u \leq n$)\
$K$ & Total number of intersections ($0 \leq K \leq n \times m$)\
$\alpha$ & Overlapping degree ($\alpha > 0$)\
$P$ & Number of processors\
Acronyms
========
[^1]: A preliminary version of this paper appeared in the proceedings of the 21st International Symposium on Distributed Simulation and Real Time Applications (DS-RT 2017). This paper is an extensively revised and extended version of the previous work in which more than 30% is new material.
[^2]: The $O(\left(d \times
f(n,m) \right)$ bound holds provided that combining the partial results can be done in time $O\left( f(n,m) \right)$; this is indeed the case for any reasonable $f(n,m)$ using hash-based set implementations, as we will discuss in Section \[sec:experimental-evaluation\].
[^3]: <http://kolntrace.project.citi-lab.fr/koln.tr.bz2>, accessed on 2019-03-29
|
---
abstract: 'We derive optimal a priori and a posteriori error estimates for Nitsche’s method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche’s method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the scalar Signorini problem and outline only the proofs of the main results since most of the auxiliary results can be traced to our previous works on the numerical approximation of variational inequalities. We end the paper by presenting results of our numerical computations which corroborate the efficiency and reliability of the a posteriori estimators.'
author:
- 'Tom Gustafsson[^1], Rolf Stenberg and Juha Videman[^2]'
bibliography:
- 'PS-ref.bib'
title: 'Nitsche’s method for unilateral contact problems'
---
Introduction
============
Unilateral contact problems are of great engineering interest and they occur in numerous areas of physics and mechanics; cf. [@KS80]. In mathematical terms, these problems are expressed as variational inequalities and most often approximated by the finite element method; cf. [@HHN96] and all the references therein.
One of the most common examples of unilateral contact problems is contact between two deformable elastic bodies. Here, we will consider the Signorini problem which consists in finding the equilibrium position of an elastic body resting on a rigid frictionless surface. For simplicity, we consider the scalar version of such problem, at times referred to as the Poisson-Signorini problem. However, our results carry over, with minor modifications, to the Signorini problem in linear elasticity.
The finite element treatment of the Signorini problem has shown to be more difficult than that of the obstacle problem (another archetypical variational inequality) due to the Signorini (no-penetration) condition at the boundary, and has led to a number of papers over the past years; cf. [@SV77; @BHR78; @HL80; @BB2000; @BB03; @BBR03; @SCH11; @HR12; @DH15] and the review papers [@W11; @Frenchsurvey]. The above mentioned works address primal and mixed formulations and focus on obtaining optimal a priori estimates based on Falk’s Lemma [@F74]. As for the a posteriori error estimates for the Signorini problem, we refer to [@BS2000; @HN05; @HN07; @WW09; @SCH09; @KVW15].
Another approach that, at the same time, imposes the contact boundary conditions weakly, avoids the additional Lagrange multiplier of the mixed formulations and is consistent in contrast to the standard penalty formulations, is the Nitsche’s formulation, first proposed for the unilateral contact problems by Chouly and Hild [@CH13], see also [@CHR15; @BHL17] for further generalisations. In [@CH13; @CHR15; @BHL17], optimal a priori estimates were derived but, to our knowledge, the only existing work on the a posteriori error estimation of unilateral contact problems approximated by Nitsche’s method is the recent work by Chouly et al. [@CFHPR18].
In this paper, we will continue to advocate (cf. [@GSV17; @GSV17a; @GSV17b; @GSV18; @GSV18b]) that Nitsche’s method is most readily analysed as a stabilised finite element method, the relation which was first suggested in [@Rolf95]. We will prove optimal a priori estimates for the stabilised mixed formulation and show the reliability and the efficiency for the a posteriori error estimators without additional saturation assumptions which are needed when Nitsche’s method is analysed directly (cf. [@CFHPR18]). We do emphasise though that the stabilised method is best implemented through the Nitsche’s formulation.
The paper is organised as follows. In Section 2, we introduce the Poisson-Signorini problem, write it in a mixed variational form and state a continuous stability result. In Section 3, we formulate the stabilised finite element method, state a discrete stability estimate and prove an a priori error estimate. In Section 4, we introduce residual-based a posteriori error estimators and establish lower and upper bounds for the error in terms of the estimators. In Section 5, we deduce the Nitsche’s formulation from the stabilised one and in Section 6 report on our numerical computations.
We note that some of the proofs have been left out since they are formally similar to the ones proven in our previous works, in particular in [@GSV17; @GSV18].
The continuous problem
======================
Let $\Omega \subset \mathbb{R}^d$, $d\in\{2,3\}$, denote a polygonal (or polyhedral) domain and $ \partial \Omega $ its boundary with $\partial\Omega = \Gamma_D \cup \Gamma_N \cup \Gamma$. We assume that the boundary parts $\Gamma_D,\Gamma_N$ and $\Gamma$ are all non-empty and non-overlapping, $\overline{\Gamma_D} \cap \overline{\Gamma}= \emptyset$ and that the part $\Gamma $ where the contact may occur coincides with one of the sides of the polygon (or the polyhedron).
The Poisson-Signorini unilateral contact problem can be written as follows: find $u$ such that $$\label{strong}
\begin{aligned}
-\Delta u &= f \quad && \text{in $\Omega$\,,} \\
u &= 0 && \text{on $\Gamma_D$,} \\
\frac{\partial u}{\partial n} &= 0 && \text{on $\Gamma_N$,} \\
u\geq 0, \ \ \frac{\partial u}{\partial n} &\geq 0, \ \ u \, \frac{\partial u}{\partial n} = 0 && \text{on $\Gamma$}\, ,
\end{aligned}$$ where $f \in L^2(\Omega)$ is a given load function and $n$ denotes the outer normal vector to $\Omega$.
For ease of exposition, we consider here the simplest version of the Signorini problem although we could easily deal with a non-homogeneous Neumann condition $ \frac{\partial u}{\partial n} = g$ on $\Gamma_N$, with a gap function $\psi$ defined on $\Gamma$, such that $u\geq \psi$ and $(u-\psi)\frac{\partial u}{\partial n} = 0 $ on $\Gamma$, and also with the linear elastic Signorini problem. Note also that by assuming that $\overline{\Gamma_D} \cap \overline{\Gamma}= \emptyset$ we avoid introducing the Lions-Magenes space $H^{1/2}_{00}(\Gamma)$; cf. [@tartar2007introduction].
To give a weak formulation for problem , we introduce the Hilbert spaces $$V =\{ v\in H^{1}(\Omega) \, :\, v=0 \ {\rm on} \ \Gamma_D \} , \quad W:= \{ w|_{\Gamma}: w\in V\} =H^{1/2}(\Gamma),$$ endowed with the norms $$\|v\|_V = \|\nabla v\|_0 \, , \qquad \|w\|_{W} = \inf_{v\in V, v|_{\Gamma}=w}\| v\|_V \, .$$ Now, defining a non-negative Lagrange multiplier by $$\lambda = \frac{\partial u}{\partial n} ,
\label{lagrmult}$$ we obtain the following (weak) mixed variational formulation of problem (cf. [@HHNL]): find $(u,\lambda)\in V\times \Lambda$ such that $$\begin{aligned}
(\nabla u, \nabla v) - \langle v,\lambda\rangle &=(f,v) \quad & \forall v\in V \, , \\
\langle u, \mu-\lambda\rangle & \geq 0 \quad &\forall \mu \in \Lambda \, .
\end{aligned}
\label{dualvf}$$ Above $$\Lambda = \{\mu \in Q : \langle w,\mu\rangle \geq 0 \ \forall w\in W, w \geq 0 \ {\rm a.e. \ on} \ \Gamma \} \, ,$$ where $Q:=W^\prime$ is the topological dual of $W$ and $\langle\cdot,\cdot\rangle:W\times Q \rightarrow \mathbb{R}$ denotes the duality pairing. The norm in $Q$ is defined as $$\|\mu\|_{Q} = \sup_{w\in W} \frac{ \langle w,\mu\rangle } {\|w\|_W} \, .$$
The corresponding primal weak formulation of problem reads: find $u\in K$ such that $$(\nabla u, \nabla (v-u))\geq (f,v-u) \quad \forall v\in K \, ,
\label{primalvf}$$ where $$K = \{ v\in V : v \geq 0 \ {\rm a.e. \ on} \ \Gamma \} \, .$$
\[bindingremark\] It is well-known that problem , equivalently , admits a unique solution $u\in H^1(\Omega)$; cf. [@KS80; @KO88]. However, even if the boundary is smooth and the data regular enough, the Signorini conditions limit the regularity of the solution. In fact, in the two-dimensional case the solution belongs only to $H^{r}, r<5/2$, in the vicinity of points at $\Gamma$ where the constraints change from binding to non-binding, cf. [@MM92] and the discussion in [@BB03]. If the boundary $\Gamma$ has corners, further singularities may occur; cf. [@ABBH13].
Defining the bilinear and linear forms, ${\mathcal{B}}: (V \times Q) \times (V \times Q) \rightarrow \mathbb{R}$ and ${\mathcal{L}}: V \rightarrow \mathbb{R}$ through $${\mathcal{B}}(w,\xi;v,\mu) = (\nabla w, \nabla v) - \langle w, \mu \rangle - \langle v, \xi \rangle, \qquad
{\mathcal{L}}(v) = (f,v) \, ,$$ the mixed variational formulation of reads: find $(u, \lambda) \in V \times \Lambda$ such that $$\label{contprob}
{\mathcal{B}}(u,\lambda; v, \mu - \lambda) \leq {\mathcal{L}}(v) \quad \forall (v, \mu) \in V \times \Lambda.$$
The proof of the following result is straightforward (see, e.g., [@GSV17]):
\[contstab\] For all $(v,\xi)\in V\times Q$ there exists $w\in V$ such that $${\mathcal{B}}(v,\xi;w,-\xi)\gtrsim \big( \Vert v \Vert_{V} + \Vert \xi \Vert_{Q} \big)^{2}$$ and $$\Vert w\Vert_{1} \lesssim \Vert v \Vert_{V} + \Vert \xi \Vert_{Q} .$$
Note that above and in the following we write $a \gtrsim b$ (or $a \lesssim b$) when $a \geq C b$ (or $a \leq C b$) for some positive constant $C$ independent of the finite element mesh.
The stabilised method
=====================
Our discretisation is based on a conforming shape-regular partitioning ${\mathcal{C}_h}$ of $\Omega$ into non-overlapping triangles or tetrahedra, with $h>0$ denoting the mesh parameter. We denote the interior edges or facets of ${\mathcal{C}_h}$ by $\mathcal{E}_h$ and let $\mathcal{G}_h$ and $\mathcal{N}_h$ be the partitioning of the boundary $\Gamma$ and $\Gamma_N$, respectively, corresponding to the partitioning ${\mathcal{C}_h}$, into line segments ($d = 2$) or triangles ($d = 3$). The finite element spaces $$V_h\subset V, \quad Q_h\subset Q,$$ are finite dimensional and consist of piecewise polynomial functions. We also define $$\Lambda_h=\{\mu_h\in Q_h : \mu_h \geq 0 \ {\rm on} \ \Gamma\}\subset \Lambda\,,$$ and introduce the discrete bilinear form $\mathcal{B}_h$ as $$\displaystyle \mathcal{B}_h(w,\xi;v,\mu)= \mathcal{B}(w,\xi;v,\mu) - \alpha\, \mathcal{S}_h(w,\xi;v,\mu)\,,\\$$ where $\alpha>0$ is a stabilisation parameter and the stabilising term $\mathcal{S}_h$ is defined as $$\mathcal{S}_h(w,\xi;v,\mu)= \sum_{E\in \mathcal{G}_h} h_E \left(\xi -\frac{\partial w}{\partial n},\mu -\frac{\partial v}{\partial n}\right)_E ,$$ with $h_E$ denoting the diameter of $E\in\mathcal{G}_h$.
The stabilised finite element method is now formulated as: find $(u_h,\lambda_h)\in V_h\times \Lambda_h$ such that $$\mathcal{B}_h(u_h,\lambda_h;v_h,\mu_h-\lambda_h) \leq \mathcal{L}(v_h) \quad \forall(v_h,\mu_h)\in V_h\times \Lambda_h.
\label{stabfem}$$
The proof of stability for method is very similar to the one given for the obstacle problem in [@GSV17 Theorem 4.1], see also [@GSV18], and is thus omitted. We only note that to estimate the stabilising term we use the following inverse estimate, proven by a scaling argument.
There exists $C_I > 0$, independent of $h$, such that $$C_I \sum_{E \in {\mathcal{G}_h}} h_E \left\| \frac{\partial v_{h}}{\partial n} \right\|_{0,E}^2 \leq \|v_{h}\|_{V}^2 \quad \forall v_{h} \in V_{h}\, .
\label{inverse}$$
\[discrstab\] Let $ 0< \alpha < C_{I}$. For all $(v_{h},\xi_{h})\in V_{h}\times Q_{h}$, there exists $w_{h}\in V_{h}$, such that $${\mathcal{B}}_{h}(v_{h},\xi_{h};w_{h},-\xi_{h}) \gtrsim \left( \| v _{h}\|_V+ \| \xi_{h} \|_{Q} + \left(\sum_{E \in {\mathcal{G}_h}} h_E \left\|\xi_h\right\|_{0,E}^2\right)^{1/2} \right)^2$$ and $$\|w _{h}\|_{V}\lesssim \| v _{h}\|_{V}+ \| \xi _{h}\|_{Q}.$$ \[discstab\]
The additional stability for $\xi_h$ in a mesh-dependent norm in Theorem \[discstab\] will be needed in the proof of Theorem \[apriori\].
We will need the following Lemma in establishing the a priori error estimate. Its detailed proof can be found in [@GSV18].
\[lem:lowresidual\] Let $f_h\in V_h$ be the $L^2$ projection of $f$, define $${\mathrm{osc}}_K(f)=h_K\| f-f_h\|_{0,K}$$ and, for each $E \in {\mathcal{G}_h}$, let $K(E) \in {\mathcal{C}_h}$ denote the element satisfying $\partial K(E) \cap E = E$. For any $(v_h, \mu_h) \in V_h \times Q_h$ it holds that $$\label{eq:lowresidual}
\begin{aligned}
&\left( \sum_{E \in {\mathcal{G}_h}} h_E \left\| \mu_h - \frac{\partial v_{h}}{\partial n} \right\|_{0,E}^2\right)^{1/2} \\
&\qquad \lesssim \|u-v_h\|_V + \|\lambda-\mu_h\|_Q + \left(\sum_{E \in {\mathcal{G}_h}} {\rm osc}_{K(E)} (f)^2\right)^{1/2}.
\end{aligned}$$
As usual, the a priori estimate now follows from the discrete stability estimate.
Let $(u, \lambda) \in V \times \Lambda$ be the solution to the continuous problem and let $(u_h,\lambda_h)\in V_h\times \Lambda_h$ be its approximation obtained by solving the discrete problem . Then the following estimate holds $$\begin{aligned}
\|u-u_h\|_V+\| \lambda -\lambda_h\|_{Q} \ \lesssim \ & \inf_{v_h\in V_h} \| u-v_h\|_V + \inf_{\mu_h\in\Lambda_h} \left( \| \lambda-\mu_h\|_{Q} + \sqrt{\langle u,\mu_h\rangle}\, \right) \\
&+ \left(\sum_{E \in {\mathcal{G}_h}} {\rm osc}_{K(E)} (f)^2\right)^{1/2} \, .
\end{aligned}$$ \[apriori\]
In view of Theorem \[discrstab\], it holds that $$\begin{aligned}
\Bigg( \left(\sum_{E \in {\mathcal{G}_h}} h_E \left\|\mu_h -\lambda_h\right\|_{0,E}^2\right)^{1/2} + & \|u_h-v_h\|_V+\|\lambda_h-\mu_h\|_{Q} \Bigg)^2 \\
& \lesssim \mathcal{B}_h(u_h-v_h,\lambda_h-\mu_h;w_h,\mu_h-\lambda_h)\, ,
\end{aligned}$$ with some $w_h\in V_h$ satisfying $$\|w_h\|_V \lesssim \|u_h-v_h\|_V+\|\lambda_h-\mu_h\|_{Q} \, .
\label{wbound}$$ It follows that $$\begin{aligned}
&\mathcal{B}_h(u_h-v_h,\lambda_h-\mu_h;w_h,\mu_h-\lambda_h)\\
&=\mathcal{B}_h(u_h,\lambda_h;w_h,\mu_h-\lambda_h)-\mathcal{B}_h(v_h,\mu_h;w_h,\mu_h-\lambda_h)\\
&\leq \mathcal{L}(w_h)+\mathcal{B}(u-v_h,\lambda-\mu_h;w_h,\mu_h-\lambda_h)-\mathcal{B}(u,\lambda;w_h,\mu_h-\lambda_h)\\
&\phantom{=}+\alpha \sum_{E \in {\mathcal{G}_h}} h_E \left(\mu_h -\frac{\partial v_h}{\partial n},\mu_h -\lambda_h-\frac{\partial w_h}{\partial n}\right)_E ,\\
\end{aligned}$$ where we have used the bilinearity of $\mathcal{B}$ and $\mathcal{B}_h$, and the discrete problem statement . Observe that $$\begin{aligned}
\mathcal{L}(w_h) -\mathcal{B}(u,\lambda;w_h,\mu_h-\lambda_h) &= (f,w_h) - (\nabla u, \nabla w_h) + \langle w_h, \lambda \rangle + \langle u,\mu_h-\lambda_h\rangle \\
& \leq \langle u,\mu_h-\lambda_h\rangle + \langle u,\lambda_h-\lambda\rangle = \langle u,\mu_h-\lambda\rangle = \langle u,\mu_h\rangle \, ,
\end{aligned}$$ where we have used the problem and recalled that $0\leq \langle u,\mu-\lambda\rangle, \ \forall \mu \in \Lambda$. Moreover $$\begin{aligned}
\sum_{E \in {\mathcal{G}_h}} & h_E \left(\mu_h -\frac{\partial v_h}{\partial n}, \mu_h -\lambda_h-\frac{\partial w_h}{\partial n}\right)_E \\
\leq \ & \left( \sum_{E \in {\mathcal{G}_h}} h_E \left\|\mu_h -\frac{\partial v_h}{\partial n}\right\|_{0,E}^2\right)^{1/2} \,\left( \sum_{E \in {\mathcal{G}_h}} h_E \left\|\mu_h -\lambda_h\right\|_{0,E}^2\right)^{1/2}
\\
& + \left( \sum_{E \in {\mathcal{G}_h}} h_E \left\|\mu_h -\frac{\partial v_h}{\partial n}\right\|_{0,E}^2\right)^{1/2} \,\left( \sum_{E \in {\mathcal{G}_h}} h_E \left\|\frac{\partial w_h}{\partial n}\right\|_{0,E}^2\right)^{1/2} \, .
\end{aligned}$$ The a priori estimate now follows from the continuity of the bilinear form $\mathcal{B}$, the inverse estimate and bound , Lemma \[lem:lowresidual\], and from the triangle inequality.
A posteriori error analysis
===========================
We will define the local error estimators corresponding to the finite element solution $(u_h,\lambda_h)$ as $$\begin{aligned}
{2}
\eta_K^2 &= h_K^2 \| \Delta u_{h} + f \|_{0,K}^2, \qquad &K \in {\mathcal{C}_h}, \\
\eta_{E,\Omega}^2 &= h_E \left \| \left\llbracket \nabla u_h \cdot n \right\rrbracket \right\|_{0,E}^2, &E \in {\mathcal{E}_h}, \\
\eta_{E,\Gamma}^2 &= h_E \left\| \lambda_h - \frac{\partial u_{h}}{\partial n} \right\|_{0,E}^2, &E \in {\mathcal{G}_h},\label{localest} \\
\eta_{E,\Gamma_N}^2 &= h_E \left\| \frac{\partial u_{h}}{\partial n} \right\|_{0,E}^2, &E \in \mathcal{N}_h ,\end{aligned}$$ where $ \left\llbracket \nabla u_h \cdot n \right\rrbracket $ denotes the jump in the normal derivative across the inter-element boundaries.
We also define the global error estimators $\eta$ and $S$ through $$\begin{aligned}
\eta^2 & = \sum_{K \in {\mathcal{C}_h}} \eta_K^2 + \sum_{E \in {\mathcal{E}_h}} \eta_{E,\Omega}^2 + \sum_{E \in {\mathcal{G}_h}} \eta_{E,\Gamma}^2 + \sum_{E \in \mathcal{N}_h} \eta_{E,\Gamma_N}^2 , \\
S & = \|u_h^{-}\|_W + \sqrt{ (\lambda_h,u_h^{+})}\, .\end{aligned}$$ where $w_{+}=\max\{w,0\}$ denotes the positive and $w_{-}=\min\{w,0\}$ the negative part of $w$. We will now establish both the efficiency and the reliability of the error estimators.
It holds that $$\label{reli}
\|u-u_h\|_V + \|\lambda-\lambda_h\|_Q\ \lesssim\,\eta + S$$ and $$\label{effi}
\eta \ \lesssim \ \|u-u_h\|_V + \|\lambda-\lambda_h\|_Q + \Bigg( \sum_{K \in {\mathcal{C}_h}} {\mathrm{osc}}_{K} (f)^2\Bigg)^{1/2}.$$
The continuous stability condition (cf. Lemma \[contstab\]) guarantees the existence of $v\in V$ such that $$\|v\|_V \lesssim \|u-u_h\|_V +\|\lambda-\lambda_h\|_Q
\label{vbound}$$ and $$\Big( \|u-u_h\|_V +\|\lambda-\lambda_h\|_Q\Big)^2 \lesssim \mathcal{B}(u-u_h,\lambda-\lambda_h;v, \lambda_h-\lambda) .$$
On the other hand, testing in with $(-\tilde{v}, \lambda_h)$, where $\tilde{v}$ is the Clément interpolant of $v$, we obtain $$0\leq - \mathcal{B}_h(u_h,\lambda_h,-\tilde{v},0)+\mathcal{L}(-\tilde{v}) \, .$$ It follows that $$\begin{aligned}
&\Big( \|u-u_h\|_V +\|\lambda- \lambda_h\|_Q\Big)^2 \\ &\lesssim
\mathcal{B}(u,\lambda;v, \lambda_h-\lambda) - \mathcal{B}(u_h,\lambda_h;v, \lambda_h-\lambda) \\ & \quad - \mathcal{B}(u_h,\lambda_h,-\tilde{v},0) +\mathcal{L}(-\tilde{v}) +
\alpha\,\mathcal{S}_h(u_h,\lambda_h,-\tilde{v},0) \\
\qquad \qquad & \lesssim \mathcal{L}(v-\tilde{v}) - \mathcal{B}(u_h,\lambda_h;v-\tilde{v}, \lambda_h-\lambda) +\alpha\,\mathcal{S}_h(u_h,\lambda_h,-\tilde{v},0) ,\end{aligned}$$ where we have used the bilinearity of $\mathcal{B}$ and the problem statement .
Observe first that $$\begin{array}{ll}
\mathcal{L}(v-\tilde{v}) - \mathcal{B}(u_h,\lambda_h;v-\tilde{v}, \lambda_h-\lambda) \vspace{2mm} \\ \qquad \qquad = (f,v-\tilde{v}) - (\nabla u_h, \nabla(v-\tilde{v})) + \langle v-\tilde{v}, \lambda_h\rangle +\langle u_h, \lambda_h-\lambda\rangle .
\label{apostterms1}
\end{array}$$ After elementwise integration by parts, the first four terms on the right-hand side of read $$\begin{aligned}
&\sum_{K\in {\mathcal{C}_h}} (\Delta u_h+f, v-\tilde{v})_K - \sum_{E\in {\mathcal{E}_h}} ( \left\llbracket \nabla u_h \cdot n \right\rrbracket , v-\tilde{v} )_E \\
&\quad + \sum_{E\in {\mathcal{G}_h}} \left( \lambda_h-\frac{\partial u_h}{\partial n} , v-\tilde{v} \right)_E - \sum_{E\in \mathcal{N}_h} \left( \frac{\partial u_h}{\partial n} , v-\tilde{v} \right)_E \, .\end{aligned}$$ The last term in can be estimated as follows $$\begin{aligned}
\langle u_h, \lambda_h-\lambda\rangle \leq (\lambda_h,u_h^{+}) + \langle u_h^{-}, \lambda_h-\lambda \rangle \leq (\lambda_h,u_h^{+}) + \|\lambda-\lambda_h\|_Q \|u_h^{-}\|_W \, ,\end{aligned}$$ given that $\langle u_h^{+}, \lambda \rangle \geq 0$. For the stabilisation term $ \mathcal{S}_h(u_h,\lambda_h,-\tilde{v},0) $, we obtain $$\begin{aligned}
\sum_{E\in \mathcal{G}_h} &h_E \left(\lambda_h -\frac{\partial u_h}{\partial n},\frac{\partial \tilde{v}}{\partial n}\right)_E \\
& \leq
\left( \sum_{E \in {\mathcal{G}_h}} h_E \left\|\lambda_h -\frac{\partial u_h}{\partial n}\right\|_{0,E}^2\right)^{1/2} \,\left( \sum_{E \in {\mathcal{G}_h}} h_E \left\| \frac{\partial \tilde{v}}{\partial n}\right\|_{0,E}^2\right)^{1/2} \, \\
& \lesssim \left( \sum_{E \in {\mathcal{G}_h}} h_E \left\|\lambda_h -\frac{\partial u_h}{\partial n}\right\|_{0,E}^2\right)^{1/2} \Vert \tilde{v} \Vert_{V} ,\end{aligned}$$ where in the last step we have used the discrete inverse estimate .
For the Clément interpolant $\tilde{v}$, it holds that $$\label{cle}
\Vert \tilde{v} \Vert_{V}^2+ \sum_{K \in {\mathcal{C}_h}} h_K^{-2} \Vert v-\widetilde{v}\Vert _{0,K }^2+ \sum_{E \in {\mathcal{E}_h}\cup {\mathcal{G}_h}\cup \mathcal{N}_h} h_E^{-1} \Vert v-\tilde{v} \Vert_{0,E}^2 \ \lesssim \Vert v \Vert_{V}^2 .$$ The reliability estimate can now be established using the Cauchy-Schwarz inequality together with and .
The efficiency follows from standard lower bounds [@Vbook] and from Lemma \[inverse\].
Nitsche’s method
================
Nitsche’s formulation can be elegantly derived from the stabilised method following the reasoning suggested in [@Rolf95]. In fact, testing with $(0,\mu_h)$ in , yields $$(u_h,\mu_h-\lambda_h) +\alpha\, \sum_{E\in {\mathcal{G}_h}} h_E \left( \lambda_h-\frac{\partial u_h}{\partial n} , \mu_h-\lambda_h \right)_E \geq 0 \qquad \forall \mu_h\in\Lambda_h \, .$$ In particular, $$(u_h,\lambda_h) +\alpha\, \sum_{E\in {\mathcal{G}_h}} h_E \left( \lambda_h-\frac{\partial u_h}{\partial n} , \lambda_h \right)_E = 0\, .$$ We can thus write $$\sum_{E\in \Gamma_C} (u_h,\mu_h)_E +\alpha\, \sum_{E\in \Gamma_C} h_E \left( \lambda_h-\frac{\partial u_h}{\partial n} , \mu_h \right)_E = 0 \qquad \forall \mu_h\in\Lambda_h ,$$ where $\Gamma_C$ denotes the contact boundary. Locally, we obtain $$\lambda_h|_{E} = \Pi_h \frac{\partial u_h}{\partial n}\Big|_E - (\alpha h_E)^{-1} \Pi_h u_h\Big|_E \qquad \forall E\in \Gamma_C ,
\label{LMN}$$ where $\Pi_h$ is the $L^2$ projection onto $\Lambda_h$. Assuming equal polynomial order for both variables, say $k\geq 1$, but with the discrete Lagrange multiplier being discontinuous from element to element, it follows that $\Pi_h=I$ and we have $$Q_h=\{ \mu_h\in Q : \mu_h\in P_k(E)~ \forall E\in {\mathcal{G}_h}\} .$$ Therefore, using the test function $(v_h,0)$ in and substituting into the resulting equation, leads to $$\begin{aligned}
(\nabla u_h,\nabla v_h) + & \sum_{E\in \Gamma_C} (\alpha h_E)^{-1} (u_h , v_h)_E - \sum_{E\in \Gamma_C} \left\{ \left( \frac{\partial u_h}{\partial n},v_h\right)_E + \left(u_h, \frac{\partial v_h}{\partial n}\right)_{E} \right\} \\
& -\alpha \sum_{E\in \Gamma \setminus\Gamma_C} h_E \left( \frac{\partial u_h}{\partial n} , \frac{\partial v_h}{\partial n} \right)_E = (f, v_h) \qquad \forall v_h \in V_h . \end{aligned}$$
Defining an $L^2(\Gamma)$ function ${\mathpzc{h}}$ through $${\mathpzc{h}}|_E = h_E \quad \forall E\in{\mathcal{G}_h},$$ the discrete Lagrange multiplier can be written globally as $$\label{lambda}
\lambda_h = \left(\frac{\partial u_h}{\partial n}\Big|_{\Gamma} - (\alpha {\mathpzc{h}})^{-1} u_h|_\Gamma\right)_+$$ and the discrete contact region becomes $$\Gamma^h_C= \{ (x,y)\in \Gamma : \lambda_h(x,y) > 0\}.$$ [*Nitsche’s method*]{} now reads: find $u_h\in V_h$ and $\Gamma^h_C=\Gamma_C^h(u_h)$ such that $$\begin{array}{l}
\displaystyle (\nabla u_h,\nabla v_h)_\Omega + \ \alpha^{-1} \big( {\mathpzc{h}}^{-1} u_h , v_h\big)_{\Gamma_C^h }- \left( \frac{\partial u_h}{\partial n},v_h\right)_{\Gamma_C^h } - \left(u_h, \frac{\partial v_h}{\partial n}\right)_{\Gamma_C^h } \vspace{2mm} \\
\displaystyle \qquad \qquad \qquad -\alpha \left( {\mathpzc{h}}\, \frac{\partial u_h}{\partial n} , \frac{\partial v_h}{\partial n} \right)_{\Gamma \setminus\Gamma_C^h } = (f, v_h) \qquad \forall v_h \in V_h .
\end{array}
\label{nitsche}$$
Note that formulation is equivalent to the Nitsche’s method introduced in [@CH13], with $\gamma = \alpha {\mathpzc{h}}$, and to the symmetric variant $(\theta_1=-1)$ of the Nitsche’s method proposed in [@CHR15].
For the implementational aspects of the Nitsche’s method , we refer to [@GSV17a] where similar method was applied to the obstacle problem.
Numerical verification
======================
We consider the problem where the domain is given by $\Omega = (0,1)^2$, the loading is $f(x,y) = x \cos(2 \pi y)$, and the boundaries are $$\begin{aligned}
\Gamma_D &= \{ (x,y) \in \mathbb{R}^2 : x = 0,~0 < y < 1 \}, \\
\Gamma_N &= \{ (x,y) \in \mathbb{R}^2 : 0 < x < 1,~y = 0\} \cup \{ (x,y) \in \mathbb{R}^2 : 0 < x < 1,~y = 1\}, \\
\Gamma &= \{ (x,y) \in \mathbb{R}^2 : x=1,~0 < y < 1 \}.\end{aligned}$$ The contact region is now a proper subset of $\Gamma$. There are two points on $\Gamma$ where the constraints change from binding to non-binding and the exact solution belongs to $H^r$, $r<5/2$, see the comments in Remark \[bindingremark\]. We employ the quadratic finite element space $$V_h = \{ w \in V : w|_K \in P_2(K) ~\forall K \in \mathcal{C}_h \}.$$ Thus, in view of the regularity of $u$, the convergence rate of the error $\|u-u_h\|_V$ with uniform mesh refinement is limited to $\mathcal{O}(N^{-3/4})$ where $N$ is the number of degrees of freedom.
To overcome the limited regularity, we consider also a sequence of adaptive meshes. Starting with an initial mesh, we compute the discrete solution $u_h$, the corresponding Lagrange multiplier through , and the respective error indicator $$\begin{aligned}
\mathcal{E}_K^2 &= h_K^2 \| \Delta u_h + f \|_{0,K}^2 + h_K \| \llbracket \nabla u_h \cdot n \rrbracket \|_{0,\partial K}^2 \\
&\quad+ h_K \left\|\lambda_h - \frac{\partial u_h}{\partial n} \right\|_{0,\partial K \cap \Gamma}^2 + h_K \left\|\frac{\partial u_h}{\partial n}\right\|_{\partial K \cap \Gamma_N}^2\end{aligned}$$ for every $K$ in the initial mesh. The mesh is then improved by splitting the elements that have large error indicators. This process is performed repeatedly until a predetermined value of $N$ is reached. Please refer to [@GSV18] or [@GSV18b] for more details on how to choose and split the elements.
The discrete solution is visualised in Figure \[fig:solution\]. The resulting sequence of adaptive meshes is given in Figure \[fig:meshes\] and the convergence of the global error estimator $\eta+S$ is compared between the uniform and adaptive mesh sequences in Figure \[fig:convergence\]. As expected, the adaptive meshes are more refined near the singular points on the contact boundary. Moreover, the observed convergence rate of the uniform refinement strategy is suboptimal, due to the limited regularity of the exact solution, whereas the adaptive method successfully recovers the optimal rate of convergence, $\mathcal{O}(N^{-1})$.
![The discrete solution after 8 adaptive refinements. The contact region on the rightmost boundary is highlighted in red.[]{data-label="fig:solution"}](solution.pdf){width="90.00000%"}
![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh14-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh13-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh12-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh11-crop.pdf "fig:"){width="23.00000%"}\
![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh10-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh9-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh8-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh7-crop.pdf "fig:"){width="23.00000%"}\
![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh6-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh5-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh4-crop.pdf "fig:"){width="23.00000%"} ![The initial mesh and the resulting sequence of adaptively refined meshes.[]{data-label="fig:meshes"}](mesh3-crop.pdf "fig:"){width="23.00000%"}
![Convergence of the error estimator $\eta + S$ as a function of the number of degrees of freedom $N$ for the uniform and the adaptive mesh sequences. The observed approximate convergence rates are given in the text box window.[]{data-label="fig:convergence"}](convergence.pdf){width="\textwidth"}
[^1]: Funding from Tekes (Decision number 3305/31/2015) and the Finnish Cultural Foundation is gratefully acknowledged.
[^2]: Work supported by the Portuguese Science Foundation (FCOMP-01-0124-FEDER-029408) and the Finnish Academy of Science and Letters.
|
---
abstract: 'Most of existing image denoising methods assume the corrupted noise to be additive white Gaussian noise (AWGN). However, the realistic noise in real-world noisy images is much more complex than AWGN, and is hard to be modeled by simple analytical distributions. As a result, many state-of-the-art denoising methods in literature become much less effective when applied to real-world noisy images captured by CCD or CMOS cameras. In this paper, we develop a trilateral weighted sparse coding (TWSC) scheme for robust real-world image denoising. Specifically, we introduce three weight matrices into the data and regularization terms of the sparse coding framework to characterize the statistics of realistic noise and image priors. TWSC can be reformulated as a linear equality-constrained problem and can be solved by the alternating direction method of multipliers. The existence and uniqueness of the solution and convergence of the proposed algorithm are analyzed. Extensive experiments demonstrate that the proposed TWSC scheme outperforms state-of-the-art denoising methods on removing realistic noise.'
author:
- 'Jun Xu$^{1}$, Lei Zhang$^{1}$[^1], David Zhang$^{1, 2}$'
bibliography:
- 'egbib.bib'
title: |
A Trilateral Weighted Sparse Coding Scheme\
for Real-World Image Denoising
---
Introduction
============
Noise will be inevitably introduced in imaging systems and may severely damage the quality of acquired images. Removing noise from the acquired image is an essential step in photography and various computer vision tasks such as segmentation [@zhu2017non], HDR imaging [@autonoisemodel], and recognition [@nguyen2015deep], etc. Image denoising aims to recover the clean image $\mathbf{x}$ from its noisy observation $\mathbf{y}=\mathbf{x}+\mathbf{n}$, where $\mathbf{n}$ is the corrupted noise. This problem has been extensively studied in literature, and numerous statistical image modeling and learning methods have been proposed in the past decades [@ksvd; @srcolor; @nlm; @bm3d; @cbm3d; @bm3dsapca; @lssc; @ncsr; @saist; @wnnm; @pgpd; @foe; @epll; @mlp; @csf; @chen2015learning; @dncnn; @Liu2008; @noiseclinic; @Zhu_2016_CVPR; @crosschannel2016; @neatimage; @mcwnnm].
Most of the existing methods [@ksvd; @srcolor; @nlm; @bm3d; @cbm3d; @bm3dsapca; @lssc; @ncsr; @saist; @wnnm; @foe; @epll; @pgpd; @mlp; @csf; @chen2015learning; @dncnn] focus on additive white Gaussian noise (AWGN), and they can be categorized into dictionary learning based methods [@ksvd; @srcolor], nonlocal self-similarity based methods [@nlm; @bm3d; @cbm3d; @bm3dsapca; @lssc; @ncsr; @saist; @wnnm; @pgpd], sparsity based methods [@ksvd; @srcolor; @bm3d; @cbm3d; @bm3dsapca; @lssc; @ncsr], low-rankness based methods [@saist; @wnnm], generative learning based methods [@foe; @epll; @pgpd], and discriminative learning based methods [@mlp; @csf; @chen2015learning; @dncnn], etc. However, the realistic noise in real-world images captured by CCD or CMOS cameras is much more complex than AWGN [@Liu2008; @noiseclinic; @Zhu_2016_CVPR; @crosschannel2016; @xuaccv2016; @mcwnnm; @gid2018], which can be signal dependent and vary with different cameras and camera settings (such as ISO, shutter speed, and aperture, etc.). In Fig. \[f1\], we show a real-world noisy image from the Darmstadt Noise Dataset (DND) [@dnd2017] and a synthetic AWGN image from the Kodak PhotoCD Dataset (<http://r0k.us/graphics/kodak/>). We can see that the different local patches in real-world noisy image show different noise statistics, e.g., the patches in black and blue boxes show different noise levels although they are from the same white object. In contrast, all the patches from the synthetic AWGN image show homogeneous noise patterns. Besides, the realistic noise varies in different channels as well as different local patches [@noiseclinic; @Zhu_2016_CVPR; @crosschannel2016; @mcwnnm]. In Fig. \[f2\], we show a real-world noisy image captured by a Nikon D800 camera with ISO=6400, its “Ground Truth” (please refer to Section 4.3), and their differences in full color image as well as in each channel. The overall noise standard deviations (stds) in Red, Green, and Blue channels are 5.8, 4.4, and 5.5, respectively. Besides, the realistic noise is inhomogeneous. For example, the stds of noise in the three boxes plotted in Fig. \[f2\] (c) vary largely. Indeed, the noise in real-world noisy image is much more complex than AWGN noise. Though having shown promising performance on AWGN noise removal, many of the above mentioned methods [@ksvd; @srcolor; @nlm; @bm3d; @cbm3d; @bm3dsapca; @lssc; @ncsr; @saist; @wnnm; @foe; @epll; @pgpd; @mlp; @csf; @chen2015learning; @dncnn] will become much less effective when dealing with the complex realistic noise as shown in Fig. \[f2\].
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\[f1\]
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In the past decade, several denoising methods for real-world noisy images have been developed [@Liu2008; @noiseclinic; @Zhu_2016_CVPR; @crosschannel2016; @mcwnnm; @neatimage]. Liu et al. [@Liu2008] proposed to estimate the noise via a “noise level function” and remove the noise for each channel of the real image. However, processing each channel separately would often achieve unsatisfactory performance and generate artifacts [@srcolor]. The methods [@noiseclinic; @Zhu_2016_CVPR] perform image denoising by concatenating the patches of RGB channels into a vector. However, the concatenation does not consider the different noise statistics among different channels. Besides, the method of [@Zhu_2016_CVPR] models complex noise via mixture of Gaussian distribution, which is time-consuming due to the use of variational Bayesian inference techniques. The method of [@crosschannel2016] models the noise in a noisy image by a multivariate Gaussian and performs denoising by the Bayesian non-local means [@kervrann2007bayesian]. The commercial software Neat Image [@neatimage] estimates the global noise parameters from a flat region of the given noisy image and filters the noise accordingly. However, both the two methods [@crosschannel2016; @neatimage] ignore the local statistical property of the noise which is signal dependent and varies with different pixels. The method [@mcwnnm] considers the different noise statistics in different channels, but ignores that the noise is signal dependent and has different levels in different local patches. By far, real-world image denoising is still a challenging problem in low level vision [@dnd2017].
Sparse coding (SC) has been well studied in many computer vision and pattern recognition problems [@wright2009robust; @yang2009linear; @yang2010image], including image denoising [@ksvd; @srcolor; @lssc; @ncsr; @pgpd]. In general, given an input signal $\mathbf{y}$ and the dictionary $\mathbf{D}$ of coding atoms, the SC model can be formulated as $$\vspace{-5mm}
\label{e1}
\min_{\mathbf{c}}
\|
\mathbf{y}-\mathbf{D}\mathbf{c}
\|_{2}^{2}
+
\lambda\|\mathbf{c}\|_{q},$$ where $\mathbf{c}$ is the coding vector of the signal $\mathbf{y}$ over the dictionary $\mathbf{D}$, $\lambda$ is the regularization parameter, and $q=0$ or $1$ to enforce sparse regularization on $\mathbf{c}$. Some representative SC based image denoising methods include K-SVD [@ksvd], LSSC [@lssc], and NCSR [@ncsr]. Though being effective on dealing with AWGN, SC based denoising methods are essentially limited by the data-fidelity term described by $\ell_{2}$ (or Frobenius) norm, which actually assumes white Gaussian noise and is not able to characterize the signal dependent and realistic noise.
In this paper, we propose to lift the SC model (\[e1\]) to a robust denoiser for real-world noisy images by utilizing the channel-wise statistics and locally signal dependent property of the realistic noise, as demonstrated in Fig. \[f2\]. Specifically, we propose a trilateral weighted sparse coding (TWSC) scheme for real-world image denoising. Two weight matrices are introduced into the data-fidelity term of the SC model to characterize the realistic noise property, and another weight matrix is introduced into the regularization term to characterize the sparsity priors of natural images. We reformulate the proposed TWSC scheme into a linear equality-constrained optimization program, and solve it under the alternating direction method of multipliers (ADMM) [@admm] framework. One step of our ADMM is to solve a Sylvester equation, whose unique solution is not always guaranteed. Hence, we provide theoretical analysis on the existence and uniqueness of the solution to the proposed TWSC scheme. Experiments on three datasets of real-world noisy images demonstrate that the proposed TWSC scheme achieves much better performance than the state-of-the-art denoising methods.
The Proposed Real-World Image Denoising Algorithm
=================================================
The Trilateral Weighted Sparse Coding Model
-------------------------------------------
The real-world image denoising problem is to recover the clean image from its noisy observation. Current denoising methods [@ksvd; @srcolor; @nlm; @bm3d; @cbm3d; @bm3dsapca; @lssc; @ncsr; @saist; @wnnm; @pgpd; @foe; @epll] are mostly patch based. Given a noisy image, a local patch of size $p\times p \times 3$ is extracted from it and stretched to a vector, denoted by $\mathbf{y}=[\mathbf{y}_{r}^{\top}\ \mathbf{y}_{g}^{\top}\ \mathbf{y}_{b}^{\top}]^{\top}\in\mathbb{R}^{3p^{2}}$, where $\mathbf{y}_{c}\in\mathbb{R}^{p^{2}}$ is the corresponding patch in channel $c$, where $c\in\{r,g,b\}$ is the index of R, G, and B channels. For each local patch $\mathbf{y}$, we search the $M$ most similar patches to it (including $\mathbf{y}$ itself) by Euclidean distance in a local window around it. By stacking the $M$ similar patches column by column, we form a noisy patch matrix $\mathbf{Y}=\mathbf{X}+\mathbf{N}\in\mathbb{R}^{3p^{2}\times M}$, where $\mathbf{X}$ and $\mathbf{N}$ are the corresponding clean and noise patch matrices, respectively. The noisy patch matrix can be written as $\mathbf{Y}=[\mathbf{Y}_{r}^{\top}\ \mathbf{Y}_{g}^{\top}\ \mathbf{Y}_{b}^{\top}]^{\top}$, where $\mathbf{Y}_{c}$ is the sub-matrix of channel $c$. Suppose that we have a dictionary $\mathbf{D}=[\mathbf{D}_{r}^{\top}\ \mathbf{D}_{g}^{\top}\ \mathbf{D}_{b}^{\top}]^{\top}$, where $\mathbf{D}_{c}$ is the sub-dictionary corresponding to channel $c$. In fact, the dictionary $\mathbf{D}$ can be learned from external natrual images, or from the input noisy patch matrix $\mathbf{Y}$.
Under the traditional sparse coding (SC) framework [@lasso], the sparse coding matrix of $\mathbf{Y}$ over $\mathbf{D}$ can be obtained by $$\vspace{-4mm}
\label{e2}
\hat{\mathbf{C}}
=
\arg\min_{\mathbf{C}}
\|\mathbf{Y}-\mathbf{D}\mathbf{C}\|_{F}^{2}
+
\lambda\|\mathbf{C}\|_{1},$$ where $\lambda$ is the regularization parameter. Once $\hat{\mathbf{C}}$ is computed, the latent clean patch matrix $\hat{\mathbf{X}}$ can be estimated as $\hat{\mathbf{X}}=\mathbf{D}\hat{\mathbf{C}}$. Though having achieved promising performance on additive white Gaussian noise (AWGN), the tranditional SC based denoising methods [@ksvd; @srcolor; @lssc; @ncsr; @pgpd] are very limited in dealing with realistic noise in real-world images captured by CCD or CMOS cameras. The reason is that the realistic noise is non-Gaussian, varies locally and across channels, which cannot be characterized well by the Frobenius norm in the SC model (\[e2\]) [@Liu2008; @crosschannel2016; @dnd2017; @jddcrf].
To account for the varying statistics of realistic noise in different channels and different patches, we introduce two weight matrices $\mathbf{W}_{1}\in\mathbb{R}^{3p^2\times3p^2}$ and $\mathbf{W}_{2}\in\mathbb{R}^{M\times M}$ to characterize the SC residual ($\mathbf{Y}-\mathbf{D}\mathbf{C}$) in the data-fidelity term of Eq. (\[e2\]). Besides, to better characterize the sparsity priors of the natural images, we introduce a third weight matrix $\mathbf{W}_{3}$, which is related to the distribution of the sparse coefficients matrix $\mathbf{C}$, into the regularization term of Eq. (\[e2\]). For the dictionary $\mathbf{D}$, we learn it adaptively by applying the SVD [@svd] to the given data matrix $\mathbf{Y}$ as $$\vspace{-3mm}
\label{e3}
\mathbf{Y} =\mathbf{D}\mathbf{S}\mathbf{V}^{\top}.$$ Note that in this paper, we are not aiming at proposing a new dictionary learning scheme as [@ksvd] did. Once obtained from SVD, the dictionary $\mathbf{D}$ is fixed and not updated iteratively. Finally, the proposed trilateral weighted sparse coding (TWSC) model is formulated as: $$\vspace{-4mm}
\label{e4}
\min_{\mathbf{C}}\|\mathbf{W}_{1}(\mathbf{Y}-\mathbf{D}\mathbf{C})\mathbf{W}_{2}\|_{F}^{2}
+
\|\mathbf{W}_{3}^{-1}\mathbf{C}\|_{1}.$$ Note that the parameter $\lambda$ has been implicitly incorporated into the weight matrix $\mathbf{W}_{3}$.
The Setting of Weight Matrices
------------------------------
In this paper, we set the three weight matrices $\mathbf{W}_{1}$, $\mathbf{W}_{2}$, and $\mathbf{W}_{3}$ as diagonal matrices and grant clear physical meanings to them. $\mathbf{W}_{1}$ is a block diagonal matrix with three blocks, each of which has the same diagonal elements to describe the noise properties in the corresponding R, G, or B channel. Based on [@Leungtip; @dnd2017; @jddcrf], the realistic noise in a local patch could be approximately modeled as Gaussian, and each diagonal element of $\mathbf{W}_{2}$ is used to describe the noise variance in the corresponding patch $\mathbf{y}$. Generally speaking, $\mathbf{W}_{1}$ is employed to regularize the row discrepancy of residual matrix ($\mathbf{Y}-\mathbf{D}\mathbf{C}$), while $\mathbf{W}_{2}$ is employed to regularize the column discrepancy of ($\mathbf{Y}-\mathbf{D}\mathbf{C}$). For matrix $\mathbf{W}_{3}$, each diagonal element is set based on the sparsity priors on $\mathbf{C}$.
We determine the three weight matrices $\mathbf{W}_{1}$, $\mathbf{W}_{2}$, and $\mathbf{W}_{3}$ by employing the *Maximum A-Posterior* (MAP) estimation technique: $$\vspace{-4mm}
\begin{split}
\label{e5}
\hat{\mathbf{C}}
&
=
\arg\max_{\mathbf{C}}\ln P(\mathbf{C}|\mathbf{Y})
=
\arg\max_{\mathbf{C}}\{\ln P(\mathbf{Y}|\mathbf{C})+\ln P(\mathbf{C})\}.
\end{split}$$ The log-likelihood term $\ln P(\mathbf{Y}|\mathbf{C})$ is characterized by the statistics of noise. According to [@Leungtip; @dnd2017; @jddcrf], it can be assumed that the noise is independently and identically distributed (i.i.d.) in each channel and each patch with Gaussian distribution. Denote by $\mathbf{y}_{cm}$ and $\mathbf{c}_{m}$ the $m$th column of the matrices $\mathbf{Y}_{c}$ and $\mathbf{C}$, respectively, and denote by $\sigma_{cm}$ the noise std of $\mathbf{y}_{cm}$. We have $$\vspace{-4mm}
\begin{split}
\label{e6}
\hspace{-3mm}
P(\mathbf{Y}|\mathbf{C})
&
=
\hspace{-3mm}
\prod_{c\in\{r, g, b\}}
\prod_{m=1}^{M}
(\pi\sigma_{cm})^{-p^{2}}
e^{-\sigma_{cm}^{-2}
\|
\mathbf{y}_{cm}
-
\mathbf{D}_{c}\mathbf{c}_{m}
\|_{2}^{2}}.
\end{split}$$ From the perspective of statistics [@glm], the set of $\{\sigma_{cm}\}$ can be viewed as a $3\times M$ contingency table created by two variables $\sigma_{c}$ and $\sigma_{m}$, and their relationship could be modeled by a log-linear model $\sigma_{cm}=\sigma_{c}^{l_{1}}\sigma_{m}^{l_{2}}$, where $l_{1}+l_{2}=1$. Here we consider $\{\sigma_{c}\}, \{\sigma_{m}\}$ of equal importance and empirically set $l_{1}=l_{2}=1/2$. The estimation of $\{\sigma_{cm}\}$ can be transferred to the estimation of $\{\sigma_{c}\}$ and $\{\sigma_{m}\}$, which will be introduced in the experimental section (Section 4).
The sparsity prior is imposed on the coefficients matrix $\mathbf{C}$, we assume that each column $\mathbf{c}_{m}$ of $\mathbf{C}$ follows i.i.d. Laplacian distribution. Specifically, for each entry $\mathbf{c}_{m}^{i}$, which is the coding coefficient of the $m$th patch $\mathbf{y}_{m}$ over the $i$th atom of dictionary $\mathbf{D}$, we assume that it follows distribution of $(2\bm{S}_{i})^{-1}\exp(-\mathbf{S}_{i}^{-1}|\mathbf{c}_{m}^{i}|)$, where $\mathbf{S}_{i}$ is the $i$th diagonal element of the singular value matrix $\mathbf{S}$ in Eq. (\[e3\]). Note that we set the scale factor of the distribution as the inverse of the $i$th singular value $\mathbf{S}_{i}$. This is because the larger the singular value $\mathbf{S}_{i}$ is, the more important the $i$th atom (i.e., singular vector) in $\mathbf{D}$ should be, and hence the distribution of the coding coefficients over this singular vector should have stronger regularization with weaker sparsity. The prior term in Eq. (\[e5\]) becomes $$\vspace{-4mm}
\label{e7}
P(\mathbf{C})
=
\prod_{m=1}^{M}
\prod_{i=1}^{3p^{2}}
(2\mathbf{S}_{i})^{-1}e^{-\mathbf{S}_{i}^{-1}|\mathbf{c}_{m}^{i}|}.$$
Put (\[e7\]) and (\[e6\]) into (\[e5\]) and consider the log-linear model $\sigma_{cm}=\sigma_{c}^{1/2}\sigma_{m}^{1/2}$, we have $$\vspace{-4mm}
\begin{split}
\label{e8}
\hspace{-0mm}
\hat{\mathbf{C}}
&
=
\arg\min_{\mathbf{C}}
\hspace{-4mm}
\sum_{c\in\{r, g, b\}}
\hspace{-1mm}
\sum_{m=1}^{M}
\hspace{-1mm}
\sigma_{cm}^{-2}
\|\mathbf{y}_{cm}
\hspace{-1mm}
-
\hspace{-1mm}
\mathbf{D}_{c}\mathbf{c}_{m}\|_{2}^{2}
\hspace{-0mm}
+
\hspace{-1mm}
\sum_{m=1}^{M}
\hspace{-1mm}
\|\mathbf{S}^{-1}\mathbf{c}_{m}\|_{1}
\\
&
=
\arg\min_{\mathbf{C}}
\hspace{-4mm}
\sum_{c\in\{r, g, b\}}
\hspace{-3mm}
\sigma_{c}^{-1}
\|(\mathbf{Y}_{c}-\mathbf{D}_{c}\mathbf{C})\mathbf{W}_{2}\|_{F}^{2}+\|\mathbf{S}^{-1}\mathbf{C}\|_{1}
\\
&
=
\arg\min_{\mathbf{C}}\|\mathbf{W}_{1}(\mathbf{Y}-\mathbf{D}\mathbf{C})\mathbf{W}_{2}\|_{F}^{2}+\|\mathbf{W}_{3}^{-1}\mathbf{C}\|_{1},
\end{split}$$ where $$\vspace{-3mm}
\begin{split}
\label{e9}
\mathbf{W}_{1}
&
=
\text{diag}(\sigma_{r}^{-1/2}\mathbf{I}_{p^2},\sigma_{g}^{-1/2}\mathbf{I}_{p^2},\sigma_{b}^{-1/2}\mathbf{I}_{p^2})
,
\\
\mathbf{W}_{2}
&
=
\text{diag}(\sigma_{1}^{-1/2},...,\sigma_{M}^{-1/2})
,
\mathbf{W}_{3}
=
\mathbf{S}
,
\end{split}$$ and $\mathbf{I}_{p^2}$ is the $p^{2}$ dimensional identity matrix. Note that the diagonal elements of $\mathbf{W}_{1}$ and $\mathbf{W}_{2}$ are determined by the noise standard deviations in the corresponding channels and patches, respectively. The stronger the noise in a channel and a patch, the less that channel and patch will contribute to the denoised output.
Model Optimization
------------------
Letting $\mathbf{C}^{*}=\mathbf{W}_{3}^{-1}\mathbf{C}$, we can transfer the weight matrix $\mathbf{W}_{3}$ into the data-fidelity term of (\[e4\]). Thus, the TWSC scheme (\[e4\]) is reformulated as $$\vspace{-3mm}
\label{e10}
\min_{\mathbf{C}^{*}}\|\mathbf{W}_{1}(\mathbf{Y}-\mathbf{D}\mathbf{W}_{3}\mathbf{C}^{*})\mathbf{W}_{2}\|_{F}^{2}
+
\|\mathbf{C}^{*}\|_{1}.$$ To make the notation simple, we remove the superscript $*$ in $\mathbf{C}^{*}$ and still use $\mathbf{C}$ in the following development. We employ the variable splitting method [@Eckstein1992] to solve the problem (\[e10\]). By introducing an augmented variable $\mathbf{Z}$, the problem (\[e10\]) is reformulated as a linear equality-constrained problem with two variables $\mathbf{C}$ and $\mathbf{Z}$: $$\vspace{-3mm}
\label{e11}
\min_{\mathbf{C},\mathbf{Z}}\|\mathbf{W}_{1}(\mathbf{Y}-\mathbf{D}\mathbf{W}_{3}\mathbf{C})\mathbf{W}_{2}\|_{F}^{2}
+
\|\mathbf{Z}\|_{1}
\
\text{s.t.}
\
\mathbf{C}=\mathbf{Z}.$$
Since the objective function is separable w.r.t. the two variables, the problem (\[e11\]) can be solved under the alternating direction method of multipliers (ADMM) [@admm] framework. The augmented Lagrangian function of (\[e11\]) is: $$\vspace{-3mm}
\label{e12}
\begin{split}
\hspace{-2mm}
\mathcal{L}
&
(\mathbf{C},\mathbf{Z},\mathbf{\Delta},\rho)
\hspace{-1mm}
=
\hspace{-1mm}
\|\mathbf{W}_{1}(\mathbf{Y}-\mathbf{D}\mathbf{W}_{3}\mathbf{C})\mathbf{W}_{2}\|_{F}^{2}
\hspace{-1mm}
+
\hspace{-1mm}
\|\mathbf{Z}\|_{1}
\hspace{-1mm}
+
\hspace{-1mm}
\langle
\mathbf{\Delta},\mathbf{C}-\mathbf{Z}
\rangle
\hspace{-1mm}
+
\hspace{-1mm}
\frac{\rho}{2}
\|\mathbf{C}-\mathbf{Z}\|_{F}^{2},
\end{split}$$ where $\mathbf{\Delta}$ is the augmented Lagrangian multiplier and $\rho>0$ is the penalty parameter. We initialize the matrix variables $\mathbf{C}_{0}$, $\mathbf{Z}_{0}$, and $\mathbf{\Delta}_{0}$ to be comfortable zero matrices and $\rho_{0}>0$. Denote by ($\mathbf{C}_{k}, \mathbf{Z}_{k}$) and $\mathbf{\Delta}_{k}$ the optimization variables and Lagrange multiplier at iteration $k$ ($k=0,1,2,...$), respectively. By taking derivatives of the Lagrangian function $\mathcal{L}$ w.r.t. $\mathbf{C}$ and $\mathbf{Z}$, and setting the derivatives to be zeros, we can alternatively update the variables as follows:\
(1) **Update $\mathbf{C}$ by fixing $\mathbf{Z}$ and $\mathbf{\Delta}$**: $$\vspace{-3mm}
\begin{split}
\label{e13}
\mathbf{C}_{k+1}
=
\arg\min_{\mathbf{C}}
&
\|\mathbf{W}_{1}(\mathbf{Y}-\mathbf{D}\mathbf{W}_{3}\mathbf{C})\mathbf{W}_{2}\|_{F}^{2}
+
\frac{\rho_{k}}{2}\|\mathbf{C} - \mathbf{Z}_{k} + \rho_{k}^{-1}\mathbf{\Delta}_{k}||_{F}^{2}.
\end{split}$$ This is a two-sided weighted least squares regression problem with the solution satisfying that $$\vspace{-3mm}
\label{e14}
\mathbf{A}\mathbf{C}_{k+1}
+
\mathbf{C}_{k+1}\mathbf{B}_{k}
=
\mathbf{E}_{k},$$ where $$\vspace{-3mm}
\begin{split}
\label{e15}
\mathbf{A}
&
=
\mathbf{W}_{3}^{\top}\mathbf{D}^{\top}\mathbf{W}_{1}^{\top}\mathbf{W}_{1}\mathbf{D}\mathbf{W}_{3}
,
\mathbf{B}_{k}
=
\frac{\rho_{k}}{2}(\mathbf{W}_{2}\mathbf{W}_{2}^{\top})^{-1}
,
\\
\mathbf{E}_{k}
&
=
\mathbf{W}_{3}^{\top}\mathbf{D}^{\top}\mathbf{W}_{1}^{\top}\mathbf{W}_{1}\mathbf{Y}+(\frac{\rho_{k}}{2}\mathbf{Z}_{k} -\frac{1}{2}\mathbf{\Delta}_{k})(\mathbf{W}_{2}\mathbf{W}_{2}^{\top})^{-1}
.
\end{split}$$ Eq. (\[e14\]) is a standard Sylvester equation (SE) which has a unique solution if and only if $\sigma(\mathbf{A})\cap\sigma(-\mathbf{B}_{k})=\emptyset$, where $\sigma(\mathbf{F})$ denotes the spectrum, i.e., the set of eigenvalues, of the matrix $\mathbf{F}$ [@simoncini2016computational]. We can rewrite the SE (\[e14\]) as $$\vspace{-3mm}
\label{e16}
(\mathbf{I}_{M}\otimes\bm{A}
+
\mathbf{B}_{k}^{\top}\otimes\bm{I}_{3p^2})\text{vec}(\mathbf{C}_{k+1})
=
\text{vec}(\mathbf{E}_{k}),$$ and the solution $\mathbf{C}_{k+1}$ (if existed) can be obtained via $\mathbf{C}_{k+1}=\text{vec}^{-1}(\text{vec}(\mathbf{C}_{k+1}))$, where $\text{vec}^{-1}(\bullet)$ is the inverse of the vec-operator $\text{vec}(\bullet)$. Detailed theoretical analysis on the existence of the unique solution is given in Section 3.1.\
(2) **Update $\mathbf{Z}$ by fixing $\mathbf{C}$ and $\mathbf{\Delta}$**: $$\vspace{-1mm}
\label{e17}
\mathbf{Z}_{k+1}
=
\arg\min_{\mathbf{Z}}\frac{\rho_{k}}{2}
\|\mathbf{Z} - (\mathbf{C}_{k+1}+\rho_{k}^{-1}\mathbf{\Delta}_{k})\|_{F}^{2}
+
\|\mathbf{Z}\|_{1}.$$ This problem has a closed-form solution as $$\vspace{-2mm}
\label{e18}
\mathbf{Z}_{k+1}
=
\mathcal{S}_{\rho_{k}^{-1}}(\mathbf{C}_{k+1}+\rho_{k}^{-1}\mathbf{\Delta}_{k}),$$ where $\mathcal{S}_{\lambda}(x) = \text{sign}(x)\cdot\max(x-\lambda, 0)$ is the soft-thresholding operator.\
(3) **Update $\mathbf{\Delta}$ by fixing $\mathbf{X}$ and $\mathbf{Z}$**: $$\vspace{-2mm}
\label{e19}
\mathbf{\Delta}_{k+1}
=
\mathbf{\Delta}_{k} + \rho_{k}(\mathbf{C}_{k+1}-\mathbf{Z}_{k+1}).$$ (4) **Update $\rho$**: $\rho_{k+1}= \mu\rho_{k}$, where $\mu\ge1$.
The above alternative updating steps are repeated until the convergence condition is satisfied or the number of iterations exceeds a preset threshold $K_{1}$. The ADMM algorithm converges when $\|\mathbf{C}_{k+1}-\mathbf{Z}_{k+1}\|_{F}\le \text{Tol}$, $\|\mathbf{C}_{k+1}-\mathbf{C}_{k}\|_{F}\le \text{Tol}$, and $\|\mathbf{Z}_{k+1}-\mathbf{Z}_{k}\|_{F}\le \text{Tol}$ are simultaneously satisfied, where $\text{Tol}>0$ is a small tolerance number. We summarize the updating procedures in Algorithm 1.
[l]{} **Algorithm 1**: Solve the TWSC Model (\[e4\]) via ADMM\
**Input:** $\mathbf{Y},\mathbf{W}_{1},\mathbf{W}_{2},\mathbf{W}_{3}$, $\mu$, $\text{Tol}$, $K_{1}$;\
**Initialization:** $\mathbf{C}_{0}=\mathbf{Z}_{0}=\mathbf{\Delta}_{0}=\mathbf{0}$, $\rho_{0}>0$, $k=0$, = ;\
**While** ( == ) **do**\
1. Update $\mathbf{C}_{k+1}$ by solving Eq. (\[e13\]);\
2. Update $\mathbf{Z}_{k+1}$ by soft thresholding (\[e18\]);\
3. Update $\mathbf{\Delta}_{k+1}$ by Eq. (\[e19\]);\
4. Update $\rho_{k+1}$ by $\rho_{k+1}=\mu\rho_{k}$, where $\mu\ge1$;\
5. $k \leftarrow k + 1$;\
**if** (Converged) or ($k\ge K_{1}$)\
6. $\leftarrow$ ;\
**end if**\
**end while**\
**Output:** Matrices $\mathbf{C}$ and $\mathbf{Z}$.\
**Convergence Analysis**. The convergence of Algorithm 1 can be guaranteed since the overall objective function (\[e11\]) is convex with a global optimal solution. In Fig. \[f3\], we can see that the maximal values in $|\mathbf{C}_{k+1}-\mathbf{Z}_{k+1}|$, $|\mathbf{C}_{k+1}-\mathbf{C}_{k}|$, $|\mathbf{Z}_{k+1}-\mathbf{Z}_{k}|$ approach to $0$ simultaneously in 50 iterations.
The Denoising Algorithm
-----------------------
Given a noisy color image, suppose that we have extracted $N$ local patches $\{\mathbf{y}_{j}\}_{j=1}^{N}$ and their similar patches. Then $N$ noisy patch matrices $\{\mathbf{Y}_{j}\}_{j=1}^{N}$ can be formed to estimate the clean patch matrices $\{\mathbf{X}_{j}\}_{j=1}^{N}$. The patches in matrices $\{\mathbf{X}_{j}\}_{j=1}^{N}$ are aggregated to form the denoised image $\hat{\mathbf{x}}_{c}$. To obtain better denoising results, we perform the above denoising procedures for several (e.g., $K_{2}$) iterations. The proposed TWSC scheme based real-world image denoising algorithm is summarized in Algorithm 2.
[l]{} **Algorithm 2**: Image Denoising by TWSC\
**Input:** Noisy image $\mathbf{y}_{c}$, $\{\sigma_{r}, \sigma_{g}, \sigma_{b}\}$, $K_{2}$;\
**Initialization:** $\hat{\mathbf{x}}_{c}^{(0)}=\mathbf{y}_{c}$, $\mathbf{y}_{c}^{(0)}=\mathbf{y}_{c}$;\
**for** $k = 1:K_{2}$ **do**\
1. Set $\mathbf{y}_{c}^{(k)}=\hat{\mathbf{x}}_{c}^{(k-1)}$;\
2. Extract local patches $\{\mathbf{y}_{j}\}_{j=1}^{N}$ from $\mathbf{y}_{c}^{(k)}$;\
**for** each patch $\mathbf{y}_{j}$ **do**\
3.Search nonlocal similar patches $\mathbf{Y}_{j}$;\
4.Apply the TWSC scheme (\[e4\]) to $\mathbf{Y}_{j}$ and obtain the estimated $\mathbf{X}_{j}=\mathbf{D}\mathbf{C}$;\
**end for**\
5. Aggregate $\{\mathbf{X}_{j}\}_{j=1}^{N}$ to form the image $\hat{\mathbf{x}}_{c}^{(k)}$;\
**end for**\
**Output:** Denoised image $\hat{\mathbf{x}}_{c}^{(K_{2})}$.\
Existence and Faster Solution of Sylvester Equation
===================================================
The solution of the Sylvester equation (SE) (\[e14\]) does not always exist, though the solution is unique if it exists. Besides, solving SE (\[e14\]) is usually computationally expensive in high dimensional cases. In this section, we provide a sufficient condition to guarantee the existence of the solution to SE (\[e14\]), as well as a faster solution of (\[e14\]) to save the computational cost of Algorithms 1 and 2.
Existence of the Unique Solution
--------------------------------
Before we prove the existence of unique solution of SE (\[e14\]), we first introduce the following theorem.
\[th1\] Assume that $\mathbf{A}\in\mathbb{R}^{3p^2\times 3p^2}$, $\mathbf{B}\in\mathbb{R}^{M\times M}$ are both symmetric and positive semi-definite matrices. If at least one of $\mathbf{A}, \mathbf{B}$ is positive definite, the Sylvester equation $\mathbf{A}\mathbf{C}
+
\mathbf{C}\mathbf{B}
=
\mathbf{E}$ has a unique solution for $\mathbf{C}\in \mathbb{R}^{3p^2\times M}$.
The proof of Theorem \[th1\] can be found in the supplementary file. Then we have the following corollary.
The SE (\[e14\]) has a unique solution.
Since $\mathbf{A},\mathbf{B}_{k}$ in (\[e14\]) are both symmetric and positive definite matrices, according to Theorem \[th1\], the SE (\[e14\]) has a unique solution.
Faster Solution
---------------
The solution of the SE (\[e14\]) is typically obtained by the Bartels-Stewart algorithm [@Bartels1972]. This algorithm firstly employs a QR factorization [@GolubMatrix], implemented via Gram-Schmidt process, to decompose the matrices $\mathbf{A}$ and $\mathbf{B}_{k}$ into Schur forms, and then solves the obtained triangular system by the back-substitution method [@bareiss1968sylvester]. However, since the matrices $\mathbf{I}_{M}\otimes\bm{A}$ and $\mathbf{B}_{k}^{\top}\otimes\bm{I}_{3p^2}$ are of $3p^2M\times 3p^2M$ dimensions, it is computationally expensive ($\mathcal{O}(p^6M^3)$) to calculate their QR factorization to obtain the Schur forms. By exploiting the specific properties of our problem, we provide a faster while exact solution for the SE (\[e14\]).
Since the matrices $\mathbf{A},\mathbf{B}_{k}$ in (\[e14\]) are symmetric and positive definite, the matrix $\mathbf{A}$ can be eigen-decomposed as $\mathbf{A}=\mathbf{U}_{\mathbf{A}}\mathbf{\Sigma}_{\mathbf{A}}\mathbf{U}_{\mathbf{A}}^{\top}$, with computational cost of $\mathcal{O}(p^6)$. Left multiply both sides of the SE (\[e14\]) by $\mathbf{U}_{\mathbf{A}}^{\top}$, we can get $
\mathbf{\Sigma}_{A}\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{C}_{k+1}
+
\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{C}_{k+1}\mathbf{B}_{k}
=
\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{E}_{k}
$. This can be viewed as an SE w.r.t. the matrix $\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{C}_{k+1}$, with a unique solution $
\text{vec}(\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{C}_{k+1})=(\mathbf{I}_{M}\otimes\bm{\Sigma}_{\mathbf{A}}
+
\mathbf{B}_{k}^{\top}\otimes\bm{I}_{3p^2})^{-1}
\text{vec}(\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{E}_{k})
$. Since the matrix $(\mathbf{I}_{M}\otimes\bm{\Sigma}_{\mathbf{A}}
+
\mathbf{B}_{k}^{\top}\otimes\bm{I}_{3p^2})$ is diagonal and positive definite, its inverse can be calculated on each diagonal element of $(\mathbf{I}_{M}\otimes\bm{\Sigma}_{\mathbf{A}}
+
\mathbf{B}_{k}^{\top}\otimes\bm{I}_{3p^2})$. The computational cost for this step is $\mathcal{O}(p^2M)$. Finally, the solution $\mathbf{C}_{k+1}$ can be obtained via $
\mathbf{C}_{k+1}=\mathbf{U}_{\mathbf{A}}\text{vec}^{-1}(\text{vec}(\mathbf{U}_{\mathbf{A}}^{\top}\mathbf{C}_{k+1}))
$. By this way, the complexity for solving the SE (\[e14\]) is reduced from $\mathcal{O}(p^6M^3)$ to $\mathcal{O}(\max(p^6,p^2M))$, which is a huge computational saving.
Experiments
===========
To validate the effectiveness of our proposed TWSC scheme, we apply it to both synthetic additive white Gaussian noise (AWGN) corrupted images and real-world noisy images captured by CCD or CMOS cameras. To better demonstrate the roles of the three weight matrices in our model, we compare with a baseline method, in which the weight matrices $\mathbf{W}_{1},\mathbf{W}_{2}$ are set as comfortable identity matrices, while the matrix $\mathbf{W}_{3}$ is set as in (\[e8\]). We call this baseline method the *Weighted Sparse Coding* (WSC).
Experimental Settings
---------------------
**Noise Level Estimation**. For most image denoising algorithms, the standard deviation (std) of noise should be given as a parameter. In this work, we provide an exploratory approach to solve this problem. Specifically, the noise std $\sigma_{c}$ of channel $c$ can be estimated by some noise estimation methods [@noiselevel; @Chen2015ICCV; @Sutournlf]. In Algorithm 2, the noise std for the $m$th patch of $\mathbf{Y}$ can be initialized as $$\vspace{-3mm}
\label{e20}
\sigma_{m}=\sigma\triangleq \sqrt{(\sigma_{r}^{2}+\sigma_{g}^{2}+\sigma_{b}^{2})/3}$$ and updated in the following iterations as $$\vspace{-2mm}
\label{e21}
\sigma_{m}=\sqrt{\max(0,\sigma^2-\|\mathbf{y}_{m}-\mathbf{x}_{m}\|_{2}^{2})},$$ where $\mathbf{y}_{m}$ is the $m$th column in the patch matrix $\mathbf{Y}$, and $\mathbf{x}_{m}=\mathbf{D}\mathbf{c}_{m}$ is the $m$th patch recovered in previous iteration (please refer to Section 2.4).
**Implementation Details**. We empirically set the parameter $\rho_{0}=0.5$ and $\mu=1.1$. The maximum number of iteration is set as $K_{1}=10$. The window size for similar patch searching is set as $60\times60$. For parameters $p$, $M$, $K_{2}$, we set $p=7$, $M=70$, $K_{2}=8$ for $0<\sigma\le20$; $p=8$, $M=90$, $K_{2}=12$ for $20<\sigma\le40$; $p=8$, $M=120$, $K_{2}=12$ for $40<\sigma\le60$; $p=9$, $M=140$, $K_{2}=14$ for $60<\sigma\le100$. All parameters are fixed in our experiments. We will release the code with the publication of this work.
\[t1\]
[1]{}[@cccccccccc]{} $\sigma_{n}$ & Metric & **BM3D-SAPCA** & **LSSC** & **NCSR** & **WNNM** & **TNRD** & **DnCNN** & **WSC** & **TWSC**\
& PSNR & 32.42 & 32.27 & 32.19 & 32.43 & 32.27 & 32.59 & 32.06 & 32.34\
& SSIM & 0.8860 & 0.8849 & 0.8814 & 0.8841 & 0.8815 & 0.8879 & 0.8673 & 0.8846\
& PSNR & 30.02 & 29.84 & 29.76 & 30.05 & 29.87 & 30.22 & 29.57 & 29.98\
& SSIM & 0.8364 & 0.8329 & 0.8293 & 0.8365 & 0.8314 & 0.8415 & 0.8179 & 0.8372\
& PSNR & 28.48 & 28.26 & 28.17 & 28.51 & 28.33 & 28.66 & 28.01 & 28.49\
& SSIM & 0.7969 & 0.7908 & 0.7855 & 0.7958 & 0.7907 & 0.8021 & 0.7765 & 0.7987\
& PSNR & 26.85 & 26.64 & 26.55 & 26.92 & 26.75 & 27.08 & 26.35 & 26.93\
& SSIM & 0.7481 & 0.7405 & 0.7391 & 0.7499 & 0.7415 & 0.7563 & 0.7258 & 0.7530\
& PSNR & 24.74 & 24.77 & 24.66 & 25.15 & 24.97 & 25.24 & 24.54 & 25.15\
& SSIM & 0.6649 & 0.6746 & 0.6793 & 0.6903 & 0.6801 & 0.6931 & 0.6612 & 0.6949\
We first compare the proposed TWSC scheme with the leading AWGN denoising methods such as BM3D-SAPCA [@bm3dsapca] (which usually performs better than BM3D [@bm3d]), LSSC [@lssc], NCSR [@ncsr], WNNM [@wnnm], TNRD [@chen2015learning], and DnCNN [@dncnn] on 20 grayscale images commonly used in [@bm3d]. Note that TNRD and DnCNN are both discriminative learning based methods, and we use the models trained originally by the authors. Each noisy image is generated by adding the AWGN noise to the clean image, while the std of the noise is set as $\sigma\in\{15,25,35,50,75\}$ in this paper. Note that in this experiment we set the weight matrix $\mathbf{W}_{1}=\sigma^{-1/2}\mathbf{I}_{p^2}$ since the input images are grayscale.
The averaged PSNR and SSIM [@ssim] results are listed in Table \[t1\]. One can see that the proposed TWSC achieves comparable performance with WNNM, TNRD and DnCNN in most cases. It should be noted that TNRD and DnCNN are trained on clean and synthetic noisy image pairs, while TWSC only utilizes the tested noisy image. Besides, one can see that the proposed TWSC works much better than the baseline method WSC, which proves that the weight matrix $\mathbf{W}_{2}$ can characterize better the noise statistics in local image patches. Due to limited space, we leave the visual comparisons of different methods in the supplementary file.
Results on Realistic Noise Removal
----------------------------------
We evaluate the proposed TWSC scheme on three publicly available real-world noisy image datasets [@ncwebsite; @crosschannel2016; @dnd2017].
**Dataset 1** is provided in [@ncwebsite], which includes around 20 real-world noisy images collected under uncontrolled environment. Since there is no “ground truth” of the noisy images, we only compare the visual quality of the denoised images by different methods.
**Dataset 2** is provided in [@crosschannel2016], which includes noisy images of 11 static scenes captured by Canon 5D Mark 3, Nikon D600, and Nikon D800 cameras. The real-world noisy images were collected under controlled indoor environment. Each scene was shot 500 times under the same camera and camera setting. The mean image of the 500 shots is roughly taken as the “ground truth”, with which the PSNR and SSIM [@ssim] can be computed. 15 images of size $512\times512$ were cropped to evaluate different denoising methods. Recently, some other datasets such as [@PolyUdataset] are also constructed by employing the strategies of this dataset.
**Dataset 3** is called the Darmstadt Noise Dataset (DND) [@dnd2017], which includes 50 different pairs of images of the same scenes captured by Sony A7R, Olympus E-M10, Sony RX100 IV, and Huawei Nexus 6P. The real-world noisy images are collected under higher ISO values with shorter exposure time, while the “ground truth” images are captured under lower ISO values with adjusted longer exposure times. Since the captured images are of megapixel-size, the authors cropped 20 bounding boxes of $512\times512$ pixels from each image in the dataset, yielding 1000 test crops in total. However, the “ground truth” images are not open access, and we can only submit the denoising results to the authors’ [Project Website](https://noise.visinf.tu-darmstadt.de/) and get the PSNR and SSIM [@ssim] results.
**Comparison Methods**. We compare the proposed TWSC method with CBM3D [@cbm3d], TNRD [@chen2015learning], DnCNN [@dncnn], the commercial software Neat Image (NI) [@neatimage], the state-of-the-art real image denoising methods “Noise Clinic” (NC) [@noiseclinic], CC [@crosschannel2016], and MCWNNM [@mcwnnm]. We also compare with the baseline method WSC described in Section 4 as a baseline. The methods of CBM3D and DnCNN can directly deal with color images, and the input noise std is set by Eq. (\[e20\]). For TNRD, MCWNNM, and TWSC, we use [@Chen2015ICCV] to estimate the noise std $\sigma_{c}$ ($c\in\{r,g,b\}$) for each channel. For blind mode DnCNN, we use its color version provided by the authors and there is no need to estimate the noise std. Since TNRD is designed for grayscale images, we applied them to each channel of real-world noisy images. TNRD achieves its best results when setting the noise std of the trained models at $\sigma_{c}=10$ on these datasets.
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
\[f4\]
**Results on Dataset 1**. Fig. \[f4\] shows the denoised images of “Dog” (the method CC [@crosschannel2016] is not compared since its testing code is not available). One can see that CBM3D, TNRD, DnCNN, NI and NC generate some noise-caused color artifacts across the whole image, while MCWNNM and WSC tend to over-smooth a little the image. The proposed TWSC removes more clearly the noise without over-smoothing much the image details. These results demonstrate that the methods designed for AWGN are not effective for realistic noise removal. Though NC and NI methods are specifically developed for real-world noisy images, their performance is not satisfactory. In comparison, the proposed TWSC works much better in removing the noise while maintaining the details (see the zoom-in window in “Dog”) than the other competing methods. More visual comparisons can be found in the supplementary file.
**Results on Dataset 2**. The average PSNR and SSIM results on the 15 cropped images by competing methods are listed in Table \[t2\]. One can see that the proposed TWSC is much better than other competing methods, including the baseline method WSC and the recently proposed CC, MCWNNM. Fig. \[f5\] shows the denoised images of a scene captured by Nikon D800 at ISO = 6400. One can see that the proposed TWSC method results in not only higher PSNR and SSIM measures, but also much better visual quality than other methods. Due to limited space, we do not show the results of baseline method WSC in visual quality comparison. More visual comparisons can be found in the supplementary file.
[cccccccccc]{} & **CBM3D** & **TNRD** & **DnCNN** & **NI** & **NC** & **CC** & **MCWNNM** & **WSC** & **TWSC**\
PSNR & 35.19 & 36.61 & 33.86 & 35.49 & 36.43 & 36.88 & 37.70 & 37.36 & **37.81**\
SSIM & 0.8580 & 0.9463 & 0.8635 & 0.9126 & 0.9364 & 0.9481 & 0.9542 & 0.9516 & **0.9586**\
**Results on Dataset 3**. In Table \[t3\], we list the average PSNR and SSIM results of the competing methods on the 1000 cropped images in the DND dataset [@dnd2017]. We can see again that the proposed TWSC achieves much better performance than the other competing methods. Note that the “ground truth” images of this dataset have not been published, but one can submit the denoised images to the project website and get the PSNR and SSIM results. More results can be found in the website of the DND dataset (<https://noise.visinf.tu-darmstadt.de/benchmark/#results_srgb>). Fig. \[f6\] shows the denoised images of a scene captured by a Nexus 6P camera. One can still see that the proposed TWSC method results better visual quality than the other denoising methods. More visual comparisons can be found in the supplementary file.
[ccccccccc]{} & **CBM3D** & **TNRD** & **DnCNN** & **NI** & **NC** & **MCWNNM** & **WSC** & **TWSC**\
PSNR & 32.14 & 34.15 & 32.41 & 35.11 & 36.07 & 37.38 & 36.81 & **37.94**\
SSIM & 0.7773 & 0.8271 & 0.7897 & 0.8778 & 0.9013 & 0.9294 & 0.9165 & **0.9403**\
[1]{}
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[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
\[f5\]
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
[1]{}
\[f6\]
**Comparison on Speed**. We compare the average computational time (second) of different methods (except CC) to process one $512\times512$ image on the DND Dataset [@dnd2017]. The results are shown in Table \[t4\]. All experiments are run under the Matlab2014b environment on a machine with Intel(R) Core(TM) i7-5930K CPU of 3.5GHz and 32GB RAM. The fastest speed is highlighted in bold. One can see that Neat Image (NI) is the fastest and it spends about 1.1 second to process an image, while the proposed TWSC needs about 195 seconds. Noted that Neat Image is a highly-optimized software with parallelization, CBM3D, TNRD, and NC are implemented with compiled C++ mex-function and with parallelization, while DnCNN, MCWNNM, and the proposed WSC and TWSC are implemented purely in Matlab.
[ccccccccc]{} & **CBM3D** & **TNRD** & **DnCNN** & **NI** & **NC** & **MCWNNM** & **WSC** & **TWSC**\
Time & 6.9 & 5.2 & 79.5 & **1.1** & 15.6 & 208.1 & 188.6 & 195.2\
Visualization of The Weight Matrices
------------------------------------
The three diagonal weight matrices in the proposed TWSC model (\[e4\]) have clear physical meanings, and it is interesting to analyze how the matrices actually relate to the input image by visualizing the resulting matrices. To this end, we applied TWSC to the real-world (estimated noise stds of R/G/B: 11.4/14.8/18.4) and synthetic AWGN (std of all channels: 25) noisy images shown in Fig. \[f1\]. The final diagonal weight matrices for two typical patch matrices ($\bm{Y}$) from the two images are visualized in Fig. \[f7\]. One can see that the matrix $\bm{W}_{1}$ reflects well the noise levels in the images. Though matrix $\bm{W}_{2}$ is initialized as an identity matrix, it is changed in iterations since noise in different patches are removed differently. For real-world noisy images, the noise levels of different patches in $\bm{Y}$ are different, hence the elements of $\bm{W}_{2}$ vary a lot. In contrast, the noise levels of patches in the synthetic noisy image are similar, thus the elements of $\bm{W}_{2}$ are similar. The weight matrix $\bm{W}_{3}$ is basically determined by the patch structure but not noise, and we do not plot it here.
[0.45]{}
[0.45]{}
\[f7\]
Conclusion
==========
The realistic noise in real-world noisy images captured by CCD or CMOS cameras is very complex due to the various factors in digital camera pipelines, making the real-world image denoising problem much more challenging than additive white Gaussian noise removal. We proposed a novel trilateral weighted sparse coding (TWSC) scheme to exploit the noise properties across different channels and local patches. Specifically, we introduced two weight matrices into the data-fidelity term of the traditional sparse coding model to adaptively characterize the noise statistics in each patch of each channel, and another weight matrix into the regularization term to better exploit sparsity priors of natural images. The proposed TWSC scheme was solved under the ADMM framework and the solution to the Sylvester equation is guaranteed. Experiments demonstrated the superior performance of TWSC over existing state-of-the-art denoising methods, including those methods designed for realistic noise in real-world noisy images.
[^1]: This project is supported by Hong Kong RGC GRF project (PolyU 152124/15E).
|
---
author:
- '*Daniel Li, Hervé Queffélec, Luis Rodr[í]{}guez-Piazza*'
date:
-
-
title: '**Composition operators with surjective symbol and small approximation numbers**'
---
[**Abstract.**]{} We give a new proof of the existence of a surjective symbol whose associated composition operator on $H^2 ({\mathbb D})$ is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the $1$-dimensional case, for $N \geq 2$, the behavior of the approximation numbers $a_n = a_n (C_\phi)$, or rather of [$\beta^-_N = \liminf_{n \to \infty} [a_n]^{1/ n^{1/ N}}$ or $\beta^+_N = \limsup_{n \to \infty} [a_n]^{1/ n^{1/ N}}$]{}, of composition operators on $H^2 ({\mathbb D}^N)$ cannot be determined by the image of the symbol.
[**MSC 2010**]{} Primary: 47B33 Secondary: 32A35 ; 46B28
[**Key-words**]{} approximation numbers ; cusp map ; composition operator ; Hardy space ; lens map ; polydisk
Introduction
============
We start by recalling some notations and facts.
Let ${\mathbb D}$ be the open unit disk, $H^2$ the Hardy space on ${\mathbb D}$, and $\varphi \colon {\mathbb D}\to {\mathbb D}$ a non-constant analytic self-map. It is well-known ([@SHA]) that $\varphi$ induces a composition operator $C_\varphi \colon H^2 \to H^2$ by the formula: $$C_{\varphi} (f) = f \circ \varphi \, ,$$ and the connection between the “symbol” $\varphi$ and the properties of the operator $C_{\varphi} \colon H^2 \to H^2$, in particular its compactness, can be further studied ([@SHA]).
We also recall that the $n$th approximation number $a_{n}(T)$, $n = 1, 2, \ldots$, of an operator $T \colon H_1 \to H_2$, between Hilbert spaces $H_1$ and $H_2$, is defined as the distance of $T$ to operators of rank $ < n$, for the operator-norm: $$a_n (T) = \inf_{{\rm rank}\, R \, < n} \| T - R\| \, .$$ The $p$-Schatten class $S_{p} (H_1, H_2)$, $p > 0$ consists of all $T \colon H_1 \to H_2$ such that $\big( a_{n} (T) \big)_n \in \ell^p$. The approximation numbers have the ideal property: $$a_{n} (A T B) \leq \Vert A\Vert \, a_{n} (T) \, \Vert B \Vert \, .$$ Let now, for $\xi \in {\mathbb T}= \partial{{\mathbb D}}$ and $h > 0$, the Carleson window $S (\xi,h)$ be defined as: $$S (\xi,h) = \{z \in {\mathbb D}\, ; \ |z - \xi| \leq h \} \, .$$ For a symbol $\varphi$, we define $m_\varphi = \varphi^{\ast} (m)$ where $m$ is the Haar measure of ${\mathbb T}$ and $\varphi^{\ast} \colon {\mathbb T}\to \overline{{\mathbb D}}$ the (almost everywhere defined) radial limit function associated with $\varphi$, namely: $$\varphi^{\ast} (\xi) = \lim_{r \to 1^{-}} \varphi (r\xi) \, .$$ Finally, we set for $h > 0$: $$\rho_{\varphi} (h) = \sup_{\xi\in {\mathbb T}} m_{\varphi} [S (\xi, h)] \, .$$ It is known ([@SHA]) that $\rho_{\varphi} (h) = {\rm O}\, (h)$ and ([@MCCL]) that $C_\varphi$ is compact if and only if $\rho_{\varphi} (h) = {\rm o}\, (h)$ as $h \to 0$. Simpler criteria ([@SHA]) exist when $\varphi$ is injective, or even $p$-valent, meaning that for any $w \in {\mathbb D}$, the equation $\varphi (z) = w$ has at most $p$ solutions.
A measure $\mu$ on ${\mathbb D}$ is called $\alpha$-Carleson, $\alpha \geq 1$, if $\sup_{|\xi| = 1} \mu [ S (\xi, h)] = {\rm O}\, (h^\alpha)$.
B. MacCluer and J. Shapiro showed in [@MCSH Example 3.12] the following result, paradoxical at first glance.
\[MacCluer-Shapiro\] \[jb\] There exists a *surjective and four-valent* symbol $\varphi \colon {\mathbb D}\to {\mathbb D}$ such that the composition operator $C_\varphi \colon H^2 \to H^2$ is compact.
Observe that such a symbol $\varphi$ cannot be one-valent (injective), because it would be an automorphism of ${\mathbb D}$, and $C_\varphi$ would be invertible and therefore not compact. In [@LLQR Theorem 4.1], we gave the following improved statement.
\[first\] For every non-decreasing function $\delta \colon (0, 1) \to (0, 1)$, there exists a two-valent symbol and nearly surjective (i.e. $\varphi ({\mathbb D}) = {\mathbb D}\setminus \{0\}$) symbol $\phi$, and $0 < h_0 < 1$, such that: $$\label{firs}
\quad m(\{ z \in {\mathbb T}\, ; \ |\phi^\ast (z)| \geq 1 - h\}) \leq \delta (h) \quad \text{for } 0 < h \leq h_0 \, .$$ As a consequence, there exists a *surjective and four-valent* symbol $\psi \colon {\mathbb D}\to {\mathbb D}$ such that the composition operator $C_\psi \colon H^2 \to H^2$ is in every Schatten class $S_{p}(H^2)$, $p > 0$.
Our proof was rather technical and complicated, and based on arguments of barriers and harmonic measures.
The goal of this paper is to give a more precise statement of Theorem \[first\] in terms of approximation numbers $a_{n} (C_\varphi)$, and not only in terms of Schatten classes, and with a simpler proof. We then apply this result to show that for the polydisk ${\mathbb D}^N$, $N \geq 2$, the nature (boundedness, compactness, asymptotic behavior of approximation numbers) of the composition operator cannot be determined by the geometry of the image $\phi ({\mathbb D}^N)$ of its symbol $\phi$. For certain asymptotic behavior of approximation numbers, this is contrary to the $1$-dimensional case (see [@SRF Theorem 3.1 and Theorem 3.14]).
The notation $A \lesssim B$ means that $A \leq C \, B$ for some positive constant $C$, and $A \approx B$ that $A \lesssim B$ and $B \lesssim A$.
Background and preliminary results
==================================
We initiated the study of approximation numbers of composition operators on $H^2$ in [@LIQUEROD], and proved the following basic results:
\[basic\] If $\varphi$ is any symbol, then, for some $\delta > 0$ and $r > 0$, or $a > 0$: $$a_{n} (C_\varphi) \geq \delta \, r^n = \delta\, {{\rm e}}^{- a n} \, .$$ Moreover, as soon as $\Vert \varphi \Vert_\infty = 1$, there exists some sequence $\varepsilon_n$ tending to $0$ such that: $$a_{n} (C_\varphi) \geq \delta \, {{\rm e}}^{- n \varepsilon_n} \, .$$
We also proved in [@LIQUEROD Theorem 5.1] that:
\[second\] For any symbol $\varphi$, we have: $$a_{n} (C_\varphi) \lesssim \inf_{0 < h < 1} \bigg[ {{\rm e}}^{- n h} + \sqrt{\frac{\rho_{\varphi}(h)}{h}} \bigg] \, .$$
We also recall (see [@LIQUEROD]) that, for $\gamma > - 1$, the weighted Bergman space $\mathcal{B}_{\gamma}$ is the space of functions $f (z) = \sum_{n = 0}^\infty a_n z^n$ such that: $$\Vert f \Vert_{\gamma}^{2} := \sum_{n = 0}^\infty \frac{|a_n|^2}{(n+1)^{\gamma+1}} < \infty \, .$$ Equivalently, $\mathcal{B}_{\gamma}$ is the space of analytic functions $f \colon {\mathbb D}\to {\mathbb C}$ such that: $$\int_{\mathbb D}|f (z)|^2 \, (\gamma + 1) (1 - |z|^2)^\gamma \, dA (z) < \infty \, ,$$ where $dA$ is the normalized area measure on ${\mathbb D}$, and then: $$\int_{\mathbb D}|f (z)|^2 \, (\gamma + 1) (1 - |z|^2)^\gamma \, dA (z) \approx \Vert f \Vert_{\gamma}^{2} \, .$$ The case $\gamma = 0$ corresponds to the usual Bergman space ${\mathcal B}^2$, and the limiting case $\gamma = - 1$ to the Hardy space $H^2$. We wish to note in passing (we will make use of that elsewhere) that the proof of Theorem 5.1 in [@LIQUEROD] easily gives the following result.
\[seconde\] Let $ \gamma > - 1$ and $\varphi$ a symbol inducing a bounded composition operator $C_\varphi \colon \mathcal{B}_{\gamma} \to H^2$. Then: $$a_{n} (C_\varphi \colon \mathcal{B}_{\gamma} \to H^2) \lesssim \inf_{0 < h < 1}
\Bigg( {(n + 1)^{(\gamma + 1)/2}} \, {{\rm e}}^{- n h} + \sup_{0 < t \leq h} \sqrt{\frac{\rho_{\phi} (t)}{t^{2 + \gamma}}} \,\, \Bigg) \, \cdot$$
Take $E = z^n \mathcal{B}_{\gamma}$; this is a subspace of $\mathcal{B}_{\gamma}$ of codimension $\leq n$. Let $f \in E$ with $\Vert f\Vert_{\gamma} = 1$. Writing $f = z^n g$ with $\Vert g \Vert_{\gamma}^2 \leq (n + 1)^{\gamma + 1}$ and splitting the integral into two parts, we have, for $0 < h < 1$: $$\| C_{\varphi} f \|_{H^2}^{2} = \int_{{\mathbb D}} |f|^2 \, dm_\phi
\leq (1 - h)^{2 n} \int_{(1 - h) {\mathbb D}} |g|^2 \, dm_\phi + \int_{{\mathbb D}\setminus (1 - h) {\mathbb D}} |f|^2 \, dm_\phi \, .$$ For the first integral, we have: $$\label{Luis 1}
\int_{(1 - h) {\mathbb D}} |g|^2 \, dm_\phi \leq \int_{{\mathbb D}} | g |^2 \, dm_\phi = \| C_\phi \, g \|_{H^2}^2
\leq \| C_\phi\|_{\mathcal{B}_{\gamma} \to H^2}^2 \| g \|_\gamma^2 \, .$$ For the second integral, we have: $$\int_{{\mathbb D}\setminus (1 - h) {\mathbb D}} |f|^2 \, dm_\phi \leq \| J \colon \mathcal{B}_{\gamma} \to L^{2} (\mu_h) \|^2 \, ,$$ where $\mu_h$ is the restriction of $m_\varphi$ to the annulus $\{z \in {\mathbb D}\, ; \ 1 - h < |z| < 1 \}$ and $J$ the canonical injection of $\mathcal{B}_{\gamma}$ into $L^{2} (\mu_h)$. Hence Stegenga’s version of the Carleson embedding theorem for $\mathcal{B}_{\gamma}$ ([@Stegenga Theorem 1.2]; see [@Hastings] for the unweighted case; see also [@DS p. 62] or [@ZH p. 167]) gives us: $$\label{Luis 2}
\int_{{\mathbb D}\setminus (1 - h) {\mathbb D}} |f|^2 \, dm_\phi \lesssim \sup_{0 < t \leq h} \frac{\rho_{\phi} (t)}{t^{2 + \gamma}} \, \cdot$$ Putting and together, that gives: $$\| C_{\varphi} f \|_{H^2} \lesssim {{\rm e}}^{- n h} (n + 1)^{(\gamma + 1)/2} + \sup_{0 < t \leq h} \sqrt{\frac{\rho_{\phi} (t)}{t^{2 + \gamma}}} \, \cdot$$ In other terms, using the Gelfand numbers $c_k$: $$c_{n + 1} (C_\phi \colon \mathcal{B}_{\gamma} \to H^2) \lesssim (n + 1)^{(\gamma + 1)/2} \, {{\rm e}}^{- n h}
+ \sup_{0 < t \leq h} \sqrt{\frac{\rho_{\phi} (t)}{t^{2 + \gamma}}} \, \cdot$$ As $a_{n + 1} = c_{n + 1}$ and as we can ignore the difference between $a_n$ and $a_{n + 1}$, that finishes the proof.
As an application, we mention the following result. We refer to [@LIQURO Section 4.1] for the definition of the cusp map, denoted $\chi$.
\[cusp\] Let $\chi \colon {\mathbb D}\to {\mathbb D}$ be the cusp map and $\Phi \colon {\mathbb D}^N\to {\mathbb D}^N$ the diagonal map defined by: $$\Phi (z_1, z_2, \ldots, z_N) = \big( \chi (z_1), \chi (z_1), \ldots, \chi (z_1) \big) \, .$$ Then, the composition operator $C_\Phi$ maps $H^{2}({\mathbb D}^N)$ to itself and: $$\label{new}
a_{n} (C_\Phi) \lesssim {{\rm e}}^{- d \sqrt{n}}$$ where $d$ is a positive constant depending only on $N$.
[**Remark.**]{} We have to compare with [@BLQR Theorem 6.2] where, for: $$\Psi (z_1, \ldots, z_N) = \big( \chi (z_1), \ldots, \chi (z_N) \big) \, ,$$ it is shown that, for constants $b \geq a > 0$ depending only on $N$: $${{\rm e}}^{- b \, (n^{1 /N} / \log n)} \lesssim a_n (C_\Psi) \lesssim {{\rm e}}^{- a \, (n^{1 /N} / \log n)} \, .$$ Note also that for $N = 1$, the estimate of Theorem \[cusp\] is very crude.
\[Proof of Theorem \[cusp\]\] Take $\gamma = N - 2$. As in [@DHL Section 4], we have thanks to the Cauchy-Schwarz inequality, and the fact that $\sum_{|\alpha|= n} 1 \approx ( n + 1)^{N - 1}$, a factorization: $$C_\Phi = J C_{\chi} M \, ,$$ where $M \colon H^{2} ({\mathbb D}^N) \to \mathcal{B}_{\gamma}$ is defined by $M \! f = g$ with: $$\label{M}
\qquad g (z) = f (z, z, \ldots, z) = \sum_{n = 0}^\infty \Bigg( \sum_{|\alpha| = n} a_{\alpha} \Bigg) \, z^n \, , \qquad z \in {\mathbb D}\, , \quad$$ for $$f (z_1, z_2, \ldots, z_N) = \sum_{\alpha} a_{\alpha} z_1^{\alpha_1} \cdots z_N^{\alpha_N} \, ,$$ and where $J \colon H^{2}({\mathbb D}) \to H^{2} ({\mathbb D}^N)$ is the canonical injection given by: $$\label{J}
(J h) (z_1, z_2, \ldots, z_N) = h (z_1) \, .$$ This corresponds to a diagram: $$\label{factorization}
H^{2} ({\mathbb D}^N) \mathop{\longrightarrow}^M \mathcal{B}_{\gamma}
\mathop{\longrightarrow}^{C_\chi} H^{2} ({\mathbb D})
\mathop{\longrightarrow}^J H^{2} ({\mathbb D}^N) \, ,$$ where $C_\chi \colon \mathcal{B}_{\gamma} = \mathcal{B}_{N - 2} \to H^{2} ({\mathbb D})$ is a bounded operator. Indeed, we have the behavior ([@LIQURO Lemma 4.2]): $$|1 - \chi^\ast ({{\rm e}}^{i \theta})| \approx \frac{1}{\log (1/|\theta|)} \, \raise 1,5 pt \hbox{,}$$ and this implies, with $c$ an absolute constant: $$\label {chi Carleson}
\begin{split}
m_\chi [S (\xi, h)] & \lesssim m_{\chi} [S (1, h)] = m (\{ |\chi^\ast ({{\rm e}}^{i \theta}) - 1 | < h) \\
& \lesssim m [ \{ c / \log (1 / |\theta|) < h \} ] \leq {{\rm e}}^{- c/h} \, ;
\end{split}$$ in particular $\rho_\chi (h) \leq {{\rm e}}^{- c/h} = {\rm O}\, (h^N)$, so $m_\chi$ is an $N$-Carleson measure and the Stengenga-Carleson theorem ([@Stegenga Theorem 1.2]) says that the operator $C_\chi \colon \mathcal{B}_{N - 2} \to H^{2} ({\mathbb D})$ is bounded.
Now Proposition \[seconde\] with give: $$a_{n} (C_{\chi} \colon \mathcal{B}_{\gamma} \to H^2)
\lesssim \inf_{0 < h < 1} \big[ (n + 1)^{(N - 1)/2} \, {{\rm e}}^{- n h} + {{\rm e}}^{- c/h} h^{- N/2} \big] \, .$$ Adjusting $h = 1/\sqrt{n}$, we get $a_{n} (C_{\chi} \colon \mathcal{B}_{\gamma} \to H^2) \lesssim {{\rm e}}^{- d\sqrt{n}}$ for some positive constant $d$. Finally, the factorization $C_\Phi = J C_{\chi} M$ and the ideal property of approximation numbers give the result.
In the case of lens maps, Proposition \[seconde\] gives very poor estimates. We avoid using this theorem in [@DHL Section 4], when $N = 2$, using the semi-group property of those lens maps. The same proof gives for arbitrary $N \geq 2$ the following result.
\[lens\] Let $\lambda_{\theta}$ the lens map with parameter $\theta$, $0 < \theta < 1$, and let $\Phi \colon {\mathbb D}^N \to {\mathbb D}^N$ be the diagonal map defined by: $$\Phi (z_1, z_2, \ldots, z_N) = \big( \lambda_{\theta} (z_1), \lambda_{\theta} (z_1), \ldots, \lambda_{\theta} (z_1) \big) \, .$$ Then:
- if $\theta >1/N$, $C_\Phi$ is unbounded on $H^{2} ({\mathbb D}^N)$;
- if $\theta = 1/N$, $C_\Phi$ is bounded and not compact on $H^{2} ({\mathbb D}^N)$;
- if $\theta < 1/N$, $C_\Phi$ is compact on $H^{2} ({\mathbb D}^N)$ and moreover: $$\label{mino lens}
a_{n} (C_\Phi) \lesssim {{\rm e}}^{- d \sqrt{n}}$$ for a constant $d > 0$ depending only on $\theta$ and $N$.
[**Remark.**]{} In [@BLQR Theorem 6.1], it is shown that, for: $$\Psi (z_1, \ldots, z_N) = \big( \lambda_\theta (z_1), \ldots, \lambda_\theta (z_N) \big) \, ,$$ we have, for constants $b \geq a > 0$, depending only on $\theta$ and $N$: $${{\rm e}}^{ - b \, n^{1 / (2 N)}} \lesssim a_n (C_\Psi) \lesssim {{\rm e}}^{ - a \, n^{1 / (2 N)}} \, .$$
\[Proof of Theorem \[lens\]\] That had been proved, for $N =2$ in [@DHL Theorem 4.2 and Theorem 4.4]. For convenience of the reader, we sketch the proof.
Assume first $\theta \leq 1/N$, and write $\lambda_\theta = \lambda_{N \theta}\circ \lambda_{1/N}$, where we set, for convenience, $\lambda_1 (z) = z$, so $C_{\lambda_1} = {\rm Id}$. As in the proof of Theorem \[cusp\] (see [@DHL Section 4]), we have a factorization: $$C_\Phi = J C_{\lambda_{N \theta}} C_{\lambda_{1/N}} M \, ,$$ where $M$ and $J$ are defined in and .
This corresponds to a diagram (recall that $\gamma = N - 2$): $$H^{2} ({\mathbb D}^N) \mathop{\longrightarrow}^M \mathcal{B}_{\gamma}
\mathop{\longrightarrow}^{C_{\lambda_{1/N}}} H^{2} ({\mathbb D})
\mathop{\longrightarrow}^{C_{\lambda_{N\theta}}} H^{2} ({\mathbb D})
\mathop{\longrightarrow}^{J} H^{2} ({\mathbb D}^N) \, .$$ The second arrow is bounded, since we know ([@LELIQURO Lemma 3.3]) that the pullback measure $m_{\lambda_{1/N}}$ is $N$-Carleson, so that $C_{\lambda_{1/N}}$ maps $\mathcal{B}_{N - 2}$ to $H^{2}({\mathbb D})$ by the Stegenga-Carleson embedding theorem ([@Stegenga Theorem 1.2]).
For $\theta < 1/ N$, we have $N \theta < 1$ and $C_{\lambda_{N \theta}}$ is compact and, for some constant $b = b (\theta)$, we have $a_n (C_{\lambda_{N \theta}}) \lesssim {{\rm e}}^{ - b \sqrt{n}}$ ([@LELIQURO Theorem 2.1]). Hence $C_\Phi$ is compact and $a_n (C_\Phi) \lesssim {{\rm e}}^{ - b \sqrt{n}}$.
Now, for $\theta \geq 1 / N$, we consider the reproducing kernels: $$K_{a_1, \ldots, a_N} (z_1, \ldots, z_N) = \prod_{j = 1}^N \frac{1}{1 - \overline{a}_ j z_j} \, \cdot$$ We have: $$\| K_{a_1, \ldots, a_N} \|^2 = \prod_{j = 1}^N \frac{1}{1 - |a_j|^2}$$ and: $$C_\Phi^\ast (K_{a_1, \ldots, a_N}) = K_{\lambda_\theta (a_1), \ldots, \lambda_\theta (a_1)} \, ,$$ so: $$\| C_\Phi^\ast (K_{a_1, \ldots, a_N}) \|^2 = \bigg( \frac{1}{1 - |\lambda_\theta (a_1) |^2} \bigg)^N \, \cdot$$ Since: $$1 - |\lambda_\theta (a_1) |^2 \approx 1 - |\lambda_\theta (a_1) | \approx (1 - |a_1|)^\theta \, ,$$ we see that $\| C_\Phi^\ast (K_{a_1, \ldots, a_N}) \| / \| K_{a_1, \ldots, a_N}\|$ is not bounded for $\theta > 1 / N$, so $C_\phi$ is then not bounded; and it does not converge to $0$ for $\theta = 1 / N$, so $C_\Phi$ is then not compact.
Surjectivity
============
Let us come back to our surjectivity issues.
Let us first remark that Theorem \[first\] gives the following result.
\[bet\] For every non-decreasing function $\delta \colon (0, 1) \to (0, 1)$, there exists a *surjective and four-valent* symbol $\psi$, and $0 < h_0 < 1$, such that, for $0 < h \leq h_0$: $$\label{be}
m (\{z \in {\mathbb T}\, ; \ |\phi^\ast (z)| \geq 1 - h\}) \leq \delta (h) \, .$$
Just observe that the passage from “$\varphi$ two-valent and nearly surjective” to “$\psi$ four-valent and surjective” is harmless: for this, consider the Blaschke product: $$B (z) = \bigg( \frac{z - a}{1 - a z} \bigg)^2 \, ,$$ where $0 < a < 1$, and take $\psi = B\circ \varphi$; we observe that $B ({\mathbb D}\setminus \{0\}) = {\mathbb D}$ since $a^2 = B \big( \frac{2 a}{1 + a^2} \big)$, and, for $z \in {\mathbb D}$: $$\frac{1 - |B (z)|}{1 - |z|}\geq \frac{\displaystyle 1 - |\frac{z - a}{1 - a z}|^2}{1 - |z|^2} = \frac{1 - a^2}{|1 - a z|^2} \geq \frac{1 - a^2}{4}
\, \raise 1pt \hbox{,}$$ so that: $$m (|\psi^\ast| > 1 - h) = m (1 - |B \circ \varphi^\ast| < h) \leq m \big( 1 - |\varphi^\ast| \leq \kappa_a h \big) \, ,$$ with $\kappa_a = 4 / (1 - a^2)$. Hence, this map $\psi$ is surjective, four-valent, and satisfies , as well, up to a change of $\delta (h)$ to $\delta (h /\kappa_a)$ for $\varphi$ at the beginning.
A more precise statement
------------------------
Our new statement is as follows.
\[third\] For every positive sequence $(\varepsilon_n)_n$ with limit $0$, there exists a *surjective and four-valent* symbol $\varphi$ such that: $$a_{n} (C_\varphi) \lesssim {{\rm e}}^{- n\varepsilon_n} \, .$$ Consequently, there exists a *surjective and four-valent* symbol $\varphi \colon {\mathbb D}\to {\mathbb D}$ such that the composition operator $C_\varphi \colon H^2 \to H^2$ is in every Schatten class $S_{p} (H^2)$, $p > 0$.
Observe first that $\Vert \varphi \Vert_\infty = 1$ when $\varphi$ is surjective, so that, in view of Theorem \[basic\], we cannot dispense with the numbers $\varepsilon_n$, even if they can tend to $0$ arbitrarily slowly.
Now, we can choose $\delta \colon (0, 1) \to (0, 1)$ non-decreasing such that $\delta (\varepsilon_n) \leq {{\rm e}}^{- n \varepsilon_n}$ for all $n$, and then, using Theorem \[bet\], we get a surjective and four-valent symbol $\varphi$, satisfying for all $h$ small enough: $$\rho_{\varphi} (h) \leq h \, \delta^{\,2} (h) \, .$$ Proposition \[second\] gives: $$a_{n} (C_\varphi) \lesssim \inf_{0 < h < 1} \big[ {{\rm e}}^{- n h} + \delta (h) \big] \, .$$ Adjusting $h = \varepsilon_n$, we get $a_{n} (C_\varphi) \lesssim {{\rm e}}^{- n \varepsilon_n}$.
To get the second part of the theorem, just take $\varepsilon_n = n^{- 1/2}$.
A simplified proof of Theorem \[first\]
---------------------------------------
We give here the announced simplified proof of Theorem \[first\]. This proof is based on the following key lemma, in which $\mathcal{H} ({\mathbb D})$ denotes the set of holomorphic functions on ${\mathbb D}$.
\[key\] There exists a numerical constant $C$ such that, if $f \in \mathcal{H}({\mathbb D})$ satisfies, for some $\alpha \in {\mathbb R}$: $$\left\{
\begin{array} {l}
{{\mathfrak I}{\rm m}\,}[f (0)] < \alpha \smallskip \\
f ({\mathbb D}) \subseteq \{z \in {\mathbb C}\, ; \ 0 < {{\mathfrak R}{\rm e}\,}z < \pi\} \cup \{z \in {\mathbb C}\, ; \ {{\mathfrak I}{\rm m}\,}z < \alpha \} \, ,
\end{array}
\right.$$ then: $$m (\{{{\mathfrak I}{\rm m}\,}f^{\ast} > y\}) \leq C \, {{\rm e}}^{\alpha - y} \, , \quad \text{for } y \geq \alpha \, .$$
We first show how this lemma allows us to conclude.
\[Proof of Theorem \[first\]\] Let $g \colon (0, \infty) \to (0, \infty)$ be a continuous decreasing function such that: $$\lim_{t \to 0^{+}} g (t) = +\infty \, , \quad g (\pi) = \pi \, ,\quad \lim_{t \to +\infty} g (t) = 0 \, .$$ Then let $\Omega$ be the simply connected region defined by: $$\Omega = \{x + i y \, ;\ x > 0 \, , \quad g (x) < y < g (x) + 4 \pi\} \, ,$$ and $f \colon{\mathbb D}\to \Omega$ be a Riemann map such that $f (0) = \pi + 3 i \pi$. Observe that we can apply Lemma \[key\] to $f$ with $\alpha = 5\pi$ since ${{\mathfrak I}{\rm m}\,}f (0) = 3\pi$ and if $f (z) = x + i y$ with $x \geq \pi$; hence: $${{\mathfrak I}{\rm m}\,}f (z) = y < g (x) + 4 \pi \leq g (\pi) + 4 \pi = 5 \pi \, .$$ Finally, consider the symbol $\varphi = {{\rm e}}^{- f}$. It is nearly surjective: $\phi ({\mathbb D}) = {\mathbb D}\setminus \{0\}$, and two-valent, as easily checked.
For $0 < h \leq 1/2$, we have for $\xi \in {\mathbb T}$ and $|\phi^\ast (\xi) | > 1 - h$: $${{\rm e}}^{ - 2 h} \leq 1 - h < |\phi^\ast (\xi) | = \exp \big( - {{\mathfrak R}{\rm e}\,}f^\ast (\xi) \big) \, ;$$ hence ${{\mathfrak R}{\rm e}\,}f^{\ast} (\xi) < 2 h$.
But if $2 h > x = {{\mathfrak R}{\rm e}\,}f^{\ast} (\xi)$, we have $g (x) > g (2 h)$. As $f^{\ast} (\xi) = x + i y \in \overline{\Omega}$, we get ${{\mathfrak I}{\rm m}\,}f^{\ast} (\xi) = y \geq g (x) > g (2h)$. Lemma \[key\] now gives: $$\label{kol}
m (\{\xi \, ; \ |\varphi^{\ast} (\xi)| > 1 - h \}) \leq m (\{\xi \, ;\ {{\mathfrak I}{\rm m}\,}f^{\ast} (\xi) > g (2 h) \}) \leq C \, {{\rm e}}^{5 \pi - g (2 h)} \, .$$ It is now enough to adjust $g$ so as to have ${{\rm e}}^{g(t)}\geq C \, {{\rm e}}^{5 \pi} / \delta (t/2)$ for $t$ small enough to get from .
\[Proof of Lemma \[key\]\] We now prove Lemma \[key\]. If ${{\rm e}}^{y - \alpha} < 2$, there is nothing to prove, since then: $$m ({{\mathfrak I}{\rm m}\,}f^{\ast} > y) \leq 1 \leq 2 \, {{\rm e}}^{\alpha - y} \, .$$ We can hence assume that ${{\rm e}}^{y - \alpha}\geq 2$. First, we make a comment. If the Riemann mapping theorem is very general and flexible, it gives very few informations on the parametrization $t \mapsto f^{\ast} ({{\rm e}}^{it})$ when $f \colon {\mathbb D}\to \Omega$ is a conformal map, except in some specific cases (lens maps, cusps, etc.: see [@LIQURO]). Here, the Kolmogorov weak type inequality provides a substitute. Write: $$f = u + i v$$ and set: $$f_1 = - i f + i \frac{\pi}{2} - \alpha = v - \alpha + i \bigg( \frac{\pi}{2} - u \bigg)$$ and: $$F_1 = 1 + {{\rm e}}^{f_1} = (1 + {{\rm e}}^{v - \alpha} \sin u) + i {{\rm e}}^{v - \alpha} \cos u \, .$$ If $v < \alpha$, then ${{\mathfrak R}{\rm e}\,}F_1 > 1 - |\sin u| \geq 0$. If $v \geq \alpha$, then $0 < u < \pi$ and ${{\mathfrak R}{\rm e}\,}F_1 \geq 1$. Hence $F_1$ maps ${\mathbb D}$ to the right half-plane ${\mathbb C}_0 = \{z \, ; \ {{\mathfrak R}{\rm e}\,}z > 0 \}$. Finally, let $F = U + i V \colon {\mathbb D}\to {\mathbb C}_0$ be defined by: $$F = F_1 - i {{\mathfrak I}{\rm m}\,}F_{1}(0) \, ,$$ so that $V (0) = 0$. By the Kolmogorov inequality for the conjugation map $U \mapsto V$, and the harmonicity of $U$, we have, for all $\lambda>0$ ($a$ designating an absolute constant): $$\label{ko}
m (|F^{\ast}| > \lambda) \leq \frac{a}{\lambda} \, \Vert U^{\ast} \Vert_1
= \frac{a}{\lambda} \int_{{\mathbb T}} U^{\ast} \, dm = \frac{a}{\lambda} \, U(0) \, .$$ Next, we claim that: $$\label{claim}
|{{\mathfrak I}{\rm m}\,}F_{1} (0)|<1 \quad \text{and} \quad U (0) < 2 \, .$$ Indeed, $v (0) < \alpha$ by hypothesis, so that $ |{{\mathfrak I}{\rm m}\,}F_{1} (0)| = {{\rm e}}^{v (0) - \alpha} |\cos u (0)| < 1$, and $U (0) = 1 + {{\rm e}}^{v (0) - \alpha} \sin u (0) < 2$. Suppose now that, for some $y > \alpha$ and $z \in {\mathbb D}$, we have $v (z) > y$. Then, $0 <u (z) < \pi$ by our second assumption, and this implies ${{\mathfrak R}{\rm e}\,}{{\rm e}}^{f_{1} (z)} = {{\rm e}}^{v (z) - \alpha} \sin u (z) > 0$, so that, using $|1 + w| \geq |w|$ if ${{\mathfrak R}{\rm e}\,}w > 0$ and , and remembering that ${{\rm e}}^{y - \alpha} \geq 2$: $$\begin{aligned}
|F (z) |
& = \big|1 + {{\rm e}}^{f_{1} (z)} - i \, {{\mathfrak I}{\rm m}\,}F_{1} (0) \big| \geq \big|1 + {{\rm e}}^{f_{1} (z)} \big| - 1 \\
& \geq \big| {{\rm e}}^{f_{1} (z)} \big| - 1 = {{\rm e}}^{v (z) - \alpha} - 1 > {{\rm e}}^{y - \alpha} - 1 \geq \frac{1}{2} \, {{\rm e}}^{y - \alpha}\, .\end{aligned}$$ Taking radial limits and using and , we get: $$m ({{\mathfrak I}{\rm m}\,}f^{\ast} > y ) \leq m (| F^{\ast}| >{{\rm e}}^{y - \alpha} / 2) \leq 4 a \, {{\rm e}}^{\alpha - y} \, .$$ This ends the proof of Lemma \[key\] with $C = \max (2, 4 a)$.
Application to the multidimensional case
========================================
In this section, we apply Theorem \[bet\] and Theorem \[third\] to show that, for $N \geq 2$, the image of the symbol cannot determine the behavior of the approximation numbers, or rather of $\beta_N (C_\phi)$, of the associated composition operator $C_\phi \colon H^2 ({\mathbb D}^N) \to H^2 ({\mathbb D}^N)$.
Recall that for an operator $T \colon H_1 \to H_2$, we set: $$\beta^-_N (T) = \liminf_{n \to \infty} [a_{n} (T)]^{1 / n^{1 / N}} \quad \text{and} \quad
\beta^+_N (T) = \limsup_{n \to \infty} [a_{n} (T)]^{1 / n^{1 / N}}\, ,$$ and write $\beta_N (T)$ when $\beta^-_N (T) = \beta^+_N (T)$.
For $N \geq 2$, there exist pairs of symbols $\Phi_1, \Phi_2 \colon {\mathbb D}^N \to {\mathbb D}^N$, such that $\Phi_1 ({\mathbb D}^N) = \Phi_2 ({\mathbb D}^N)$ and:
- $C_{\Phi_1}$ is not bounded, but $C_{\Phi_2}$ is compact, and even $\beta_N (C_{\Phi_2}) = 0$;
- $C_{\Phi_1}$ is bounded but not compact, so $\beta_N (C_{\Phi_1}) = 1$, and $C_{\Phi_2}$ is compact, with $\beta_N (C_{\Phi_2}) = 0$;
- $C_{\Phi_1}$ is compact, with $\beta^-_N (C_{\Phi_1}) > 0$ and $\beta^+_N (C_{\Phi_1}) < 1$, and $C_{\Phi_2}$ is compact, with $\beta_N (C_{\Phi_2}) = 0$;
- $C_{\Phi_1}$ is compact, with $\beta_N (C_{\Phi_1}) = 1$, and $C_{\Phi_2}$ is compact, but with $\beta_N (C_{\Phi_2}) = 0$.
Let $\sigma \colon {\mathbb D}\to {\mathbb D}$ be a surjective symbol such that $\rho_\sigma (h) \leq h^N \, {{\rm e}}^{- 2 / h^2}$ given by Theorem \[bet\]. By Proposition \[seconde\], we have, with $\gamma = N - 2$: $$a_{n} (C_\sigma \colon \mathcal{B}_{\gamma} \to H^2) \lesssim \inf_{0 < h < 1} ({n^{(N - 1)/2}} {{\rm e}}^{- n h} + {{\rm e}}^{- 1 / h^2} ) \, \raise 1 pt \hbox{,}$$ and, with $h = 1 / n^{1/3}$, we get $a_{n} (C_\sigma \colon \mathcal{B}_{\gamma} \to H^2) \lesssim {{\rm e}}^{- d \, n^{2 / 3}}$.
We choose the exponent $2/3$ for fixing the ideas, but every exponent $\alpha > 1/2$, with $\alpha < 1$, (i.e. $a_{n} (C_\sigma \colon \mathcal{B}_{\gamma} \to H^2) \lesssim {{\rm e}}^{- d \, n^\alpha}$) would be suitable.
$1)$ We take $\Phi_1 (z_1, z_2, z_3, \ldots, z_N) = (z_1, z_1, \ldots, z_1)$. The composition operator $C_{\Phi_1}$ is not bounded because if $f_n (z_1, \ldots, z_N) = \big( \frac{z_1 + z_2}{2} \big)^n$, then $\| f_n \|_2^2 = 4^{- n} \sum_{k = 0}^n \binom{n}{k}^2 = 4^{- n} \binom{2 n}{n} \approx 1 / \sqrt{n}$, though $(C_{\Phi_1} f_n) (z_1, \ldots, z_N) = z_1^n$ and $\| C_{\Phi_1} f_n \|_2 = 1$.
We define $\Phi_2$ by: $$\Phi_2 (z_1, z_2, \ldots, z_N) = \big( \sigma (z_1), \sigma (z_1), \ldots, \sigma (z_1) \big) \, .$$ Since $\sigma$ is surjective, we have $\Phi_2 ({\mathbb D}^N) = \Phi_1 ({\mathbb D}^N)$. Now, as in the proof of Theorem \[cusp\], we have $C_{\Phi_2} = J C_\sigma M$, so: $$a_n (C_{\Phi_2}) \leq a_{n} (C_\sigma \colon \mathcal{B}_{N - 2} \to H^2) \lesssim {{\rm e}}^{- d \, n^{2 / 3}} \, ,$$ by the ideal property. Hence $[ a_n (C_{\Phi_2}) ]^{1 / n^{1 / N}} \lesssim {{\rm e}}^{- d \, n^{\frac{2}{3} - \frac{1}{N}}}$ and therefore $\beta_N (C_{\Phi_2}) = 0$ since $\frac{2}{3} - \frac{1}{N} > 0$.
$2)$ We consider the lens map $\lambda = \lambda_{1/N}$ of parameter $1 / N$. We define: $$\left\{
\begin{array}{l}
\Phi_1 (z_1, \ldots, z_N) = \big( \lambda (z_1), \lambda (z_1), \ldots, \lambda (z_1) \big) \smallskip \\
\Phi_2 (z_1, \ldots, z_N) = \big( \lambda [\sigma (z_1)], \lambda [\sigma (z_1)], \ldots, \lambda [\sigma (z_1)] \big) \, .
\end{array}
\right.$$ Since $\sigma$ is surjective, we have $\Phi_1 ({\mathbb D}^N) = \Phi_2 ({\mathbb D}^N)$ and we saw in Theorem \[lens\] that $C_{\Phi_1}$ is bounded but not compact.
On the other hand, we have the factorization $C_{\Phi_2} = J C_\sigma C_\lambda M$. Hence $C_{\Phi_2}$ is compact, and, as in $1)$, $\beta_N (C_{\Phi_2}) = 0$.
$3)$ For this item, the map $\sigma$ does not suffice, and we will use another surjective symbol $s \colon {\mathbb D}\to {\mathbb D}$. By Theorem \[bet\], there exists such a map $s$ with: $$\label{s1}
\rho_s (t) \leq t^2 {{\rm e}}^{- 2/t^2}$$ and $$\label{s2}
\rho_s (t) \leq t \, \delta^{\,2} (t)$$ for $t$ small enough, where $\delta \colon (0, 1) \to (0, 1)$ is a non-decreasing function such that $\delta ({\varepsilon}_n) \leq {{\rm e}}^{- n {\varepsilon}_n}$ and: $$\label{eps_n}
{\varepsilon}_n = n^{- \frac{1}{4 N - 7}} \, .$$ By the proof of Theorem \[third\], implies that: $$\label{a_n (s)}
a_n (C_s) \leq {{\rm e}}^{- n {\varepsilon}_n} \, .$$ We also consider a lens map $\lambda = \lambda_\theta$, with parameter $\theta < 1 / N$, and we set: $$\left\{
\begin{array}{l}
\displaystyle \Phi_1 (z_1, \ldots, z_N) = \Big( \lambda (z_1), \lambda (z_1), \raise -1,5 pt \hbox{$\displaystyle \frac{z_3}{2}$} , \ldots ,
\raise -1,5 pt \hbox{$\displaystyle \frac{z_N}{2}$ }\Big) \smallskip \\
\displaystyle \Phi_2 (z_1, \ldots, z_N) = \Big( \lambda [s (z_1)], \lambda [s (z_1)], \raise -1,5 pt \hbox{$\displaystyle \frac{s (z_3)}{2}$} , \ldots ,
\raise -1,5 pt \hbox{$\displaystyle \frac{s (z_N)}{2}$} \Big) \, .
\end{array}
\right.$$ Since $s$ is surjective, we have $\Phi_1 ({\mathbb D}^N) = \Phi_2 ({\mathbb D}^N)$.
a\) Let us prove that $\beta_N^- (C_{\Phi_1}) > 0$ and $\beta_N^+ (C_{\Phi_1}) < 1$.
Note that: $$C_{\Phi_1} = C_u \otimes C_{v_3}\otimes \cdots \otimes C_{v_N} \, ,$$ where $u \colon {\mathbb D}^2 \to {\mathbb D}^2$ is defined by $u (z_1, z_2) = \big( \lambda (z_1), \lambda (z_1) \big)$ and $v_j \colon \mathbb{D} \to \mathbb{{\mathbb D}}$ is defined by $v_{j} (z_j) = z_j /2$. In fact, if $f \in H^2 ({\mathbb D}^2)$ and $g_j \in H^2 ({\mathbb D})$, $3 \leq j \leq N$, we have: $$\begin{aligned}
[C_{\Phi_1} & (f \otimes g_3 \otimes \cdots \otimes g_N )] (z_1, z_2, z_3, \ldots, z_N) \\
& = (f \otimes g_3 \otimes \cdots \otimes g_N ) \big( u (z_1, z_2), v_3 (z_3), \ldots, v_N (z_N) \big) \\
& = f [ \lambda (z_1), \lambda (z_1) ] \, g_3 [v_3 (z_3)] \cdots g_N [v_N (z_N)] \\
& = (C_u f) (z_1, z_2) \, (C_{v_3} g_3) (z_3) \cdots (C_{v_N} g_N) (z_N) \\
& = [(C_u \otimes C_{v_3} \otimes \cdots \otimes C_{v_N}) (f \otimes g_3 \otimes \cdots \otimes g_N)] (z_1, z_2, z_3, \ldots, z_N) \, ,\end{aligned}$$ hence the result since $H^2 ({\mathbb D}^2) \otimes H^2 ({\mathbb D}) \otimes \cdots \otimes H^2 ({\mathbb D})$ is dense in $H^2 ({\mathbb D}^N)$. That proves in particular that $C_{\Phi_1}$ is compact since $C_u$ and $C_{v_3}, \ldots, C_{v_N}$ are (by Theorem \[lens\] for $C_u$).
By the supermultiplicativity of singular numbers of tensor products (see [@DHL Lemma 3.2]), it ensues that: $$a_{n^N} (C_{\Phi_1}) \geq a_{n^2} (C_u) \prod_{j = 3}^N a_{n} (C_{v_j}) = a_{n^2} (C_u) \, \Big( \frac{1}{2} \Big)^{n (N - 2)} \, .$$ By [@DHL Remark at the end of Section 4], we have $a_{n^2} (C_u) \gtrsim {{\rm e}}^{ - b n}$ for some positive constant $b = b (\theta)$. Indeed, if $J = J_2 \colon H^2 ({\mathbb D}) \to H^2 ({\mathbb D}^2)$ is the canonical injection defined by $(J h) (z_1, z_2) = h (z_1)$ and $Q \colon H^2 ({\mathbb D}^2) \to H^2 ({\mathbb D})$ is defined by $(Qf) (z_1) = f (z_1, 0)$, we have $C_\lambda = Q C_u J$. Hence $a_k (C_u) \gtrsim a_k (C_\lambda) \gtrsim {{\rm e}}^{- b \sqrt{k}}$.
Therefore we get: $$a_{n^N} (C_{\Phi_1}) \gtrsim {{\rm e}}^{ - c n}$$ for some positive constant depending only on $\theta$ and $N$. It follows that $\beta^-_N (C_{\Phi_1}) > 0$.
To see that $\beta^+_N (C_{\Phi_1}) < 1$, we need the following lemma, whose proof is postponed.
\[uno\] Let $S \colon H_1 \to H_1$ and $T \colon H_2 \to H_2$ be two operators between Hilbert spaces and $A, B$ a pair of positive numbers. Then, whenever: $$a_{[n^A]}(S) \leq {{\rm e}}^{- c n} \quad \text{and} \quad a_{[n^B]} (T) \leq {{\rm e}}^{- c n} \, ,$$ where $[\, . \,]$ stands for the integer part, we have, for some constant integer $M = M (A, B) > 0$: $$a_{M \, [n^{A + B}]} (S \otimes T) \leq {{\rm e}}^{- c n} \, .$$
Let $S = C_u$ and $T = C_{v_3} \otimes \cdots \otimes C_{v_N}$. For $c$ small enough, we have $a_{n^{N - 2}} (T) \leq C \, (1/2)^n \leq {{\rm e}}^{- c n}$ and, using , $a_{n^2} (S) \leq {{\rm e}}^{- d n} \leq {{\rm e}}^{- c n}$. Hence, with $A = 2$, $B = N - 2$, Lemma \[uno\] gives: $$a_{M n^N} (C_{\Phi_1}) \lesssim {{\rm e}}^{- c n} \, .$$ Therefore $\beta^+_N (C_{\Phi_1}) \leq {{\rm e}}^{- c / M^{1/N}} < 1$.
b\) Define $\Psi \colon {\mathbb D}^N \to {\mathbb D}^N$ by: $$\Psi (z_1, z_2, z_3, \ldots, z_N) = \big( s (z_1), s (z_1), s (z_3), \ldots, s (z_N) \big) \,.$$ If $\tau_1 \colon {\mathbb D}^2 \to {\mathbb D}^2$ is defined by $\tau_{1} (z_1, z_2) = \big( s (z_1), s (z_1) \big)$ and the map $\tau_2 \colon {\mathbb D}^{N - 2} \to {\mathbb D}^{N - 2}$ by $\tau_{2} (z_3, \ldots, z_N) = \big( s (z_3), \ldots, s (z_N) \big)$, we have: $$C_\Psi = C_{\tau_1} \otimes C_{\tau_2} \, .$$ As in the proof of Theorem \[cusp\], we have the factorization: $$\tau_1 \colon H^{2} ({\mathbb D}^2) \mathop{\longrightarrow}^M \mathcal{B}_{0} = {\mathcal B}^2
\mathop{\longrightarrow}^{C_s} H^{2} ({\mathbb D}) \mathop{\longrightarrow}^J H^{2} ({\mathbb D}^2) \, .$$ Hence $a_n (C_{\tau_1}) \leq \| M \|\, \| J \|\, a_n (C_s \colon \mathcal{B}^2 \to H^2)$.
By Proposition \[seconde\], we have: $$a_n (C_s \colon \mathcal{B}^2 \to H^2) \lesssim \inf_{0 < h < 1} \bigg( \sqrt{n} \, {{\rm e}}^{- n h} + \sup_{0 < t \leq h} \sqrt{\frac{\rho_s (t)}{t^2}} \, \bigg)
\, ;$$ so implies that $a_n (C_s \colon \mathcal{B}^2 \to H^2) \lesssim \inf_{0 < h < 1} ( \sqrt{n} \, {{\rm e}}^{- n h} + {{\rm e}}^{- 1 / h^2})$ and, taking $h = n^{- 1/3}$, we get, with some $c$ small enough: $$a_n (C_s \colon \mathcal{B}^2 \to H^2) \lesssim {{\rm e}}^{- c n^{2 / 3}} \, .$$ It follows that $a_n (C_{\tau_1}) \lesssim {{\rm e}}^{- c \, n^{2 / 3}}$ and hence: $$\label{tau1}
a_{[n^{3 / 2}]} (C_{\tau_1}) \lesssim {{\rm e}}^{- c \, n} \, .$$ On the other hand, [@BLQR Theorem 5.5] says that: $$a_n (C_{\tau_2}) \leq 2^{N - 3} \| C_s\|^{N - 2} \inf_{n_3 \cdots n_N \leq n} \big( a_{n_3} (C_s) + \cdots + a_{n_N} (C_s) \big) \, .$$ Taking $n_3 = \cdots = n_N = n^{\frac{1}{N - 2}}$, we get, using : $$a_n (C_{\tau_2}) \leq K^N N \, \exp \Big( - n^{\frac{1}{N - 2}} \, {\varepsilon}_{n^{\frac{1}{N - 2}}} \Big)\, .$$ Using , that gives: $$a_n (C_{\tau_2}) \, \lesssim \exp \big( - n^{\frac{1}{N - 2} ( 1 - \frac{1}{4N - 7} ) } \big) =
\exp \big( - n^{\frac{4}{4 N - 7}} \big) \, ,$$ or: $$\label{tau2}
a_{\big[n^{N - \frac{7}{4}} \big]} (C_{\tau_2}) \lesssim {{\rm e}}^{ - n} \leq {{\rm e}}^{- c n} \, .$$ Now, and allow to use Lemma \[uno\] with $A = 3/2$ and $B = N - 7/4$, and we get: $$a_{M \, \big[n^{N - \frac{1}{4}} \big] } (C_\Psi) \lesssim {{\rm e}}^{- c n} \, .$$ Equivalently: $$a_k (C_\Psi) \lesssim \exp \big( - c' k^{\frac{4}{4 N - 1}} \big)$$ and: $$\big( a_k (C_\Psi) \big)^{1/ k^{1 / N}} \lesssim \exp \big( - c' k^{\frac{4}{4 N - 1} - \frac{1}{N}} \big)
= \exp \big( - c' k^{\frac{1}{N (4 N - 1)}} \big) \, ,$$ which gives $\beta_N (C_\Psi) = 0$.
To end the proof, it suffices to remark that $C_{\Phi_2} = C_\Psi \circ C_{\Phi_1}$, since $\Phi_2 = \Phi_1 \circ \Psi$, and hence $\beta_N^+ (C_{\Phi_2}) \leq \beta_N^+ (C_\Psi) = 0$, so $\beta_N (C_{\Phi_2}) = 0$.
$4)$ We use a Shapiro-Taylor map. This one-parameter map $\varsigma_\theta\,$, $\theta > 0$, was introduced by J. Shapiro and P. Taylor in 1973 ([@SHTA]) and was further studied, with a slightly different definition, in [@JFA Section 5]. J. Shapiro and P. Taylor proved that $C_{\varsigma_\theta} \colon H^2 \to H^2$ is always compact, but is Hilbert-Schmidt if and only if $\theta > 2$. Let us recall their definition.
For $0 < {\varepsilon}< 1$, we set $V_{\varepsilon}= \{ z\in {\mathbb C}\, ; \ {{\mathfrak R}{\rm e}\,}z > 0 \text{ and } |z | < {\varepsilon}\}$. For ${\varepsilon}= {\varepsilon}_\theta > 0$ small enough, one can define: $$f_\theta (z) = z (- \log z )^\theta ,$$ for $z \in V_{\varepsilon}$, where $\log z$ will be the principal determination of the logarithm. Let now $g_\theta$ be the conformal mapping from ${\mathbb D}$ onto $V_{\varepsilon}$, which maps ${\mathbb T}= \partial {\mathbb D}$ onto $\partial V_{\varepsilon}$, defined by $g_\theta (z) = {\varepsilon}\, \phi_0 (z)$, where $\phi_0$ is given by: $$\phi_0 (z) = \frac{\displaystyle \Big( \frac{z - i}{i z - 1} \Big)^{1/2} - i} {\displaystyle - i \, \Big( \frac{z - i}{i z - 1} \Big)^{1/2} + 1} \, \cdot$$ Then, we define: $$\varsigma_\theta = \exp ( - f_\theta \circ g_\theta) .$$ We proved in [@LIQURO Section 4.2] (though it is not sharp) that: $$a_n (C_{\varsigma_\theta}) \gtrsim \frac{1}{n^{\theta/2}} \, \cdot$$ We define $\Phi_1 \colon {\mathbb D}^N \to {\mathbb D}^N$ as: $$\Phi_1 (z_1, z_2, \ldots, z_N) = \big( \varsigma_\theta (z_1), 0, \ldots, 0\big) \, .$$ If $J = J_N \colon H^2 ({\mathbb D}) \to H^2 ({\mathbb D}^N)$ is the canonical injection defined by $(Jh) (z_1, \ldots, z_N) = h (z_1)$ and $Q = Q_N \colon H^2 ({\mathbb D}^N) \to H^2 ({\mathbb D})$ is defined by $(Q f) (z_1) = f (z_1, 0, \ldots, 0)$, then $C_{\Phi_1} = J C_{\varsigma_\theta} Q$; hence $C_{\Phi_1}$ is compact. On the other hand, we also have $Q C_{\Phi_1} J = C_{\varsigma_\theta}$, which implies that $a_n (C_{\Phi_1}) \gtrsim a_n (C_{\varsigma_\theta}) \gtrsim n^{- \theta/2}$. It follows that: $$\beta_N (C_{\Phi_1}) \geq \lim_{n \to \infty} (n^{- \theta/2})^{1/ n^{1/ N}} = 1 \, ,$$ and hence $\beta_N (C_{\Phi_1}) = 1$.
Now, if: $$\Phi_2 (z_1, \ldots, z_N) = \big( \varsigma_\theta [\sigma (z_1)], 0, \ldots, 0\big) \, ,$$ since $\sigma$ is surjective, we have $\Phi_1 ({\mathbb D}^N) = \Phi_2 ({\mathbb D}^N)$. Moreover, we have $C_{\Phi_2} = J C_{\varsigma_\theta \circ \sigma} Q = J C_\sigma C_{\varsigma_\theta} Q$, so $a_n (C_{\Phi_2}) \lesssim a_n (C_\sigma)$. Since $\rho_\sigma (h) \leq h^{N + 1} \, {{\rm e}}^{- 2 / h^2}$, Proposition \[second\] gives, with $h = 1/ n^{1/3}$: $$a_n (C_\sigma) \lesssim {{\rm e}}^{- c n^{2/3}} \, ,$$ so $[a_n (C_{\Phi_2})]^{1/n^{1/N}} \lesssim \exp ( - c \, n^{\frac{2}{3} - \frac{1}{N}})$ and $\beta_N (C_{\Phi_2}) = 0$.
\[Proof of Lemma \[uno\]\] In [@DHL], we observed that the singular numbers of $S \otimes T$ are the non-increasing rearrangement of the numbers $s_j t_k$, where $s_j$ and $t_k$ denote respectively the $j$-th and the $k$-th singular number of $S$ and $T$. We can assume $s_1 = t_1 = 1$. Using this observation, we will majorize the number of pairs $(j, k)$ such that $s_j t_k > {{\rm e}}^{- c n}$. Let $(j, k)$ be such a pair. Since $s_j \leq s_1 = 1$, we have $t_k \geq {{\rm e}}^{- c n}$ so that $k \leq [n^B] \leq n^B$. Hence, for some $2 \leq l \leq n$, we have $(l - 1)^B < k \leq l^B$. Then, due to the assumption on $T$, $t_k < {{\rm e}}^{- c (l - 1)}$ and $s_j \geq {{\rm e}}^{- c n} t_{k}^{- 1} \gtrsim {{\rm e}}^{- c (n - l + 1)}$, implying that $j \lesssim (n - l + 1)^{A}$, thanks to the assumption on $S$. As a consequence, since the number of integers $k$ such that $(l - 1)^B < k \leq l^B$ is dominated by $l^{B - 1}$, the number $\nu_n$ of pairs $(j, k)$ such that $s_j t_k > {{\rm e}}^{- c n}$ is dominated by: $$\sum_{l = 1}^n (n - l + 1)^{A} l^{B - 1} \sim n^{A+B} \int_{0}^1 t^{A} (1 - t)^{B} \, dt \, ,$$ by a Riemann sum argument. Next, let $M \in {\mathbb N}$ big enough to have: $$\sum_{l = 1}^n (n - l + 1)^A l^{B - 1} \leq M n^{A + B} - 1 \, , \quad \text{for all } n \, .$$ By definition, $a_{M [n^{A + B}]} (S \otimes T) \leq a_{\nu_{n} + 1} (S\otimes T) \leq {{\rm e}}^{- c n}$, giving the result.
[**Acknowledgement.**]{} This paper was made when the two first-named authors visited the University of Sevilla in February 2018. It is their pleasure to thank this university and all colleagues therein for their warm welcome.
The third-named author is partially supported by the project MTM2015-63699-P (Spanish MINECO and FEDER funds).
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Daniel Li\
Univ. Artois, Laboratoire de Mathématiques de Lens (LML) EA 2462, & Fédération CNRS Nord-Pas-de-Calais FR 2956, Faculté Jean Perrin, Rue Jean Souvraz, S.P. 18 F-62 300 LENS, FRANCE\
[email protected]
Hervé Queffélec\
Univ. Lille Nord de France, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération CNRS Nord-Pas-de-Calais FR 2956 F-59 655 VILLENEUVE D’ASCQ Cedex, FRANCE\
[email protected]
Luis Rodr[í]{}guez-Piazza\
Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Calle Tarfia s/n\
41 012 SEVILLA, SPAIN\
[email protected]
|
---
abstract: 'A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $\Sigma$. We study proper affine actions of the double extension of $\pi_1 (M) \cong \pi_1 (\Sigma)$ when $\Sigma$ is homeomorphic to a projective plane minus two discs. We classify such actions and show that there exist proper actions that do not admit crooked fundamental domains.'
author:
- 'William M. Goldman and Gregory D. Laun'
bibliography:
- 'master.bib'
title: |
Affine Coxeter Extensions of the Two-Holed\
Projective Plane
---
The goal of this paper is to classify affine orbifolds that are double-covered by a Margulis spacetime whose associated hyperbolic surface is homeomorphic to a two-holed cross surface (topologically, a plane minus two discs).
This is part of a larger project to study proper actions of non-solvable discrete groups on affine spaces. The case of a free group acting properly on $\R^3$ without fixed points is now fairly well understood. The quotients of such an action are geodesically complete affine three-manifolds called *Margulis spacetimes*. Margulis spacetimes arise as infinitesimal deformations of hyperbolic surfaces with free fundamental group and such that the deformation uniformly lengthens or shortens all closed geodesics [@Goldman2009; @goldman2000flat]. In this way, each Margulis spacetime is canonically associated with a non-compact complete hyperbolic surface. If $\Sigma$ is such a surface, and $M$ is the associated Margulis spacetime, we say that $M$ is an $\emph{affine deformation}$ of $\Sigma$. Similarly, we say that the holonomy representation of $\pi_1 (M)$ is an affine deformation of the holonomy representation of $\pi_1 (\Sigma)$.
In this paper, we study proper actions by the involution group $\Psi \defeq \Z_2 \ast \Z_2 \ast \Z_2$. As an abstract group, $\Psi$ naturally contains a free group $\F_2$ of rank two as an index two subgroup. Moreover, every irreducible representation $\rho_0: \F_2 \to \Isom (\H^2)$ into the isometries of the hyperbolic plane admits a unique extension to a representation $\Psi \to \SL (2,\C)$, which we call a *Coxeter extension*.
The case of orbifold quotients of affine actions is less well-studied than the manifold case. Charette [@charette2009groups] investigated affine deformations of reflection groups when the index two subgroup is the holonomy group of a three-holed sphere. She showed the that there exist involution groups that act properly on affine three space which do not admit crooked fundamental domains, but whose index-two subgroups do admit crooked fundamental domains. This is in contrast to the case of Margulis spacetimes: every proper action of a free discrete group of affine transformations on $\R^3$ admits a crooked fundamental domain [@danciger2014margulis]. The present paper is the first to investigate orbifold quotients when the corresponding hyperbolic manifold is non-orientable.
The basic strategy of this paper is similar to [@CDG1HT]. Let $\Sigma$ be a two-holed projective plane. Given a hyperideal triangulation of $\Sigma$, we can use the theory of crooked ideal triangulations from [@Burelle2012; @CDG1HT] to realize this hyperideal triangulation as a configuration of crooked planes.
This process parametrizes the deformation space associated with a particular ideal triangulation of $\Sigma$. In order to describe the full proper affine deformation space, we also need to describe how the deformation space changes when we change ideal triangulations. Following [@CDG1HT], we use the *flip graph* to encode an algebraic structure on the space of ideal triangulations on $\Sigma$. By [@danciger2014margulis] the image of the deformation space is actually the dual complex to the flip graph, the arc complex. In [@charette2011finite], it was shown that every proper affine deformation of a hyperbolic two-holed cross surface admits a crooked fundamental domain. Moreover, the projectivized space of crooked fundamental domains was shown to be a quadrilateral $\mathbf{Q}$ in $\P ( H^1 (\Gamma_0,\R^{2,1})) \cong \RP^2$. The main goal of the present paper is to describe the projectivized space of crooked fundamental domains for a Coxeter extension of such a group. Up to a choice of geodesic representatives for the linearized domain, this space is hexagon $\mathbf{H}$ inscribed in $\mathbf{Q}$.
Let $\Sigma$ be homeomorphic to a two-holed cross surface, and let $\pi \defeq \pi_1 (S)$ be the image of a choice of holonomy representation. Let $\pi^{\prime}$ be a Coxeter extension of $\pi$. Choose a triangular fundamental domain for the action of $\pi^{\prime}$ on $\H^2$ such that the sides of the triangle are pairwise ultraparallel. Two sides are determined by the two reflections that generate $\pi^{\prime}$: they are necessarily the fixed geodesics of these reflections. The homotopy class of the remaining arc is fixed, but there is an interval’s worth of choice in the geodesic representative of this class.
Fix some choice of geodesic representative by picking a $\theta$ in the interval. Call the corresponding fundamental domain $\mathscr{D}_{\theta}$. Every other triangular fundamental domain for $\pi^{\prime}$ (with possibly a different set of generators) is given by $\tau \cdot \mathscr{D}_{\theta}$ for some $\tau$ a mapping class of $\Sigma$ and some choice of $\theta$. Every crooked fundamental domain for a proper affine deformation of $\pi^{\prime}$ linearizes to a fundamental domain of the form $\tau \cdot \mathscr{D}_{\theta}$. In what follows, we fix the parameter $\theta$.
\[main\] The the space of proper affine deformations of $\pi^{\prime}$ that admit a crooked fundamental domain for fixed $\theta$ is a six-sided cone over the moduli space of hyperbolic structures on the orbifold quotient of $S$. The cone projectivizes to a hexagon $\mathbf{H}$ inscribed in $\mathbf{Q}$.
Varying $\theta$ gives an octagon instead of a hexagon, as two of the vertices are replaced by intervals. See Figure \[fig:octagon\].
As a corollary, there are proper affine deformations of $\pi$ that do not admit a crooked fundamental domain; namely those corresponding to points in $\mathbf{Q} \setminus \mathbf{H}$. The corresponding fact was proved in the context of the three-holed sphere in [@charette2009groups].
Notation {#sec:notation}
========
Following Charette, Drumm, and Goldman [@charette2011finite] and John H. Conway, we call the topological surface underlying a projective plane a “cross surface”. This is to avoid any confusion with the notion of the projective plane as a space carrying parabolic geometry.
We work in three-dimensional affine space. Every Margulis spacetime is a quotient $\R^{2,1}/\Gamma$ where $\R^{2,1}$ is Minkowski space and $\Gamma$ is a free discrete subgroup of $\Aff (\R^{2,1}) = O (2,1)\semidirect \R^{2,1}$. Let $G$ be a free group, and $\phi: G \to \Aff (\R^{2,1})$ the holonomy representation of a Margulis spacetime. Projection onto the first factor gives a representation into $\SO (2,1) \cong \Isom^{\pm} (\H^2)$. Call its image $\Gamma_0$. Then $\Gamma_0$ can be identified with the holonomy group of a hyperbolic surface $\Sigma \defeq \H^2/\Gamma_0$. In this paper we fix $\Sigma$ to be homeomorphic to a two-holed cross-surface.
Affine deformations $\Gamma$ of a fixed linear $\Gamma_0$ are classified by the cohomology $H^1 (\Gamma_0, \R^{2,1})$. We view $[u] \in H^1 (\Gamma_0,\R^{2,1})$ as assigning to each hyperbolic isometry $X \in \Gamma_0 \subset \Isom (\H^2)$ a translational part $u (X) \in \R^{2,1}$. We call an element of $H^1 (\Gamma_0,\R^{2,1})$ an *affine deformation*. We call $[u]$ *proper* if the semidirect product $\Gamma$ associated to $[u]$ acts properly on $\R^{2,1}$. Additional details can be found in [@Charette2010; @Burelle2012; @CDG1HT].
We identify $\R^{2,1}$ with the Lie algebra $\mathfrak{psl} (2,\R)$. The signature $(2,1)$ inner product, denoted by $\cdot$, is given by $1/2$ the trace form: $$v \cdot u \defeq \frac{1}{2}\tr (vu)$$ We also need the Lorentzian cross product $\boxtimes$, defined as the unique map satisfying $$(u \boxtimes v) \cdot w = \det (u,v,w)$$ In particular, $u \boxtimes v$ is Lorentz-orthogonal to both $u$ and $v$.
As a Lie group, $SO (2,1)^0$ is isomorphic to $\PSL (2,\R)$. Since $\Sigma$ is non-orientable, we need orientation-reversing isometries as well. These can be identified with matrices $iP$, $P \in \GL (2,\R)$, $\det (P) = -1$. See [@Goldm2009; @charette2011finite].
We identify $\H^2$ with the space of timelike subspaces in $\R^{2,1}$ in the standard way. That is, each point in $\H^2$ defines a class $[t] \in \P (\R^{2,1} \setminus \left\{ 0 \right\})$ with $t \cdot t < 0$. For a vector $u \in \R^{2,1}$, let $u^{\perp}$ denote the associated Lorentz-orthogonal subspace. If $u$ is a spacelike vector, then $u^{\perp}$ is a linear plane that intersects the lightcone transversely, and so may be identified with a hyperbolic geodesic in $\H^2$. The intersection of $u^{\perp}$ with the lightcone determines two future-pointing unit lightlike vectors $u^{\pm}$, which we think of as points on the ideal boundary $\partial \H^2$. Specifically, we choose $u^{\pm}$ such that $\left\{ u^-, u^+,u \right\}$ is a right-handed basis of $\R^{2,1}$. If $v$ and $w$ are spacelike vectors, we say that $v$ and $w$ are *ultraparallel* if the corresponding hyperbolic geodesics defined by $v^{\perp}$ and $w^{\perp}$ are ultraparallel. We say that spacelike vectors $v_1, v_2, v_3$ are *consistently oriented* if
- $v_i \cdot v_j < 0$
- $v_i \cdot v_j^{\pm} \leq 0$
whenever $i \neq j$.
Let $X \in \Isom (\H^2)$ be a hyperbolic or parabolic isometry. The linear map defined by $A \mapsto XAX^{-1} = \Ad_X (A)$ has a 1-eigenspace that is spacelike if $X$ is hyperbolic and lightlike if $X$ is elliptic. If $X$ is hyperbolic, choose a $1$-eigenvector $X^0$ of $X$ satisfying $ X^0\cdot X^0 = 1$.
Let $(X, u (X)) \in \Gamma$ with $X \in \Gamma_0$ hyperbolic. The *Margulis invariant* of the affine deformation $(X,u (X))$ is the neutral projection of the translational part $u$: $$\alpha_{[u]} (X) \defeq u (X) \cdot X^0$$ The map $g \mapsto \alpha_{[u]} (g)$ depends only on the cohomology class $[u]$ of $u$. For any nonzero point $p \in \R^{2,1}$, the Margulis invariant of $X$ can be computed as $(XpX^{-1} - p) \cdot X^0$.
Every proper affine action by a non-solvable discrete group on $\R^{3}$ admits a fundamental domain bounded by *crooked planes* [@danciger2014margulis]. A crooked plane is a piecewise-linear surface invented by Drumm [@Drumm1990] to enable ping-pong arguments in $\R^{2,1}$. A crooked plane is determined by a spacelike vector, called its *direction vector*, and a point, called its *vertex*. Specifically, for a point $p$ and a spacelike vector $v$, define the crooked plane $\mathsf{C} (v,p)$ as follows. It is the union of two *wings* $$\begin{aligned}
p &+ \R_{+} v^+ + \R_+ v\\
p &+ \R_+ v^- - \R_+ v\end{aligned}$$ and a *stem* $$p + \left\{ x \in \R^{2,1} \mid v \cdot x =0, x \cdot x \leq 0 \right\}$$ By duality the direction vector corresponds to a hyperbolic geodesic $\ell$. In the language of the Lie group $\PSL (2,\R)$ and its Lie algebra $\mathfrak{psl} (2,\R)$, a crooked plane is the set of all Killing fields with a non-repelling fixed point on $\ell$ (cf. [@danciger2014margulis]).
In order to build fundamental domains, we need to know when two crooked planes are disjoint. The following criterion provides this information. Let $w_1, w_2, w_3$ be unit spacelike vectors defining crooked planes. Let $u_i^{\pm} \in \R_{\geq 0}$ for $i \in \left\{ 1,2,3 \right\}$, and define the points $$\begin{aligned}
q_1 &= u_1^{-} w_1^{-} - u_1^{+} w_1^{+}\\
q_2 &= u_2^- w_2^{-} - u_2^{+} w_2^{+}\\
q_{0} &= u_3^{-} w_3^{-} - u_3^+ w_3^{+}\\\end{aligned}$$ The following proposition follows from [@Charette2010].
When all the coefficients $u_i^+$, $u_i^-$ are positive, then the crooked planes $\mathscr{C} (w_1,q_1)$, $\mathscr{C} (w_2,q_2)$, and $\mathscr{C} (w_{3},q_3)$ are disjoint.
The parameters $u_i^{\pm}$ form a translation semigroup, called the *stem quadrant*. For a spacelike vector $w$, denote the stem quadrant by $V (w)$. $$\label{cocycle}V (w) = \R_+^{*} w^{-} - \R_+^{*} w^{+}$$ See [@Burelle2012] for additional details. The disjointness criterion can also be interpreted in the language of strip deformations, as in [@danciger2014margulis].
Finally, we need an analog of geodesic reflections in the language of affine deformations. These are provided by *spine reflections*. Given a spacelike vector $u$, the corresponding spine reflection is a map ${\operatorname{Spine}}(u) \in \PSL (2,\R)$ defined as: $${\operatorname{Spine}}(u): v \mapsto -v + 2 \frac{v \cdot u}{u \cdot u}u$$ Charette studied spine reflections in [@charette2009groups].
The space of hyperideal triangulations {#sec:space-ideal-triang}
======================================
In this section, we build fundamental domains for the action of the involution group $\Psi$.
A Fundamental Domain for the Action of $\Psi$ {#sec:2hx-fund-doma}
---------------------------------------------
Let $\pi \cong \pi_1 (\Sigma)$ be the holonomy of a hyperbolic structure on $\Sigma$. Then $\pi$ is a free group generated by two glide reflections $X,Y$ that intersect in a distinguished point $p_0$. Additionally, $\Sigma$ has two boundary components $A,B \in \PSL (2,\R)$ which we can choose so that $A \defeq XY$, $B \defeq Y^{-1}X$. This gives a redundant presentation $$\pi = \langle X, Y, A,B \mid A = XY, B = Y^{-1}X\rangle$$
Let $\iota_0$ be the (orientation-preserving) point symmetry in $p_{0}$. Then $\iota_0$ reverses the orientation of every geodesic passing through $p_{0}$. Since $p_{0} = \Axis (X) \cap \Axis (Y)$, $\iota_0 X \iota_0$ is a glide reflection with the same translation distance and axis as $X$, but in the opposite direction. It follows that $\iota_0 X \iota_0 = X^{-1}$. Similarly, $\iota_0 Y \iota_0 = Y^{-1}$. Following [@Goldman2009], we see that $\langle X, Y, \iota_0 \rangle \cong \Z_2 \ast \Z_2 \ast \Z_2$ is a Coxeter extension of $\pi$.
We can see better the structure of the involution group by defining $R_X \defeq X \iota_0$, $R_Y \defeq \iota_0 Y$. The index two subgroup $\pi$ can then be recovered as: $$\begin{aligned}
\label{eq:index2}
X &=& R_X \iota_0\\
Y &=& \iota_0\ R_Y\\
A &=& R_X R_Y\\
B &=& R_Y \iota_0 R_X \iota_0\end{aligned}$$ While $\iota_0$ is a symmetry in a point, $R_X$ and $R_Y$ are reflections in hyperbolic geodesics. Call these geodesics $\ell_X$ and $\ell_Y$ respectively. Then $\ell_Y$ is the mutual perpendicular of $\Axis (A)$ and $\Axis (B)$, and $\ell_X$ is the mutual perpendicular of $\Axis (A)$ and $\Axis (XBX^{-1})$. See Figure \[fig:core\].
We depict two types of fundamental domain for the action of this Coxeter group in Figure \[fig:cox-options\]. Each is a hyperideal triangle bounded by geodesics that project in the quotient to $\ell_X$, and $\ell_Y$ and by a third hyperbolic geodesic $\ell_0$ through $p_{0}$. The two triangulations differ by a diagonal flip that sends $\ell_0$ to a the line $\ell_0^{\prime}$ orthogonal to $\ell_0$ through $p_{0}$.
Arc Complex of $\Sigma$ {#sec:arc-complex-sigma}
-----------------------
Recall the definition of the *flip graph* of a surface $\Sigma$ with boundary. The vertices of the flip graph are ideal triangulations of $\Sigma$, and there is an edge between two triangulations if and only if the two differ by a diagonal flip. For surfaces with fundamental group free of rank 2, the dual complex to the flip graph is the *arc complex*. The vertices of the arc complex are homotopy classes of properly embedded arcs in $\Sigma$, and $k$ arcs span a simplex if and only if they can be realized disjointly. The duality expresses the fact that a maximal collection of disjoint arcs defines an hyperideal triangulation of the surface.
Charette, Drumm, and Goldman [@CDG1HT] used the flip graph (although using different language) to parametrize the proper affine deformation space of a one-holed torus. Danciger, Gu[é]{}ritaud, and Kassel [@danciger2014margulis] generalized this approach to use the arc complex to parametrize the proper affine deformation space of all convex-cocompact surfaces.
The arc complex for the two-holed cross-surface is depicted in Figure 3. In the diagram, the antipodal points on the circle’s boundary are identified. The black semidisks (antipodally identified) indicate the two removed discs. For some of the arc classes, we show two representatives for clarity.
An hyperideal triangle corresponds to a top-dimensional simplex of the arc complex (or alternately, to a point in the flip graph). In the present paper, we consider the hyperideal triangle of type $I$. This forms a fundamental domain for the action of $\Psi$. There is an element of the mapping class group that interchanges $I$ with $II$. Since this induces an automorphism of the fundamental group, $II$ is also a fundamental domain for the Coxeter extension. The remaining simplices do not arise as fundamental domains for the Coxeter extension.
Algebraically, the flip between the two triangulations is achieved by an automorphism of $\pi$ that fixes $X$ and sends $Y$ to its inverse. See the proof of Proposition \[pentagon-flip\].
Parametrization of Hyperideal Triangles {#sec:param-ultr-triangl}
---------------------------------------
In this section, we work in a fundamental domain in configuration $I$. We first parametrize hyperideal triangles in this configuration.
Without loss of generality, we assume that the axes of $X$ and $Y$ intersect at the origin $p_0 = 0$. Let $\ell_0$ be a hyperbolic geodesic through $p_0$ making angle $\theta$ with the horizontal axis. Let $m$ be the common perpendicular between $\ell_X$ and $\ell_Y$. Let $d$ be the distance between $m$ and $p_0$, and let $u_1$ (respectively $u_2$) be the distance between $m$ and $\ell_X$ (respectively $\ell_{Y})$.
Given $u_1, u_2$, $d$, and $\theta$, define the spacelike vectors $$w_{i} =\begin{pmatrix} \cosh u_i\\ \sinh u_{i} \sinh d \\ \sinh u \cosh u_{i}\end{pmatrix}$$ for $i = 1,2$, and $$w_{0} = \begin{pmatrix}\cos \theta\\ \sin \theta\\ 0\end{pmatrix}.$$ With appropriate bounds on the parameters, the geodesics $\left\{ w_i^{\perp} \right\}$ are disjoint and define a hyperideal triangle in configuration $I$. The group $\Gamma_0 = \rho (\F_2)$ does not depend on $\theta$, only the choice of fundamental domain does.
Define the spine reflections $$\begin{aligned}
\label{eq:spine-refl}
R_X &\defeq {\operatorname{Spine}}(w_{1})\\
R_Y &\defeq {\operatorname{Spine}}(w_2)\end{aligned}$$ and define $\iota_0$ to be the point symmetry in $p_{0}$. Then $R_X$ and $R_Y$ are glide reflections whose axes intersect at $p_{0}$.
Statement of Theorems {#sec:statement-theorems}
=====================
From now on, fix the parameters $d, u_1, u_2, \theta$. This also fixes $\Gamma_0$. In order to determine which cocycles $[u] \in H^1 (\Gamma_0, \R^{2,1})$ act properly, we need the following facts about the Margulis invariant.
For fixed $g \in \Gamma$, the Margulis invariant determines a well-defined functional on $H^1 (\Gamma_0,\R^{2,1})$ $$\mu_g: H^1 (\Gamma_0,\R^{2,1}) \to \R$$ defined by $$\mu_g ([u]) \defeq \mu_{[u]} (g)$$ The invariants $X, Y$, and $A$ determine an isomorphism of vector spaces: $$H^1 (\Gamma_0,\R^{2,1}) \cong \R^3,$$ given by $$[u] \mapsto \bmat{\mu_{[u]} (X)\\ \mu_{[u]} (Y) \\ \mu_{[u]} (A)}$$ By abuse of notation, for each $g \in \Gamma_0$ denote its image $\rho (g) \in
\Aff (2,1)$ by $g$ as well. The following proposition was proved in [@charette2011finite].
\[CDG2HX\] Let $\Sigma$ be a two-holed cross surface with holonomy group $\pi = \rho \left (\pi_{1} (\Sigma) \right )$ with presentation as above. Then an affine deformation $[u] \in H^1 (\Gamma_{0}, \R^{2,1})$ of $\pi$ acts properly on $\R^{2,1}$ if and only if the Margulis invariants $\alpha_{[u]} (X)$,$\alpha_{[u]} (Y)$,$\alpha_{[u]} (A)$, and $\alpha_{[u]} (B)$ all are nonzero with the same sign.
A triple of points $q_1, q_2, q_3$ in the stem quadrants $V (w_i)$ as in (\[cocycle\]) gives a cocycle $[u] \in H^1 (\Gamma_0, \R^{2,1})$. Let $v \in \R^6$ be given by $v =
\begin{pmatrix}
u_1^-\\ u_1^{+} \\ u_2^-\\ u_2^{+} \\ u_3^-\\ u_3^{+} \\
\end{pmatrix}$. Define the map $M: \R^6 \to \R^3$ by $$Mv \defeq
\begin{pmatrix}
\label{alpha-matrix}
\alpha_{[u]} (X)\\ \alpha_{[u]} (Y)\\ \alpha_{[u]} (A)
\end{pmatrix}$$ Then $M$ is a linear map $M: \R^6 \to H^1 (\Gamma_0,\R^{2,1}) \cong \R^3$.
By Proposition \[CDG2HX\], the image under $M$ of the positive orthant in $\R^6$ identifies with the space of crooked fundamental domains for the action of the affine deformation of $\pi$ corresponding to $[u]$ for a fixed choice of $\theta$. This image is a cone in $\R^3$. Projectivized, this cone becomes a polygon in $\RP^2$.
In the present paper, we are interested not in the proper affine deformation space for $\Sigma$, but for its quotient $\Sigma^{\prime}$. Define $\pi^{\prime}$ to be the Coxeter extension of $\pi$. Since $\pi^{\prime}$ is a finite extension of a discrete group, it acts properly on $\R^{2,1}$ if and only if $\pi$ does. By [@CDG1HT], $\pi$ admits a crooked fundamental domain. However, even if the action of $\pi$ admits a crooked fundamental domain, the action of $\pi^{\prime}$ may not. In particular, we show
\[hexagon\] Let $\Gamma_0^{\prime}$ be a Coxeter group whose index two subgroup $\Gamma_0$ is the holonomy group for a two-holed cross surface. Fix the parameter $\theta$. This fixes a fundamental domain $\kappa$ for the linear Coxeter group.
Then the projectivized space of crooked fundamental domains for $\Gamma^{\prime}$ that linearize to $\kappa$ is a hexagon $\mathbf{H}$ in $\P (H^1 (\Gamma_0,\R^{2,1})) \subset \RP^2$. The hexagon $\mathbf{H}$ is inscribed in the quadrilateral $\mathbf{Q}$ that parametrizes the space of crooked fundamental domains for the action of $\Gamma_0$.
The hexagon is depicted in Figure \[fig:hexagon\]. Allowing $\theta$ to vary gives an octagon, with the additional sides corresponding to the extra degree of freedom in choosing a geodesic representative for the arc preserved by the point symmetry. See Figure \[fig:octagon\].
The hexagon is the union of two pentagons. Each pentagon is the projectivized image of the space of crooked fundamental domains in $\RP^2$ corresponding to a fixed ideal triangulation. The map that flips the two ideal triangulations induces a map on the space of crooked fundamental domains that interchanges the pentagons. We thus prove Theorem \[hexagon\] in two steps.
\[pentagon\] Define the map $M$ as above. The image of the positive orthant in $\R^6$ projectivizes to a pentagon $\mathbf{P_1}$ in $\RP^2$.
\[fig:hexagon\]
. \[fig:octagon\]
Notice here the asymmetry between $\alpha (B)$ and $\alpha (A)$. Specifically, $\alpha (A)$ can vanish, but there is no situation in which $\alpha (B) = 0$ while the other Margulis invariants remain nonzero. This is an artifact of working with a fundamental domain in the configuration $I$. Such a fundamental domain is asymmetric with respect to $A$ and $B$: it contains a self-loop at $B$ but not one at $A$. We can recover the inherent symmetry of the problem by considering the map $\phi$ that achieves the diagonal flip $I \to II$. The fundamental domain $II$ contains a self-loop at $A$ but not at $B$.
\[pentagon-flip\] Consider an automorphism of the fundamental group $\phi \in \Aut (\pi)$ defined on generators by $\phi (X) = X$, $\phi(Y) = Y^{-1}$. The automorphism extends to the Coxeter group as $$\begin{aligned}
\label{eq:1}
\phi(R_X) &= R_X\\
\phi(R_Y) &= \iota_0 R_Y \iota_0\\
\phi(\iota_0) &= \iota_{0}\end{aligned}$$ Then the image of $\phi$ is another inscribed pentagon $\mathbf{P_2}$ such that $\mathbf{P_1} \cap \mathbf{P_2}$ is a quadrilateral $\mathbf{Q_{small}}$ inscribed in the larger quadrilateral $\mathbf{Q}$.
This automorphism switches the boundary components: $\phi (A) = B$, $\phi (B) = A$.
Together with the parametrization of hyperideal triangles, Theorem \[hexagon\] establishes Theorem \[main\].
Proofs of the Theorems {#sec:proofs-two-theorems}
======================
Denote the action of $\PSL (2,\R)$ on $\mathfrak{psl} (2,\R)$ by $(g,p) \mapsto g.p = gpg^{-1}$.
We now prove Theorem \[pentagon\] by explicitly computing $M$. Recall the vectors $q_1,q_{2}, q_0$ defined above.
\[alpha-compute\] We can compute the Margulis invariants as $$\begin{aligned}
\alpha (X) &= 2 (q_1 - q_0) \cdot X^0\\
\alpha (Y) &= 2 (q_0 - q_2) \cdot Y^0\\
\alpha (A) &= 2(q_1 - q_2) \cdot A^0\\
\alpha (B) &= 2 (q_2 - \iota_0^{-1}q_1 \iota_0) \cdot B^0\\\end{aligned}$$
We prove the result for $\alpha (B)$. The rest are similar, but simpler. Recall $$B = Y^{-1}X = (\iota_0 R_y)^{-1}X = R_y\iota_0^{-1}R_x\iota_0.$$ Also note that $\alpha (B) = (B.x - x, B^0)$ where $x \in \R^3$ is any nonzero vector. It is convenient to compute $B.x$ where $$x = \iota_0^{-1} .q_1 = \iota_0^{-1}q_1 \iota_0.$$ This is because since $R_x$ fixes $q_{1}$, $\iota_0^{-1}R_x \iota_0$ fixes $\iota_0^{-1} q_1 \iota_{0}$.
$$\begin{aligned}
(Y^{-1}X) (\iota_0^{-1}.q_1) &= \left( R_y \iota_0^{-1}R_x \iota_0 \right) (\iota_0^{-1}.q_1)\\
&= (R_y) (\iota_0^{-1}.q_1)\\
&= (R_y)\left( q_2 + (\iota_0^{-1}.q_1 - q2)\right)\\
&= q_2 - (\iota_0^{-1}.q_1 - q2) \mod w_2\end{aligned}$$
Compare with [@CDG1HT]. Since $w_2 \perp B^{0}$, this term vanishes in the computation of the Margulis invariant.
Then $$\begin{aligned}
\alpha (B) &= \left (B.(\iota_0^{-1}.q_1) - \iota_0^{-1}.q_1 \right )
\cdot B^0\\
&= \left ( q_2 - \iota_0^{-1}.q_1 + q_{2} - \iota_0^{-1}.q_{1}\right ) \cdot B^0\\
&= 2 (q_2 - \iota_0^{-1}.q_1) \cdot B^0\end{aligned}$$ as desired.
View the direct sum of stem quadrants $V (w_1) \oplus V (w_2) \oplus V (w_0)$, as the positive orthant in $\R^6$. We can decompose $M$ as $$M = M_1
\begin{pmatrix}
u_1^-\\ u_1^{+} \\
\end{pmatrix}
+ M_2 \begin{pmatrix}
u_2^-\\ u_2^{+} \\
\end{pmatrix}
+ M_3 \begin{pmatrix}
u_3^-\\ u_3^{+} \\
\end{pmatrix}$$ We now compute the matrices $M_1$, $M_2$, and $M_3$.
If $u_2^{\pm} = 0$ and $u_3^{\pm} = 0$, then $Mv = M_1 \begin{pmatrix}
u_1^-\\ u_1^{+} \\
\end{pmatrix}$. Let $e_1$, $e_2$ be the standard basis vectors of $\R^2$. Then $$\begin{aligned}
\textsc{M}_{1}e_1 =
\begin{pmatrix}
\alpha (X)\\
\alpha (Y)\\
\alpha (A)
\end{pmatrix}
& =
\begin{pmatrix}
2 (q_1 - q_0) \cdot X^0\\
2 (q_0 - q_2) \cdot Y^0\\
2(q_1 - q_2) \cdot A^0
\end{pmatrix}\\
&=
\begin{pmatrix}
2 q_1 \cdot X^0\\
0\\
2 q_1 \cdot A^0\\
\end{pmatrix}
\\
&=\begin{pmatrix}
2 w_1^- \cdot X^0\\
0\\
2w_1^- \cdot A^0
\end{pmatrix}\end{aligned}$$
Similarly, $M_{1}e_2 =
\begin{pmatrix}
-2w_1^+ \cdot X^{0}\\
0\\
-2w_1^+ \cdot A^0
\end{pmatrix}
$.
For a general vector $
$ $u_1^- e_1 + u_1^{+} e_2$. We compute $M_1$ $$\begin{aligned}
M_1
&=
2 \begin{pmatrix}
w_1^- \cdot X^0 & -w_1^+ \cdot X^0\\
0 & 0 \\
w_1^- \cdot A^0 & - w_1^+ \cdot A^0
\end{pmatrix}\end{aligned}$$ The remaining matrices $M_2$ and $M_3$ are analogous. Explicitly: $$M_2 = 2
\begin{pmatrix}
0 & 0 \\
- W_2^- \cdot Y^0 & w_2^+ \cdot Y^0\\
- w_2^- \cdot A^0 & w_2^+ \cdot A^0\\
\end{pmatrix}$$
$$M_3 = 2\begin{pmatrix}
-w_3^- \cdot X^0 & w_3^+ \cdot X^0\\
w_3^- \cdot Y^0& -w_3^+ \cdot Y^0\\
0 & 0
\end{pmatrix}$$
Define $$v_{1 } \defeq \begin{pmatrix}
u_1^-\\ u_1^{+} \\
\end{pmatrix}; v_{2 } \defeq \begin{pmatrix}
u_2^-\\ u_2^{+} \\
\end{pmatrix}; v_{3 } \defeq \begin{pmatrix}
u_3^-\\ u_3^{+} \\
\end{pmatrix};$$
As above, define $M$ by $$Mv = M_1v_1 + M_2v_2 + M_3 v_3$$ Then $M_1$ and $M_2$ have rank 2 and $M_3$ has rank 1.
We first prove that $M_3$ has rank 1. Recall the definition of $M_3$ as $$M_3 = 2
\begin{pmatrix}
-w_3^- \cdot X^0 & w_3^+ \cdot X^0\\
w_3^- \cdot Y^0& -w_3^+ \cdot Y^0\\
0 & 0
\end{pmatrix}.$$ Let $M_3^{\prime}$ be the $2 \times 2$ submatrix consisting of the nonzero entries in $M_3$. Then we can write it as the product $$M_3^{\prime} = 2\begin{pmatrix}
X^0\\ Y^0
\end{pmatrix} \cdot
\begin{pmatrix}
w_3^- &- w_3^+
\end{pmatrix}.$$ The determinant is $$\begin{aligned}
4(w_3^- \cdot X^0) (w_3^+ \cdot Y^0) - (w_3^- \cdot Y^0) (w_3^+ \cdot X^0) &= -4(X^0 \boxtimes Y^0 ) \cdot (w_3^- \boxtimes w_3^+) \\
&= -4(X^0 \boxtimes Y^0) \cdot w_3\end{aligned}$$ By construction, $w_3$ is a spacelike vector whose orthogonal space $w_3^{\perp}$ contains the timelike vector $p_0 =
\begin{pmatrix}
0\\0\\1
\end{pmatrix}
$ that spans the intersection of $(X^0)^{\perp} \cap (Y^0)^{\perp}$. In particular, $w_3 \cdot (X^0 \boxtimes Y^0) = 0 $, and $M_3$ has determinant $0$. Since $M_3$ is not the $0$ matrix, it has rank $1$.
We will show that $M_1$ has rank $2$. The $M_2$ case is entirely similar. As before, let $M_1^{\prime}$ be the $2 \times 2$ submatrix of $M_1$ consisting of nonzero entries. As in the computation of $M_3^{\prime}$, $$\frac{1}{4} \det M_1^{\prime} = -w_1 \cdot (X^{0} \boxtimes A^0).$$ Note that $X^{0} \boxtimes A^0$ is a spacelike vector since $X^0$ and $A^0$ are ultraparallel. The spacelike vector spans the one-dimensional space $(X^0)^{\perp} \cap (A^0)^{\perp}$. But as we noted above, the hyperbolic geodesic defined by the subspace $w_1^{\perp}$ is mutually perpendicular to the geodesics defined by $(X^0)^{\perp}$ and $(A^0)^{\perp}$. Thus $$X^0 \boxtimes A^0 = \lambda w_1$$ for some nonzero $\lambda$. Hence $$\frac{1}{4} \det M_1 = \lambda \|w_1\| \neq 0$$ and $M_1^{\prime}$ has full rank, so $M_1$ has rank $2$.
Because $M_1$ and $M_2$ are matrices of full rank, the image of $e_1, \ldots, e_4$ are distinct. Because $M_{3}$ is a rank-1 matrix, $\mathsf{M}e_5 = \mathsf{M}e_6$. As a result, the image of the positive orthant in $\R^6$ projectivizes to a pentagon $\mathbf{P}_1$ in $\RP^2$.
It is clear that $Me_5 = Me_6$, but the images $Me_i$ for $i \neq 5,6$ are distinct. This gives $5$ distinct vectors in $\R^3$, which projectivize to a pentagon in $\RP^2$.
$\mathbf{P}_1$ is inscribed in $\mathbf{Q}$:
Let $\left\{ e_i \right\}$ be the standard basis vectors for $\R^6$. Then
- $M e_1$ and $M e_2$ lie in $\ker \alpha_{[u]} (Y)$
- $M e_3$ and $M e_4$ lie in $\ker \alpha_{[u]} (X)$
- $M e_5$ and $M e_6$ lie in $\ker \alpha_{[u]} (A)$
In particular, the pentagon $\mathbf{P}_1$ is inscribed in the quadrilateral $\mathbf{Q}$ defined by the projectivized images of the kernels of $\alpha_u (X)$, $\alpha_u (Y)$, $\alpha_u (A)$, $\alpha_u (B)$.
This follows easily from the formulas.
In the above proposition $Me_5 = Me_6$, and this point corresponds to the top of the pentagon in the diagram. The image of the other basis vectors gives a smaller quadrilateral $\mathbf{Q_{small}}$ inscribed in $\mathbf{Q}$. To form the hexagon, we need to recover the corresponding singular point at the bottom. We do this by proving Proposition \[pentagon-flip\].
\[pentagon-flip\] Using a fundamental domain in configuration $II$ corresponds to using $\phi (R_X)$, $\phi (R_Y)$ and $\phi (\iota_0)$ as the generators. This, in turn, corresponds to considering the image of the positive orthant in $\R^6$ under the matrix $$M^{\phi } v \defeq \begin{pmatrix}
\label{alpha-phi-matrix}
\alpha_{[u]} \phi (X)\\ \alpha_{[u]} \phi ( Y)\\ \alpha_{[u]} \phi ( A)
\end{pmatrix} =
\begin{pmatrix}
\alpha_{[u]} (X)\\ \alpha_{[u]} (Y)\\ \alpha_{[u]} (B)
\end{pmatrix}$$ The equality on $Y$ is due to the fact that the Margulis invariant satisfies $\alpha (Y^{-1}) = \alpha (Y)$ for any hyperbolic element $Y$.
Using $M^{\phi}$ in place of $M$ swaps the roles of $A$ and $B$ in Lemma \[alpha-compute\]. Following the argument for $\mathbf{P_1}$, we get maps $M^{\phi}_1$, $M^{\phi}_2$ and $M^{\phi}_3$. For any vectors $v,w$ $M_1^{\phi}v + M_2^{\phi}w = M_1v + M_2 w$. This image is the quadrilateral $\mathbf{Q_{small}}$.
Like $M_3$, $M_3^{\phi}$ has a one-dimensional image. However, its image corresponds to a point on the line $\alpha_{[u]} (B) = 0$, giving another pentagon $\mathbf{P_2}$ with a vertex on the line in $\RP^2$ corresponding to the image of $\ker \alpha_{[u]} (B)$ and such that $\mathbf{P_1} \cap \mathbf{P_2} = \mathbf{Q_{small}}$.
|
---
abstract: 'Early diagnosis is important for type 2 diabetes (T2D) to improve patient prognosis, prevent complications and reduce long-term treatment costs. We present a novel risk profiling approach based exclusively on health expenditure data that is available to Belgian mutual health insurers. We used expenditure data related to drug purchases and medical provisions to construct models that predict whether a patient will start glucose-lowering pharmacotherapy in the coming years, based on that patient’s recent medical expenditure history. The design and implementation of the modeling strategy are discussed in detail and several learning methods are benchmarked for our application. Our best performing model obtains between $74.9\%$ and $76.8\%$ area under the ROC curve, which is comparable to state-of-the-art risk prediction approaches for T2D based on questionnaires. In contrast to other methods, our approach can be implemented on a population-wide scale at virtually no extra operational cost. Possibly, our approach can be further improved by additional information about some risk factors of T2D that is unavailable in health expenditure data.'
author:
- |
Marc Claesen [email protected]\
KU Leuven, Department of Electrical Engineering (ESAT)\
STADIUS Center for Dynamical Systems, Signal Processing and Data Analytics\
iMinds, Department of Medical Information Technologies\
Kasteelpark Arenberg 10 - box 2446, 3001 Leuven, Belgium\
Frank De Smet [email protected]\
National Alliance of Christian Mutualities\
Haachtsesteenweg 579, 1031 Schaarbeek, Belgium\
KU Leuven, Department of Public Health and Primary Care, Environment and Health\
Kapucijnenvoer 35 block d - box 7001, 3000 Leuven, Belgium Pieter Gillard [email protected]\
Chantal Mathieu [email protected]\
KU Leuven, Department of Clinical and Experimental Endocrinology\
UZ Herestraat 49 - box 902, 3001 Leuven, Belgium Bart De Moor [email protected]\
KU Leuven, Department of Electrical Engineering (ESAT)\
STADIUS Center for Dynamical Systems, Signal Processing and Data Analytics\
iMinds, Department of Medical Information Technologies\
Kasteelpark Arenberg 10 - box 2446, 3001 Leuven, Belgium
bibliography:
- 'bibliography.bib'
title: 'Building Classifiers to Predict the Start of Glucose-Lowering Pharmacotherapy Using Belgian Health Expenditure Data'
---
diabetes, public health, insurance data, classification, risk analysis
Introduction
============
Type 2 diabetes mellitus (T2D) is a chronic metabolic disorder characterized by hyperglycemia and is considered one of the main threats to human health [@zimmet2001global]. In developed countries, T2D makes up about 85% of diabetes mellitus patients and occurs when either insufficient insulin is produced, the body becomes resistant to insulin or both [@world1994prevention]. Prediabetes and less severe cases of T2D are initially managed by lifestyle changes, specifically increasing physical exercise, dietary change and smoking cessation [@tuomilehto2001prevention; @diabetes2002reduction; @american2014standards]. If this yields insufficient glycemic control, pharmacotherapy with glucose-lowering agents (GLAs) like metformin or insulin is started [@turner1999glycemic; @american2014standards].
Several studies have indicated that one third to one half of T2D patients are undiagnosed [@harris1998prevalence; @king1998global; @rubin1994health]. Additionally, patients often remain undiagnosed for extended periods of time, with average diagnose-free intervals ranging from 4 to 7 years [@harris1992onset]. The prognosis of untreated patients can deteriorate rapidly as prolonged hyperglycemia can cause serious damage to many of the body’s systems. Timely diagnosis of T2D proves challenging in contemporary medicine, as many patients already present signs of complications of the disease at the time of clinical diagnosis of T2D [@harris1998risk; @rajala1998prevalence; @kohner1998united; @ballard1988epidemiology; @harris2000early; @hu2002elevated].
Earlier diagnosis and subsequent treatment is believed to prevent or delay complications and improve prognosis [@pauker1993deciding; @engelgau2000screening]. When impaired glucose tolerance is diagnosed early, initial treatment can often be limited to lifestyle changes [@pan1997effects; @tuomilehto2001prevention; @diabetes2002reduction]. Compared to pharmacotherapy, lifestyle changes are simple, fully manageable by the patient and far less likely to cause serious treatment-induced complications like hypoglycemia [@seltzer1989drug; @zammitt2005hypoglycemia]. Complementary to health benefits, early diagnosis of T2D poses a health economical advantage, as patients that do not require acute or intensive long-term treatment are far less demanding on the health care system.
Universal screening for T2D is cost-prohibitive [@wareham2001should; @engelgau2000screening], but many organizations advise opportunistic screening of high-risk subgroups [@world1994prevention; @alberti1998report; @engelgau2000screening; @american2014standards]. Several risk profiling strategies have been developed to aid in the timely diagnosis of T2D [@baan1999performance; @stern2002identification; @lindstrom2003diabetes; @mcneely2003comparison; @charlone2004danish; @heikes2008diabetes; @schwarz2009finnish]. Risk profiling is typically done by assessing some of the key risk factors for T2D, which include obesity [@mokdad2003prevalence], genetic predisposal [@shai2006ethnicity; @interact2013link], lifestyle [@reis2011lifestyle] and various clinical parameters. Existing risk profiling approaches are implemented via questionnaires, potentially augmented with clinical information that is available to the patient’s general practitionner [@griffin2000diabetes; @spijkerman2004performance; @lindstrom2003diabetes; @glumer2004danish; @schulze2007accurate; @heikes2008diabetes]. Commonly required information includes BMI, family history, exercise and smoking habits and various clinical parameters.
In this work, we present an alternative approach for risk profiling which only requires data that is already available to Belgian mutual health insurers. This work was done in collaboration with the National Alliance of Christian Mutualities (NACM). NACM is the largest Belgian mutual health insurer with over four million members. Our approach does not require any questionnaires or additional clinical information and predicts whether a patient will start taking GLAs in the next few years. Interestingly, our approach works well despite the fact that Belgian health insurer data contains little direct information regarding key risk factors of T2D, that is weight, lifestyle and family history are all unavailable.
Existing type 2 diabetes risk profiling approaches {#sec:stateoftheart}
==================================================
The Cambridge Risk Score (CRS) was developed to assess the probability of undiagnosed T2D based on data that is routinely available in primary care records, including age, sex, medication use, family history of diabetes, BMI and smoking status [@griffin2000diabetes], The CRS has been shown to be useful on multiple occasions [@griffin2000diabetes; @park2002performance; @spijkerman2004performance], though its AUC seems to depend heavily on the population in which it is used, ranging between $67\%$ [@spijkerman2004performance] and $80\%$ [@griffin2000diabetes]. The information used in the CRS is comparable to another approach which obtained AUCs ranging between $70\%$ and $78\%$ [@baan1999performance].
The FINDRISC score is based on a 10-year follow-up using age, BMI, waist circumference, history of antihypertensive drugs and high blood glucose, physical activity and diet with reported AUCs of $85\%$ and $87\%$ in predicting drug-treated diabetes [@lindstrom2003diabetes]. The strongest reported predictors in this study were BMI, waist circumference, history of high blood glucose and physical activity. @glumer2004danish developed a risk score based on age, sex, BMI, known hypertension, physical activity and family history of diabetes with AUC ranging from $72\%$ to $87.6\%$. The German diabetes risk score reached AUCs ranging from $75\%$ to $83\%$ on validation data and is based on age, waist circumference, height, history of hypertension, physical activity, smoking, and diet [@schulze2007accurate].
@heikes2008diabetes developed a decision tree for risk prediction achieving $82\%$ AUC in a cross-validation setting, based on weight, age, family history and various clinical parameters. Various other approaches based on routine clinical information have demonstrated similarly accurate predictions of type 2 diabetes [@stern2002identification; @mcneely2003comparison].
Health expenditure data
=======================
The Belgian health care insurance is a broad solidarity-based form of social insurance. Mutual health insurers such as NACM are the legally-appointed bodies for managing and providing the Belgian compulsory health care and disability insurance, among other things. To implement their operations, Belgian mutual health insurers dispose of large databases containing health expenditure records of all their respective members.
These expenditure records hold all financial reimbursements of drugs, procedures and contacts with health care professionals. Each record comprises a timestamp, financial details and a description of the claim. The financial aspect is irrelevant from a medical point of view, but the type of resource-use as indicated by the description can contain medical information about the patient. These types belong to one of two main categories:
1. **Drug purchases** are recorded per package. The coding of packages contains information about the active substances in the drug along with the volume of the package.
2. **Medical provisions** are identified by a national encoding along with an identifier of the associated medical caregiver. Each provision has a distinct code number.
In addition to resource-use data, some biographical information is available about each patient including age, gender, place of residence and social parameters. In the remainder of this Section we will elaborate on expenditure records related to drugs and provisions. Subsequently we will briefly summarize the main strengths and limitations of using health expenditure data for predictive modeling.
Records related to drug purchases
---------------------------------
Expenditure records concerning drug purchases contain information about the active substances in the drug and the purchased volume. We mapped all active substances onto the anatomical therapeutic chemical (ATC) classification system maintained by the @world1996guidelines. The ATC classification system divides active substances into different groups based on the organ or system on which they act and their therapeutic, pharmacological and chemical properties. Each drug is classified in groups at 5 levels in the ATC hierarchy: fourteen main groups (1st level), pharmacological/therapeutic subgroups (2nd level), chemical subgroups (3rd and 4th level) and the chemical substance (5th level).
After mapping records onto the ATC classification system, a patient’s medication history consists of specific ATC codes (5th level) along with the associated number of defined daily doses (DDD). In the period of interest, purchases of 4,580 distinct active substances were recorded in the NACM database. Table \[table:atc-example\] shows an example of the classification of active substance on all levels in the ATC system.
level ATC code description
------- ---------- --------------------------------------------------
1 A alimentary tract and metabolism
2 A10 drugs used in diabetes
3 A10B blood glucose lowering drugs, excluding insulins
4 A10BA biguanides
5 A10BA02 metformin
: Example of the ATC classification system: classification of metformin per level.[]{data-label="table:atc-example"}
Records related to medical provisions
-------------------------------------
Expenditure records concerning medical provisions can be considered tuples containing time-stamped identifiers of the patient, physician and medical provision. A single patient-physician interaction may yield multiple such records, one for each specific provision that occurred.
In the Belgian health care system, medical provisions are encoded via the Belgian nomenclature of medical provisions [@van2008financing], which is maintained by the National Institute for Health and Disability Insurance (NIHDI).[^1] This nomenclature is an unstructured list of unique codes (numbers) for each provision that is being refunded. Nomenclature numbers are added when new provisions are defined or when revisions are made. A single provision may correspond to multiple numbers for various reasons.
Advantages of health expenditure data
-------------------------------------
The key benefit of expenditure databases is that they centralize structured medical information across all medical stakeholders to yield a comprehensive, longitudinal overview of each patient’s medical history. Other health data sources are fragmented, e.g. medical records maintained by the patient’s general practitioner or hospital often contain only a subset of the patient’s medical history. This fragmentation hampers the identification of patterns that may indicate elevated risk for diseases like type 2 diabetes. The NACM database comprises claims records of over four million Belgians, which enables complex modeling. Additionally, claims data have few omissions due to the financial incentive for patients and medical stakeholders (hospitals) to claim refunds. While other health data sources may contain more detailed information, the strength of NACM’s data is in its volume, both in terms of number of patients and the amount of information that is recorded per individual. Finally, as most people tend to stay affiliated with the same mutual health insurer, their expenditure records provide long-term information.
Limitations of health expenditure data
--------------------------------------
Belgian health expenditure data is strictly limited to what is required for mutual health insurers to implement their operations, which are mainly administrative in nature. Detailed health information such as diagnoses and test results are not directly available. In some other countries, health insurers dispose of more detailed information, such as ICD-10 codes which include diagnoses and symptoms [@world2012international]. Including such information is out of scope of this work as we focus exclusively on data that is already available to Belgian mutual health insurers. Biographical information about patients does not contain direct information about some important risk factors such as lifestyle, family history and BMI, though this may be partially embedded indirectly in medical resource-use.
Methods
=======
In this Section we define the prediction task and describe all its aspects: the overall setup (Section \[setup\]), the data and its representation (Section \[data\]) and the learning algorithms (Section \[learning-methods\]). Briefly, our aim is to predict which patients will start glucose-lowering pharmacotherapy within the next 4 years, based on expenditure records of the previous 4 years.
Our key hypothesis is that patients with increased risk for T2D or those that are already afflicted but not diagnosed have a different medical expenditure history than patients without impaired glycemic control. We essentially use the start of GLA therapy as a proxy for diagnosis of (advanced) type 2 diabetes. This is reasonable since most patients that start GLA therapy above 40 years old have T2D [@world1994prevention].
We posed this task as a binary classification problem. Our classifiers produce a numeric level of confidence that a given patient will start glucose-lowering pharmacotherapy. When predicting a population, the outputs can be used to rank patients according to decreasing confidence that the patients will start glucose-lowering therapy. Highly ranked patients represent a high-risk subgroup which can be targetted for clinical screening.
The full learning setup is described in Section \[setup\], involving different learning methods and representations of patients’ expenditure data. Briefly, we used nested cross-validation to obtain unbiased estimates of the predictive performance of each vectorization and learning approach. Predictive performance of all models was quantified via (area under) receiver operating characteristic (ROC) curves.
#### Data
Our work is based on a subset of the expenditure records of NACM. All data extractions and analyses were performed at the Medical Management Department of the NACM under supervision of the Chief Medical Officer. The other research partners received no personally identifiable information (including small cells) from NACM. The patient selection and vector representations are described in detail in Section \[data\].
#### Class definitions
The positive class was defined as patients that require GLAs for long-term glycemic control.[^2] The negative class is then defined as patients that do not need GLAs. Expenditure records related to GLAs were used to identify a set of known positives. However, the absence of such records in a patient’s resource use history is not proof that this patient has no need for GLAs. This subtle difference is crucial, because it is well known that patients with impaired glycemic control or T2D often remain undiagnosed and hence untreated for a very long time [@harris1998prevalence; @king1998global; @american2014standards]. As we cannot identify negatives, we had to build models from positive and unlabeled data.
#### PU learning
Learning binary classifiers from positive and unlabeled data (PU learning) is a well-studied branch of semi-supervised learning [@Lee03learningwith; @Elkan:2008:LCO:1401890.1401920; @MORDELET-2010-523336; @Claesen2015resvm]. PU learning is more challenging than fully supervised binary classification, since it requires special learning approaches and quality metrics for hyperparameter optimization that account for the lack of known negatives. We benchmarked three PU learning methods, which are discussed in more detail in Section \[learning-methods\].
#### Software
The entire data analysis pipeline was implemented using open-source software. For general data transformations and preprocessing we used *SciPy* and *NumPy* [@scipy; @van2011numpy]. The learning algorithms we used are available in *scikit-learn* and *EnsembleSVM* [@pedregosa2011scikit; @JMLR:v15:claesen14a] . Finally, we used *Optunity* for automated hyperparameter optimization [@claesen2014easy].
Experimental setup {#setup}
------------------
We gathered all expenditure records during the 4-year interval of 2008 up to 2012. The selection protocol and representations of patients’ medical resource-use are discussed in detail in Section \[data\]. All vector representations of patients include age (in years), an indicator variable for gender and positive entries related to the patient’s medical resource-use. A patient vector $\mathbf{p}$ can be written in the following general form, where ${d_{\rm meds}}$ and ${d_{\rm provs}}$ denote the number of features in the vectorization of medication and provision use, respectively:
$$\mathbf{p} \in \mathbb{R}^{2 + {d_{\rm meds}}+ {d_{\rm provs}}}_+ =
\begin{bmatrix}
age & gender & medication & provisions \\
\mathbb{R}_+ & \{0, 1\} & \mathbb{R}^{{d_{\rm meds}}}_+ & \mathbb{R}^{{d_{\rm provs}}}_+
\end{bmatrix}.$$
In Sections \[ATC-vectorization\] and \[nomenclature\] we explain how records related to medication purchases and provisions were represented in vector form. All entries in the vector representations were consistently normalized to the interval $[0, 1]$ by dividing feature-wise by the $99^{th}$ percentile and subsequently clipping where necessary. These normalized vector representations are used as inputs for the learning algorithms described in Section \[learning-methods\].
Figure \[fig:flowchart\] summarizes the full machine learning pipeline, which starts from expenditure records and ends with models to predict whether a patient will start glucose-lowering pharmacotherapy along with an estimate of their generalization performance. We used nested cross-validation to estimate generalization performance of different learning configurations [@varma2006bias]. The outer 3-fold cross-validation is used to estimate generalization performance of the full learning approach. Internally, twice iterated 10-fold cross-validation was used to find optimal hyperparameters for every learning method.
![Overview of the full learning approach: data set vectorization, normalization and the nested cross-validation setup. Per iteration, hyperparameter optimization and model training is done based exclusively on $\mathbf{X}_{train}^{(outer)}$.[]{data-label="fig:flowchart"}](flowchart.pdf){width="\textwidth"}
#### Hyperparameter search
We used Optunity’s particle swarm optimizer to identify suitable hyperparameters for each approach based on the given training set as defined by the outer cross-validation procedure [@claesen2014easy]. Every tuple of hyperparameters was evaluated using twice iterated 10-fold cross-validation on the training set. Per technique, the hyperparameters that maximized cross-validated performance were selected and used to train a model on the full training set.
#### Model evaluation
Models are compared based on area under the ROC curve. ROC curves visualize a classifier’s performance spectrum by depicting its true positive rate (TPR)[^3] as a function of its false positive rate (FPR)[^4] while varying the decision threshold to decide on positives. Area under the ROC curve (AUROC) is a useful summary statistic of a classifier’s performance. AUROC is equal to the probability that the classifier ranks a random positive higher than a random negative and is known to be equivalent to the Wilcoxon test of ranks [@hanley1982meaning].
#### Computing ROC curves
Full label knowledge is required to compute ROC curves. In previous work, we introduced a method to compute bounds on ROC curves based on positive and unlabeled data [@claesen2015icml]. Briefly, it is based on the positions of known positives in a ranking produced by a given classifier and requires two things:
- The rank distributions of labeled and latent positives must be comparable. This holds when known and latent positives follow the same distribution in input space (ie. the vector representation of patients). This is a fair assumption in our application, since we specifically ignore records after the start of glucose-lowering pharmacotherapy while identifying the set of positives (see Section \[data\]), so the medication regimen of known positives has not yet diverged from the regimen of untreated patients.
- An estimate $\hat{\beta}$ of the fraction of latent positives in the unlabeled set is needed, that is the fraction of members that have never used GLAs but are likely to start glucose-lowering pharmacotherapy. In the period $2010$–$2014$ roughly $8\%$ of members of NACM aged 40 or higher started using GLAs. Underestimating $\hat{\beta}$ results in an underestimated ROC curve and vice versa [@claesen2015icml]. We opted to be conservative and used $\hat{\beta}_{lo} = 5\%$ to estimate lower bounds and $\hat{\beta}_{up}=10\%$ for upper bounds.
We consistently used the *lower* bounds for hyperparameter search. All our performance reports contain lower and upper bounds, based on $\hat{\beta}_{lo}$ and $\hat{\beta}_{up}$, respectively.
#### Diagnosing overfitting
In addition to measuring performance, we diagnosed overfitting via the concept of rank distributions as defined by @claesen2015icml. The rank distribution of a subset of test instances is defined as the distribution of the positions of these test instances in a ranking of the full test set based on a model’s predicted decision values. We diagnose overfitting based on the rank distributions of known positive training instances ($\mathcal{P}_{train}$) and known positives in the independent test fold ($\mathcal{P}_{test}$) after predicting the full data set. If the model overfits, the rank distribution of $\mathcal{P}_{train}$ is inconsistent with the rank distribution of $\mathcal{P}_{test}$. Specifically, ranks in $\mathcal{P}_{test}$ are worse than those in $\mathcal{P}_{train}$ when the model overfits. This can be quantified via the Mann-Whitney U test [@mann1947test] based on ranks of $\mathcal{P}_{train}$ and $\mathcal{P}_{test}$ after predicting the full data set (that is all outer folds). The Mann-Whitney U test is expected to yield a non-significant result when the rank distributions of $\mathcal{P}_{train}$ and $\mathcal{P}_{test}$ are comparable. We report the average $p$-values of the test across outer cross-validation folds for each model (low $p$-values indicate overfitting).
Data Set Construction {#data}
---------------------
We constructed a data set containing records of patients born before 1973 (e.g. $40$ or more years old in 2012). Patients with records of glucose-lowering agents (GLAs) during less than 30 days were discarded. Patients with records of glucose-lowering therapy prior to 2012 were discarded. Patients that joined NACM after 2005 were also discarded, as we cannot determine whether these patients used GLAs in the recent past.
All patients that started glucose-lowering pharmacotherapy in 2012 or later are included as known positives ($n=31,066$), along with unlabeled patients that were sampled at random from the remaining NACM members ($n=79,243$). Known positives have a minimum of 30 days between the first and last purchase of GLAs to avoid contaminating the data set with false positives, for instance due to insulin use in surgical and medical ICUs [@van2001intensive; @van2006intensive]. It must be noted that some false positives remain, that is patients that use GLAs but not for glycemic control.
In Sections \[ATC-vectorization\] and \[nomenclature\] we describe the vector representations of records regarding medication and medical provisions, respectively.
### Representation of medication records {#ATC-vectorization}
The simplest way to represent medication purchases during a time interval is by having one input dimension per active substance (level 5 ATC codes) and counting the purchased volume in terms of DDDs. This representation is easy to construct but fails to capture any similarity between active substances, such as the system or organ on which they act.
#### Imposing structure
We can directly use the hierarchical structure of the ATC system to define a measure of similarity between drugs. To impose structure between drugs we included input dimensions related to more generic levels of the ATC hierarchy (levels 1 to 4). On more generic levels we summed all DDD counts of active substances per category (level 5). This redundancy allowed us to express similarity between different active substances with a standard inner product. By normalizing every feature to the unit interval, we obtained the desired effect that patients with comparable drug use on ATC level 5 are more similar than patients that only share coefficients on more generic levels. Figure \[fig:medication-vectorization\] illustrates this vector representation of trees and the effect of normalization.
![Visualization and vectorization of trees. In the tree representation, the value of internal nodes is the sum of the values of its children. The unnormalized vector representations $\mathcal{V}_A$ and $\mathcal{V}_B$ contain the values per node in the tree representation in some fixed order. Inner products between unnormalized representations $\mathcal{V}_A$ and $\mathcal{V}_B$ are mainly influenced by the top level nodes, since those have the largest value by construction. This undesirable effect can be fixed through feature-wise scaling. The scaling vector $\mathcal{S}$ was constructed using node-wise maxima. The normalized vector representations $\mathcal{V}_A^\star$ and $\mathcal{V}_B^\star$ are obtained by dividing the vector representations ($\mathcal{V}_A$, $\mathcal{V}_B$) element-wise by entries in the scaling vector $\mathcal{S}$. $\mathcal{V}_A^\star$ and $\mathcal{V}_B^\star$ are used as input to classifiers in the remainder of this work. As desired, the inner product of normalized vector representations is increasingly influenced by similarities at higher depths in the tree representations.[]{data-label="fig:medication-vectorization"}](medication-vectorization.pdf){width="\textwidth"}
#### Summary
All vectorizations related to drug purchases are described in Table \[table:meds-vects\].
vectorization description ${d_{\rm meds}}$
------------------------------------------------------- ------------------------------------------------------- ------------------
<span style="font-variant:small-caps;">atc 5</span> counts of DDDs per medication class in ATC level 5 4,580
<span style="font-variant:small-caps;">atc 1–4</span> counts of DDDs per medication class in ATC levels 1–4 1,257
<span style="font-variant:small-caps;">atc 1–5</span> counts of DDDs per medication class in ATC levels 1–5 5,837
: Summary of vectorization schemes used for records of drug purchases.[]{data-label="table:meds-vects"}
### Representation of provision records {#nomenclature}
When considering a specific time period, we can describe records by a (sparse) three-dimensional tensor containing frequency counts as illustrated in Figure \[fig:provisions-tensor\]. We filtered all provisions with a description containing *diabetes*, *insulin* and *glucose* and provisions not recorded with a physician identifier. After filtering, 5,799 distinct provision codes remain (denoted by $\# provisions$).
![Tensor formulation of medical provisions with three components: patients, physicians and provisions. Each entry in the tensor is the frequency of the given tuple. This provision tensor is very sparse. The patient matrix is obtained by summing counts over all physicians (transposed). The physician matrix is obtained by summing counts over all patients. These matrices capture complementary information.[]{data-label="fig:provisions-tensor"}](provisions-tensor.pdf){width="\textwidth"}
Each patient is modelled by a histogram of their provisions in the period of interest. This essentially means we compute the sum over the $physician$-component of the tensor representation to obtain a matrix, in which rows and columns represent patients and provisions, respectively. Unfortunately, the encoding of provisions has no medically relevant structure in contrast to the ATC hierarchy for drugs as discussed in Section \[ATC-vectorization\].
#### Imposing structure
In order to define a reasonable similarity measure between patients, we first had to impose a structure onto the nomenclature that captures similarity between provisions. To structure provisions, we should not use information originating from the patient matrix, as this may cause information leaks (since the patient matrix is used directly in our models for prediction). Instead, we used the complementary physician matrix as a basis to define similarity between provisions, which essentially serves as a proxy for the medical specializations to which each provision belongs. We started from cosine similarity between nomenclature codes based on the physician matrix. We used cosine similarity because it is known to work well for text mining with bag-of-words representations, which is comparable to our use case as it also features sparse, high dimensional input spaces. The cosine similarity $\kappa_{cos}$ between two row vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as: $$\kappa_{cos}(\mathbf{u}, \mathbf{v}) = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \cdot \|\mathbf{v}\|} = \frac{\mathbf{u}\mathbf{v}^T}{\|\mathbf{u}\| \cdot \|\mathbf{v}\|}.
\label{eq:cosine}$$ Using cosine similarity we can construct a pair-wise similarity matrix $\mathbf{S}_{prov}$ between provisions based on the rows of the physician matrix $\mathbf{x}_i, i=1..\# provisions$: $$\mathbf{S}_{prov} = \big(\kappa_{cos}(\mathbf{x}_i,\mathbf{x}_j)\big)_{ij} \in \mathbb{R}^{\# provisions \times \# provisions}.
\label{eq:cosine-kernel}$$ $\mathbf{S}_{prov}$ expresses similarity between provision codes based on the physicians that provide them and can be regarded as a proxy for the medical subdomain each provision frequently occurs in. In our context, its entries range from $0$ (completely orthogonal) to $+1$ (exact similarity). To impose sparsity we set all entries of $\mathbf{S}_{prov}$ below $0.05$ to $0$. Its structure is visualized in Figure \[fig:Sprovs\], which clearly indicates that our approach successfully identifies some coherent groups of provisions.
![Structure of the provision similarity matrix $\mathbf{S}_{prov}$ based on providing physicians.[]{data-label="fig:Sprovs"}](Sprovs.pdf){width="50.00000%"}
Finally, the structured representation of provisions $\mathbf{P}_{struct}$ is defined as the matrix product between the patient matrix $\mathbf{P}_{flat}$ and the provision similarity matrix $\mathbf{S}_{prov}$: $$\mathbf{P}_{struct} = \mathbf{P}_{flat} \times \mathbf{S}_{prov} \in \mathbb{R}^{\# patients \times \# provisions}.$$ $\mathbf{P}_{struct}$ approximately captures which provisions occur in a patient’s history with redundancy based on medical specializations.
#### Summary
All vectorizations related to medical provisions are described in Table \[table:prov-vects\].
vectorization symbol description ${d_{\rm provs}}$
------------------------------------------------------------ --------------------------------------------- ---------------------------------------- -------------------
<span style="font-variant:small-caps;">provs flat</span> $\mathbf{P}_{flat}$ entries taken from the patient matrix $5,799$
<span style="font-variant:small-caps;">provs struct</span> $\mathbf{P}_{struct}$ captures similarity between provisions $5,799$
<span style="font-variant:small-caps;">provs both</span> $\mathbf{P}_{flat}\ |\ \mathbf{P}_{struct}$ concatenation of flat & structured $11,598$
: Summary of vectorization schemes used for records of medical provisions.[]{data-label="table:prov-vects"}
Modeling approaches {#learning-methods}
-------------------
Having only positive and unlabeled data (PU learning) presents additional challenges for learning algorithms. Two broad classes of approaches exist to tackle these problems: (i) two-phase methods that first attempt to identify likely negatives from the unlabeled set and then train a supervised model on the positives and inferred negatives [@liu02partially; @Yu:2005:SCM:1108759.1108762] and (ii) approaches that treat the unlabeled set as negatives with label noise [@Elkan:2008:LCO:1401890.1401920; @Lee03learningwith; @MORDELET-2010-523336; @Claesen2015resvm].
We have tested three approaches from the latter category in this work, namely class-weighted SVM [@Liu:2003:BTC:951949.952139], bagging SVM [@MORDELET-2010-523336] and the robust ensemble of SVM models [@Claesen2015resvm]. All of these approaches are based on support vector machines. We used the linear kernel on vector representations of patients as described in Section \[data\].[^5] We will briefly introduce each method in the following subsections.
### Class-weighted SVM {#bsvm}
Class-weighted SVM (CWSVM) uses a misclassification penalty per class. CWSVM was first applied in a PU learning context by @Liu:2003:BTC:951949.952139, by considering the unlabeled set to be negative with noise on its labels. A CWSVM is trained to distinguish positives ($\mathcal{P}$) from unlabeled instances ($\mathcal{U}$), leading to the following optimization problem: $$\begin{aligned}
\min_{\alpha,\xi,b}\ & \frac{1}{2}\sum_{i=1}^N\sum_{j=1}^N \alpha_i\alpha_j y_i y_j \kappa(\mathbf{x}_i,\mathbf{x}_j)+C_{\mathcal{P}}\sum_{i \in\mathcal{P}} \xi_i + C_{\mathcal{U}}\sum_{i\in\mathcal{U}} \xi_i, \label{eq:bsvm} \\
\text{s.t. } &y_i(\sum_{j=1}^N \alpha_j y_j \kappa(\mathbf{x}_i,\mathbf{x}_j)+b)\geq 1-\xi_i, &i=1,\ldots,N, \nonumber \\
&\xi_i \geq 0, &i=1,\ldots,N, \nonumber\end{aligned}$$ where $\alpha \in \mathbb{R}^N$ are the support values, $\mathbf{y} \in \{-1,+1\}^N$ is the label vector, $\kappa(\cdot,\cdot)$ is the kernel function, $b$ is the bias term and $\xi \in \mathbb{R}^N$ are the slack variables for soft-margin classification. The misclassification penalties $C_{\mathcal{P}}$ and $C_{\mathcal{U}}$ require tuning. We used the implementation available in scikit-learn [@pedregosa2011scikit] based on LIBSVM [@CC01a].
### Bagging SVM {#baggingsvm}
In bagging SVM, random resamples are drawn from the unlabeled set and CWSVM classifiers are trained to discriminate all positives from each resample [@MORDELET-2010-523336]. Resampling the unlabeled set induces variability in the base models which is exploited via bagging. Base model predictions are aggregated via majority voting.
Bagging SVM with linear base models has two hyperparameters, namely the size of resamples of the unlabeled set $n_\mathcal{U}$ and the misclassification penalty on unlabeled instances $C_\mathcal{U}$. The misclassification penalty on positives $C_\mathcal{P}$ is fixed via the following rule: $$C_{\mathcal{P}} = \frac{n_{\mathcal{U}} \times C_{\mathcal{U}}}{|\mathcal{P}|}, \label{eq:bagpenalties}$$ where $|\mathcal{P}|$ denotes the number of known positives. The heuristic rule in Equation \[eq:bagpenalties\] is common in imbalanced settings [@cawley2006leave; @daemen2009kernel]. We implemented bagging SVM using the EnsembleSVM library [@JMLR:v15:claesen14a].
### Robust ensemble of SVM models
The robust ensemble of SVM models (RESVM) is a modified version of bagging SVM in which both the positive and unlabeled sets are resampled when constructing base model training sets [@Claesen2015resvm]. The extra resampling induces additional variability between base models which improves performance when combined with a majority vote aggregation scheme. @Claesen2015resvm demonstrated that resampling the positive set provides robustness against false positives, which makes RESVM appealing for our application since our data set is known to contain a small fraction of false positives (as explained in Section \[data\]).
When using linear base models, the RESVM approach has four hyperparameters that must be tuned, namely resample sizes and misclassification penalties per class. This approach was implemented based on EnsembleSVM [@JMLR:v15:claesen14a].
Results and discussion
======================
Section \[benchmark\] shows the predictive performance per learning configuration and compares these performances to the current state-of-the-art in large-scale risk assessment for T2D. Section \[resvmroc\] shows performance curves of the best configuration, which enable us to determine suitable cutoffs to identify target groups in practice. Finally, Section \[features\] describes a simple approach to assess which features contribute most to risk according to our best models.
Benchmark of learning methods {#benchmark}
-----------------------------
Table \[table:results\] summarizes the performance of each learning configuration. The <span style="font-variant:small-caps;">age,gender</span> feature set provides a baseline for comparison, all other feature sets include these as well. As shown in the results, this two-dimensional representation already carries some information.
------------------------------------------------------------- ----------------------------------- -------- -- ----------------- -------- -- ----------------------------------- --------
features AUROC ($\%$) $p$ AUROC ($\%$) $p$ AUROC ($\%$) $p$
<span style="font-variant:small-caps;">age, gender</span> $55.74$–$56.64$ $*$ $58.61$–$59.67$ $*$ $\mathbf{60.96}$–$\mathbf{62.21}$ $0.04$
<span style="font-variant:small-caps;">atc 5</span> $\mathbf{72.55}$–$\mathbf{74.43}$ $0.17$ $70.83$–$72.62$ $0.09$ $71.89$–$73.74$ $0.01$
<span style="font-variant:small-caps;">atc 1–4</span> $\mathbf{73.12}$–$\mathbf{75.07}$ $0.07$ $69.57$–$71.24$ $*$ $73.05$–$74.91$ $0.04$
<span style="font-variant:small-caps;">atc 1–5</span> $\mathbf{74.34}$–$\mathbf{76.27}$ $0.13$ $71.50$–$73.27$ $0.05$ $72.13$–$73.94$ $*$
<span style="font-variant:small-caps;">provs flat</span> $58.45$–$59.51$ $*$ $60.74$–$61.92$ $*$ $\mathbf{63.01}$–$\mathbf{64.31}$ $*$
<span style="font-variant:small-caps;">provs struct</span> $57.40$–$58.39$ $0.02$ $59.53$–$60.58$ $0.01$ $\mathbf{62.53}$–$\mathbf{63.81}$ $0.01$
<span style="font-variant:small-caps;">provs both</span> $58.89$–$59.75$ $*$ $61.72$–$62.87$ $*$ $\mathbf{63.45}$–$\mathbf{64.75}$ $*$
<span style="font-variant:small-caps;">atc $|$ provs</span> $\mathbf{74.89}$–$\mathbf{76.82}$ $0.04$ $69.72$–$71.40$ $*$ $73.77$–$75.64$ $*$
------------------------------------------------------------- ----------------------------------- -------- -- ----------------- -------- -- ----------------------------------- --------
: Average bounds on area under the ROC curve and $p$-value of the Mann-Whitney U test over all folds for different feature sets per learning approach in a long-term prediction setup. The lower and upper bounds on AUC were computed with $\hat{\beta}_{lo}=0.05$ and $\hat{\beta}_{up}=0.10$, respectively. The feature set is the concatenation of the best performing sets per aspect, namely <span style="font-variant:small-caps;">atc 1–5</span> and <span style="font-variant:small-caps;">provs both</span>. Stars ($*$) denote $p$-values below $0.005$. []{data-label="table:results"}
Based on Table \[table:results\] we can conclude that a patient’s medication history is highly informative to predict the start of GLA therapy. Using features based on ATC level 5, the RESVM model obtained an AUC between $72.55\%$ and $74.43\%$. By adding redundancy as described in Section \[ATC-vectorization\] the performance based on medication history alone was further increased to between $74.34\%$ and $76.27\%$ for the best learning approach (RESVM).
Predictive performance based on provisions alone turned out fairly poor, showing only a mild improvement compared to models based exclusively on age and gender for all learning algorithms. Interestingly, the best approach for representations based on provisions was class-weighted SVM, with RESVM being worst of all three learning methods. It appears that for these representations, large training sets are better: class-weighted SVM uses the full training set, bagging SVM uses all positives and a subset of unlabeleed instances per base model and RESVM uses (small) subsets of both positives and unlabeled instances per base model.
The best representation included age, gender, and structured information about drugs and provision history of the patient. The best learning method on this representation was RESVM, achieving an AUC between $74.89\%$ and $76.82\%$. In Section \[stateoftheart\] we compare the performance of our approach to competing screening methods.
Finally, RESVM appears most resistant to overfitting in the hyperparameter optimization stage as it consistently exhibits the highest average $p$-values in our diagnostic test (higher is better, see Section \[setup\]). We believe this to be attributable to the use of small resamples of both positives and unlabeled instances when training base models in RESVM, since this makes it unlikely to obtain a structural overfit of the ensemble model on the full training set. In contrast, bagging SVM is far more prone to overfitting because every base model is trained on all positives.
### Comparison to state-of-the-art {#stateoftheart}
Our best approach obtained cross-validated AUC between $74.89\%$ and $76.82\%$ (exact numbers are unknown due to the lack of known negatives). This is comparable to many competing approaches, based on questionnaires and some clinical information such as the Cambridge Risk Score (AUC 67%–80%, [@spijkerman2004performance; @griffin2000diabetes]), the Danish risk score (AUC $72\%$–$87.6\%$, [@glumer2004danish]), the German diabetes risk score (AUC $75\%$–$83\%$, [@schulze2007accurate]) and a Dutch approach (AUC 74%, [@baan1999performance]). Approaches using detailed clinical information generally perform better, but are more expensive to maintain [@stern2002identification; @mcneely2003comparison; @lindstrom2003diabetes; @heikes2008diabetes]. They key advantage of our approach is the fact it is easy to implement on a population wide scale at virtually no operational cost.
The target class we used in this work is stricter than in the risk prediction methods mentioned in Section \[sec:stateoftheart\], namely patients that require GLAs for glycemic control versus patients with impaired glycemic control, respectively (except for @lindstrom2003diabetes, which also predicted drug-treated T2D). It is reasonable to assume that our models generally rank patients with impaired glycemic control but without a need for GLAs higher than patients without impaired glycemic control. In our performance assessment both of these patient groups are essentially treated as negatives, in contrast to the screening programmes mentioned previously which treat patients with impaired glycemic control as positives. Hence, we believe the performance of our models would appear higher when evaluated against a target class comprising all patients with impaired glycemic control, as is done in the evaluation of other screening approaches. Unfortunately, we are unable to accurately identify patients with impaired glycemic control but without need for GLAs.
All competing methods use either clinical information or direct knowledge of risk factors that is unavailable to us. Furthermore, the characteristics that are lacking in our data have been reported to be the most informative to assess risk for T2D [@lindstrom2003diabetes; @stern2002identification; @mcneely2003comparison]. We obtained generalization performances that are comparable to existing approaches, despite these missing predictors. Finally, our approach is the only one that is based exclusively on existing data that is always available, without requiring additional patient contacts or clinical tests.
Receiver Operating Characteristic curves for RESVM {#resvmroc}
--------------------------------------------------
The RESVM model based on <span style="font-variant:small-caps;">atc $|$ provs</span> vectorization had the best overall performance. Figure \[fig:performance-curves\] shows bounds on the ROC and PR curves for this model. These bounds were computed using the technique described by @claesen2015icml. The true curve is unknown because we do not dispose of negative labels.
ROC curves enable us to determine a cutoff to use in practice, based on a suitable balance between true and false positive rate (sensitivity and 1-specificity, respectively). Determining a suitable balance requires a tradeoff between the relative importance of identifying undiagnosed patients (true positives) vis-à-vis increased amounts of screening tests on patients that are in fact healthy (false positives).
\
It should be noted that precision depends on class balance, and therefore the PR curve shown in Figure \[fig:pr\] is not representative for screening an overall population, since the overall population has a higher fraction of negatives than our custom data set (i.e. precision would be lower in practice). In contrast, the bounds in ROC space are representative because ROC curves are insensitive to changes in class distribution [@Fawcett:2006:IRA:1159473.1159475].
Feature importance analysis for the RESVM model {#features}
-----------------------------------------------
The RESVM model is implicitly nonlinear due to its majority voting rule to aggregate base model decisions, which poses problems in assessing the importance of each predictor. However, our use of linear base models enables a simple approximation. The decision value for base model $i \in \{1,\ldots,n_{\rm models}\}$ for a test instance $\mathbf{z}$ can be written as follows: $$f_i(\mathbf{z}) = \langle \mathbf{w}_i, \mathbf{z}\rangle + \rho_i,$$ where $\mathbf{w}_i$ is the separating hyperplane and $\rho_i$ is a bias term. A simple linear approximation of such ensemble models can be computed as the average of all base model hyperplanes: $$\bar{\mathbf{w}} = \sum_{i=1}^{n_{\rm models}} \mathbf{w}_i / n_{\rm models}.$$ Feature importance can then be determined based on the coefficients in $\bar{\mathbf{w}}$. Since we normalized all features to the unit interval $[0,1]$ we can conclude that the features with largest (positive) coefficients contribute most to risk as identified by our model.
Via this approach, the risk associated to use of cardiovascular medication (ATC main category <span style="font-variant:small-caps;">c</span>) far outweights all other ATC main categories. This is not surprising, as diabetes is known to be strongly related to cardiovascular problems [@doi:10.1001/jama.1979.03290450033020; @grundy1999diabetes; @hu2002elevated]. The relative importance of features will be discussed in detail in a subsequent medical paper.
Conclusion
==========
In this work we have demonstrated the ability to predict clinical outcomes based solely on readily available health expenditure data. We successfully built proof-of-concept classifiers to predict the start of glucose-lowering pharmacotherapy in patients above 40. Our experiments show that accurate predictions can be made based on historical medication purchases. These predictions can be further improved by incorporating information about medical provisions and the use of appropriate vectorization schemes.
Since adult patients starting glucose-lowering pharmacotherapy are mainly afflicted with type 2 diabetes (T2D), our models can be used for T2D risk assessment. Our approach presents a novel method for case finding which can be easily incorporated in modern healthcare, since all required data is already available. The associated operational costs are very low as the entire workflow can be fully automated without any need for patient contacts or medical tests. As such, our work provides an efficient and cost-effective method to identify a high risk subgroup, which can then be screened using decisive clinical tests.
Interestingly, our approach works well even though health expenditure data contains very limited direct information on some important known risk factors. In that sense, our approach is fundamentally different from the current state-of-the-art which mainly focuses on quantifying known risk factors directly, either by asking the patient or through clinical tests. The performance of our approach is expected to improve further when additional information about these risk factors can be obtained, e.g. family history and lifestyle.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors wish to thank Bernard Debbaut, Frie Niesten, Koen Cornelis and Michiel Callens for their valuable input in various aspects of this study.
This study was supported by the Flemish Government (FWO: projects: G.0871.12N (Neural circuits); IWT: TBM-Logic Insulin(100793), TBM Rectal Cancer(100783), TBM IETA(130256); PhD grants; Industrial Research fund (IOF): IOF Fellowship 13-0260; iMinds Medical Information Technologies SBO 2015, ICON projects (MSIpad, MyHealthData); VLK Stichting E. van der Schueren: rectal cancer) and the Belgian Federal Government (FOD: Cancer Plan 2012-2015 KPC-29-023 (prostate); COST: Action: BM1104: Mass Spectrometry Imaging). M.C. is funded by a PhD grant awarded by IWT (\#111065). P.G is funded by a clinical research foundation of the University Hospitals Leuven-KUL.
[^1]: The website of NIHDI is available at <http://www.riziv.fgov.be>.
[^2]: GLAs are defined as any drug in ATC category `A10`, which includes metformin, sulfonylurea and insulin.
[^3]: TPR measures the fraction of true positives that are correctly identified by the classifier.
[^4]: FPR measures the fraction of true negatives that are incorrectly identified by the classifier.
[^5]: Though it must be noted that the ensemble methods are always implicitly nonlinear.
|
---
abstract: |
In this paper we study the boundary value problem $$\left\{
\begin{array}{ll}
-\Delta u+ {\varepsilon}q\Phi f(u)=\eta|u|^{p-1}u & \text{in } \Omega, \\
- \Delta \Phi=2 qF(u)& \text{in } \Omega, \\
u=\Phi=0 & \text{on }\partial \Omega,
\end{array}
\right.$$ where $\Omega \subset \mathbb{R}^3$ is a smooth bounded domain, $1 < p < 5$, ${\varepsilon},\eta= \pm 1$, $q>0$, $f:{\mathbb{R}}\to{\mathbb{R}}$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.
author:
- 'Antonio Azzollini [^1] & Pietro d’Avenia[^2] & Valeria Luisi'
title: ' Generalized Schrödinger-Poisson type systems [^3]'
---
[*Keywords:* Schrödinger-Poisson equations, variational methods, mountain pass.*2000 MSC:* 35J20, 35J57, 35J60.]{}
Introduction
============
This paper deals with the following problem
$$\label{eq:P}\tag{$\mathcal P$}
\left\{
\begin{array}{ll}
-\Delta u+ {\varepsilon}q\Phi f(u)=\eta|u|^{p-1}u &\text{in } \Omega, \\
-\Delta \Phi=2 qF(u) & \text{in }\Omega, \\
u=\Phi=0 & \text{on }\partial \Omega,
\end{array}
\right.$$
where $\Omega \subset \mathbb{R}^3$ is a bounded domain with smooth boundary $\partial{\Omega}$, $1 < p < 5$, $q >0$, ${\varepsilon},\eta = \pm 1$, $f:{\mathbb{R}}\to{\mathbb{R}}$ is a continuous function and $F(s)=\int_0^s f(t)\, dt.$\
When the function $f(t)=t$ and ${\varepsilon}=\eta =1$, this system represents the well known Schrödinger-Poisson (or Schrödinger-Maxwell) equations, briefly SPE, that have been widely studied in the recent past. In the pioneer paper of Benci and Fortunato [@BF], the *linear* version of SPE (where $\eta=0$) has been approached as an eigenvalue problem. In [@PS] the authors have proved the existence of infinitely many solutions for SPE when $p>4$, whereas in [@RS] an analogous result has been found for almost any $q>0$ and $p\in\, ]2,5[$. A multiplicity result has been obtained in [@S] for any $q>0$ and $p$ sufficiently close to the critical exponent 5, by using the abstract Lusternik-Schnirelmann theory. For the sake of completeness we mention also [@PS0] where Neumann condition on $\Phi$ is assumed on $\partial \Omega$, [@A] and the references within for results on SPE in unbounded domains and [@DPS1; @DPS2] for the Klein-Gordon-Maxwell system in a bounded domain.\
If $f(t)=t$ and ${\varepsilon}=-1$, the system is equivalent to a nonlocal nonlinear problem related with the following well known Choquard equation in the whole space ${\mathbb{R}}^3$ $$\Delta u + u -\left(\frac 1 {|x|}*u^2\right)u=0.$$ We refer to [@Lieb; @Lions] for more details on the Choquard equation and to [@Mu] for a recent result on a system in ${\mathbb{R}}^3$ strictly related with ours.\
Up to our knowledge, problem has not been investigated when a more general function $f$ appears instead of the identity. Since problem possesses a variational structure, our aim is to find weak assumptions on $f$ in order to apply the usual variational techniques. In particular, the first step in a classical approach to such a type of systems consists in the use of the reduction method. To this end, we need to assume suitable growth conditions on $f$ which allow us to invert the Laplace operator and thus to solve the second equation of the system. Then, after we have reduced the problem to *a single equation*, we find critical points of a one variable functional, checking geometrical and compactness assumptions of the Mountain Pass Theorem. If on one hand a suitable use of some [*a priori*]{} estimates makes quite immediate to show that geometrical hypotheses are verified (at least for small $q$), on the other some technical difficulties arise in getting boundedness for the Palais-Smale sequences. If $\eta=1$, we use a suitable truncation argument based on an idea of Berti and Bolle [@BB] and Jeanjean and Le Coz [@JC] (see also [@ADP; @K2]) and we are able to show the existence of a bounded Palais-Smale sequence of the functional taking $q$ sufficiently small. If we had the sufficient compactness, we would conclude by extracting any strongly convergent subsequence from this bounded Palais-Smale sequence. However the growth hypothesis we assume on $f$ does not permit to deduce compactness on the variable $\Phi$. Indeed the exponent $4$ turns out to be *critical* and in this sense we are justified to refer to as the [*critical growth condition*]{} for the function $f.$\
The first result in this paper is the following.
\[main\] Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $$\label{ipotesi} \tag{$\mathbf{f}$}
|f(s)| {\leqslant}c_1+c_2 |s|^{4}$$ for all $s\in{\mathbb{R}}.$ Then, there exists $\bar{q}>0$ such that for all $0 < q {\leqslant}\bar{q}$ problem $$\tag{$\mathcal P_f$}
\left\{
\begin{array}{ll}
-\Delta u+ {\varepsilon}q\Phi f(u)=|u|^{p-1}u &\text{in } \Omega, \\
-\Delta \Phi=2 qF(u) & \text{in }\Omega, \\
u=\Phi=0 & \text{on }\partial \Omega,
\end{array}
\right.
\label{P}$$ has at least a nontrivial solution.
Inspired by [@Mu], in the second part of this paper we study with ${\varepsilon}=\eta=-1$ and $f(s)= |s|^{r-2}s.$ The system becomes $$\label{Pq}\tag{$\mathcal P_r$}
\left\{
\begin{array}{ll}
-\Delta u - q |u|^{r-2} u \Phi + |u|^{p-1} u=0
&
\hbox{in } \Omega,\\
-r\Delta \Phi=2q |u|^r
&
\hbox{in } \Omega,\\
u=\Phi=0
&
\hbox{on } \partial \Omega.
\end{array}
\right.$$ where $1< r $ and $1<p<5$.
In this situation the contrasting nonlocal and local nonlinear terms perturb the functional in a way which in some sense recalls the typical concave-convex power-like nonlinearity. As a consequence, the functional’s geometry depends on the ratio of magnitude between $r$ and $p$. We get the following result.
\[th:r\] If $\frac {p+1}2<r<5$, problem has infinitely many solutions for any $q>0.$\
If $r=\frac{p+1}2,$ then there exists an increasing sequence $(q_n)_n$ such that, if $q{\geqslant}q_n$, problem has at least $n$ couples of solutions.\
If $1<r<\frac{p+1}2,$ then there exists an increasing sequence $(q_n)_n$ such that, if $q{\geqslant}q_n$, problem has at least $2n$ couples of solutions.\
Finally, if $p<5{\leqslant}r$, then the problem has no nontrivial solution.
The paper is organized as follows: in Section \[VT\] we introduce the functional setting where we study the problem and the variational tools we use; in Section \[proof\] we provide the proof of Theorem \[main\]; in Section \[sec:positive\] we consider the system and prove Theorem \[th:r\].
Throughout the paper we will use the symbols $H^{-1}$ to denote the dual space of $H^1_0(\Omega)$, $\langle \cdot,\cdot \rangle$ to denote the duality between $H^1_0(\Omega)$ and $H^{-1}$ and $\|u\|_{H^1_0}:=\left(\int_{{\Omega}}|\nabla u|^2\right)^{\frac 1 2}$ for the norm on $H^1_0(\Omega)$. Moreover $\|\cdot\|_p$ will denote the usual $L^p({\Omega})$-norm.
We point out the fact that in the sequel we will use the symbols $C,$ $C_1,$ $C_2,$ $C_3$ and so on, to denote positive constants whose value might change from line to line.
Variational tools {#VT}
=================
Standard arguments can be used to prove that problem is variational and the related $C^1$ functional $J_q:H^1_0(\Omega) \times
H^1_0(\Omega) \rightarrow \mathbb{R}$ is given by $$J_q(u,\Phi)=\frac{1}{2}\int_{\Omega} |\nabla
u|^2dx-\frac{{\varepsilon}}{4}\int_{\Omega} |\nabla \Phi|^2dx+{\varepsilon}q \int_{\Omega}
F(u)\Phi dx-\frac{1}{p+1}\int_{\Omega} |u|^{p+1}dx.$$
Since, by , $F:L^{6}(\Omega) \to L^{\frac{6}{5}}(\Omega)\hookrightarrow H^{-1}$, then certainly we have that for all $u \in
H^1_0(\Omega)$ there exists a unique $\Phi_u \in
H^1_0(\Omega)$ which solves $$-\Delta \Phi=2 qF(u) \label{2eq}$$ in $H^{-1}.$ In particular, we are allowed to consider the following map $$u \in L^{6}(\Omega) \mapsto \Phi_u \in H^1_0(\Omega)$$ which is continuously differentiable by the Implicit Function Theorem applied to $\partial_\Phi H$ where, for any $(u,\Phi)\in L^{6}(\Omega)\times{H^1_0({\Omega})}$, $$H(u,\Phi) :=
\frac 1 4 \int_{{\Omega}}|{\nabla }\Phi|^2 dx -q\int_{{\Omega}}F(u)\Phi dx.$$
Since, for every $u\in{H^1_0({\Omega})}$, $$\partial_{\Phi} J_q(u,\Phi_{u})=- {\varepsilon}\partial_\Phi H(u,\Phi_{u})
=0,
\label{eq:J'}$$ then $$\int_{\Omega}|{\nabla }\Phi_u|^2\, dx=2 q\int_{\Omega}F(u)\Phi_u\, dx \label{svista}$$ and $$\int_{\Omega} F(u)\Phi_u dx {\geqslant}0.
\label{eq:pos}$$ Moreover, we have the following estimates.
For every $u\in{H^1_0({\Omega})}$ $$\left(\int_{\Omega} |\n\Phi_u|^2dx\right)^{1/2}{\leqslant}Cq
\left(\int_{\Omega} |F(u)|^{6/5}dx\right)^{5/6} \label{Holder2}$$ and $$\label{eq:control}
\int_{\Omega}F(u)\Phi_u dx
{\leqslant}q( C_1\|u\|_{ 6/5}^2+C_2\|u\|^{10}_{6}).$$
By Holder inequality we have $$\int_{\Omega} F(u)\Phi_u dx{\leqslant}\left(\int_{\Omega}
|F(u)|^{6/5}dx\right)^{5/6}\left(\int_{\Omega}
|\Phi_u|^{6}dx\right)^{1/6} .\label{Holder1}$$ Then, from , by using Sobolev embedding ${H^1_0({\Omega})}\hookrightarrow L^6({\Omega})$, we get $$\begin{aligned}
\int_{\Omega} |\nabla \Phi_u |^2 dx&= 2 q \int_{\Omega} F(u) \Phi_u dx
{\leqslant}2 q \left(\int_{\Omega}
|F(u)|^{6/5}dx\right)^{5/6}\left(\int_{\Omega}
|\Phi_u|^{6}dx\right)^{1/6}
\\ &
{\leqslant}C q \left(\int_{\Omega} |F(u)|^{6/5}dx\right)^{5/6}\left(\int_{\Omega}
|\nabla \Phi_u|^{2}dx \right)^{1/2}\end{aligned}$$ and then we get .\
By and we have $$ \int_{\Omega} F(u) \Phi_u dx {\leqslant}C q \left(\int_{\Omega}|F(u)|^{6/5}dx\right)^{5/3}$$ and then, since for it is $$|F(s)|^{6/5} {\leqslant}C_1 |s|^{6/5}+C_2|s|^{6}, \label{crescita}$$ for all $s \in \mathbb{R}$, we get .
Equation allows us to define on $H^1_0(\Omega)$ the $C^1$ one variable functional $$I_q(u):=J_q(u,\Phi_u)=
\frac{1}{2}\int_{\Omega} |\nabla u|^2dx+ {\varepsilon}\frac{q}{2}
\int_{\Omega} F(u)\Phi_u dx-\frac{1}{p+1}\int_{\Omega} |u|^{p+1}dx.
$$
By using standard variational arguments as those in [@BF], the following result can be easily proved.
Let $(u,\Phi)\in H^1_0(\Omega)\times
H^1_0(\Omega)$, then the following propositions are equivalent:
1. $(u,\Phi)$ is a critical point of functional $J_q$;
2. $u$ is a critical point of functional $I_q$ and $\Phi=\Phi_u$.
So we are led to look for critical points of $I_q.$ To this end, we need to investigate the compactness property of its Palais-Smale sequences.\
It is easy to see that the standard arguments used to prove boundedness do not work. Indeed, assuming that $(u_n)_n\in ({H^1_0({\Omega})})^{{\mathbb{N}}}$ is a Palais-Smale sequence, namely $(I_q(u_n))_n$ is bounded and $I_q'(u_n)\to 0$ in $H^{-1}$, we obtain the following inequality $$\label{eq:p-s}
I_q(u_n) - \frac{1}{p+1}\langle I_q'(u_n),u_n\rangle
{\leqslant}C_1 + C_2 \|u_n\|.$$ Since, by , $$\langle I'_q(u_n),u_n\rangle = \langle \partial_u
J_q(u_n,\Phi_{u_n}),u_n\rangle,$$ from we get $$\left(\frac 1 2 -\frac 1 {p+1}\right)\int_{\Omega}|{\nabla }u_n|^2 dx+
{\varepsilon}\frac q 2\int_{\Omega}F(u_n)\Phi_{u_n} dx
-{\varepsilon}\frac q {p+1} \int_{\Omega}f(u_n)u_n
\Phi_{u_n } dx {\leqslant}C_1+ C_2\|u_n\|.$$ In the classical SPE (${\varepsilon}=1$ and $f(t)=t$) we should deduce the boundedness of the sequence $(u_n)_n$ for $p{\geqslant}3$. In our general situation we need a different approach.
Let $T>0$ and $\chi :[0,+\infty[ \rightarrow [0,1]$ be a smooth function such that $\|\chi^{\prime}\|_{L^{\infty}} {\leqslant}2$ and $$\chi(s)=\left\{
\begin{array}{ll}
1 &\text{if }0 {\leqslant}s {\leqslant}1 \\
0 &\text{if }s {\geqslant}2.
\end{array} \label{chi}
\right.$$ We define a new functional $I^T_q:H^1_0(\Omega) \rightarrow
\mathbb{R}$ as follows $$I^T_q(u)=\frac{1}{2}\int_{\Omega} |\nabla u|^2dx+ {\varepsilon}\frac{q}{2}\chi \left(\frac{\|u\|_{H^1_0}}{T}\right) \int_{\Omega}
F(u)\Phi_u dx-\frac{1}{p+1}\int_{\Omega} |u|^{p+1}dx
\label{functional2}$$ for all $u \in H^1_0(\Omega)$. We are going to find a critical point $u \in H^1_0(\Omega)$ of this new functional such that $\|u\|_{H^1_0} {\leqslant}T$ in order to get solutions of our problem.
Proof of Theorem \[main\] {#proof}
=========================
We prove that the functional $I^T_q$ satisfies Mountain Pass geometrical assumptions. More precisely, we have the following result.
\[lemma1\] Under hypothesis , there exists $\bar{q}\in
{\mathbb{R}}_+\cup\{+\infty\}$ such that for all $0<q < \bar{q}$ functional $I^T_q$ satisfies:
1. \[31i\]$I^T_q(0)=0$;
2. \[31ii\]there exist constants $\rho,\alpha>0$ such that $$I^T_q(u){\geqslant}\alpha \qquad \text{for all } u \in H^1_0(\Omega) \quad
\text{with } \|u\|_{H^1_0}=\rho ;$$
3. \[31iii\] there exists a function $\bar{u} \in H^1_0(\Omega)$ with $\|\bar{u}\|_{H^1_0}> \rho$ such that $I^T_q(\bar{u}) < 0$.
Property \[31i\] is trivial.\
To prove \[31ii\] we distinguish two cases.\
If ${\varepsilon}=1$, by , for every $q>0$ we deduce that $$I^T_q(u){\geqslant}\frac{1}{2} \rho^2 - C \rho^{p+1} {\geqslant}{\alpha }>0$$ for suitable $\rho, {\alpha }>0$.\
If ${\varepsilon}=-1$, by using and the immersion of $H^1_0(\Omega)$ into $L^p(\Omega)$ spaces, we get $$\begin{aligned}
I^T_q(u) & {\geqslant}\frac{1}{2} \|u\|^2_{H^1_0}-\frac{q}{2} \int_{\Omega}
F(u)\Phi_u dx-\frac{1}{p+1}\int_{\Omega}
|u|^{p+1}dx \\
& {\geqslant}\frac{1}{2} \|u\|^2_{H^1_0}-\frac{q^2}{2}(C_1\|u\|_{
6/5}^2+C_2\|u\|^{10}_{6}) -\frac{1}{p+1}
\|u\|^{p+1}_{p+1} \\
& {\geqslant}\frac{1}{2} \|u\|^2_{H^1_0}-\frac{q^2}{2}(C_1
\|u\|^{2}_{H^1_0}+C_2 \|u\|^{10}_{H^1_0}) -\frac{C}{p+1}
\|u\|^{p+1}_{H^1_0}.\end{aligned}$$ Thus, if $q$ is such that $q^2C_1<1$ and $\rho$ is small enough, there exists $\alpha
>0$ such that $I^T_q(u) {\geqslant}\alpha$ for all $u \in H^1_0(\Omega)$ with $\|u\|_{H^1_0}=\rho$.\
To prove \[31iii\], let us consider $u \in H^1_0(\Omega)$, $u \neq 0$ and $t >
\frac{2T}{\|u\|_{H^1_0}}$ such that $\chi \left(\frac{ \|tu\|_{H^1_0}}{T}\right)=\chi \left(\frac{t \|u\|_{H^1_0}}{T}\right)=0$. Then, we have $$I^T_q(tu) =\frac{t^2}{2} \int_{\Omega} |\nabla u|^2 dx -
\frac{t^{p+1}}{p+1}\int_{\Omega} |u|^{p+1}dx$$ and so, for $t$ large enough, $I^T_q(tu)$ is negative.
Thus we can complete the proof of Theorem \[main\].
Lemma \[lemma1\] allows us to define, for $q<\bar q$, $$m^T_q=\inf_{\gamma \in \Gamma}\sup_{t \in [0,1]}I^T_q(\gamma(t))>0
\label{infsup2}$$ where $$\Gamma=\{ \gamma \in \mathcal{C}([0,1],H^1_0(\Omega))\mid
\gamma(0)=0, I^T_q(\gamma(1))<0 \} .$$ Certainly there exists a Palais-Smale sequence at mountain pass level $m^T_q$, that is a sequence $(u_n)_n$ in $H^1_0(\Omega)$ such that $$I^T_q(u_n) \rightarrow m^T_q \label{PS1}$$ and $$(I^T_q)^{\prime}(u_n) \rightarrow 0 .\label{PS2}$$ As a first step we prove that there exists $\bar T>0$ and $\tilde q >0$ such that for any $0<q{\leqslant}\tilde q$ there exists a Palais-Smale sequence $(u_n)_n$ of $I_q$ at the level $m^{\bar T}_q$ such that, up to a subsequence, $\|u_n\|_{{H^1_0({\Omega})}}{\leqslant}\bar T$ for any $n\in{\mathbb{N}}$.\
Let $T>0$, $q>0$ and $(u_n)_n$ in $H^1_0(\Omega)$ be a Palais-Smale sequence of $I^T_q$ at level $m^{T}_q$.\
We make some preliminary computations. The first is an estimate on the mountain pass level $m_q^{T}$.\
Let $u \in
H^1_0(\Omega), u \neq 0$ and $\bar{t}>0$ be such that the path $\bar\g(t)=t\bar t u$ belongs to ${\Gamma}.$ For all $t \in [0,1]$ it is $$I^{T}_q(\bar\g(t)) = \frac{t^2}{2}\int_{\Omega} |\n\bar{t}
u|^2
dx
+{\varepsilon}\frac{q}{2}\chi \left(\frac{t \|\bar{t}u\|_{H^1_0}}{\bar
T}\right) \int_{\Omega} F(t\bar{t}u)\Phi_{t\bar{t}u}
dx-\frac{t^{p+1}}{p+1}\int_{\Omega} |\bar{t}u|^{p+1}dx.$$ From we obtain $$\begin{aligned}
\max_{t\in[0,1]}I^{T}_q(\bar\g(t)) &{\leqslant}\max_{t\in
[0,1]}\left(C_1 t^2\int_{\Omega}|{\nabla }u|^2\, dx -
C_2 t^{p+1} \int_{\Omega}|u|^{p+1}\, dx\right)\\
&\qquad +q^2\max_{t\in [0,1]}\left[\chi\left(\frac{t\bar t
\|u\|_{H^1_0}}{{T}}\right)\left(C_3 t^2\|\bar t
u\|^2_{H^1_0}+C_4
t^{10}\|\bar t u\|^{10}_{H^1_0}\right)\right]\\
&{\leqslant}C + q^2 (C_5 {T}^2 + C_6 {T}^{10}).\end{aligned}$$ Then, from we get $$m^{T}_q {\leqslant}C + q^2 (C_5 {T}^2 + C_6 {T}^{10})
.\label{stimalivello}$$ From we deduce that $$\langle (I^{\bar T}_q)^{\prime}(u_n), u_n
\rangle=\int_{\Omega}|\nabla u_n|^2 dx+{\mathcal A}_n+{\mathcal
B}_n+{\mathcal C}_n-\int_{\Omega}|u_n|^{p+1}dx$$ where $${\mathcal A}_n={\varepsilon}\frac q 2
\chi^{\prime}\left(\frac{\|u_n\|_{H^1_0}}{T}\right)\frac{\|u_n\|_{H^1_0}}{T}\int_{\Omega}F(u_n)\Phi_{u_n}dx$$ $${\mathcal B}_n={\varepsilon}\frac{q}{2}\chi
\left(\frac{\|u_n\|_{H^1_0}}{{\bar
T}}\right)\int_{\Omega}f(u_n)u_n\Phi_{u_n} dx$$ and $${\mathcal C}_n={\varepsilon}\frac{q}{2}\chi
\left(\frac{\|u_n\|_{H^1_0}}{{\bar T}}\right)\int_{\Omega}F(u_n)
\Phi_{u_n}^{\prime}[u_n] dx .$$ Since the functional $$S(u)=\int_{\Omega}|{\nabla }\Phi_u|^2 dx - 2 q\int_{\Omega}F(u)\Phi_u dx, \; u\in{H^1_0({\Omega})}$$ is identically equal to 0, we have that $$0=\frac{1}{2}\langle S'(u),u\rangle = \int_{{\Omega}} ({\nabla }\Phi_u|{\nabla }\Phi'_u[u])dx - q \int_{\Omega}f(u)\Phi_u u dx- q \int_{\Omega}F(u) \Phi'_u[u]dx.$$ On the other hand, multiplying the second equation of by $\Phi'_u[u]$ and integrating, we have that $$\int_{\Omega}({\nabla }\Phi_u|{\nabla }\Phi'_u[u])dx=2q \int_{\Omega}F(u) \Phi'_u[u]dx$$ and then $$\int_{\Omega}F(u) \Phi'_u[u]dx=\int_{\Omega}f(u)\Phi_u u dx.$$ We deduce that ${\mathcal B}_n={\mathcal C}_n$ for any $n\in \mathbb{N}.$ By using and we get $$\begin{aligned}
m^{T}_q+o_n(1)\|u_n\|_{H^1_0}+o_n(1)&=I^{T}_q(u_n)-\frac{1}{p+1}\langle (I^{T}_q)^{\prime}(u_n), u_n
\rangle\nonumber\\
&=\frac{p-1}{2(p+1)}\|u_n\|^2_{H^1_0}+{\mathcal D}_n
-\frac{1}{p+1}{\mathcal A}_n-\frac{2}{p+1}{\mathcal B}_n,
\label{eq:ps}\end{aligned}$$ where $${\mathcal D}_n={\varepsilon}\frac{q}{2}\chi \left(\frac{ \|u_n\|_{H^1_0}}{{T}}\right)
\int_{\Omega} F(u_n)\Phi_{u_n} dx.$$ For and we have also the following estimate $$\max \left( |{\mathcal A}_n|,|{\mathcal B}_n|,|{\mathcal
D}_n|\right){\leqslant}q^2(C_1 {T}^2 + C_2 {T}^{10}).$$ We show that, if ${T}$ is sufficiently large, then $\limsup_{n}\|u_n\|_{H^1_0}{\leqslant}{T}$.\
By contradiction, we will assume that there exists a subsequence (relabeled $(u_n)_n$) such that for all $n\in\mathbb{N}$ we have $\|u_n\|_{H^1_0}> {T}$. By our contradiction hypothesis, and , we obtain, for $n$ large enough, $${T}^2 -{\sigma }{T} {\leqslant}\|u_n\|^2_{H^1_0} - \s
\|u_n\|_{H^1_0}{\leqslant}C + q^2 (C_1 {T}^2+C_2 {T}^{10}),$$ with ${\sigma }>0$ small. If ${T}^2 - {\sigma }{T} >C$, we can find $\tilde q$ such that for any $q{\leqslant}\tilde q$ the previous inequality turns out to be a contradiction. The contradiction arises from the assumption that $\limsup_{n}\|u_n\|_{H^1_0}{\geqslant}{T}$. So we have that the sequence $(u_n)_n$ possesses a subsequence which is bounded in the ${H^1_0({\Omega})}$ norm by ${T}$ and such that, for every $n\in\mathbb{N}$, $I^{T}_q(u_n)$ coincides with $I_q(u_n)$.\
The last step is to prove that there exists $\bar q >0$ such that for any $0<q{\leqslant}\bar q$ there exists a Palais-Smale sequence $(u_n)_n$ of $I_q$ which is, up to a subsequence, weakly convergent to a nontrivial critical point of $I_q$.\
Let $\bar T$ and $\tilde q$ be given by the first step, and consider any $0< q {\leqslant}\tilde q$. We know that there exists a Palais-Smale sequence of the functional $I_q$ at the level $m_q:= m_q^{\bar T}$, such that $$\label{eq:bou}
\|u_n\|_{{H^1_0({\Omega})}}{\leqslant}\bar T, \hbox{ for any $n\in{\mathbb{N}}$.}$$ Up to subsequences, there exist $u_0\in{H^1_0({\Omega})}$ and $\Phi_0\in{H^1_0({\Omega})}$ such that $$\begin{aligned}
u_n &\rightharpoonup u_0\text{ in } H^1_0(\Omega)\label{eq:uzero}\\
\Phi_{u_n} &\rightharpoonup {\Phi_0} \text{ in }
H^1_0(\Omega)\label{fibarra} .\end{aligned}$$ By and we also have $$\begin{aligned}
F(u_n) &\rightharpoonup F(u_0) \quad \text{in } L^{\frac{6}{5}}({\Omega}),\label{convl}\\
f(u_n)u_n&\rightharpoonup f(u_0)u_0 \quad \text{in }
L^{\frac{6}{5}}({\Omega}).\label{convv2}\end{aligned}$$ Now we show that $\Phi_0=\Phi_{u_0}$ and that $(u_0,\Phi_0)$ is a weak nontrivial solution of .\
Let us consider a test function $\psi \in {C^{\infty}_{0}}(\Omega)$. From the second equation of our problem we obtain $$\int_{\Omega}(\nabla \Phi_{u_n}| \nabla \psi ) dx= 2q \int_{\Omega}
F(u_n) \psi dx.$$ Passing to the limit and using and , we have that $$\int_{\Omega}(\nabla {\Phi_0}| \nabla \psi ) dx= 2q \int_{\Omega}
F(u_0) \psi dx.$$ So ${\Phi_0}$ is a weak solution of $- \Delta \Phi =2q F(u_0)$, and then, by uniqueness, it is ${\Phi_0}=\Phi_{u_0}$.\
Since $(u_n)_n$ is a Palais-Smale sequence, for any $\psi \in
{C^{\infty}_{0}}(\Omega)$ we obtain that $$\int_{{\Omega}} ({\nabla }u_n|{\nabla }\psi)dx+{\varepsilon}q \int_{{\Omega}} \Phi_{u_n}
f(u_n)\psi dx=\int_{{\Omega}}
|u_n|^{p-1}u_n\psi dx+ o_n(1).$$ Passing to the limit, by and we have $$ \int_{{\Omega}} ({\nabla }u_0|{\nabla }\psi)dx+{\varepsilon}q \int_{{\Omega}} \Phi_{u_0}
f(u_0)\psi dx=\int_{{\Omega}}
|u_0|^{p-1}u_0\psi dx$$ that is $(u_0,\Phi_{u_0})$ is a weak solution of .\
It remains to prove that $u_0\neq 0.$\
Assume by contradiction that $u_0=0.$ By compactness we obtain that $u_n\to 0$ in $L^{p+1}({\Omega}).$ On the other hand, since $\langle
I_q'(u_n),u_n\rangle \to 0,$ we deduce that, up to subsequences,
$$\label{psseq}
\lim_n \int_{\Omega}|{\nabla }u_n|^2\, dx=-\lim_n {\varepsilon}q\int_{\Omega}f(u_n)u_n\Phi_{u_n}\, dx =:l_q{\geqslant}0.$$
Of course $l_q>0$. Otherwise from we would deduce that $u_n\to 0$ in ${H^1_0({\Omega})}$ and then $0<m_q=\lim_n I_q(u_n)=0$.\
By we also have that $l_q{\leqslant}\bar T.$ By using Holder inequality, Sobolev inequalities, , and we have that $$\begin{aligned}
-{\varepsilon}q \int_{\Omega}f(u_n)u_n\Phi_{u_n}dx&{\leqslant}Cq^2
\left(\int_{\Omega} |f(u_n)u_n|^{6/5}dx\right)^{5/6}
\left(\int_{\Omega}
|F(u_n)|^{6/5}dx\right)^{5/6}\\
&{\leqslant}q^2\left( C_1\|u_n\|_{{H^1_0({\Omega})}}^2+C_2\|u_n\|^{10}_{{H^1_0({\Omega})}}\right).
\end{aligned}$$ Passing to the limit, by we deduce that $$l_q{\leqslant}q^2 \left( C_1l_q^2+C_2l_q^{10}\right)$$ that is $$1{\leqslant}q^2\left( C_1l_q+C_2l_q^{9}\right){\leqslant}q^2\left( C_1\bar T +C_2\bar T^{9}\right).$$ We conclude observing that the previous inequality does not hold if we take $q{\leqslant}\bar q< \min\left\{\tilde q,(\frac 1 {C_1\bar T +C_2\bar T^{9}})^{\frac 1
2}\right\}$.
Proof of Theorem \[th:r\] {#sec:positive}
=========================
This section deals with the study of the system . We divide the proof of Theorem \[th:r\] in two parts, concerning respectively the existence and the non existence of a solution according to the value of $r$.
The existence result
--------------------
Here, we suppose that $1<r<5$. In analogy to problem , we use a variational approach finding the solutions as critical points of the $C^1$ functional $$I_{q,r} (u)=\frac{1}{2}\int_{\Omega} |\nabla u|^2dx
-\frac{q}{2r}\int_{\Omega} |u|^r \Phi_u dx
+\frac{1}{p+1}\int_{\Omega} |u|^{p+1}dx,$$ where, for any $u\in{H^1_0({\Omega})}$, $\Phi_u\in{H^1_0({\Omega})}$ is the unique positive solution of $$\left\{
\begin{array}{ll}
-r\Delta \Phi=2q |u|^r
&
\hbox{in } \Omega,\\
\Phi=0
&
\hbox{on } \partial \Omega.
\end{array}
\right.
\label{eq:seceq}$$ We have the following estimates.
For every $u \in H^1_0(\Omega)$ $$\label{eq:estimate}
\| \Phi_u \|_{{H^1_0({\Omega})}} {\leqslant}C \|u \|^r_{\frac{6}{5}r}$$ and for any $k>0$ it is $$\label{eq:upperest}
\frac{kq}{r}\int_{\Omega} |u|^r \Phi_u dx{\geqslant}\frac{2q}{r}\int_{\Omega}|u|^{r+1}dx-\frac 1 {2k} \int_{\Omega}|{\nabla }u|^2 dx.$$
The first part of the lemma is a consequence of the fact that, since $1< r< 5$, then for any $u\in{H^1_0({\Omega})}$ the function $|u|^r\in L^{\frac 6 5}({\Omega})$ and we can argue as in Section \[VT\] to define the map $\Phi$ and to deduce .\
In order to prove , we proceed as in [@Ru]: multiplying by $|u|$ and integrating we get $$\begin{aligned}
\frac{2q}{r}\int_{\Omega}|u|^{r+1}dx&= \int_{\Omega}({\nabla }\Phi_u|{\nabla }|u|)\,dx=\int_{\Omega}\left( \sqrt k\n\Phi_u\;\vline\;\frac1{\sqrt k}\n
|u|\right)\,dx\\
&{\leqslant}\frac k 2 \int_{\Omega}|\n\Phi_u|^2 dx+\frac 1 {2k} \int_{\Omega}|{\nabla }|u||^2 dx.
\end{aligned}$$ Inequality follows since it is $$\int_{\Omega}|\n\Phi_u|^2\,dx=\frac{2q}r\int_{\Omega}|u|^r\Phi_u\,
dx.$$
In the following lemma we establish the compactness of the Palais-Smale sequences of the functional $I_{q,r}$.
\[le:ps\] If $1<r<5$ the functional $I_{q,r}$ satisfies the Palais-Smale condition.
Let $(u_n)_n$ be a a Palais-Smale sequence for the functional $I_{q,r}$, namely $(I_{q,r}(u_n))_n$ is bounded and $I'_{q,r}(u_n)$ converges to zero in $H^{-1}$.\
We distinguish two cases.\
If $r{\geqslant}(p+1)/2$, then $$\begin{aligned}
I_{q,r} (u_n)- \frac{I'_{q,r} (u_n)[u_n]}{p+1} &=\frac{p-1}{2(p+1)} \int_{\Omega}|\nabla u_n|^2dx + q \frac{2r-p-1}{2r(p+1)} \int_{\Omega}|u_n|^r
\Phi_{u_n} dx
\\
&{\leqslant}C + o_n(1)\|u_n\|_{{H^1_0({\Omega})}}\end{aligned}$$ and then $(u_n)_n$ is bounded.\
If $r < (p+1)/2$, then the boundedness of $(u_n)_n$ comes from the boundedness of $(I_q(u_n))_n$ since the functional is coercive. Indeed, if we suppose that $(u_n)_n$ diverges in the $H^1_0 ({\Omega})-$norm, then, by and the Lebesgue embedding $L^{p+1}(\Omega)\hookrightarrow L^{\frac{6}{5}r}(\Omega)$, $$I_{q,r}(u_n){\geqslant}C_1 \| u_n \|_{{H^1_0({\Omega})}}^2 - C_2 \| u_n
\|_{p+1}^{2r} + C_3 \| u_n \|_{p+1}^{p+1} \rightarrow +\infty.$$
Thus we can complete the proof of the Theorem \[th:r\].
We first suppose that $r$ is *subcritical* and we deal with each case separately.
*case 1: $\displaystyle\frac{p+1}{2} < r< 5$*
We show that for any $q>0$ and for any finite dimensional subspace of ${H^1_0({\Omega})}$, the functional $I_{q,r}$ satisfies the assumptions of [@AR Theorem 2.23].\
By and the Sobolev embedding ${H^1_0({\Omega})}\hookrightarrow
L^{\frac 6 5 r}({\Omega})$ it is $$I_{q,r}(u){\geqslant}\frac{1}{2} \int_{\Omega}|\nabla u|^2dx - C
\|u\|^{2r}_{\frac{6}{5}r}
{\geqslant}\frac{1}{2} \|u\|_{{H^1_0({\Omega})}}^2 -
\bar C \|u\|_{{H^1_0({\Omega})}}^{2r}, $$ then assumption $(I_1)$ holds since $I_{q,r}(u) {\geqslant}\a>0$ if $\|u\|_{{H^1_0({\Omega})}}$ is sufficiently small. By Lemma \[le:ps\], also assumption $(I_3)$ holds. $(I_4)$ can be checked by an easy computation. In order to prove $(I_7),$ we show that for any finite dimensional subspace $E$ of ${H^1_0({\Omega})}$ there exists a ball $B_{\bar \rho}$ such that $I_{q,r}|_{E\cap
\partial B_{\bar\rho}}<0$.\
Let $E$ be a finite dimensional subspace of ${H^1_0({\Omega})}.$ It is easy to see that for $\rho>0$ and $u\in{H^1_0({\Omega})}$ $$\Phi_{\rho u}=\rho ^r \Phi_u.$$ Since in $E$ all the norms are equivalent, if $u\in E\cap
\partial B_1,$ by we have $$\begin{aligned}
I_{q,r}(\rho u)&=\frac{\rho^2}{2} \int_{\Omega}|\nabla u|^2dx - \frac{q}{2r}
\rho^{2r} \int_{\Omega}|u|^r \Phi_{u} dx +
\frac{\rho^{p+1}}{p+1}\int_{\Omega} |u|^{p+1}dx\nonumber\\
&{\leqslant}\frac{\rho^2}{2}- \frac{q}{kr}\rho^{2r}\int_{\Omega}|u|^{r+1}\,dx+\frac 1 {4k^2}\rho^{2r} + \frac{\rho^{p+1}}{p+1}\int_{\Omega}
|u|^{p+1}dx\nonumber\\ &{\leqslant}\frac{\rho^2}{2}- (c_1(k) -c_2(k))\rho^{2r} +
c_3\rho^{p+1}.\label{eq:negative}\end{aligned}$$ Since $c_1(k)= O(1/k)$ and $c_2(k)= O(1/{k^2})$ for $k\to + \infty,$ we have that for a $k$ sufficiently large $c_1(k)- c_2(k)>0.$ So we can take $\bar\rho>0$ as large as needed to have that is negative.
*case 2: $\displaystyle\frac{p+1}{2}= r$*
The proof is the same as in the previous case, except for $(I_7).$ In particular we can only prove that for any $E\subset{H^1_0({\Omega})}$ finite dimensional subspace there exists $\bar q>0$ such that for every $
q>\bar q$ and $\bar\rho$ large enough, $I_{q,r}|_{E\cap
\partial B_{\bar\rho}}<0$. Indeed, as in , we have that for any $u\in E\cap
\partial B_1,$ it is $$\begin{aligned}
I_{q,r}(\rho u)&{\leqslant}\frac{\rho^2}{2}- \frac{q}{rk}\rho^{2r}
\|u\|_{r+1}^{r+1}+\frac {\rho^{2r}} {4k^2} + \frac{\rho^{2r}}{2r}
\|u\|_{2r}^{2r}\\
&{\leqslant}\frac{\rho^2}{2}- (q c_1(k) -c_2(k)- c_3)\rho^{2r}
\end{aligned}$$ so that if $q$ and $\rho$ are sufficiently large, $I_{q,r}(\rho u)<0$.
*case 3: $\displaystyle 1 < r < \frac{p+1}{2}$*
In this case it is easy to check that for any $E\subset{H^1_0({\Omega})}$ finite dimensional subspace there exists $q>0$ such that $I_{q,r}|_{E\cap
\partial B_1}<0$. In fact, for any $u\in E\cap
\partial B_1,$ we have that $$\begin{aligned}
I_{q,r}(\rho u)&{\leqslant}\frac{\rho^2}{2}- \frac{q}{kr}\rho^{2r}\int_{\Omega}|u|^{r+1}\,dx+\frac 1 {4k^2}\rho^{2r} + \frac{\rho^{p+1}}{p+1}\int_{\Omega}
|u|^{p+1}dx\\
&{\leqslant}\frac{\rho^2}{2}- (q c_1(k) -c_2(k)- c_3)\rho^{p+1}.
\end{aligned}$$ To complete the proof, by [@AR Corollary 2.24] we have just to show that for any $q$ the functional $I_{q,r}$ is bounded from below. Indeed, by , $$\begin{aligned}
I_{q,r}(u){\geqslant}&
\frac{1}{2} \int_{\Omega}|\nabla u|^2dx
- C \| u \|_{\frac{6}{5}r}^{2r}
+ \frac{1}{p+1} \| u \|_{p+1}^{p+1}
\\
{\geqslant}&
\frac{1}{2} \int_{\Omega}|\nabla u|^2dx - C \| u \|_{p+1}^{2r} + \frac{1}{p+1} \| u \|_{p+1}^{p+1}\\
=&\frac{1}{2} \int_{\Omega}|\nabla u|^2dx + \| u \|_{p+1}^{2r}
\left(\frac{1}{p+1} \| u \|_{p+1}^{p+1-2r} - C \right)\end{aligned}$$ where the last quantity cannot diverge negatively.\
It is quite natural to wonder if some nonexistence result can be proved in the case $1 < r {\leqslant}\frac{p+1}{2}$ and $q$ small. Actually, it can easily be observed that the existence of at least a solution is guaranteed also for small $q$ when a ball with a sufficiently large radius $R$ is contained in ${\Omega}.$ Indeed, consider $u\in C_0^\infty({\Omega})$ such that $\|u\|_\infty {\leqslant}\sigma$ where $\sigma>0$ and $\frac q {kr}|s|^{r+1}- \frac 1 {p+1}|s|^{p+1}>0$ for any $s\in ]0,\sigma[.$ We set $u_t=u(\frac\cdot t)$, and we suppose that $\operatorname{Supp}(u_t)=t \operatorname{Supp} (u)\subset {\Omega}$. By a straight computation, using , we have that $$I_{q,r}(u_t){\leqslant}t\left(\frac{1}{2}+\frac{1}{4k^2}\right) \int_{\Omega}|\nabla u|^2dx
- t^3 \left(\frac{q}{kr}\int_{\Omega}|u|^{r+1}\,dx-
\frac{1}{p+1}\int_{\Omega} |u|^{p+1}dx\right)$$ where this last sum is negative for $t$ sufficiently large. As a consequence, we should have a mountain pass solution for $2r=p+1$ and also a minimum solution for $2r<p+1.$
The nonexistence result
-----------------------
Here we assume that $1<p<5{\leqslant}r.$ Following [@St], we adapt the Pohozaev arguments in [@P] to our situation (for a similar result see also [@DM]).\
Actually the proof is the same as in [@AD], but we report it here for completeness.
Let ${\Omega}\subset {{\mathbb{R}^3}}$ be a star shaped domain and $u,\Phi\in C^2({\Omega})\cap C^1(\bar{\Omega})$ be a nontrivial solution of (\[Pq\]). If we multiply the first equation of (\[Pq\]) by $x\cdot{\nabla }u$ and the second one by $x\cdot\n\Phi$ we have that $$\begin{aligned}
0=&(\Delta u + q \Phi |u|^{r-2} u - |u|^{p-1} u )(x\cdot{\nabla }u)\\
=&\operatorname{div}\left[({\nabla }u)(x\cdot{\nabla }u)\right] - |{\nabla }u|^2
- x\cdot {\nabla }\left(\frac{|{\nabla }u|^2}{2}\right)
+ \frac{q}{r} x\cdot\n\left(\Phi |u|^r\right) - \frac{q}{r} (x\cdot\n\Phi)|u|^r
- \frac{1}{p+1} x \cdot {\nabla }(|u|^{p+1})\\
=&\operatorname{div}\left[({\nabla }u)(x\cdot{\nabla }u) - x \frac{|{\nabla }u|^2}{2}
+\frac{q}{r} x \Phi |u|^r - \frac{1}{p+1} x |u|^{p+1}\right] \\
&\qquad +\frac{1}{2} |{\nabla }u|^2
- \frac{3}{r} q \Phi |u|^r - \frac{q}{r} (x\cdot\n\Phi) |u|^r
+ \frac{3}{p+1} |u|^{p+1}\end{aligned}$$ and $$\begin{aligned}
0=&(r\Delta \Phi + 2q |u|^r)(x\cdot{\nabla }\Phi)\\
=&r\operatorname{div}\left[({\nabla }\Phi)(x\cdot{\nabla }\Phi)\right] - r|{\nabla }\Phi|^2
- \frac{r}{2} x\cdot {\nabla }\left(|{\nabla }\Phi|^2\right) + 2q (x\cdot\n\Phi) |u|^r\\
=&r\operatorname{div}\left[({\nabla }\Phi)(x\cdot{\nabla }\Phi) - x \frac{|{\nabla }\Phi|^2}{2}
\right] + \frac{r}{2} |{\nabla }\Phi|^2 + 2q (x\cdot\n\Phi) |u|^r.\end{aligned}$$ Let ${\bf n}$ be the unit exterior normal to $\partial{\Omega}$. Integrating on ${\Omega}$, since by boundary conditions $\nabla u = \frac{\partial u}{\partial {\bf n}} {\bf n}$ and $\nabla \Phi = \frac{\partial \Phi}{\partial {\bf n}} {\bf n}$ on $\partial \Omega$, we obtain $$\label{eq:Poho1}
-\frac{1}{2} \|{\nabla }u \|_2^2 - \frac{1}{2} \int_{\partial{\Omega}} \left| \frac{\partial u}{\partial {\bf n}}\right|^2 x\cdot {\bf n} = -\frac{3}{r} q \int_{\Omega}\Phi |u|^r -\frac{q}{r} \int_{\Omega}(x\cdot\n\Phi) |u|^r + \frac{3}{p+1} \|u\|_{p+1}^{p+1}$$ and $$\label{eq:Poho2}
-\frac{r}{2} \|\n\Phi\|_2^2 -\frac{r}{2} \int_{\partial{\Omega}} \left| \frac{\partial \Phi}{\partial {\bf n}}\right|^2 x\cdot {\bf n}
= 2q \int_{\Omega}(x \cdot {\nabla }\Phi) |u|^r.$$ Substituting into we have $$\label{eq:Pohoc}
-\frac{1}{2} \|{\nabla }u \|_2^2 - \frac{1}{2} \int_{\partial{\Omega}} \left| \frac{\partial u}{\partial {\bf n}}\right|^2 x\cdot {\bf n} = -\frac{3}{r} q \int_{\Omega}\Phi |u|^r + \frac{1}{4} \|\n\Phi\|_2^2 + \frac{1}{4} \int_{\partial{\Omega}} \left| \frac{\partial \Phi}{\partial {\bf n}}\right|^2 x\cdot {\bf n} + \frac{3}{p+1} \|u\|_{p+1}^{p+1}.$$ Moreover, multiplying the first equation of (\[Pq\]) by $u$ and the second one by $\Phi$ we get $$\label{eq:Ne1}
\|{\nabla }u \|_2^2= q \int_{\Omega}\Phi |u|^r - \|u\|_{p+1}^{p+1}$$ and $$\label{eq:Ne2}
r\|\n\Phi\|_2^2 = 2q \int_{\Omega}\Phi |u|^r.$$
Hence, combining , and , we have $$\frac{r-5}{4}\|\n\Phi\|_2^2+\frac{5-p}{2(p+1)} \|u\|_{p+1}^{p+1} + \frac{1}{2} \int_{\partial{\Omega}} \left| \frac{\partial u}{\partial {\bf n}}\right|^2 x\cdot {\bf n} + \frac{1}{4} \int_{\partial{\Omega}} \left| \frac{\partial \Phi}{\partial {\bf n}}\right|^2 x\cdot {\bf n} = 0$$ and we get a contradiction.
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[^1]: Dipartimento di Matematica ed Informatica, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy, e-mail: [[email protected]]{}
[^2]: Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona 4, I-70125 Bari, Italy, e-mail: [[email protected]]{}
[^3]: The first two authors are supported by M.I.U.R. - P.R.I.N. “Metodi variazionali e topologici nello studio di fenomeni non lineari”
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---
abstract: 'Military and civilian applications of nuclear energy have left a significant amount of spent nuclear fuel over the past 70 years. Currently, in many countries world wide, the use of nuclear energy is on the rise. Therefore, the management of highly radioactive nuclear waste is a pressing issue. In this letter, we explore antineutrino detectors as a tool for monitoring and safeguarding nuclear waste material. We compute the flux and spectrum of antineutrinos emitted by spent nuclear fuel elements as a function of time, and we illustrate the usefulness of antineutrino detectors in several benchmark scenarios. In particular, we demonstrate how a measurement of the antineutrino flux can help to re-verify the contents of a dry storage cask in case the monitoring chain by conventional means gets disrupted. We then comment on the usefulness of antineutrino detectors at long-term storage facilities such as Yucca mountain. Finally, we put forward antineutrino detection as a tool in locating underground “hot spots” in contaminated areas such as the Hanford site in Washington state.'
author:
- Vedran Brdar
- Patrick Huber
- Joachim Kopp
bibliography:
- 'nuclear-waste.bib'
title: Antineutrino monitoring of spent nuclear fuel
---
Introduction
============
With carbon dioxide induced climate change and the scarceness of fossil fuels becoming imminent problems for humanity, nuclear energy is undergoing a renaissance. However, nuclear technology comes with a number of intrinsic problems, such as the limited availability of nuclear fuel, the danger of proliferation of nuclear weapons technology, the risk of major accidents, and the management of highly radioactive waste. As a result, nuclear energy is relatively expensive compared to many other energy sources.
In this letter, we will in particular focus on the waste issue: we will argue that a measurement of the antineutrino flux emitted by beta-decaying isotopes can be a unique component in a multi-faceted approach to monitoring and safeguarding nuclear waste repositories. The unique advantage of antineutrinos is that they penetrate the shielding surrounding the repository and thus offer a direct method for remotely probing the nuclear material inside. Other probes like gamma rays or neutrons, see for instance Ref. [@Ziock], are heavily attenuated by the materials they need to traverse on the way to a detector[^1]. Therefore, relating their measured fluxes to the actual content of the repository requires a sophisticated propagation model, which in turn relies on an accurate knowledge of the contents of the repository. This cyclic dependence on information is one of the major limitations of conventional monitoring methods. On the downside, the very fact that antineutrinos are not attenuated even by a whole mountain, implies that antineutrino detection has to deal with very small cross sections $\lesssim
10^{-41}$ cm$^2$ [@Vogel:1999zy]. Any meaningful flux measurement thus requires the deployment of a large detector with at least several tons of active material for a time period of order months.
Nevertheless, thanks to advances in detector technology, this appears feasible at a comparatively reasonable cost. In fact, practical applications of antineutrino detectors in the nuclear industry have been discussed for a long time, mostly in the context of monitoring power reactors [@Mikaelian:1978; @Bernstein:2001cz; @Nieto:2003wd; @Huber:2004xh; @Bernstein:2010; @Christensen:2013eza; @Christensen:2014pva]. Several detectors have been built to demonstrate the feasibility of such applications [@Klimov:1994a; @Bowden:2006hu], and further studies are planned in current and future experiments [@Bowden:2016ntq].
In the following, we will first compute the antineutrino flux and spectrum emitted by spent nuclear fuel, and then consider several scenarios in which antineutrino detectors can be used in the context of radioactive waste repositories.
Antineutrino emission from spent nuclear fuel
=============================================
For the first 1,000–10,000 years after discharge from a reactor, the total activity of spent nuclear fuel is nearly exclusively caused by beta decays (and the associated gamma emission). Therefore, a large number of antineutrinos is produced. However, detection by inverse beta decay, $\bar\nu_e + p \to n + e^+$, the main detection reaction for electron antineutrinos, requires antineutrino energies of at least 1.8MeV. The lifetime of a beta decaying nucleus scales roughly like $Q^5$, where $Q$ is the energy released in the decay. Therefore, the detectable antineutrino signal for most fission fragments decays within hours to days after fission ends. There is, however, a handful of isotopes that have a two stage decay, where the first decay has very small $Q$ and thus a resulting long lifetime, followed by a fast decay with $Q>1.8\,\text{MeV}$. The most notable example is strontium-90, which decays with a half-life of 28.90yrs to yttrium-90, which in turn decays within hours to the stable zirconium-90 with $Q=2.22801$MeV [@Browne:1997cbp]. Strontium-90 is produced in around 5% of all fission events [@Koning:2006; @SCALE; @ORIGEN]. The isotopes with the next longest lifetimes with antineutrino emission above 1.8MeV in their decay chains are ruthenium-106 (371.8days [@DeFrenne:2008kmy]) and cerium-144 (284.91days [@NuDat]). As a result, the detectable antineutrino emission of spent nuclear fuel after more than a few years is entirely given by strontium-90. It is worth noting that strontium-90 (like all other fission fragments) remains in the high-level waste resulting from reprocessing using the widely employed PUREX process. In \[fig:spectra\], we plot the number of electron antineutrinos emitted per second, per MeV, and per ton of spent nuclear fuel as a function of antineutrino energy for fuel elements of different age. We assume a burnup[^2] of 45 GWdays. As expected, we observe a softening of the spectrum over time, as short-lived isotopes with large $Q$ values decay away. Note, however, that even after 100 yrs, a non-zero flux remains above the energy threshold of 1.8 MeV for inverse beta decay.
![The spectrum of electron antineutrinos emitted by spent nuclear fuel as a function of the time after discharge from the reactor. We also indicate in gray the area below the threshold for inverse beta decay, the dominant antineutrino detection process, at 1.8 MeV. The data underlying this plot is available in the supplemental material [@Huber:2016supp].[]{data-label="fig:spectra"}](spectra){width="0.8\columnwidth"}
Dry cask storage facilities
===========================
As long term storage facilities for spent nuclear fuel are becoming available only slowly, temporary storage solutions have become a necessity. Once fuel elements have been allowed to cool in a spent fuel pool for $\sim 10$ yrs [@Alvarez:2011; @NRCweb] after discharge from the reactor, they are typically transferred to dry storage casks, large shielded steel cylinders several meters tall, each of them holding $\sim
14$–$24$ tons of spent nuclear fuel elements with a uranium content of 10–17 tons [@Greene:2013; @IDB1996; @NRCweb]. The layout of a typical dry storage facility is shown in \[fig:surry\]. Even though safety and security measures are in place to protect such facilities, manipulations are imaginable. The core of the IAEA’s (International Atomic Energy Agency’s) methodology for spent fuel is so-called continuity of knowledge (CoK): the amount and type of fuel loaded into a cask is monitored and recorded, the cask is closed, and a tamper-proof seal is applied. As long as the seal is intact and the records are available, the resulting CoK allows to infer with a great deal of certainty the contents of the cask. However, even during routine operations it is conceivable that records are inaccurate or lost or that seals are compromised. Several methods based on on neutron or gamma ray detection are under development to restore CoK in this case, see for instance [@Ziock].
Here, we envision instead the deployment of an antineutrino detector, with a fiducial target mass[^3] of order $\sim 20$ tons, close to the storage casks for several months. Using as an example the storage facility at the Surry Nuclear Power Station in the U.S., where casks hold 9–16 metric tons of uranium (MTU), we assume that 50% of the radioactive material from two of the 15 MTU casks (colored in red in \[fig:surry\]) goes missing. This roughly corresponds to removing 3% of the total amount of nuclear waste stored at Surry. We make no claim that an actual diversion case would have any similarity to this scenario nor that this could occur as part of routine operations, it merely serves to indicate the general level of sensitivity we might expect from antineutrino monitoring.
To determine what it takes to discover such an anomaly, we simulate the expected number of detected antineutrino events as a function of the detector position for the two hypothesis “all storage casks full” ($F$) and “50% of nuclear material missing in two casks” ($M$). We use the antineutrino spectrum given by the blue dashed curve in \[fig:spectra\] (10 years after discharge) and the inverse beta decay cross sections from [@Vogel:1999zy]. Neutrino oscillation effects, though small, are taken into account, with the oscillation parameters given in [@Gonzalez-Garcia:2014bfa]. The rate of antineutrino events per ton of fiducial detector mass and per MTU of source mass is $$\begin{aligned}
N_\nu = 5.17\;\text{yr}^{-1} \, \text{ton}^{-1} \, \text{MTU}^{-1}
\times \bigg( \frac{\text{10\,\text{m}}}{d} \bigg)^2 \,,\end{aligned}$$ where $d$ is the distance between the source and the detector (both treated as point-like). This number depends mildly on the time after discharge and is for instance reduced by $\sim
5\%$ one year later. In the following, we will always assume measurement campaigns lasting one year or less and therefore neglect this small effect.
![The dry storage facility at the Surry Nuclear Power Plant in Virginia, USA [@Greene:2013]. Filled storage casks, highlighted in yellow, contain 9–16 MTU each. In the benchmark scenario discussed in the text, we assume that 50% of the spent fuel in two 15 MTU casks (marked in red) have gone missing. Colored contours indicate the exposure (in ton yrs) required to establish the loss of nuclear material at the 90% confidence level.[]{data-label="fig:surry"}](surry-contours){width="0.95\columnwidth"}
The irreducible background to the measurement includes antineutrinos from running nuclear reactors with an expected event rate of $$\begin{aligned}
N_\text{bg} = 359\;\text{yr}^{-1} \, \text{ton}^{-1} \, \text{GWth}^{-1}
\times \bigg( \frac{\text{\text{km}}}{d} \bigg)^2 \,.
\label{eq:}\end{aligned}$$ For the 5.2 GWth (thermal power) reactor in Surry, located $d \sim 1$ km away from the envisioned 20 ton detector, this leads to $\sim 37\,300$ antineutrino events per year. We take this background into account in our simulations. Backgrounds from other power stations and from radioactive decays in the Earth (geo-neutrinos) are smaller by at least a factor $\sim 10^{-4}$, and we therefore neglect them.
The dominant reducible backgrounds arise from radioactive decays and cosmic ray interactions mimicking an antineutrino signal. With current single-volume liquid scintillator detectors like Double Chooz, RENO, and Daya Bay, when deployed at the surface, these backgrounds would be a factor 10–10000 larger than the anticipated antineutrino signal. Current detectors identify signal candidates by looking for a delayed coincidence between a primary particle and a delayed neutron capture. However, they are not able to exploit the spatial correlations between the primary and delayed signals, nor can they tell whether the primary particles is a positron, as in inverse beta decay, or a photon or electron, as in most background events. Fortunately, these shortcomings could be overcome in a detector with sufficient spatial resolution to tag positrons by resolving the two 511keV x-rays from their annihilation [@Safdi:2014hwa]. Prototypes of detectors with this capability exist and have been successfully operated in particular by the SoLid and CHANDLER collaborations [@Bowden:2016ntq]. Currently, an improvement of the signal-to-background ratio by a factor of 1000 is achievable, and further improvements appear feasible with improved shielding and an increased concentration of neutron capture targets like lithium-6. It thus appears plausible that within a few years even the low rate of antineutrinos from nuclear waste will become detectable in surface detectors. In the following, we will assume that this has been achieved by the time the proposed measurements are carried out, and we will neglect reducible backgrounds.
Events are divided into 0.2MeV wide energy bins. Denoting the number of signal events expected under the two alternative hypotheses by $F_i$ and $M_i$, and the number of background events by $B_i$, we define the test statistic $$\begin{aligned}
\hspace{-0.8cm}
\chi^2 \equiv 2 \sum_i \left\{ F_i - M_i
+ (M_i+B_i)\,\log\left[\frac{M_i+B_i}{F_i+B_i}\right]\right\},
\label{eq:chi2}\end{aligned}$$ which follows a $\chi^2$ distribution.
The results of the analysis are represented by the contours in \[fig:surry\] which indicate where the antineutrino detector should be placed in order to establish the flux deficit at the 90% confidence level with 20tonyrs, 40tonyrs, and 80tonyrs of exposure, respectively. We see that the detector needs to be placed within $\sim 50$ meters of the affected casks in order to collect the $\sim 4000$ events needed for the measurement.
Application to long-term storage facilities
===========================================
Above-ground storage of spent nuclear fuel, while widely used, is only a temporary solution, and the long-term goal must be to establish underground repositories that can keep radioactive material out of the biosphere for $10^4$–$10^6$ years [@NRC:1995]. The usefulness of antineutrino detectors at such geological repositories is limited by the low antineutrino fluxes after strontium-90 has decayed away (half-life 28.8yrs). Moreover, in order not to disturb the repository, construction of antineutrino detectors seems feasible and useful only at distances of order 100 meters or larger.
![The planned long term storage facility at Yucca mountain. The yellow grid indicates the drifts holding the radioactive material at a depth of 300m below the surface, while red and orange contours show the expected antineutrino count rates for a detector at the surface.[]{data-label="fig:yucca"}](yucca){width="0.8\columnwidth"}
To illustrate the prospects of detecting antineutrinos from a geological nuclear waste repository, we show in \[fig:yucca\] the signal event rates expected at the proposed Yucca mountain repository in Nevada, which would hold 70000MTU of radioactive material, stored $\sim 300$m underground. We see that even a small detector ($\sim 10\,\text{tons}$) located at the surface would see an appreciable event rate. With a kiloton-scale instrument like KamLAND [@Alivisatos:1998it] or the planned JUNO experiment [@Djurcic:2015vqa], count rates would be significantly larger, especially when such a detector is placed in an underground location closer to the repository. Even then, however, it would only be possible to detect cataclysmic disruptions of the repository. More typical (but nevertheless highly dangerous) failure scenarios that involve the leakage of only a small amount of nuclear material into the surrounding soil would not be detectable. This may change, however, once detector technology with better directional sensitivity becomes available (see below).
Leakage of nuclear material at the Hanford site
===============================================
Sometimes, nuclear oversight agencies are faced with the challenge to secure or decommission a nuclear waste repository in which the contents, and perhaps even the underground location, of storage casks are not known. An example is the Hanford site in the state of Washington (USA), where plutonium for military purposes was produced from 1944 to 1987. At Hanford, a major problem is the leakage of storage containers for high level nuclear waste, leading to radioactive contamination of ground water [@Fuller:2005; @Rockhold:2012].
Consider first a scenario where the location of storage tanks is known, but their precise content is not. We will focus on one particular array of storage tanks at Hanford, the T tank farm [@Fuller:2005], which consists of 16 tanks, arranged in a $4
\times 4$ grid measuring $\sim 120\,\text{m} \times 80\,\text{m}$ and originally containing between 0.2 and 5MTU of spent fuel each. We assume the nuclear material in the tanks was discharged from a reactor 50 years ago. With a detector placed 30 meters from the most massive (5MTU) tank, and taking into account background antineutrinos from other storage tanks and from the Columbia nuclear power plant (30km away, 3.5GW thermal power), the amount of material in that tank can be measured with an uncertainty of $\pm
2.1\,\text{MTU}$ for an exposure of 20tonyrs ($35$ signal events) and with an uncertainty of $\pm 1.1\,\text{MTU}$ for an exposure of 80tonyrs ($140$ signal events). The age of the nuclear material (i.e. the time after discharge) can be determined to lie between 44yrs and 54yrs with 80tonyrs of exposure, assuming the true age is 50yrs.
Assume now that a fraction of the radioactive material in the most massive tank is slowly leaking out. We model this situation by reducing the inventory of the tank and introducing a secondary point source containing the leaked material 20 meters below its original location. Detecting such leakage seems unfeasible with established detector technologies, but requires antineutrino detectors that not only measure energy, but also the direction of incoming antineutrinos. Some preliminary efforts in this direction have been undertaken [@Tanaka:2014; @Safdi:2014hwa; @Li:2016yey], but a working detector is still far off. As one of the goals of the present study is to motivate further R&D in this field, we will in the following assume the availability of a compact detector with an expected angular resolution down to $\mathcal{O}(10)$ degrees [@Safdi:2014hwa; @Brdar:2017vxy]. We bin events in the cosine of the zenith angle, $\cos\theta$, (5 bins) and the azimuth angle $\phi$ (9 bins). We use a highly simplified model of angular smearing in terms of a Gaussian with a width of 20 degrees. For simplicity, we integrate over energy, assuming that the time of discharge and thus the antineutrino energy spectrum are known already. We estimate that by deploying a directionally sensitive 20ton detector at a distance of 30 meters from the damaged tank, leakage of 55% of the tank’s content can be discovered at 90% CL after $12$ months ($30$ signal events). With an 80ton detector, detection of 25% leakage is possible.
Radioactive spill at Hanford building 324
=========================================
As a further application scenario, we consider an actual spill of radioactive material that happened in October 1986 in a radiochemical plant at Hanford known as Building 324. At the time, a large amount of strontium-90 and caesium-137, with a total activity of 1.3MCi was released from a hot cell. Half of it leaked into the ground and is now presumed to be located within a $10\,\text{meter} \times 10\,\text{meter}$ area about 2 meters below ground level [@Rockhold:2012]. In 2010, a pit was excavated $\sim 20$ meters from the spill to drive long steel pipes into the affected area, thus allowing the deployment of temperature and activity sensors. This method has the disadvantage that it allows moisture to enter the contaminated soil, which may ultimately allow radioactive material to seep further into the ground, possibly reaching ground water levels. For future incidents of this type, we therefore consider the deployment of antineutrino detectors for remote sensing. Modeling the spill as a point source at a depth of 2m, and assuming the availability of an 80ton antineutrino detector with angular sensitivity located 30m away at the same depth, we find that a further downward shift of the nuclear material by 3.5m is detectable at the 90% CL after one year of exposure.
Localizing nuclear waste containers
===================================
Let us now turn to a more speculative scenario where neither the exact location nor the contents of storage casks are known. This could happen, for instance, when documentation is lost and localization using other methods like ground-penetrating radar is not feasible, for instance in a scenario where many casks are buried underground, but only few contain high level nuclear waste. We envision successive or simultaneous deployment of 80ton antineutrino detectors on a two-dimensional grid with a spacing of 250m. We again assume angular sensitivity, but since the distance between detectors and sources is large, the zenith angle measurement is irrelevant and can be discarded. illustrates the outcome of such an analysis for four randomly placed storage casks of unknown content and for an exposure of 80tonyrs per detector. Using a stochastic optimization method, we fit the positions $(x_j, y_j)$ and activities $m_j$ of the four sources. Colored contours show the dependence of the test statistic $\chi^2$ (defined in analogy to \[eq:chi2\]) on the fit values of $(x_1, y_1)$, with the other $(x_j, y_j)$ as well as all $m_j$ allowed to float. We see that storage casks can be localized to within tens of meters. For further refinement of their position (for instance in order to guide cleanup efforts), the procedure can be repeated with detectors moved closer to the source positions determined in the initial scan. With a $\sim 30$m spacing between detectors, the position of each source can be determined to $\mathcal{O}(\text{m})$ accuracy.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Using antineutrino detectors (brown $\oplus$ symbols) to localize nuclear waste storage casks (radioactive hazard symbols). Colored contours indicate the accuracy with which casks can be localized (see text for details). We have assumed an exposure of $80\,\text{t}\,\text{yrs}$ per detector, and we have used the antineutrino spectrum expected 50 years after discharge from a reactor. []{data-label="fig:hanford"}](hanford-wide "fig:"){width=".7\columnwidth"}
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Summary
=======
We have calculated the antineutrino flux and spectrum from spent nuclear fuel, and have used these results to outline possible applications of antineutrino detectors in monitoring and managing nuclear waste repositories. We have shown that, in a specific diversion scenario at a dry cask storage facility, installation of an antineutrino detector could allow oversight agencies to remotely detect leakage or theft of the stored nuclear waste. Further study is needed to assess the applicability of the method to other diversion scenarios. At long term geological repositories, a significant antineutrino flux is expected, but detecting realistic anomalies such as leakage of a small amount of radioactive material would require advanced detector technologies with angular sensitivity. Such detectors could also help in the decommissioning of nuclear installations like the Hanford site, where they would allow for the localization of nuclear material and for the characterization of spills.
Acknowledgments {#acknowledgments .unnumbered}
===============
PH thanks Fermilab for hospitality during the completion of this manuscript. This work was in part supported by the U.S. Department of Energy under contracts DE-SC0013632 and DE-SC0009973. The work of VB and JK is supported by the German Research Foundation (DFG) under Grant Nos. and FOR 2239 and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 637506, “$\nu$Directions”). VB is also supported by the DFG Graduate School Symmetry Breaking in Fundamental Interactions (GRK 1581).
[^1]: For a recent study on muon radiography applied to the problem at hand, see [@Poulson:2016fre].
[^2]: Burnup is a measure of how much energy per unit mass has been extracted from nuclear fuel. It is directly proportional to the total number of fissions and thus to the strontium-90 content and the antineutrino emission rate.
[^3]: The fiducial detector mass is the effective mass, after accounting for fiducial volume cuts and efficiency factors introduced in event reconstruction and analysis.
|
---
abstract: 'Paradigms of bilinear maps $\beta\colon E_1\times E_2\to F$ between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of $\beta$ with Keller $C^n_c$-maps (on suitable domains) are $C^n_c$. Our main applications concern holomorphic families of operators, and the foundations of locally convex Poisson vector spaces.'
---
[**Applications of hypocontinuous bilinear maps\
in infinite-dimensional differential calculus**]{}\
[**Helge Glöckner[^1]**]{}
[: 26E15, 26E20 (primary); 17B63, 22E65, 46A32, 46G20, 46T25 (secondary).\
[*Key words*]{}: hypocontinuity, bilinear map, differentiability, infinite-dimensional calculus, smoothness,analyticity, analytic map, holomorphic map, family of operators, Poisson vector space, Poisson bracket, Hamiltonian vector field]{}
[**Introduction**]{}
If $\beta\colon E_1\times E_2\to F$ is a continuous bilinear map between locally convex spaces, then $\beta$ is smooth and hence $\beta\circ f\colon U\to F$ is smooth for each smooth map $f\colon U\to E_1\times E_2$ on an open subset $U$ of a locally convex space.\
Unfortunately, many bilinear mappings of interest are discontinuous. For example, it is known that the evaluation map $E'\times E\to {{\mathbb R}}$, $(\lambda,x){\mapsto}\lambda(x)$ is discontinuous for each locally convex vector topology on $E'$, if $E$ is a non-normable locally convex space (cf. [@Mai]). Hence also the composition map $L(F,G) \times L(E,F) \to L(E,G)$, $(A,B ){\mapsto}A \circ B$ is discontinuous, for any non-normable locally convex space $F$, locally convex spaces $E,G\not=\{0\}$, and any locally convex vector topologies on $L(F,G)$, $L(E,F)$ and $L(E,G)$ such that the maps $F\to L(E,F)$, $y{\mapsto}y{\otimes}\lambda$ and $L(E,G)\to G$, $A{\mapsto}A(x)$ are continuous for some $\lambda\in E'$ and some $x\in E$ with $\lambda(x)\not=0$, where $(y{\otimes}\lambda)(z):= \lambda(z)y$ (see Remark \[nachtrg\]).\
Nonetheless, both evaluation and composition do show a certain weakened continuity property, namely *hypocontinuity*. So far, hypocontinuity arguments have been used in differential calculus on Fréchet spaces in some isolated cases (cf. [@FAM], [@HYP] and [@Wur]). In this article, we distill a simple, but useful general principle from these arguments (which is a variant of a result from [@Tho]). Let us call a Hausdorff topological space $X$ a *$k^\infty$-space* if $X^n$ is a $k$-space for each $n\in {{\mathbb N}}$. Our observation (recorded in Theorem \[mainobs\]) is the following:\
*If a bilinear map $\beta\colon E_1\times E_2\to F$ is hypocontinuous with respect to compact subsets of $E_1$ or $E_2$ and $f\colon U\to E_1\times E_2$ is a $C^n$-map on an open subset $U$ of a locally convex space $X$ which is a $k^\infty$-space, then $\beta\circ f\colon U\to F$ is a $C^n$-map.*
As a byproduct, we obtain an affirmative solution to an old open problem by Serge Lang (see Corollary \[corlang\]). Our main applications concern two areas.\
[**Application 1: Holomorphic families of operators.**]{}\
In Section \[secholfam\], we apply our results to holomorphic families of operators, i.e., holomorphic maps $U\to L(E,F)$ on an open set $U{\subseteq}{{\mathbb C}}$ (or $U{\subseteq}X$ for a suitable locally convex space $X$). We obtain generalizations (and simpler proofs) for various results formulated in [@BaO].\
[**Application 2: Locally convex Poisson vector spaces.**]{}\
Finite-dimensional Poisson vector spaces are encountered naturally in finite-dimensional Lie theory as the dual spaces ${{\mathfrak g}}^*$ of finite-dimensional Lie algebras. They give rise to a distribution on ${{\mathfrak g}}^*$ whose maximal integral manifolds are the coadjoint orbits of the corresponding simply connected Lie group $G$. These are known to play an important role in the representation theory of $G$ (by Kirillov’s orbit philosophy).\
The study of infinite-dimensional Poisson vector spaces (and Poisson manifolds) only began recently with works of Odzijewicz and Ratiu concerning the Banach case (see [@OR1], [@OR2]). On Neeb’s initiative, a setting of locally convex Poisson vector spaces (which need not be Banach spaces) has recently been developed ([@GLN], [@Ne2]). In Section \[secpoisson\], we preview this framework and explain how hypocontinuity arguments can be used to overcome the analytic problems arising beyond the Banach case. In particular, hypocontinuity is the crucial tool needed to define the Poisson bracket and Hamiltonian vector fields. In Section \[nwsec1\], we describe situations where the Poisson bracket is hypocontinuous or even continuous. In the most relevant case, we also prove continuity of the linear map which takes a smooth function to the corresponding Hamiltonian vector field (Section \[nwsec2\]).\
Having thus secured the foundations, natural next steps will be the investigation of coadjoint orbits for examples of non-Banach, infinite-dimensional Lie groups and geometric quantization in this context. These will be undertaken in [@GLN], [@Ne2] and later research.\
Let us remark that an alternative path leading to Theorem \[mainobs\] is to prove, in a first step, smoothness of hypocontinuous bilinear maps (with respect to compact sets) in an alternative sense, replacing “continuity” by “continuity on each compact subset” in the definition of a $C^n$-map (see [@Tho Theorem 4.1]).[^2] The second step is to observe that such $C^n$-maps coincide with ordinary $C^n$-maps (i.e., Keller $C^n_c$-maps) if the domain is a \
*Acknowledgement.* The referee of [@FAM] suggested some valuable references to the literature.
Preliminaries and basic facts {#scprelim}
=============================
Throughout this article, ${{\mathbb K}}\in \{{{\mathbb R}},{{\mathbb C}}\}$ and ${{\mathbb D}}:=\{z\in {{\mathbb K}}\colon
|z|\leq 1\}$. As the default, the letters $E$, $E_1$, $E_2$, $F$ and $G$ denote locally convex topological ${{\mathbb K}}$-vector spaces. When speaking of linear or bilinear maps, we mean ${{\mathbb K}}$-linear (resp., ${{\mathbb K}}$-bilinear) maps. A subset $U{\subseteq}E$ is called *balanced* if ${{\mathbb D}}U{\subseteq}U$. The locally convex spaces considered need not be Hausdorff, but whenever they serve as the domain or range of a differentiable map, we tacitly assume the Hausdorff property. We are working in a setting of infinite-dimensional differential calculus known as Keller’s $C^n_c$-theory (see, e.g., [@RES] or [@GaN] for streamlined introductions).
\[C11\] Let $E$ and $F$ be locally convex spaces over ${{\mathbb K}}\in \{{{\mathbb R}},{{\mathbb C}}\}$, $U{\subseteq}E$ be open and $f\colon U\to F$ be a map. We say that $f$ is $C^0_{{\mathbb K}}$ if $f$ is continuous. The map $f$ is called $C^1_{{\mathbb K}}$ if it is continuous, the limit $$df(x,y)\;=\;\lim_{t\to 0}\frac{f(x+ty)-f(x)}{t}$$ exists for all $x\in U$ and all $y\in E$ (with $0\not=t\in {{\mathbb K}}$ sufficiently small), and the map $df\colon U\times E\to F$ is continuous. Given $n\in {{\mathbb N}}$, we say that $f$ is $C^{n+1}_{{\mathbb K}}$ if $f$ is $C^1_{{\mathbb K}}$ and $df\colon U\times E\to F$ is $C^n_{{\mathbb K}}$. We say that $f$ is $C^\infty_{{\mathbb K}}$ if $f$ is $C^n_{{\mathbb K}}$ for each $n\in {{\mathbb N}}_0$. If ${{\mathbb K}}$ is understood, we simply write $C^n$ instead of $C^n_{{\mathbb K}}$, for $n\in {{\mathbb N}}_0\cup\{\infty\}$.
If $f\colon E\supseteq U\to F$ is $C^1_{{\mathbb K}}$, then $f'(x):=df(x,{{\scriptscriptstyle \bullet}})\colon E\to F$ is a continuous ${{\mathbb K}}$-linear map, for each $x\in U$. It is known that compositions of composable $C^n_{{\mathbb K}}$-maps are $C^n_{{\mathbb K}}$. Furthermore, continuous linear (or multilinear) maps are $C^\infty_{{\mathbb K}}$ (see [@GaN Chapter 1], or [@RES] for all of this).
Keller’s $C^n_c$-theory is used as the basis of infinite-dimensional Lie theory by many authors (see [@FUN], [@DL3], [@GaN], [@Mil], [@Ne1], [@Wur]). Others use the “convenient calculus” [@KaM].\
For some purposes, it is useful to impose certain completeness properties on the locally convex space $F$ involved. These are, in decreasing order of strength: Completeness (every Cauchy net converges); quasi-completeness (every bounded Cauchy net converges); sequential completeness (every Cauchy sequence converges); and Mackey completeness (every Mackey-Cauchy sequence converges, or equivalently: the Riemann integral $\int_0^1\gamma(t)\, dt$ exists in $F$, for each smooth curve $\gamma\colon {{\mathbb R}}\to F$; see [@KaM Theorem 2.14] for further information).
\[cxanacompa\] If ${{\mathbb K}}={{\mathbb C}}$, then a map $f\colon E\supseteq U\to F$ is $C^\infty_{{\mathbb C}}$ if and only if it is *complex analytic* in the usual sense (as in [@BaS]), i.e., $f$ is continuous and for each $x\in U$, there exists a $0$-neighbourhood $Y{\subseteq}U-x$ and continuous homogeneous polynomials $p_n\colon E\to F$ of degree $n$ such that $f(x+y)=\bigcup_{n=0}^\infty p_n(y)$ for all $y\in Y$. Such maps are also called holomorphic. If $F$ is Mackey complete, then $f$ is $C^1_{{\mathbb C}}$ if and only if it is $C^\infty_{{\mathbb C}}$ (see [@BGN Propositions 7.4 and 7.7] or [@GaN Chapter 1] for all of this; cf. [@RES]). For suitable non-Mackey complete $F$, there are $C^n_{{\mathbb C}}$-maps ${{\mathbb C}}\to F$ for all $n\in {{\mathbb N}}$ which are not $C^{n+1}_{{\mathbb C}}$ ([@COM], [@Gro]).
If $f\colon U\to F$ is a map from an open subset of ${{\mathbb K}}$ to a locally convex space, then $f$ is $C^1_{{\mathbb K}}$ in the above sense if and only if the (real, resp. complex) derivative $f^{(1)}(x)=f'(x)=\frac{df}{dx}(x)$ exists for each $x\in U$, and $f'\colon U\to F$ is continuous. Likewise, $f$ is $C^n_{{\mathbb K}}$ if it has continuous derivatives $f^{(k)}\colon U\to F$ for all $k\in {{\mathbb N}}_0$ such that $k\leq n$ (where $f^{(k)}:=(f^{(k-1)})'$). This is easy to see (and spelled out in [@GaN Chapter 1]). If ${{\mathbb K}}={{\mathbb C}}$ here, then complex analyticity of $f$ simply means that $f$ can be expressed in the form $f(z)=\sum_{n=0}^\infty (z-z_0)^na_n$ close to each given point $z_0\in U$, for suitable elements $a_n\in F$.
\[complements\] Consider a map $f\colon U\to F$ from an open set $U{\subseteq}{{\mathbb C}}$ to a Mackey complete locally convex space $F$. Replacing sequential completeness with Mackey completeness in [@BaS Theorem 3.1] and its proof, one finds that also each of the following conditions is equivalent to $f$ being a $C^\infty_{{\mathbb C}}$-map (see also [@Gro Chapter II], notably Theorems 2.2, 2.3 and 5.5):[^3]
- $f$ is weakly holomorphic, i.e. $\lambda\circ f\colon U\to {{\mathbb C}}$ is holomorphic for each $\lambda\in F'$;
- $\int_{\partial\Delta} f(\zeta)\,d\zeta=0$ for each triangle $\Delta{\subseteq}U$;
- $f(z)=\frac{1}{2\pi i}\int_{|\zeta-z_0|=r}\frac{f(\zeta)}{\zeta-z}\,d\zeta$ for each $z_0\in U$, $r>0$ such that $z_0+r{{\mathbb D}}{\subseteq}U$, and each $z$ in the interior of the disk $z_0+r{{\mathbb D}}$.
Given locally convex spaces $E$ and $F$, we let $L(E,F)$ be the vector space of all continuous linear maps $A\colon E\to F$. If ${{\mathcal S}}$ is a set of bounded subsets of $E$, we write $L(E,F)_{{\mathcal S}}$ for $L(E,F)$, equipped with the topology of uniform convergence on the sets $M\in {{\mathcal S}}$. Finite intersections of sets of the form $$\lfloor M,U\rfloor\; :=\;
\{A\in L(E,F)\colon A(M){\subseteq}U\}$$ (for $M\in {{\mathcal S}}$ and $U{\subseteq}F$ a $0$-neighbourhood) form a basis for the filter of $0$-neighbourhoods of this vector topology. See [@BTV Chapter III, §3] for further information. Given $M{\subseteq}E$ and $N{\subseteq}E'$, we write $M^\circ:=\lfloor M,{{\mathbb D}}\rfloor{\subseteq}E'$ and ${}^\circ N:=\{x\in E\colon
(\forall\lambda\in N)\; \lambda(x)\in {{\mathbb D}}\}$ for the polar in $E'$ (resp., in $E$).
If $F$ is Hausdorff and $F\not=\{0\}$, then $L(E,F)_{{\mathcal S}}$ is Hausdorff if and only $\bigcup_{M\in {{\mathcal S}}}M$ is total in $E$, i.e., it spans a dense vector subspace.\
In fact, totality of $\bigcup{{\mathcal S}}$ is sufficient for the Hausdorff property by Proposition 3 in [@BTV Chapter III, §3, no.2]. If $V:=\operatorname{span}_{{\mathbb K}}(\bigcup{{\mathcal S}})$ is not dense in $E$, the Hahn-Banach Theorem provides a linear functional $0\not= \lambda\in E'$ such that $\lambda|_V=0$. We pick $0\not = y\in F$. Then $y{\otimes}\lambda\in
\lfloor M,U\rfloor$ for each $M\in {{\mathcal S}}$ and $0$-neighbourhood $U{\subseteq}F$, whence $y{\otimes}\lambda\in W$ for each $0$-neighbourhood $W{\subseteq}L(E,F)_{{\mathcal S}}$. Since $y{\otimes}\lambda\not=0$, $L(E,F)_{{\mathcal S}}$ is not Hausdorff.
\[prehypo\] Given a separately continuous bilinear map $\beta\colon E_1\times E_2\to F$ and a set ${{\mathcal S}}$ of bounded subsets of $E_2$, consider the following conditions:
- For each $M \in {{\mathcal S}}$ and each $0$-neighbourhood $W {\subseteq}F$, there exists a $0$-neighbourhood $V{\subseteq}E_1$ such that $\beta(V\times M){\subseteq}W$.
- The mapping $\beta^\vee\colon
E_1\to L(E_2,F)_{{\mathcal S}}$, $x{\mapsto}\beta(x,{{\scriptscriptstyle \bullet}})$ is continuous.
- $\beta|_{E_1\times M}\colon E_1\times M\to F$ is continuous, for each $M\in {{\mathcal S}}$.
Then [(a)]{} and [(b)]{} are equivalent, and [(a)]{} implies [(c)]{}. If $$\label{simplf}
(\forall M\in {{\mathcal S}})\;(\exists N\in {{\mathcal S}})\;\;\;
{{\mathbb D}}M{\subseteq}N\,,$$ then all of [(a)–(c)]{} are equivalent.
\[defhypo\] A bilinear map $\beta$ which is separately continuous and satisfies the equivalent conditions (a) and (b) of Proposition \[prehypo\] is called *${{\mathcal S}}$-hypocontinuous* (in the second argument), or simply *hypocontinuous* if ${{\mathcal S}}$ is clear from the context. Hypocontinuity in the first argument with respect to a set of bounded subsets of $E_1$ is defined analogously.
[**Proof of Proposition \[prehypo\].**]{} For the equivalence (a)${\Leftrightarrow}$(b) and the implication (b)${\Rightarrow}$(c), see Proposition 3 and 4 in [@BTV Chapter III, §5, no.3], respectively.\
We now show that (c)${\Rightarrow}$(a) if (\[simplf\]) is satisfied. Given $M\in {{\mathcal S}}$ and $0$-neighbourhood $W{\subseteq}F$, by hypothesis we can find $N\in {{\mathcal S}}$ such that ${{\mathbb D}}M{\subseteq}N$. By continuity of $\beta|_{E_1\times N}$, there exist $0$-neighbourhoods $V$ in $E_1$ and $U$ in $E_2$ such that $\beta(V\times (N\cap U)){\subseteq}W$. Since $M$ is bounded, $M {\subseteq}nU$ for some $n\in {{\mathbb N}}$. Then $\frac{1}{n}M{\subseteq}N\cap U$. Using that $\beta$ is bilinear, we obtain $\beta((\frac{1}{n}V)\times M)=\beta(V \times(\frac{1}{n}M))
{\subseteq}\beta(V\times (N\cap U)){\subseteq}W$.[$\Box$]{}
\[rembalanc\] By Proposition \[prehypo\](b), ${{\mathcal S}}$-hypocontinuity of $\beta\colon E_1\times E_2\to F$ only depends on the topology on $L(E_2,F)_{{\mathcal S}}$, not on ${{\mathcal S}}$ itself. Given ${{\mathcal S}}$, define ${{\mathcal S}}':=\{{{\mathbb D}}M\colon M\in {{\mathcal S}}\}$. Then the topologies on $L(E_2,F)_{{\mathcal S}}$ and $L(E_2,F)_{{{\mathcal S}}'}$ coincide (as is clear), and hence $\beta$ is ${{\mathcal S}}$-hypocontinuous if and only if $\beta$ is ${{\mathcal S}}'$-hypocontinuous. After replacing ${{\mathcal S}}$ with ${{\mathcal S}}'$, we can therefore always assume that (\[simplf\]) is satisfied, whenever this is convenient.
Each continuous bilinear map is hypocontinuous (as condition (a) in Proposition \[prehypo\] is easy to check), but the converse is false. The next proposition compiles various useful facts.
\[somefcts\] Let $\beta\colon E_1\times E_2\to F$ be an ${{\mathcal S}}$-hypocontinuous bilinear map, for some set ${{\mathcal S}}$ of bounded subsets of $E_2$. Then the following holds.
- $\beta(B\times M)$ is bounded in $F$, for each bounded subset $B{\subseteq}E_1$ and each $M\in {{\mathcal S}}$.
- Assume that, for each convergent sequence $(y_n)_{n\in {{\mathbb N}}}$ in $E_2$, with limit $y$, there exists $M\in {{\mathcal S}}$ such that $\{y_n\colon n\in {{\mathbb N}}\}\cup\{y\}{\subseteq}M$. Then $\beta$ is sequentially continuous.
The condition described in [(b)]{} is satisfied, for example, if ${{\mathcal S}}$ is the set of all bounded subsets of $E_2$, or the set of all compact subsets of $E_2$.
\(a) See Proposition 4 in [@BTV Chapter III, §5, no.3].
\(b) See [@Koe p.157, Remark following §40,1.,(5)].
In many cases, separately continuous bilinear maps are automatically hypocontinuous. To make this precise, we recall that a subset $B$ of a locally convex space $E$ is called a *barrel* if it is closed, convex, balanced and absorbing. The space $E$ is called *barrelled* if every barrel is a $0$-neighbourhood. See Proposition 6 in [@BTV Chapter III, §5, no.3] for the following fact:
\[auothypo\] If $\beta \colon E_1\times E_2\to F$ is a separately continuous bilinear map and $E_1$ is barrelled, then $\beta$ is hypocontinuous with respect to any set ${{\mathcal S}}$ of bounded subsets of $E_2$.[$\Box$]{}
A simple fact will be useful.
\[pla\] Let $M$ be a topological space, $F$ be a locally convex space, and $BC(M,F)$ be the space of bounded $F$-valued continuous functions on $M$, equipped with the topology of uniform convergence. Then the evaluation map $\mu\colon BC(M,F)\times M\to F$, $\mu(f,x):=f(x)$ is continuous.
Let $(f_\alpha,x_\alpha)$ be a convergent net in $BC(M,F)\times X$, convergent to $(f,x)$, say. Then $\mu(f_\alpha,x_\alpha)-\mu(f,x)=
(f_\alpha(x_\alpha)-f(x_\alpha))+(f(x_\alpha)-f(x))$, where $f_\alpha(x_\alpha)-f(x_\alpha)\to 0$ as $f_\alpha\to f$ uniformly and $f(x_\alpha)-f(x)\to 0$ as $f$ is continuous.
We now turn to two paradigmatic bilinear maps, namely evaluation and composition.
\[resulteval\] Let $E$ and $F$ be locally convex spaces and ${{\mathcal S}}$ be a set of bounded subsets of $E$ which covers $E$, i.e., $\bigcup_{M\in {{\mathcal S}}}M=E$. Then the evaluation map $${\varepsilon}\colon L(E,F)_{{\mathcal S}}\times E\to F\, ,\quad
{\varepsilon}(A, x)\, :=\, A(x)$$ is hypocontinuous in the second argument with respect to ${{\mathcal S}}$. If $E$ is barrelled, then ${\varepsilon}$ is also hypocontinuous in the first argument, with respect to any locally convex topology ${{\mathcal O}}$ on $L(E,F)$ which is finer than the topology of pointwise convergence, and any set ${{\mathcal T}}$ of bounded subsets of $(L(E,F),{{\mathcal O}})$.
By Remark \[rembalanc\], we may assume that ${{\mathcal S}}$ satisfies (\[simplf\]). Given $A \in L(E,F)$, we have ${\varepsilon}(A,{{\scriptscriptstyle \bullet}})=A$, whence ${\varepsilon}$ is continuous in the second argument. It is also continuous in the first argument, as the topology on $L(E,F)_{{\mathcal S}}$ is finer than the topology of pointwise convergence, by the hypothesis on ${{\mathcal S}}$. Let $M\in {{\mathcal S}}$ now. As $L(E,F)$ is equipped with the topology of uniform convergence on the sets in ${{\mathcal S}}$, the restriction map $\rho\colon L(E,F)\to BC(M,F)$, $A {\mapsto}A|_M$ is continuous. By Lemma \[pla\], the evaluation map $\mu\colon BC(M,F)\times M\to F$ is continuous. Now ${\varepsilon}|_{L(E,F)\times M}=\mu\circ (\rho\times \operatorname{id}_M)$ shows that ${\varepsilon}|_{L(E,F)\times M}$ is continuous. Since we assume (\[simplf\]), the implication “(c)${\Rightarrow}$(a)” in Proposition \[prehypo\] shows that ${\varepsilon}$ is ${{\mathcal S}}$-hypocontinuous.\
Since ${{\mathcal O}}$ is finer than the topology of pointwise convergence, the map ${\varepsilon}$ remains separately continuous in the situation described at the end of the proposition. Hence, if $E$ is barrelled, Proposition \[auothypo\] ensures hypocontinuity with respect to ${{\mathcal T}}$.
While it was sufficient so far to consider an individual set ${{\mathcal S}}$ of bounded subsets of a given locally convex space, we now frequently wish to select such a set ${{\mathcal S}}$ simultaneously for each space. The following definition captures the situations of interest.
\[bsetfun\] A *bounded set functor* is a functor ${{\mathcal S}}$ from the category of locally convex spaces to the category of sets, with the following properties:
- ${{\mathcal S}}(E)$ is a set of bounded subsets of $E$, for each locally convex space $E$.
- If $A\colon E\to F$ is a continuous linear map, then $A(M)\in {{\mathcal S}}(F)$ for each $M\in {{\mathcal S}}(E)$, and ${{\mathcal S}}(A)\colon {{\mathcal S}}(E)\to{{\mathcal S}}(F)$ is the map taking $M\in {{\mathcal S}}(E)$ to its image $A(M)$ under $A$.
Given a bounded set functor ${{\mathcal S}}$ and locally convex spaces $E$ and $F$, we write $L(E,F)_{{\mathcal S}}$ as a shorthand for $L(E,F)_{{{\mathcal S}}(E)}$. Also, an ${{\mathcal S}}(E_2)$-hypocontinuous bilinear map $\beta\colon E_1\times E_2\to F$ will simply be called ${{\mathcal S}}$-hypocontinuous in the second argument.
\[threemain\] Bounded set functors are obtained if ${{\mathcal S}}(E)$ denotes the set of all bounded, (quasi-)compact, or finite subsets of $E$, respectively. We then write $b,c$, resp., $p$ for ${{\mathcal S}}$.
Further examples abound: For instance, we can let ${{\mathcal S}}(E)$ be the set of precompact subsets of $E$, or the set of metrizable compact subsets (if only Hausdorff spaces are considered).
If ${{\mathcal S}}$ is a bounded set functor, and $A\colon E\to F$ a continuous linear map between locally convex spaces, then also its adjoint $A'\colon F'_{{\mathcal S}}\to E'_{{\mathcal S}}$, $\lambda{\mapsto}\lambda\circ A$ is continuous, because $A'(\lfloor A(M),U\rfloor){\subseteq}\lfloor
M,U\rfloor$ for each $M\in {{\mathcal S}}(E)$ and $0$-neighbourhood $U{\subseteq}{{\mathbb K}}$.
The double use of $f'(x)$ (for differentials) and $A'$ (for adjoints) should not cause confusion.
If ${{\mathcal S}}(E)$ contains all finite subsets of $E$, then $\eta_E(x)\colon E'_{{\mathcal S}}\to {{\mathbb K}}$, $\lambda{\mapsto}\lambda(x)$ is continuous for each $x\in E$ and we obtain a linear map $\eta_E\colon E\to (E'_{{\mathcal S}})'$, called the *evaluation homomorphism*. We say that $E$ is *${{\mathcal S}}$-reflexive* if $\eta_E\colon E\to (E'_{{\mathcal S}})'_{{\mathcal S}}$ is an isomorphism of topological vector spaces. If ${{\mathcal S}}=b$, we simply speak of a *reflexive* space; if ${{\mathcal S}}=c$, we speak of a *Pontryagin reflexive* space. Occasionally, we call $E'_b$ the *strong* dual of $E$.
See Proposition 9 in [@BTV Chapter III, §5, no.5] for the following fact in the three cases described in Example \[threemain\]. It might also be deduced from [@Koe §40,5.,(6)].
\[resultcomp\] Let $E$, $F$, and $G$ be locally convex spaces and ${{\mathcal S}}$ be a bounded set functor such that ${{\mathcal S}}(E)$ covers $E$ and $$\label{strnge}
\forall M\in {{\mathcal S}}(L(E,F)_{{\mathcal S}})\;\, \forall N \in {{\mathcal S}}(E)\;\,
\exists K\in {{\mathcal S}}(F)\colon \quad
{\varepsilon}(M\times N){\subseteq}K\,,$$ where ${\varepsilon}\colon L(E,F)\times E\to F$, $(A,x){\mapsto}A(x)$. Then the composition map $$\Gamma\colon L(F,G)_{{\mathcal S}}\times L(E,F)_{{\mathcal S}}\to L(E,G)_{{\mathcal S}}\,,\quad
\Gamma(\alpha,\beta):=\alpha\circ \beta$$ is ${{\mathcal S}}(L(E,F)_{{\mathcal S}})$-hypocontinuous in the second argument.[$\Box$]{}
If ${{\mathcal S}}=b$, then condition (\[strnge\]) is satisfied by Proposition \[somefcts\](a). If ${{\mathcal S}}=p$, then ${\varepsilon}(M\times N)$ is finite and thus (\[strnge\]) holds. If ${{\mathcal S}}=c$, then ${\varepsilon}|_{M\times N}$ is continuous since ${\varepsilon}\colon L(E,F)_{{\mathcal S}}\times E\to F$ is ${{\mathcal S}}(E)$-hypocontinuous by Proposition \[resulteval\]. Hence ${\varepsilon}(M\times N)$ is compact (and thus (\[strnge\]) is satisfied).
[**Proof of Proposition \[resultcomp\].**]{} *$\Gamma$ is continuous in the second argument*: Let $M\in {{\mathcal S}}(E)$, $U{\subseteq}G$ be a $0$-neighbourhood, and $A\in L(F,G)$. Then $A^{-1}(U)$ is a $0$-neighbourhood in $F$. For $B \in L(E,F)$, we have $A(B(M)){\subseteq}U$ if and only if $B(M){\subseteq}A^{-1}(U)$, showing that $\Gamma(A,\lfloor M,A^{-1}(U)\rfloor)
{\subseteq}\lfloor M,U\rfloor$. Hence, $\Gamma(A,{{\scriptscriptstyle \bullet}})$ being linear, it is continuous.\
*Continuity in the first argument*: Let $U{\subseteq}G$ be a $0$-neighbourhood, $B \in L(E,F)$ and $M\in {{\mathcal S}}(E)$. Then $B(M)\in {{\mathcal S}}(F)$ by Definition \[bsetfun\](b) and $\Gamma(\lfloor B(M),U\rfloor,B){\subseteq}\lfloor
M,U\rfloor$.\
To complete the proof, let $M\in {{\mathcal S}}(L(E,F)_{{\mathcal S}})$, $U{\subseteq}G$ be a $0$-neighbourhood and $N\in {{\mathcal S}}(E)$. By (\[strnge\]), there exists $K\in {{\mathcal S}}(F)$ such that ${\varepsilon}(M\times N){\subseteq}K$. Note that for all $B\in M$ and $A\in \lfloor K ,U\rfloor$, we have $\Gamma(A,B).N=(A\circ B)(N)
=A(B(N)){\subseteq}A(K){\subseteq}U$. Thus $\Gamma(\lfloor K ,U\rfloor\times M){\subseteq}\lfloor N,U\rfloor$. Since $\lfloor K,U\rfloor$ is a $0$-neighbourhood in $L(F,G)_{{\mathcal S}}$, condition (a) of Proposition \[prehypo\] is satisfied.[$\Box$]{}
\[nachtrg\] Despite the hypocontinuity of the composition map $\Gamma$, it is discontinuous in the situations specified in the introduction. To see this, pick $\lambda\in E'$ and $x\in E$ as described in the introduction. Let $0\not=z\in G$ and give $F'$ the topology induced by $F'\to L(F,G)$, $\zeta{\mapsto}z{\otimes}\zeta$. There is $\mu\in G'$ such that $\mu(z)\not=0$. If $\Gamma$ was continuous, then also the following map would be continuous: $F'\times F\to {{\mathbb K}}$, $(\zeta, y){\mapsto}\mu(\Gamma(z{\otimes}\zeta, y{\otimes}\lambda)(x))
=\mu(z)\lambda(x)\zeta(y)$. But this map is a non-zero multiple of the evaluation map and hence discontinuous [@Mai].
Differentiability properties of compositions with\
hypocontinuous bilinear mappings {#mainsec}
==================================================
In this section, we introduce a new class of topological spaces (“$k^\infty$-spaces”). We then discuss compositions of hypocontinuous bilinear maps with $C^n$-maps on open subsets of locally convex spaces which are $k^\infty$-spaces.\
Recall that a Hausdorff topological space $X$ is called a *$k$-space* if, for every subset $A{\subseteq}X$, the set $A$ is closed in $X$ if and only if $A\cap K$ is closed in $K$ for each compact subset $K{\subseteq}X$. Equivalently, a subset $U{\subseteq}X$ is open in $X$ if and only if $U\cap K$ is open in $K$ for each compact subset $K{\subseteq}X$. It is clear that closed subsets, as well as open subsets of $k$-spaces are again $k$-spaces when equipped with the induced topology. If $X$ is a $k$-space, then a map $f\colon X\to Y$ to a topological space $Y$ is continuous if and only if $f|_K\colon K\to Y$ is continuous for each compact subset $K{\subseteq}X$, as is easy to see.[^4] This property is crucial for the following. It can also be interpreted as follows: $X={{\displaystyle \lim_{\longrightarrow}}}_K\, K$ as a topological space.
\[defkinfty\] We say that a topological space $X$ is a *$k^\infty$-space* if it is Hausdorff and its $n$-fold power $X^n=X\times\cdots \times X$ is a $k$-space, for each $n\in {{\mathbb N}}$.
\[exmetr\] It is well known (and easy to prove) that every metrizable topological space is a $k$-space. Finite powers of metrizable spaces being metrizable, we see: *Every metrizable topological space is a $k^\infty$-space.*
\[komeg\] A Hausdorff topological space $X$ is called a *$k_\omega$-space* if it is a $k$-space and *hemicompact*,[^5] i.e., there exists a sequence $K_1{\subseteq}K_2{\subseteq}\cdots$ of compact subsets of $X$ such that $X=\bigcup_{n\in {{\mathbb N}}}K_n$ and each compact subset of $X$ is contained in some $K_n$. Since finite products of $k_\omega$-spaces are $k_\omega$-spaces (see, e.g., [@GGH Proposition 4.2(c)]), it follows that each $k_\omega$-space is a $k^\infty$-space. For an introduction to $k_\omega$-spaces, the reader may consult [@GGH].
\[supplykom\] We remark that $E'_c$ is a $k_\omega$-space (and hence a $k^\infty$-space), for each metrizable locally convex space $E$ (see [@Aus Corollary 4.7 and Proposition 5.5]). In particular, every Silva space $E$ is a $k_\omega$-space (and hence a $k^\infty$-space), i.e., every locally convex direct limit $E={{\displaystyle \lim_{\longrightarrow}}}\,E_n$ of an ascending sequence $E_1{\subseteq}E_2{\subseteq}\cdots$ of Banach spaces, such that the inclusion maps $E_n\to E_{n+1}$ are compact operators (see [@DL3 Example 9.4]).
Having set up the terminology, let us record a simple, but useful observation.
\[mainobs\] Let $n\in {{\mathbb N}}_0\cup\{\infty\}$. If $n=0$, let $U=X$ be a topological space. If $n\geq 1$, let $X$ be a locally convex space and $U{\subseteq}X$ be an open subset. Let $\beta\colon E_1\times E_2\to F$ be a bilinear map and $f\colon U\to E_1\times E_2$ be a $C^n$-map. Assume that at least one of [(a), (b)]{} holds:
- $X$ is metrizable and $\beta$ is sequentially continuous; or:
- $X$ is a $k^\infty$-space and $\beta$ is hypocontinuous in the second argument with respect to a set ${{\mathcal S}}$ of bounded subsets of $E_2$ which contains all compact subsets of $E_2$.
Then $\beta\circ f\colon U\to F$ is $C^n$.
It suffices to consider the case where $n<\infty$. The proof is by induction.\
We assume (a) first. If $n=0$, let $(x_k)_{k\in {{\mathbb N}}}$ be a convergent sequence in $U$, with limit $x$. Then $f(x_k)\to x$ by continuity of $f$ and hence $\beta(f(x_k))\to \beta(f(x))$, since $\beta$ is sequentially continuous.\
Now let $n\geq 1$ and assume that the assertion holds if $n$ is replaced with $n-1$. Given $x\in U$ and $y\in X$, let $(t_k)_{k\in {{\mathbb N}}}$ be a sequence in ${{\mathbb K}}\setminus \{0\}$ such that $x+t_ky\in U$ for each $k\in {{\mathbb N}}$, and $\lim_{k\to\infty}\,t_k=0$. Write $f=(f_1,f_2)$ with $f_j\colon U\to E_j$. Then $$\begin{aligned}
\hspace*{-2mm}\lefteqn{\frac{\beta(f(x+t_ky))-\beta(f(x))}{t_k}}\qquad\\
&=&
\beta\left(\frac{f_1(x+t_ky)-f_1(x)}{t_k},f_2(x+t_ky)\right)
+
\beta\left(f_1(x),\frac{f_2(x+t_ky)-f_2(x)}{t_k}\right)\\
&\to & \beta(df_1(x,y),f_2(x))+\beta(f_1(x),df_2(x,y))\quad
\mbox{as $k\to\infty$,}\end{aligned}$$ by continuity of $f$ and sequential continuity of $\beta$. Hence the limit $d(\beta\circ f)(x,y)=
{\displaystyle \lim_{t\to 0}}\,
\frac{\beta(f(x+ty))-\beta(f(x))}{t}$ exists, and is given by $$\label{prodrul}
d(\beta\circ f)(x,y)\;=\;
\beta(df_1(x,y),f_2(x))+\beta(f_1(x),df_2(x,y))\,.$$ The mappings $g_1, g_2 \colon U\times X \to E_1\times E_2$ defined via $g_1(x,y):=(df_1(x,y),f_2(x))$ and $g_2(x,y):=(f_1(x),df_2(x,y))$ are $C^{n-1}_{{\mathbb K}}$. Since $$\label{abstrprod}
d(\beta\circ f)\; =\; \beta\circ g_1+\beta\circ g_2$$ by (\[prodrul\]), we deduce from the inductive hypotheses that $d(\beta\circ f)$ is $C^{n-1}_{{\mathbb K}}$ and hence continuous. Thus $\beta\circ f$ is $C^1_{{\mathbb K}}$ with $d(\beta\circ f)$ a $C^{n-1}_{{\mathbb K}}$-map and hence $\beta\circ f$ is $C^n_{{\mathbb K}}$, which completes the inductive proof in the situation of (a).\
In the situation of (b), let $K{\subseteq}U$ be compact. Then $f_2(K){\subseteq}E_2$ is compact and hence $f_2(K)\in{{\mathcal S}}$, by hypothesis. Since $\beta|_{E_1\times f_2(K)}$ is continuous by Proposition \[prehypo\](c), we see that $(\beta\circ f)|_K=\beta|_{E_1\times f_2(K)}\circ f|_K$ is continuous. Since $X$ and hence also its open subset $U$ is a $k$-space, it follows that $\beta\circ f$ is continuous, settling the case $n=0$.\
Now let $n\geq 1$ and assume that the assertion holds if $n$ is replaced with $n-1$. Since $\beta$ is sequentially continuous by Proposition \[somefcts\](b), we see as in case (a) that the directional derivative $d(\beta\circ f)(x,y)$ exists, for all $(x,y)\in U\times X$, and that $d(\beta\circ f)$ is given by (\[abstrprod\]). Since $g_1$ and $g_2$ are $C^{n-1}_{{\mathbb K}}$, the inductive hypothesis can be applied to the summands in (\[abstrprod\]). Thus $d(\beta\circ f)$ is $C^{n-1}_{{\mathbb K}}$, whence $\beta\circ f$ is $C^1_{{\mathbb K}}$ with $d(\beta\circ f)$ a $C^{n-1}_{{\mathbb K}}$-map, and so $\beta\circ f$ is $C^n_{{\mathbb K}}$.
Combining Proposition \[resultcomp\] and Theorem \[mainobs\], as a first application we obtain an affirmative answer to an open question formulated by Serge Lang [@Lan p.8, Remark].
\[corlang\] Let $U$ be an open subset of a Fréchet space, $E$, $F$ and $G$ be Fréchet spaces, and $f\colon U\to L(E,F)_b$, $g\colon U \to L(F,G)_b$ be continuous maps. Then also the mapping$U\to L(E, G)_b$, $x{\mapsto}g(x)\circ f(x)$ is continuous.[$\Box$]{}
Holomorphic families of operators {#secholfam}
=================================
In this section, we compile conclusions from the previous results and some useful additional material. Specializing to the case $X:={{\mathbb K}}:={{\mathbb C}}$ and $n:=\infty$, we obtain results concerning holomorphic families of operators, i.e., holomorphic maps $U\to L(E,F)_{{\mathcal S}}$, where $U$ is an open subset of ${{\mathbb C}}$. Among other things, such holomorphic families are of interest for representation theory and cohomology (see [@BaO] and [@BO2]).
\[cor2\] Let $X$, $E$, $F$ and $G$ be locally convex spaces over ${{\mathbb K}}$, such that $X$ is a $k^\infty$-space $($e.g. $X={{\mathbb K}}={{\mathbb C}})$. Let $U{\subseteq}X$ be open, $n\in {{\mathbb N}}_0\cup\{\infty\}$ and $f\colon U\to L(E,F)_{{\mathcal S}}$ as well as $g\colon U\to L(F,G)_{{\mathcal S}}$ be $C^n_{{\mathbb K}}$-maps, where ${{\mathcal S}}=b$ or ${{\mathcal S}}=c$. Then also the map $U\to L(E, G)_{{\mathcal S}}$, $z{\mapsto}g(z)\circ f(z)$ is $C^n_{{\mathbb K}}$.
The composition map $L(F,G)_{{\mathcal S}}\times L(E,F)_{{\mathcal S}}\to L(E,G)_{{\mathcal S}}$ is hypocontinuous with respect to ${{\mathcal S}}(L(E,F)_{{\mathcal S}})$, by Proposition \[resultcomp\]. Hence Theorem \[mainobs\](b) applies.
The remainder of this section is devoted to the proof of the following result. Here ${{\mathcal S}}$ is a bounded set functor such that ${{\mathcal S}}(E)$ covers $E$, for each locally convex space $E$.
\[cor1\] Let $E$, $F$ and $X$ be locally convex spaces over ${{\mathbb K}}$. If $\eta_F\colon F\to (F_{{\mathcal S}}')'_{{\mathcal S}}$ is continuous, then $g\colon U\to L(F'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}$, $z{\mapsto}f(z)'$ is $C^n_{{\mathbb K}}$, for each $n\in {{\mathbb N}}_0\cup\{\infty\}$ and $C^n_{{\mathbb K}}$-map $f\colon U\to L(E,F)_{{\mathcal S}}$ on an open subset $U{\subseteq}X$.
The proof of Proposition \[cor1\] exploits the continuity of the formation of adjoints.
\[adj\] Let $E$, $F$ be locally convex spaces and ${{\mathcal S}}$ be a bounded set functor such that ${{\mathcal S}}(F)$ covers $F$. If the evaluation homomorphism $\eta_F\colon F\to (F'_{{\mathcal S}})'_{{\mathcal S}}$ is continuous, then $$\label{dfnPsi}
\Psi\colon L(E,F)_{{\mathcal S}}\to L(F'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}\,,
\quad
\alpha{\mapsto}\alpha'$$ is a continuous linear map.
After replacing ${{\mathcal S}}(V)$ with $\{r_1M_1\cup\cdots\cup r_nM_n\colon r_1,\ldots,r_n\in {{\mathbb K}},
M_1,\ldots,M_n\in {{\mathcal S}}(V)\}$ for each locally convex space $V$ (which does not change ${{\mathcal S}}$-topologies), we may assume that ${{\mathcal S}}(E)$ is closed under finite unions and multiplication with scalars. Let $M\in {{\mathcal S}}(F'_{{\mathcal S}})$ and $U{\subseteq}E'_{{\mathcal S}}$ be a $0$-neighbourhood; we have to show that $\Psi^{-1}(\lfloor M,U\rfloor)$ is a $0$-neighbourhood in $L(E,F)_{{\mathcal S}}$. After shrinking $U$, without loss of generality $U=N^\circ$ for some $N\in {{\mathcal S}}(E)$ (by our special hypothesis concerning ${{\mathcal S}}(E)$). For $\alpha\in L(E,F)$, we have $$\begin{aligned}
\alpha'\in\lfloor M,U\rfloor & {\Leftrightarrow}& (\forall \lambda\in M)\;\;
\lambda\circ\alpha=\alpha'(\lambda)\in U=N^\circ \\
&{\Leftrightarrow}& (\forall \lambda\in M)\,(\forall x\in N)\;\;
|\lambda (\alpha(x))|\leq 1\\
&{\Leftrightarrow}& \alpha(N){\subseteq}{}^\circ M\\
&{\Leftrightarrow}& \alpha\in\lfloor N, {}^\circ M\rfloor.\end{aligned}$$ Since $M\in {{\mathcal S}}(F'_{{\mathcal S}})$ and $\eta_F$ is continuous, the polar ${}^\circ M=\eta_F^{-1}(M^\circ)$ is a $0$-neighbourhood in $F$. Thus $\lfloor N,{}^\circ M \rfloor =\Psi^{-1}(\lfloor M,U\rfloor)$ is a $0$-neighbourhood in $L(E,F)_{{\mathcal S}}$.
[**Proof of Proposition \[cor1\].**]{} Since $f$ is a $C^n_{{\mathbb K}}$-map and $\Psi$ in Proposition \[adj\] is continuous linear and hence a $C^\infty_{{\mathbb K}}$-map, also $g=\Psi\circ f$ is $C^n_{{\mathbb K}}$.[$\Box$]{}
The locally convex spaces $E$ such that $\eta_E\colon E\to (E_b')'_b$ is continuous are known as “quasi-barrelled” spaces. They can characterized easily. Recall that a subset $A$ of a locally convex space $E$ is called *bornivorous* if it absorbs all bounded subsets of $E$. The space $E$ is called *bornological* if every convex, balanced, bornivorous subset of $E$ is a $0$-neighbourhood. See Proposition 2 in [@Jar §11.2] (and the lines following it) for the following simple fact:\
*The evaluation homomorphism $\eta_E\colon E\to (E'_b)'_b$ is continuous if and only if each closed, convex, balanced subset $A{\subseteq}E$ which absorbs all bounded subsets of $E$ $($i.e., each bornivorous barrel $A)$ is a $0$-neighbourhood in $E$.*\
Thus $\eta_E\colon E\to (E'_b)'_b$ is continuous if $E$ is bornological or barrelled. It is also known that $\eta_E\colon E\to (E'_c)'_c$ is continuous if $E$ is a $k$-space (cf. [@Aus Corollary 5.12 and Proposition 5.5]).\
We mention that Proposition \[cor2\] generalizes [@BaO Lemma 2.4], where $X={{\mathbb K}}={{\mathbb C}}$, ${{\mathcal S}}=b$, $E$ is assumed to be a Montel space and $E$, $F$ $G$ are complete (see last line of [@BaO p.637]). The method of proof used in *loc.cit.*depends on completeness properties of $L(E,G)_b$, because the characterization of holomorphic functions via Cauchy integrals (as in Remark \[complements\](c)) requires Mackey completeness. Proposition \[cor1\] generalizes [@BaO Lemma 2.3], where $X={{\mathbb K}}={{\mathbb C}}$, ${{\mathcal S}}=b$ and continuity of $\eta_F$ is presumed as well (penultimate sentence of their proof), and whose proof is valid whenever $L(F',E')_b$ is at least Mackey complete (since only weak holomorphicity of $g$ is checked there).
Locally convex Poisson vector spaces {#secpoisson}
====================================
We now consider locally convex Poisson vector spaces in a framework which arose from [@GLN]. Fundamental facts concerning such spaces will be proved, using hypocontinuity as a tool.
\[thesett\] Throughout this section, we let ${{\mathcal S}}$ be a bounded set functor such that the following holds for each locally convex space $E$:
- ${{\mathcal S}}(E)$ contains all compact subsets of $E$; and:
- For each $M\in {{\mathcal S}}(E'_{{\mathcal S}})$ and $N\in {{\mathcal S}}(E)$, the set ${\varepsilon}(M\times N){\subseteq}{{\mathbb K}}$ is bounded, where ${\varepsilon}\colon E'\times E\to{{\mathbb K}}$ is the evaluation map.
Since ${{\mathcal S}}({{\mathbb K}})$ contains all compact sets and each bounded subset of ${{\mathbb K}}$ is contained in a compact set, condition (b) means that there exists $K\in {{\mathcal S}}({{\mathbb K}})$ such that ${\varepsilon}(M\times N){\subseteq}K$.
\[defnposp\] An *${{\mathcal S}}$-reflexive locally convex Poisson vector space* is a locally convex space $E$ which is ${{\mathcal S}}$-reflexive and a $k^\infty$-space, together with an ${{\mathcal S}}$-hypocontinuous bilinear map $[.,.] \colon E'_{{\mathcal S}}\times E'_{{\mathcal S}}\to E'_{{\mathcal S}}$, $(\lambda,\eta){\mapsto}[\lambda,\eta]$ which makes $E'_{{\mathcal S}}$ a Lie algebra.
Of course, we are mostly interested in the case where $[.,.]$ is continuous, but only ${{\mathcal S}}$-hypocontinuity is required for the basic results described below.
\[remtypi\] We mainly have two choices of ${{\mathcal S}}$ in mind.
- The case ${{\mathcal S}}=b$. If $E$ is a Hilbert space, a reflexive Banach space, a nuclear Fréchet space, or the strong dual of a nuclear Fréchet space, then both reflexivity is satisfied and also the $k^\infty$-property (by Example \[exmetr\] and Remark \[supplykom\]).[^6]
- If ${{\mathcal S}}=c$, then the scope widens considerably. For example, every Fréchet space $E$ is both Pontryagin reflexive (see [@Ban Propositions 15.2 and 2.3]) and a $k^\infty$-space (see Example \[exmetr\]), and the same holds for $E'_c$ (see [@Aus Proposition 5.9] and Remark \[supplykom\]).
\[remtypl\] The most typical examples of ${{\mathcal S}}$-reflexive locally convex Poisson vector spaces are dual spaces of topological Lie algebras. More precisely, let ${{\mathcal S}}=b$ or ${{\mathcal S}}=c$, and $({{\mathfrak g}},[.,.]_{{\mathfrak g}})$ be a locally convex topological Lie algebra. If ${{\mathfrak g}}$ is ${{\mathcal S}}$-reflexive and ${{\mathfrak g}}'_{{\mathcal S}}$ happens to be a $k^\infty$-space, then $E:={{\mathfrak g}}'_{{\mathcal S}}$ is an ${{\mathcal S}}$-reflexive locally convex Poisson vector space with Lie bracket defined via $[\lambda,\mu]:=[\eta_{{\mathfrak g}}^{-1}(\lambda),
\eta_{{\mathfrak g}}^{-1}(\mu)]_{{\mathfrak g}}$ for $\lambda, \mu\in E'=({{\mathfrak g}}'_{{\mathcal S}})'$, using the isomorphism $\eta_{{\mathfrak g}}\colon {{\mathfrak g}}\to ({{\mathfrak g}}'_{{\mathcal S}})_{{\mathcal S}}'$. Here are typical examples.
- If ${{\mathfrak g}}$ is a Banach-Lie algebra whose underlying Banach space is reflexive, then ${{\mathfrak g}}'_b$ is a *reflexive* locally convex Poisson vector space (i.e., w.r.t. ${{\mathcal S}}=b$); see Remark \[remtypi\](a).
- If ${{\mathfrak g}}$ is a Fréchet-Lie algebra (a topological Lie algebra which is a Fréchet space), then ${{\mathfrak g}}'_c$ is a *Pontryagin reflexive* locally convex Poisson vector space (i.e., with respect to ${{\mathcal S}}=c$), by Remark \[remtypi\](b).
- If ${{\mathfrak g}}$ is a Silva-Lie algebra, then ${{\mathfrak g}}$ is reflexive (hence also Pontryagin reflexive), and ${{\mathfrak g}}_b'={{\mathfrak g}}'_c$ is a Fréchet-Schwartz space (see [@Flo]) and hence a $k^\infty$-space. Therefore ${{\mathfrak g}}'_b={{\mathfrak g}}'_c$ is a reflexive and Pontryagin reflexive locally convex Poisson vector space.
If a topological group $G$ is a projective limit ${{\displaystyle \lim_{\longleftarrow}}}\,G_n$ of a projective sequence of finite-dimensional Lie groups, then ${{\mathfrak g}}:={{\displaystyle \lim_{\longleftarrow}}}\,L(G_n){\cong}{{\mathbb R}}^{{\mathbb N}}$ can be considered as the Lie algebra of $G$ and coadjoint orbits of $G$ in ${{\mathfrak g}}'$ can be studied [@Ne2], where ${{\mathfrak g}}'_c$ ($={{\mathfrak g}}'_b$) is a Pontryagin reflexive (and reflexive) locally convex Poisson vector space, by (b).\
If a group $G$ is a union $\bigcup_{n\in {{\mathbb N}}}\, G_n$ of finite-dimensional Lie groups $G_1{\subseteq}G_2{\subseteq}\cdots$, then $G$ can be made an infinite-dimensional Lie group with Lie algebra ${{\mathfrak g}}={{\displaystyle \lim_{\longrightarrow}}}\, L(G_n){\cong}{{\mathbb R}}^{({{\mathbb N}})}$ (see [@FUN]), where ${{\mathfrak g}}'_c$ ($={{\mathfrak g}}'_b$) is a Pontryagin reflexive (and reflexive) locally convex Poisson vector space, by (c). Again coadjoint orbits can be studied [@GLN]. Manifold structures on them do not pose problems, since all homogeneous spaces of $G$ are manifolds [@FUN Proposition 7.5].
Given a Lie algebra $({{\mathfrak g}},[.,.])$ and $x\in {{\mathfrak g}}$, we write $\operatorname{ad}_x:=\operatorname{ad}(x):=[x,.]\colon
{{\mathfrak g}}\to{{\mathfrak g}}$, $y{\mapsto}[x,y]$.\
Definition \[dfnpoivec\] can be adapted to spaces which are not ${{\mathcal S}}$-reflexive, along the lines of [@OR1], [@OR2]:
\[nonrcase\] A *locally convex Poisson vector space* with respect to ${{\mathcal S}}$ is a locally convex space $E$ that is a $k^\infty$-space and whose evaluation homomorphism $\eta_E\colon E\to (E'_{{\mathcal S}})'_{{\mathcal S}}$ is a topological embedding, together with an ${{\mathcal S}}$-hypocontinuous bilinear map $(\lambda,\eta){\mapsto}[\lambda,\eta]$ which makes $E'_{{\mathcal S}}$ a Lie algebra, and such that $$\label{compcond}
\eta_E(x)\circ \operatorname{ad}_\lambda\; \in \; \eta_E(E)\quad
\mbox{for all $x\in E$ and $\lambda\in E'$.}$$
Identifying $E$ with $\eta_E(E){\subseteq}(E'_{{\mathcal S}})_{{\mathcal S}}'$, we can rewrite (\[compcond\]) as $$\label{compcond2}
(\operatorname{ad}_\lambda)'(E)\; {\subseteq}\; E\quad
\mbox{for all $\lambda\in E'$.}$$
Every ${{\mathcal S}}$-reflexive Poisson vector space $(E,[.,.])$ in the sense of Definition \[defnposp\] also is a Poisson vector space with respect to ${{\mathcal S}}$, in the sense of Definition \[nonrcase\]. In fact, since $[.,.]$ is separately continuous, the linear map $\operatorname{ad}_\lambda=[\lambda ,.]\colon E'_{{\mathcal S}}\to E'_{{\mathcal S}}$ is continuous, for each $\lambda \in E'_{{\mathcal S}}$. Hence $\alpha\circ \operatorname{ad}_\lambda \in (E_{{\mathcal S}}')'=\eta_E(E)$ for each $\alpha \in (E_{{\mathcal S}}')'$, and thus (\[compcond\]) is satisfied.
If ${{\mathcal S}}=b$, then $\eta_E\colon E\to (E'_b)'_b$ is a topological embedding if and only $\eta_E$ is continuous, i.e., if and only if $E$ is quasi-barrelled in the sense recalled in Section \[secholfam\] (see [@Jar §11.2]). Most locally convex spaces of practical interest are bornological or barrelled and hence quasi-barrelled.\
If ${{\mathcal S}}=c$, then $\eta_E$ is a topological embedding *automatically* in the situation of Definition \[nonrcase\] as we assume that $E$ is a $k^\infty$-space (and hence a $k$-space). In fact, $\eta_E\colon E\to (E'_c)'_c$ is injective (by the Hahn-Banach theorem) for each locally convex space $E$, and open onto its image (cf. [@Aus Proposition 6.10] or [@Ban Lemma 14.3]). Hence $\eta_E\colon E\to (E'_c)'_c$ is an embedding if and only if it is continuous, which holds if $E$ is a $k$-space (cf. [@Ban Lemma 14.4]).
Since reflexive Banach spaces are rather rare, the more complicated non-reflexive theory cannot be avoided in the study of Banach-Lie-Poisson vector spaces (as in [@OR1], [@OR2]). By contrast, typical non-Banach locally convex spaces are reflexive and hence fall within the simple, basic framework of Definition \[defnposp\]. And the class of Pontryagin reflexive spaces is even more comprehensive.
\[dfnpoivec\] Let $(E,[.,.])$ be a locally convex Poisson vector space with respect to ${{\mathcal S}}$, and $U{\subseteq}E$ be open. Given $f,g\in C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$, we define a function $\{f,g\}\colon U\to{{\mathbb K}}$ via $$\label{poissonbr}
\{f,g\}(x)\; :=\; \langle [f'(x),g'(x)], x\rangle\quad
\mbox{for $x\in U$,}$$ where $\langle .,.\rangle\colon E'\times E\to {{\mathbb K}}$, $\langle \lambda,x\rangle :=\lambda(x)$ is the evaluation map and $f'(x)=df(x,.)$.\
Condition (\[compcond\]) in Definition \[nonrcase\] enables us to define a map $X_f\colon U \to E$ via $$\label{defhamil}
X_f(x)\, :=\, \eta_E^{-1}\bigl(\eta_E(x)
\circ \operatorname{ad}(f'(x))\bigr)\quad
\mbox{for $\,x\in U$,}$$ where $\eta_E\colon E\to (E'_{{\mathcal S}})'_{{\mathcal S}}$ is the evaluation homomorphism.
\[mainappoi\] Let $(E,[.,.])$ be a locally convex Poisson vector space with respect to ${{\mathcal S}}$ and $U{\subseteq}E$ be an open subset. Then
- $\{f,g\}\in C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$, for all $f,g\in C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$.
- For each $f\in C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$, the map $X_f\colon U \to E$ is $C^\infty_{{\mathbb K}}$.
The following fact will help us to prove Theorem \[mainappoi\].
\[frombdls\] Let $E$ and $F$ be locally convex spaces, $U{\subseteq}E$ be open and $f\colon U\to F$ be a $C^\infty_{{\mathbb K}}$-map. Then also the map $f'\colon U\to L(E,F)_{{\mathcal S}}$, $x{\mapsto}f'(x)=df(x,{{\scriptscriptstyle \bullet}})$ is $C^\infty_{{\mathbb K}}$, for eachset ${{\mathcal S}}$ of bounded subsets of $E$ such that $L(E,F)_{{\mathcal S}}$ is Hausdorff.
For ${{\mathcal S}}=b$, see [@HYP]. The general case is a trivial consequence of the case ${{\mathcal S}}=b$.
[**Proof of Theorem \[mainappoi\].**]{} (a) The maps $f'\colon U\to L(E,{{\mathbb K}})_{{\mathcal S}}=E'_{{\mathcal S}}$ and $g'\colon U\to E'_{{\mathcal S}}$ are $C^\infty_{{\mathbb K}}$ by Lemma \[frombdls\], $E$ is a $k^\infty$-space by hypothesis, and $[.,.]$ is ${{\mathcal S}}$-hypocontinuous. Hence $h:= [.,.]\circ (f',g')\colon U\to E'_{{\mathcal S}}$, $x{\mapsto}[f'(x),g'(x)]$ is $C^\infty_{{\mathbb K}}$, by Theorem \[mainobs\](b). The evaluation map ${\varepsilon}\colon E'_{{\mathcal S}}\times E\to {{\mathbb K}}$ is ${{\mathcal S}}$-hypocontinuous in the second argument by Proposition \[resulteval\], and the inclusion map $\iota\colon U\to E$ is $C^\infty_{{\mathbb K}}$. Hence $\{f,g\}={\varepsilon}\circ (h,\iota)$ is $C^\infty_{{\mathbb K}}$, by Theorem \[mainobs\](b).
\(b) Since the bilinear map $[.,.]\colon E'_{{\mathcal S}}\times E'_{{\mathcal S}}\to E'_{{\mathcal S}}$ is ${{\mathcal S}}$-hypocontinuous, the linear map $[.,.]^\vee\colon E'_{{\mathcal S}}\to L(E'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}$, $\lambda{\mapsto}[\lambda ,.]=\operatorname{ad}(\lambda)$ is continuous, by Proposition \[prehypo\](b). Hence $h\colon U \to L(E'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}$, $h(x):=
\operatorname{ad}(f'(x))$ is $C^\infty_{{\mathbb K}}$, using Lemma \[frombdls\]. The composition map $\Gamma\colon (E'_{{\mathcal S}})'_{{\mathcal S}}\times L(E'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}\to
(E'_{{\mathcal S}})'_{{\mathcal S}}$, $(\alpha ,A){\mapsto}\alpha\circ A$ is ${{\mathcal S}}$-hypocontinuous in the second argument, because condition (b) in §\[thesett\] ensures that Proposition \[resultcomp\] can be applied. Then $V:=\{A\in L(E'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}\colon \mbox{$(\forall
x\in E)\; \eta_E(x)\circ A\in \eta_E(E)$}\}$ is a vector subspace of $L(E'_{{\mathcal S}},E'_{{\mathcal S}})_{{\mathcal S}}$ and the bilinear map $\Theta\colon E\times V\to E$, $\Theta(x,A):=\eta_E^{-1}(\Gamma(\eta_E(x),A))$ is ${{\mathcal S}}$-hypocontinuous in its second argument, using that $\eta_E$ is an isomorphism of topological vector spaces onto its image. The inclusion map $\iota\colon U\to E$, $x{\mapsto}x$ being $C^\infty_{{\mathbb K}}$, Theorem \[mainobs\] shows that $X_f=\Theta \circ (\iota ,h)$ is $C^\infty_{{\mathbb K}}$.[$\Box$]{}
\[defhampost\] Let $(E,[.,.])$ be a locally convex Poisson vector space with respect to ${{\mathcal S}}$, and $U{\subseteq}E$ be an open subset.
- The map $\{.,.\}\colon C^\infty_{{\mathbb K}}(U,{{\mathbb K}})\times
C^\infty_{{\mathbb K}}(U,{{\mathbb K}})\to C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$ taking $(f,g)$ to $\{f,g\}$ (as in Definition \[dfnpoivec\]) is called the *Poisson bracket*.
- Given $f\in C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$, the map $X_f\colon U\to E$ is a smooth vector field on $U$, by Theorem \[mainappoi\](b). It is called the *Hamiltonian vector field* associated with $f$.
\[ispbr\] Basic differentiation rules entail that $\{.,.\}$ makes $C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$ a Poisson algebra, i.e., $(C^\infty_{{\mathbb K}}(U,{{\mathbb K}}),\{.,.\})$ is a Lie algebra[^7] and $\{f,.\}\colon$ $C^\infty_{{\mathbb K}}(U,{{\mathbb K}})\to
C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$, $g{\mapsto}\{f,g\}$ is a derivation for the commutative, associative ${{\mathbb K}}$-algebra $(C^\infty_{{\mathbb K}}(U,{{\mathbb K}}),\cdot)$, for each $f\in C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$.
Continuity properties of the Poisson bracket {#nwsec1}
============================================
If $E$ and $F$ are locally convex spaces, $U{\subseteq}E$ is an open set and $n\in {{\mathbb N}}_0\cup\{\infty\}$, then $C^n_{{\mathbb K}}(U,F)$ carries a natural topology (the “$C^n$-topology”), namely the initial topology with respect to the maps $$C^\infty_{{\mathbb K}}(U,F)\to C(U\times E^k,F)_{c.o.}\,\quad
f {\mapsto}d^kf$$ for $k\in {{\mathbb N}}_0$ such that $k\leq n$, where the right hand side is equipped with the compact-open topology, $d^0f:=f$ and $d^kf(x,y_1,\ldots, y_k):=
(D_{y_k}\cdots D_{y_1}f)(x)$ is defined as an iterated directional derivative, if $k\geq 1$. Our goal is the following result:
\[poiss2\] Let $(E,[.,.])$ be a locally convex Poisson vector space with respect to ${{\mathcal S}}=c$. Let $U{\subseteq}E$ be open. Then the Poisson bracket $$\{.,.\}\colon C^\infty_{{\mathbb K}}(U,{{\mathbb K}})\times
C^\infty_{{\mathbb K}}(U,{{\mathbb K}})\to C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$$ is hypocontinuous with respect to compact subsets of $C^\infty_{{\mathbb K}}(U,{{\mathbb K}})$. If $[.,.]\colon E'_c\times E'_c\to E'_c$ is continuous, then also the Poisson bracket is continuous.
The topology on spaces of smooth maps goes along well with the ${{\mathcal S}}$-topology on spaces of operators if ${{\mathcal S}}=c$, since it enables to control smooth functions and their differentials on compact sets. For ${{\mathcal S}}\not=c$ (notably, for ${{\mathcal S}}=b$), there is no clear connection between the topologies, and one cannot hope for an analogue of Theorem \[poiss2\].
Various auxiliary results are needed to prove Theorem \[poiss2\]. With little risk of confusion with subsets of spaces of operators, given a $0$-neighbourhood $W{\subseteq}F$ and a compact set $K{\subseteq}U$ we shall write $\lfloor K, W\rfloor:=\{f \in C(U,F)\colon
f(K){\subseteq}W\}$.
\[hlp0\] Let $E, F$ be locally convex spaces and $U{\subseteq}E$ be open. Then the linear map $$D\colon C^\infty_{{\mathbb K}}(U,F)\to C^\infty_{{\mathbb K}}(U, L(E,F)_c)\,,\quad
f{\mapsto}f'$$ is continuous.
The map $D$ is linear and also $C^\infty(U,L(E,F)_c)\to
C(U\times E^k,L(E,F)_c)$, $f {\mapsto}d^k f$ is linear, for each $k\in {{\mathbb N}}_0$. Hence $$\label{djD}
d^k\circ D\colon C^\infty(U,F)\to C(U\times E^k,L(E,F)_c)_{c.o.}$$ is linear, whence it will be continuous if it is continuous at $0$. We pick a typical $0$-neighbourhood in $C(U\times E^k,L(E,F)_c)_{c.o.}$, say $\lfloor K,V\rfloor$ with a compact subset $K{\subseteq}U\times E^k$ and a $0$-neighbourhood $V{\subseteq}L(E,F)_c$. After shrinking $V$, we may assume that $V=\lfloor A ,W\rfloor$ for some compact set $A{\subseteq}E$ and $0$-neighbourhood $W{\subseteq}F$.\
We now recall that for $f\in C^\infty_{{\mathbb K}}(U,F)$, we have $$\label{usfrec}
d^k(f')(x,y_1,\ldots,y_k)
\; =\; d^{k+1}f(x,y_1,\ldots, y_k,{{\scriptscriptstyle \bullet}})\colon E\to F$$ for all $k\in {{\mathbb N}}_0$, $x\in U$ and $y_1,\ldots, y_k\in E$ (see [@HYP]). Since $\lfloor K\times A, W\rfloor$ is an open $0$-neighbourhood in $C(U\times E^{k+1},F)$ and the map $C^\infty(U,F)
\to C(U\times E^{k+1},F)_{c.o.}$, $f {\mapsto}d^{k+1}f$ is continuous, we see that the set $\Omega$ of all $f \in C^\infty(U,F)$ such that $d^{k+1}f \in\lfloor K\times A, W\rfloor$ is a $0$-neighbourhood in $C^\infty(U,F)$. In view of (\[usfrec\]), we have $d^k(f')\in \lfloor K,\lfloor A,W\rfloor\rfloor$ for each $f \in\Omega$. Hence $d^k\circ D$ from (\[djD\]) is continuous at $0$, as required.
\[wannerf\] Let $X$ be a Hausdorff topological space, $F$ be a Hausdorff locally convex space, $K{\subseteq}X$ be compact and also $M{\subseteq}C(X,F)_{c.o.}$ be compact. Let $\operatorname{eval}\colon C(X,F)\times X\to F$, $(f,x){\mapsto}f(x)$ be the evaluation map. Then $\operatorname{eval}(M\times K)$ is compact.
The restriction map $\rho\colon C(X,F)_{c.o.}
\to C(K,F)_{c.o.}$, $f{\mapsto}f|_K$ being continuous by [@Eng §3.2(2)], the set $\rho(M)$ is compact in $C(K,F)_{c.o.}$. The evaluation map ${\varepsilon}\colon C(K,F)\times K\to F$, $(f,x){\mapsto}f(x)$ is continuous by [@Eng Theorem 3.4.2]. Hence also $\operatorname{eval}(M\times K)
={\varepsilon}(\rho(M)\times K)$ is compact.
\[hlp1\] Let $E$, $F_1$, $F_2$ and $G$ be locally convex spaces, and $\beta\colon F_1\times F_2\to G$ be a bilinear map which is hypocontinuous with respect to compact subsets of $F_2$. Then $$C^n_{{\mathbb K}}(U,\beta )\colon C^n_{{\mathbb K}}(U,F_1)\times C^n_{{\mathbb K}}(U,F_2)\to
C^n_{{\mathbb K}}(U,G)\,,\quad (f,g){\mapsto}\beta\circ (f, g)$$ is hypocontinuous with respect to compact subsets of $C^n_{{\mathbb K}}(U,F_2)$, for each $n\in {{\mathbb N}}_0\cup\{\infty\}$. If $\beta$ is continuous, then also $C^n_{{\mathbb K}}(U,\beta)$ is continuous.
If $\beta$ is continuous and hence smooth, then $C^n(U,\beta)$ is smooth and hence continuous, as a very special case of [@ZOO Proposition 4.16].[^8]\
If $\beta$ is hypocontinuous, it suffices to prove hypocontinuity of $C^n(U,\beta)$ for each finite $n$. To see this, let $i_{n,G}\colon \colon C^\infty(U,G)\to C^n(U,G)$ be the inclusion map for $n\in {{\mathbb N}}_0$, and define $i_{n,F_1}$ and $i_{n,F_2}$ analogously. Since the topology on $C^\infty(U,G)$ is initial with respect to the maps $i_{n,G}$, the restriction $C^\infty(U,\beta)|_Y$ to a subset $Y{\subseteq}C^\infty(U,F_1)\times C^\infty(U,F_2)$ is continuous if and only if $i_{n,G}\circ C^\infty(U,\beta)|_Y
=C^n(U,\beta)|_{{\overline}{Y}}$ is continuous, where ${\overline}{Y}:=(i_{n,F_1}\times i_{n,F_2})(Y)$. Applying this with $Y=\{f\}\times C^\infty(U,F_2)$, $Y=C^\infty(U,F_1)\times \{g\}$ and $Y=C^\infty(U,F_1)\times M$ with $M{\subseteq}C^\infty(U,F_2)$ compact, we see that hypocontinuity of each $C^n(U,\beta)$ implies hypocontinuity of $C^\infty(U,\beta)$ (using the characterization given in Proposition \[prehypo\](c)).\
The case $n=0$. Let $M{\subseteq}C(U,F_2)$ be compact and consider a typical $0$-neighbourhood in $C(U,G)$, say $\lfloor K, W\rfloor$ with $K{\subseteq}U$ compact and a $0$-neighbourhood $W{\subseteq}G$. By Lemma \[wannerf\], the set $N:=\operatorname{eval}(M\times K){\subseteq}F_2$ is compact, where $\operatorname{eval}\colon C(U,F_2)\times U\to F_2$ is the evaluation map. Since $\beta$ is hypocontinuous, there exists a $0$-neighbourhood $V{\subseteq}F_1$ with $\beta(V\times N ){\subseteq}W$. Then $\beta \circ (\lfloor K, V \rfloor\times M) {\subseteq}\lfloor K, W\rfloor$ and hence $C(U,\beta)$ is hypocontinuous.\
Induction step. Given $n\in {{\mathbb N}}$, assume that $C^{n-1}(U,\beta)$ is hypocontinuous in the second argument for each $U$ and $\beta$. The topology on $C^n(U,G)$ being initial with respect to the linear maps $\lambda_1\colon C^n(U,G)\to C(U,G)_{c.o.}$, $f {\mapsto}f$ and $\lambda_2\colon C^n(U,G)\to C^{n-1}(U\times E,G)$, $f{\mapsto}df$ (by [@ZOO Lemma A.1(d)]), we only need to show that $\lambda_j\circ C^n(U,\beta)$ is hypocontinuous for $j\in \{1,2\}$. We have $\lambda_1\circ C^n(U,\beta)=C(U,\beta)\circ (i_1\times i_2)$, where $i_j\colon C^n(U,F_j) \to C(U,F_j)$ is the inclusion map which is continuous and linear. Since $C(U,\beta)$ is hypocontinuous by the case $n=0$, we readily deduce that $\lambda_1\circ C^n(U,\beta)$ is hypocontinuous. The map $\delta_j \colon C^n(U,F_j)\to C^{n-1}(U\times E,F_j)$, $f{\mapsto}df$ is continuous linear and $\pi\colon U\times E\to U$, $(x,y){\mapsto}x$ is smooth, whence $\rho_j \colon C^n(U,F_j)\to C^{n-1}(U\times E,F_j)$, $\rho_j(f):=f\circ \pi$ is continuous linear (cf. [@ZOO Lemma 4.4]). By (\[abstrprod\]), we have $$\label{git}
\lambda_2\circ C^n(U,\beta)\;=\;
C^{n-1}(U\times E,\beta)\circ (\delta_1\times \rho_2)
\,+\, C^{n-1}(U\times E,\beta)\circ (\rho_1\times \delta_2)\,.$$ Since $C^{n-1}(U\times E,\beta)\colon C^{n-1}(U\times E,F_1)
\times C^{n-1}(U\times E,F_2)\to C^{n-1}(U\times E,G)$ is hypocontinuous by the inductive hypothesis and each summand in (\[git\]) is a composition thereof with a direct product of continuous linear maps, we deduce that each summand and hence also $\lambda_2\circ C^n(U,\beta)$ is hypocontinuous in the second argument. This completes the proof.
\[hlp2\] Let $E$, $F$ and $G$ be locally convex spaces, $U{\subseteq}E$ be open and $\beta\colon E \times F \to G$ be a bilinear map which is hypocontinuous with respect to compact subsets of $E$. Then $$\beta_*\colon C^n_{{\mathbb K}}(U,F)
\to C^n_{{\mathbb K}}(U,G)\,,\quad
(\beta_* (f))(x)\, :=\, \beta(x,f(x))
\quad\mbox{for $\,f \in C^n_{{\mathbb K}}(U,F)$, $x\in U$}$$ is a continuous linear map, for each $n\in {{\mathbb N}}_0\cup\{\infty\}$.
It is clear that $\beta_*$ is linear. We only need to prove the assertion for finite $n$, by an argument similar to that in the proof of Lemma \[hlp1\]. The proof is by induction.
The case $n=0$. Suppose we are given a $0$-neighbourhood in $C(U,G)$, say $\lfloor K,W\rfloor$ with $K{\subseteq}U$ compact and a $0$-neighbourhood $W{\subseteq}G$. By Proposition \[prehypo\](a), there exists a $0$-neighbourhood $V{\subseteq}F$ such that $\beta(K\times V){\subseteq}W$. Then $\beta_*(\lfloor K,V\rfloor){\subseteq}\lfloor K,W\rfloor$. Thus $\beta_*$ is continuous at $0$ and hence continuous, being linear.
Induction step. Let $n\in {{\mathbb N}}$ and assume that the assertion holds for $n-1$ in place of $n$. Let us write $\beta_{*,n}\colon
C^n(U,F)\to C^n(U,G)$, for added clarity. The topology on $C^n(U,G)$ being initial with respect to the linear maps $\lambda_1\colon C^n(U,G)\to C(U,G)_{c.o.}$, $f {\mapsto}f$ and $\lambda_2\colon C^n(U,G)\to C^{n-1}(U\times E,G)$, $f{\mapsto}df$, we only need to show that $\lambda_j\circ \beta_{*,n}$ is continuous for $j\in \{1,2\}$. We have $\lambda_1\circ \beta_{*,n}
=\beta_{*,0}\circ i$, where $i \colon C^n(U,F) \to C(U,F)$ is the continuous linear inclusion map and $\beta_{*,0}$ is continuous by the above. To tackle $\lambda_2\circ \beta_{*,n}$, note that $$A\colon (E\times E)\times F\to G\,,\quad
A((x,y),z)\,:=\,\beta(y,z)\quad\mbox{and}$$ $$B\colon (E\times E)\times F\to G\,,\quad
B((x,y),z)\,:=\,\beta(x,z)\quad\quad\;$$ are hypocontinuous with respect to compact subsets of $E\times E$. Let $\pi_1\colon U\times E\to U$ be the projection on the first component. Then the map $p\colon C^n(U,F)\to C^{n-1}(U\times E,F)$, $f {\mapsto}f \circ \pi_1$ is continuous linear (cf. [@ZOO Lemma 4.4]). By (\[abstrprod\]), we have $$\label{lasttm}
\lambda_2\circ \beta_{*,n}\, =\,
A_*\circ p \, +\, B_*\circ d\,,$$ where $d\colon C^n(U,F)\to C^{n-1}(U\times E,F)$, $f {\mapsto}df$ is continuous linear and also the maps $A_*\colon C^{n-1}(U\times E,F)\to
C^{n-1}(U\times E,G)$ and $B_*\colon C^{n-1}(U\times E,F)\to
C^{n-1}(U\times E,G)$ are continuous linear, by the inductive hypothesis. Hence $\lambda_2\circ \beta_{*,n}$ is continuous linear. This completes the proof.
[**Proof of Theorem \[poiss2\].**]{} By Lemma \[hlp0\], the mapping $D\colon C^\infty(U,{{\mathbb K}})\to C^\infty(U, E'_c)$,$f{\mapsto}f'$ is continuous and linear. By Lemma \[hlp1\], the bilinear map $$C^\infty(U,[.,.])\colon C^\infty(U,E')\times
C^\infty(U,E')\to C^\infty(U,E')\,,\quad
(f,g){\mapsto}(x{\mapsto}[f(x),g(x)])$$ is hypocontinuous with respect to compact subsets of the second factor; and if $[.,.]$ is continuous, then also $C^\infty(U,[.,.])$. The evaluation map $\beta \colon E\times E'_c\to {{\mathbb K}}$, $\beta(x,\lambda):=\lambda(x)$ is hypocontinuous with respect to compact subsets of $E$ by Proposition \[resulteval\]. Hence $\beta_*\colon C^\infty(U,E'_c)\to C^\infty(U,{{\mathbb K}})$, $f {\mapsto}\beta_*(f)=\beta\circ (\operatorname{id}_U,f)$ is continuous linear by Lemma \[hlp2\]. Since $$\{.,.\}\, =\, \beta_*\circ C^\infty(U,[.,.])
\circ (D\times D)$$ by definition, we see that $\{.,.\}$ is a composition of continuous maps if $[.,.]$ is continuous, and hence continuous. In the general case, $\{.,.\}$ is a composition of a hypocontinuous bilinear map and continuous linear maps and hence hypocontinuous.[$\Box$]{}
Continuity of the map taking [$f$]{} to [$X_f$]{} {#nwsec2}
=================================================
In this section, we show continuity of the mapping which takes a smooth function to the corresponding Hamiltonian vector field, in the case ${{\mathcal S}}=c$.
\[poiss3\] Let $(E,[.,.])$ be a locally convex Poisson vector space with respect to ${{\mathcal S}}=c$. Let $U{\subseteq}E$ be an open subset. Then the map $$\label{thmap}
\Psi \colon C^\infty_{{\mathbb K}}(U,{{\mathbb K}})\to C^\infty_{{\mathbb K}}(U,E)\,,
\quad f{\mapsto}X_f$$ is continuous and linear.
Let $\eta_E\colon E\to (E'_c)'_c$ be the evaluation homomorphism and $V:= \{A\in L(E'_c,E'_c)\colon$ $(\forall x\in E)\; \eta_E(x)\circ A\in
\eta_E(E)\}$. Then $V$ is a vector subspace of $L(E'_c,E'_c)$ and $\operatorname{ad}(E'){\subseteq}V$. The composition map $\Gamma\colon (E'_c)'_c\times L(E'_c,E'_c)_c\to
(E'_c)'_c$, $(\alpha ,A){\mapsto}\alpha\circ A$ is hypocontinuous with respect to equicontinuous subsets of $(E'_c)'_c$, by Proposition 9 in [@BTV Chapter III, §5, no.5]. If $K{\subseteq}E$ is compact, then the polar $K^\circ$ is a $0$-neighbourhood in $E'_c$, entailing that $(K^\circ)^\circ{\subseteq}(E'_c)'$ is equicontinuous. Hence $\eta_E$ takes compact subsets of $E$ to equicontinuous subsets of $(E'_c)'$, and hence $$\beta\colon E \times V \to E\,,\quad
(x,A) {\mapsto}\eta_E^{-1}(\Gamma(\eta_E(x), A))$$ is hypocontinuous with respect to compact subsets of $E$. Using Lemma \[hlp2\], we see that $\beta_*\colon
C^\infty(U,V)\to C^\infty(U,E)$ is continuous linear. Also the map $D\colon C^\infty(U,{{\mathbb K}})\to C^\infty(U,E'_c)$, $f{\mapsto}f'$ is continuous linear by Lemma \[hlp0\]. Furthermore, $\operatorname{ad}=[.,.]^\vee \colon E'_c\to L(E'_c,E'_c)_c$ is continuous linear since $[.,.]$ is hypocontinuous (see Proposition \[prehypo\](b)), entailing that also $$C^\infty(U,\operatorname{ad})\colon C^\infty(U,E'_c)\to
C^\infty(U,L(E'_c,E'_c)_c)\,,\quad
f {\mapsto}\operatorname{ad}\circ \, f$$ is continuous linear (see, e.g., [@ZOO Lemma 4.13]). Hence $\Psi= \beta_*\circ C^\infty(U,\operatorname{ad})\circ D$ is continuous and linear.
[99]{} Außenhofer, L., [*Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups*]{}, Diss. Math. [**384**]{}, 1999. Banaszczyk, W., “Additive Subgroups of Topological Vector Spaces,” Springer, 1991. Bertram, W., H. Glöckner and K.-H. Neeb, *Differential calculus over general base fields and rings*, Expo. Math. [**22**]{} (2004), 213–282. Bochnak, J. and J. Siciak, *Analytic functions in topological vector spaces*, Studia Math. [**39**]{} (1971), 77–112. Bourbaki, N., “Topological Vector Spaces, Chapters 1-5,” Springer-Verlag, 1987. Bunke, U. and M. Olbrich, [*Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group*]{}, Ann. Math. [**149**]{} (1999), 627–689. Bunke, U. and M. Olbrich, *The spectrum of Kleinian manifolds*, J. Funct. Anal. [**172**]{} (2000), 76–164. Engelking, R., “General Topology,” Heldermann Verlag, Berlin, 1989. Floret, K., *Lokalkonvexe Sequenzen mit kompakten Abbildungen*, J. Reine Angew.Math. [**247**]{} (1971), 155–195. Glöckner, H., *Infinite-dimensional Lie groups without completeness restrictions*, pp.43–59 in: A. Strasburger et al. (Eds.), “Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups,” Banach Center Publications [**55**]{}, Warszawa, 2002. Glöckner, H., *Remarks on holomorphic families of operators*, TU Darmstadt Preprint [**2256**]{}, December 2002. Glöckner, H., *Lie groups over non-discrete topological fields*, preprint, arXiv:math.FA/0408008. Glöckner, H., [*Fundamentals of direct limit Lie theory*]{}, Compos. Math. [**141**]{} (2005), 1551–1577. Glöckner, H., *Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories*, to appear in J. Funct. Anal.(cf. arXiv:math.FA/0606078). Glöckner, H., *Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces*, preprint, arXiv:math.FA/0701197. Glöckner, H., *Bundles of locally convex spaces, group actions, and hypocontinuous bilinear mappings*, manuscript, Darmstadt 2002 (currently undergoing revision). Glöckner, H., R. Gramlich and T. Hartnick, *Final group topologies, Phan systems and Pontryagin duality*, preprint, cf. arXiv:math.GR/0603537. Glöckner, H., R.L. Lovas and K.-H. -Neeb, *Locally convex Poisson vector spaces and coadjoint orbits* (provisional title), work in progress. Glöckner, H. and K.-H. -Neeb, “Infinite-Dimensional Lie Groups. Vol.I: Basic Theory and Main Examples,” to appear 2007 or 2008 in Springer Verlag. Große-Erdmann, K.-G., “The Borel-Okada Theorem Revisited,” Habilitationsschrift, FernUniversität Hagen, 1992. Jarchow, H., “Locally Convex Spaces,” B.G. Teubner, Stuttgart, 1981. Köthe, G., “Topological Vector Spaces II,” Springer-Verlag, New York, 1979. Kriegl, A. and P.W. Michor, “The Convenient Setting of Global Analysis,” AMS, Providence, 1997. Lang, S., “Fundamentals of Differential Geometry,” Springer-Verlag, 1999. Maissen, B., *Über Topologien im Endomorphismenraum eines topologischen Vektorraumes*, Math. Ann. [**151**]{} (1963), 283–285. Milnor, J., [*Remarks on infinite-dimensional Lie groups*]{}, pp.1007–1057 in: DeWitt, B. and R. Stora (Eds.), “Relativité, Groupes et Topologie II,” 1984. Neeb, K.-H., *Towards a Lie theory of locally convex groups*, Japan J. Math. [**1**]{} (2006), 291–468. Neeb, K.-H., *Poisson structures on pro-Lie groups*, manuscript in preparation. Odzijewicz, A. and T.S. Ratiu, [*Banach Lie-Poisson spaces and reduction*]{}, Comm. Math. Phys. [**243**]{} (2003), 1–54. Odzijewicz, A. and T.S. Ratiu, [*Extensions of Banach Lie-Poisson spaces*]{}, J. Funct. Anal. [**217**]{} (2004), 103–125. Seip, U., “Kompakt erzeugte Vektorräume und Analysis,” Springer Lecture Notes in Math., Springer, Berlin,1972. Thomas, E.G.F., “Calculus on Locally Convex Spaces,” Preprint [**W-9604**]{} (unpublished), Department of Mathematics, University of Groningen, 1996. Treves, F., “Topological Vector Spaces, Distributions and Kernels,” Academic Press, New York, 1967. Wurzbacher, T., *Fermionic second quantization and the geometry of the restricted Grassmannian*, pp.287–375 in: A. Huckleberry and T. Wurzbacher (Eds.), “Infinite-Dimensional Kähler Manifolds,” Birkhäuser, Basel, 2001.
[, TU Darmstadt, Fachbereich Mathematik AG 5, Schlossgartenstr.7,\
64289 Darmstadt, Germany. E-Mail: [[email protected]]{}]{}
[^1]: This article is partially based on the unpublished preprint [@FAM], the preparation of which was supported by the German Research Foundation (DFG, FOR 363/1-1).
[^2]: Compare also [@Sei] for the use of Kelleyfications in differential calculus.
[^3]: Considering $f$ as a map into the completion of $F$, we see that (a), (b) and (c) remain equivalent if $F$ is not Mackey complete. But (a)–(c) do not imply that $f$ is $C^1_{{\mathbb C}}$ ([@Gro Example II.2.3], [@COM Theorem 1.1]).
[^4]: Given a closed set $A{\subseteq}Y$, the intersection $f^{-1}(A)\cap K=(f|_K)^{-1}(A)$ is closed in $K$ in the latter case and thus $f^{-1}(A)$ is closed, whence $f$ is continuous.
[^5]: An equivalent definition runs as follows: A Hausdorff space $X$ is a $k_\omega$-space if and only if $X={{\displaystyle \lim_{\longrightarrow}}}\,K_n$ for an ascending sequence $(K_n)_{n\in {{\mathbb N}}}$ of compact subsets of $X$ with union $X$.
[^6]: Let $E$ be a nuclear Fréchet space. Then $E$ is Pontryagin reflexive [@Ban Propositions 15.2 and 2.3].By [@Ban Theorem 16.1] and [@Aus Proposition 5.9], also $E'_c$ is nuclear and Pontryagin reflexive. Since $E$ is metrizable and hence a $k$-space, $E'_c$ is complete [@Aus Proposition 4.11]. We now see with [@Tre Proposition 50.2] that every closed, bounded subset of $E$ (and $E'_c$) is compact. Hence $E_b'=E_c'$, $(E_b')'_b=(E'_c)'_c$ and both $E$ and $E'_b=E'_c$ are also reflexive. By Remark \[supplykom\], $E'_b=E'_c$ is a $k^\infty$-space.
[^7]: The Jacobi identity can be established as in the proof of [@OR1 Theorem 4.2].
[^8]: Note that the ordinary $C^n$-topology is used there, by [@ZOO Proposition 4.19(d) and Lemma A2].
|
---
abstract: 'We present a preliminary calculation of the electromagnetic form factors of $^3$He and $^3$H, performed within the Light-Front Hamiltonian Dynamics. Relativistic effects show their relevance even at the static limit, increasing at higher values of momentum transfer, as expected.'
author:
- 'F. A. Baroncini, E. Pace and G. Salmè'
title: 'Relativistic Hamiltonian Dynamics and Few-Nucleon Systems'
---
Introduction
============
[The *standard model*]{} of Few-Nucleon Systems, where Nucleon and pion degrees of freedom are taken into account, is already at a very sophisticated stage, and many efforts are presently carried on to retain all the [*general principles*]{} of a theory with a fixed number of constituents. In particular, to satisfy Poincaré covariance appears a quite reachable goal and at the same time a very compelling requirement in view of the forthcoming measurements of electromagnetic (em) form factors (ff’s) for A=3,4, in the region of few GeV’s [@Makis]. In order to extract unambiguous signatures of effects beyond the [*standard model*]{} of Few-Nucleon Systems one should develop a fully field-theoretical approach, based on the Bethe-Salpeter equation, but the difficulties are well-known, and therefore one has to consider alternative approaches, like 3-D reductions (see, e.g., [@Gross] for A=2 and [@Stadler] for the work in progress for A=3) or the Relativistic Hamiltonian Dynamics (RHD) framework, suggested by Dirac in a seminal paper [@Dirac].
Our aim is to construct, within the Light-Front form of RHD, a relativistic approach for Few-Nucleon System that i) retains the whole [*successful phenomenology*]{} already developed and ii) includes, [*in a non perturbative way*]{}, the relativistic features requested by Poincaré covariance. In order to have a strong and immediate comparison with the experiments we have focused our efforts on the development of Poincaré covariant calculations of the electromagnetic ff’s, extending our approach from the Deuteron [@LPS2K] to the Trinucleon. The adopted Bakamjian-Thomas (BT) procedure (see, e.g., [@KP91]) allows us to exploit realistic wave functions for Few-Nucleon Systems (for A=3, see, e.g., [@KVR94]) in order to evaluate matrix elements of a Poincaré covariant current operator [@LPS98] for an interacting system. The relativistic effects imposed by Poincaré covariance materialize in the relativistic kinematics and in the presence of the so-called Melosh rotations (see, e.g., [@KP91]), that allows one to use the standard Clebsh-Gordan machinery to obtain many-nucleon wave functions with the correct angular coupling.
Formalism
=========
As well-known [@KP91], Light-Front (LF) RHD has some appealing features, like the largest number of kinematical Poincaré generators (given the symmetry of the [*initial*]{} hypersurface $x^+=0$) and the simplest procedure for separating out the center of mass motion from the intrinsic one, in strict analogy with the nonrelativistic procedure. Moreover, it shares with the other two RHD’s (Instant and Point forms) the rigorous fulfillment of the Poincaré covariance, for a system with a fixed number of constituents. In some sense, RHD’s fall between non-relativistic quantum mechanics and the local, relativistic field theory.
For an interacting system, an em current operator, $J^\mu$, that fulfills the extended Poincaré covariance (i.e. including parity and time reversal) and Hermiticity, can be constructed by a suitable auxiliary operator, $j^\mu$, that fulfills rotational covariance around the $z$-axis in a Breit frame (${\bf P}_f+{\bf P}_i=0$), where the $\perp$ component of the momentum transfer is vanishing (${\bf q}_\perp=0$) [@LPS98]. Note that such a frame is [*different*]{} from the Drell-Yan one, where $q^+ = 0$. In general [@LPS98], the matrix elements $\langle P_f|J^\mu| P_i \rangle$, still acting on internal variables, are directly given by the matrix elements of the auxiliary operator $j^\mu$, evaluated in the chosen Breit frame. A minimal Ansatz for a [*many-body*]{} auxiliary operator is built from i) the free current (a one-body operator) and ii) the $\perp$ component of the angular momentum operator $\vec S$ (a many-body operator in LF) as follows j\^\_[fi]{}(q\_z)=[1 2 ]{} \[curlf\]with $
{\cal{J}}^{\mu}_{fi}(q\hat{e}_z) =
\Pi_f ~J_{free}^{\mu}(0) ~\Pi_i
$, $\Pi \equiv$ projector onto the states of the (initial or final) system and $\vec S \equiv$ the LF-spin operator of the system as whole: it acts on the “internal” space and is unitarily related to the standard angular momentum operator through the Melosh operators. Let us remind that $ J_{free}^{\mu}(0)$ is the proper sum over A=2,3 free Nucleon current given by $J^\mu_{N}= -F_{2}[(p^{\prime
\mu}-p^\mu)^2](p^\mu+p^{\prime
\mu})/2M +\gamma^\mu (F_{1}[(p^{\prime
\mu}-p^\mu)^2]+F_{2}[(p^{\prime
\mu}-p^\mu)^2])$ , with $F_{1(2)}[(p^{\prime
\mu}-p^\mu)^2]$ the Dirac (Pauli) Nucleon ff. In our Breit frame, charge normalization and current conservation (for $M_f=M_i$) can be fulfilled by imposing $ {\cal{J}}^{-}(q\hat{e}_z)={\cal{J}}^{+}(q\hat{e}_z)$[@LPS98; @LPS2K].
For evaluating matrix elements of $j^\mu(x)$, eigenstates of the interacting system are needed. To this end one can use the “non relativistic solutions”, but with Melosh Rotations in the angular part, if the interaction $V\equiv M_{int} -M_0$ (where $M_{int} (M_0)$ is the mass operator of the interacting (free) system) can be embedded in a BT framework. The BT construction for obtaining interacting Poincaré generators suggests a necessary (not sufficient) condition [@KP91] on the interaction: $V$ must depend upon intrinsic variables combined in scalar products, i.e. $ [\vec{\cal B}_{LF}, V ] = [\vec S_{0}, V ] =
[P_{\perp}, V ] = [P^+, V ]= 0
$, where $\vec{\cal B}_{LF}$ are the LF boosts, $\vec S_{0}\equiv$ the angular momentum operator for the non interacting case (since $S^2_{0}=S^2_{int}$ and $S_{0,z}=S_{int,z}$, the eigenvalues of $S^2_{0}$ and $S_{0,z}$ can label the eigenstates of the interacting system). The non relativistic interaction fulfills the above requirements.
EM observables for A=2 and A=3 nuclei
=====================================
First, let us briefly review our results for the Deuteron, and then we present preliminary calculations for the trinucleon case.
Magnetic and quadrupole moments of the Deuteron (see Table 1), as well as the ff’s, $A(Q^2)$ and $B(Q^2)$ (see Fig. 1), and the tensor polarization $T_{20}(Q^2)$ have been calculated [@LPS2K], showing the great relevance of the Poincaré covariance, even at low $Q^2$. From our results, one could argue that the role of MEC and pair contributions should shrink, but clearly a direct evaluation of two-body contributions is mandatory for a closer comparison with experiments.
Interaction $P_D$ $\mu _D^{NR}$ $\mu^{LFD}_D$ $Q_D^{NR}$ $Q^{LFD}_D$
------------- ------- --------------- --------------- ------------ -------------
CD-Bonn 4.83 0.8523 0.8670 0.2696 0.2729
Nijm1 5.66 0.8475 0.8622 0.2719 0.2758
RSC93 5.70 0.8473 0.8637 0.2703 0.2750
Av18 5.76 0.8470 0.8635 0.2696 0.2744
: Magnetic moment (in nuclear magnetons) and quadrupole moment (in $fm^2$) for the Deuteron [@LPS2K]; $P_D$ is the $D$-state percentage. The corresponding exp. values are: $\mu _{exp}=0.857406(1)$ and $Q _{exp}= 0.2859(3)$
![Left panel: the $^2$H structure function, $F_2^D(x)$, vs. the Bjorken variable $x$. Solid lines: LF calculations, see Eq. (\[fLF\]). Dashed lines: standard approach calculations. Right panel: the same as in the left panel but for the ratio $r(x)=F_2^n(x)/F_2^p(x)$. AV18 NN interaction and the model of Ref. [@Aubert] for the Nucleon structure functions are adopted. (After Ref. [@PSCor]).](f2xh2.eps){width="12.cm"}
A very interesting topic, related indeed to both A=2 and A=3 nuclei, given the absence of free neutron targets, is the extraction of the ratio of Nucleon structure functions, $r(x)=F_2^n(x)/F_2^p(x)$, in the Bjorken limit. This extraction has been analyzed adopting a standard approach [@KPSS], and here we would simply recall that a LF approach could give remarkably different results. By using the Impulse Approximation, the Deuteron structure function can be expressed through a convolution of the Nucleon structure functions and the distribution probability, $f^{D}(z)$, to find inside $^2$H a nucleon with LF momentum $z$. In the usual approach, $f^D(z)$ is basically an adapted Instant-form approach with an off-mass-shell struck Nucleon, while within a rigorous LF approach [@PSCor] it becomes f\^[D]{}\_[LF]{}(z) = d [p]{} n\^D( |[p]{} |) ( z - ) \[fLF\]where $n^D( |{\vec p} |)$ is the Nucleon momentum distribution in the Deuteron, $M ( M_D )$ the Nucleon (Deuteron) mass, $ \xi = p^+/P^+=(\sqrt{M^2 +|\vec{p}|^2} +p_z)/2
\sqrt{M^2 +|\vec{p}|^2} $.
Following [@KPSS] one can extract the ratio $r(x)$ from the experimental data for the Deuteron structure function and a suitable recurrence relation, where $f_D(x)$ has to be considered. In Fig. 2, one sees the interesting effect on the Deuteron structure function and on $r(x)$ (cf. $x\to 1$), produced by a LF $f^{D}(z)$, Eq. (\[fLF\]).
Trinucleon em observables, see Table 2 and Figs. 3-4, can be evaluated analogously to $^2$H case, but with a very cumbersome angular coupling. In this preliminary calculation ($S$, $P$ and $D$ waves only for observables in Table 2) the A=3 wave function [@KVR94] without Coulomb interaction and corresponding to AV18 [@AV18] NN interaction has been used. The charge and magnetic ff’s are calculated from the matrix elements of a Poincaré covariant current as follows $$F_{ch}^{ T_z}(Q^2) = {1 \over 2} ~ Tr[{\cal{I}}^+(T_z)]
\quad \quad \quad F_{mag}^{ T_z}(Q^2) = - i { M \over Q} ~ Tr[{\sigma}_y ~
{\cal{I}}_x(T_z)]$$ where $
{\cal{I}}^{r}_{\sigma' \sigma }(T_z) \equiv
\langle \Psi_{{1 \over 2} \sigma'}^{{1 \over 2} T_z},
{P}_f | ~ {\cal J}^{r} ~ |\Psi_{{1 \over 2} \sigma}^{{1 \over 2} T_z},
{P}_i \rangle
$ with $r=+,1$ (cf Eq. (\[curlf\])).
[|c||c| c|c| c|]{} Theory & $\mu(^3\rm{He})$ & $\mu(^3\rm{H})$ & $r_{ch}(^3\rm{He})$ &$r_{ch}(^3\rm{H})$\
NR(S) &-1.723(2) & 2.515(2) &1.841(3) fm &1.798(3) fm\
LFD(S) & -1.7860(2) & 2.6034(2)&1.867(3) fm & 1.821(3) fm\
NR(S+S’) & -1.7093(2) & 2.515(2)& 1.896(3) &1.726(3) fm\
LFD(S+S’) & -1.768(2) & 2.600(2) & 1.919(3) &1.772(3) fm\
NR(S+S’+P+D) & -1.769(2) & 2.579(2) & 1.882(4) fm & 1.714(4) fm\
LFD(S+S’+P+D) & -1.839(2)& 2.674(2) & 1.906(4) fm & 1.754(4) fm\
Exp. & -2.1276 & 2.9789 & 1.959(30)fm & 1.755(86) fm\
Conclusions & Perspectives
==========================
In order to construct a [*standard model*]{} for Few-Nucleon Systems it is necessary to consider relativistic effects. For a fixed number of constituents, one can be a Poincaré covariant approach by adopting a Light-Front RHD and the Bakamjian-Thomas procedure. Within such an approach and the em current operator suggested in Ref. [@LPS98], the em observables of the A=3 nuclei have been calculated for the first time ($S$, $P$ and $D$ waves for the observables in Table 2 and $S$-wave only for ff’s in Figs. 3-4). The few % effect for em observables at $Q^2=0$, in the correct direction, is very encouraging. Moreover, the sizable effect at high $Q^2$ indicates the essential role played by relativity for analyzing the em ff’s in the region of few GeV’s. A full calculation, with a systematic analysis of the pair contribution (Z-diagram) and current operators fulfilling the Ward-Takahashi Identity, will be presented elsewhere.
We gratefully thank Alejandro Kievsky for providing us the trinucleon wave function corresponding to the AV18 NN interaction.
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|
---
abstract: 'A physical system exposes to us in a real space, while its description often refers to its reciprocal momentum space. A connection between them can be established by exploring patterns of quasiparticles interference (QPI), which is experimentally accessible by Fourier transformation of the scanning tunneling spectroscopy (FT-STS). We here investigate how local and global features of QPI patterns are related to the geometry and topology of electronic structure in the considered physical system. A reduced response function (RRF) approach is developed that can analyze QPI patterns with clear physical pictures. It is justified that the generalized joint density of states, which is the imaginary part of RRF, for studying QPI. Moreover, we reveal that global patterns of QPI may be indicators of topological numbers for gapless systems, and demonstrate that robustness of such indicators against distractive local features of QPI for topological materials with complicated band structures.'
author:
- 'Dan-Bo Zhang'
- Qiang Han
- 'Z. D. Wang'
title: 'Local and global patterns in quasiparticle interference: a reduced response function approach'
---
Introduction
============
Interference of quasiparticles in the presence of impurities leads to a modulation of the local density of states (LDOS). The LDOS is accessible experimentally with scanning tunneling spectroscopy (STS)[@crommie_93; @hasegawa_93; @sprunger_97], and the pattern of modulation is further extracted by its Fourier transformation. There are local and global patterns of QPI that depend on the band structure of the underlying physical system. Concisely, local patterns of QPI depend on the geometry of the dispersion [@capriotti_03], and are further modified (suppressed or enhanced) by the internal structure of quasiparticles [@wang_03; @hanaguri_09]. Global patterns of QPI, on the other hand, depend on the topology of band structure, e.g., a global structure of spin-momentum locking [@guo_10]. Consequently, we can use QPI to infer the geometry and topology of the band structure, rendering QPI and FT-STS as facilities bridging the $\bf{r}$-space observations with the $\bf{k}$-space description of the underlying system.
There have been intensive studies of QPI for quantum materials [@avraham_18], including metals [@crommie_93; @sprunger_97], superconductors [@hoffman_02; @capriotti_03; @wang_03; @pereg-barnea_03; @hanaguri_09; @lee_09; @zhang_13], graphene [@rutter_07; @pereg-barnea_08; @brihuega_08; @dombrowski_17; @jolie_18], surface states of topological insulators [@zhou_09; @lee_09_TI; @beidenkopf_11; @kohsaka_15; @kohsaka_17], Weyl semimetals [@batabyal_16; @inoue_16; @mitchell_16; @zheng_18], and nonsymmorphic materials [@topp_17; @queiroz_18; @zhu_18]. The connection between QPI and ${{\bf k}}$-space spectral information is established through the joint density of states (JDOS) and its generalization (GJDOS) that takes internal structures of quasiparticles into consideration. The JDOS approach has been proposed at an earlier stage to explain hot spots in patterns of QPI for d-wave superconductor [@hoffman_02; @capriotti_03; @wang_03; @mcelroy_03], and has been applied for gapless topological systems in terms of spin-selective scattering probability (SSP) recently [@beidenkopf_11; @inoue_16]. A more rigorous treatment, however, should refer to Fourier transform of LDOS (FT-LDOS). Ref. [@derry_15; @kohsaka_17] point out that GJDOS may give some false features, and is not appliable in general. Nevertheless, as a convenient tool GJDOS allows us to intuitively analyze and understand QPI patterns from band structures directly. It is desirable for a clarification of the applicability of GJDOS. Moreover, a systematic treatment that can relate local and global patterns of QPI to the geometry and topology of the band structure is still awaited.
In this paper, we adopt a reduced response function (RRF) approach for analyzing QPI patterns under different scattering and probing channels. The RRF includes GJDOS as its imaginary component, and is a faithful encoding of the QPI information. We will prove that GJDOS shares the same singularities with FT-LDOS, with few exceptions that can be excluded at first hand. Moreover, we reveal the existence of higher-degree singularities, which can give rise to more significant features (hot spots) in QPI. With the justified GJDOS, we derive its analytical expression that clearly shows how its singularities and coherent factors jointly determine the QPI patterns. Based on this expression, we propose indicators of topological numbers in ideal topological systems. We then testify and demonstrate the robustness of such indicators of topological numbers against distracting local features in QPI arising from complicated geometry of the band structure, by exploring the QPI patterns in several representative topological materials, including topological insulators $\text{Bi}_2\text{Te}_3$ and BiTeI, as well as the graphene family.
The paper is organized as follows. We first propose the reduced response function and derive its singular behavior in Sec. . Then, we give an analytical expression for GJDOS and apply it for analyzing global patterns in Sec. . We then numerally study QPI for topological materials in Sec. , including surface states of topological insulators and the graphene family.
Reduced Response function for quasiparticle interference {#sec:rrf}
========================================================
The reduced response function
-----------------------------
For simplicity without loss of generality, we restrict our discusses of physical systems described by two-band Hamiltonians $\mathcal{H}({{\bf k}})=E_0({{\bf k}})+\mathbf{d}({{\bf k}})\cdot\boldsymbol{\sigma}$. Here $\boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ denotes a vector of Pauli matrix for a spin (or a pseudospin), which can represent systems with two internal degrees of freedom, including spin-half, sublattice, or particle-hole. Impurity is described as $V(r)=\sum_{\beta}V^\beta(r)\sigma_\beta$, where $\beta=0,x,y,z$ stands for scattering channels. In the presence of an impurity, interference between scattered-in and out quasiparticles leads to a perturbation to the local density of states. The local density is related to the Green function, $$n_{\alpha}({{\bf r}},\omega)=-\frac{1}{\pi}{{\mathrm{Im}}\, }\{{{\rm Tr}}[\sigma_\alpha G({{\bf r}},{{\bf r}},\omega)]\},$$ where $\sigma_\alpha$ ($\alpha=0,x,y,z$) represents probe channels [@guo_10]. The Green function in ${{\bf k}}$-space can be written as $$G({{\bf k}}',{{\bf k}},\omega)=G_0({{\bf k}},\omega)\delta_{{{\bf k}}{{\bf k}}'}+G_0({{\bf k}}',\omega)T_{{{\bf k}}'{{\bf k}}}(\omega) G_0({{\bf k}},\omega),$$ where $G_0({{\bf k}},\omega)=[\omega-\mathcal{H}({{\bf k}})+i0^+]^{-1}$ is the unperturbed Green function with positive infinitesimal $0^+$. For a $\sigma_\beta$ impurity and under Born approximation, the T-matrix takes the form $T_{{{\bf k}}'{{\bf k}}}(\omega)=V^{\beta}_{{{\bf k}}'{{\bf k}}}\sigma_\beta$, where $V^{\beta}_{{{\bf k}}',{{\bf k}}}$ is a Fourier transform of the potential strength for scattering channel $\sigma_\beta$. The scattering potential may be independent of ${{\bf k}}'$ and ${{\bf k}}$, e.g., for a impurity with a delta potential $\delta({{\bf r}})$. Or it can depend on ${{\bf k}}'-{{\bf k}}$ [@capriotti_03]. We mention that in general it can depend on both ${{{{\bf k}}+{{\bf p}}}}$ and ${{\bf k}}$, e.g., for spin-orbit scattering the scattering potential is $V_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}}=V_0\{1+ic[({{{{\bf k}}+{{\bf p}}}})\times{{\bf k}}]\cdot\boldsymbol{\sigma}\}$ [@lee_09_TI; @kohsaka_17].
The perturbed local density for probing channel $\sigma_\alpha$ and scattering channel $\sigma_\beta$ turns to be $$\delta n_{\alpha\beta}({{\bf r}},\omega)=-\frac{1}{\pi}{{\mathrm{Im}}\, }\{{{\rm Tr}}[\sum_{{{\bf k}}'{{\bf k}}}\sigma_\alpha G_0({{\bf k}}',\omega)V^{\beta}_{{{\bf k}}',{{\bf k}}}\sigma_\beta G_0({{\bf k}},\omega)]\}.$$
QPI patterns can be captured by a Fourier transformation of the perturbed local density, $$\begin{aligned}
\label{def:ldos_q}
\delta n_{\alpha\beta}({{\bf p}},\omega)&\equiv& \sum_{{{\bf r}}} e^{-i{{\bf p}}\cdot{{\bf r}}} \delta n_{\alpha\beta}({{\bf r}},\omega)\nonumber \\
&=&-{1\over 2\pi i}
[\Lambda_{\alpha\beta}({{\bf p}},\omega)-\Lambda^*_{\alpha\beta}(-{{\bf p}},\omega)],\end{aligned}$$ where the response function reads $$\label{lam1}
\Lambda_{\alpha\beta}({{\bf p}},\omega) =\sum_{{\bf k}}V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}}{{\rm Tr}}\left[\sigma_\alpha G_0({{\bf k}}+{{\bf p}},\omega)\sigma_\beta G_0({{\bf k}},\omega)\right].$$ In the presence of centrosymmetry, $\Lambda_{\alpha\beta}({{\bf p}},\omega)=\Lambda_{\alpha\beta}(-{{\bf p}},\omega)$, Eq. reduces to $\delta n_{\alpha\beta}({{\bf p}},\omega)=-\frac{1}{\pi}{{\mathrm{Im}}\, }[\Lambda_{\alpha\beta}({{\bf p}},\omega)]$. We consider the centrosymmetry in this paper since it is applicable for many QPI, while the case $\Lambda_{\alpha\beta}({{\bf p}},\omega)\neq\Lambda_{\alpha\beta}(-{{\bf p}},\omega)$ is left for further investigation.
To establish a connection between FT-LDOS and GJDOS, we first decompose the response function $\Lambda_{\alpha\beta}({{\bf p}},\omega)$. Note $H({{\bf k}}){|\psi_{{{\bf k}}}^{\pm}\big>}=E_{{{\bf k}}}^\pm{|\psi_{{{\bf k}}}^{\pm}\big>}$, with eigenstates $\psi_{{{\bf k}}}^{\pm}(\mathbf{r})\sim u_{{{\bf k}},\pm}\text{exp}(i{{\bf k}}\cdot\mathbf{r})$ and eigenvalues $E_{{{\bf k}}}^\pm=E_0\pm d_{{\bf k}}$. Here we write $\mathbf{d(\bf{k})}=d_{{\bf k}}\hat{\bf{d}}_{{\bf k}}$ with $d_{{\bf k}}=|\mathbf{d(\bf{k})}|$ . Then $\Lambda_{\alpha\beta}$ can be explicitly written as, $$\begin{aligned}
\Lambda_{\alpha\beta}({{\bf p}},\omega)=\sum_{{{\bf k}}ss'}\frac{P_{\alpha\beta}F^{ss'}_{\alpha\beta}({{{{\bf k}}+{{\bf p}}}},{{\bf k}},\omega)}{(\omega_{{{{\bf k}}+{{\bf p}}}}-s'd_{{{{\bf k}}+{{\bf p}}}}+i0^+)(\omega_{{\bf k}}-sd_{{\bf k}}+i0^+)},\nonumber \\\end{aligned}$$ where $$P_{\alpha\beta}F^{ss'}_{\alpha\beta}={{\rm Tr}}\left[ V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}}(u_{{{\bf k}}+{{\bf p}},s'}^{\dagger} \sigma_\alpha u_{{{\bf k}},s}) (u_{{{\bf k}},s}^{\dagger}\sigma_\beta u_{{{\bf k}}+{{\bf p}},s'})\right].$$ Here we have briefly written $\omega_{{\bf k}}=\omega-E_0({{\bf k}})$. $F^{ss'}_{\alpha\beta}$ is the spin coherent factor and we set as a real function (see Sec. for more details), and the complex component has been put in $P_{\alpha\beta}$, which is momentum independent if we assume $V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}}=e^{i\phi_\beta}|V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}}|$, e.g., the phase shift in the scattering is independent of ${{\bf p}}$. The dominator in $\Lambda_{\alpha\beta}({{\bf p}},\omega)$ contributes to singularities and the numerator accounts for further enhancements or depressions, and a combination of both can explain hot-spot features in QPI. Introducing $$\begin{aligned}
&&A_s({{\bf k}},\omega)\equiv-{1\over \pi}
{{{\mathrm{Im}}\, }} [\frac{1}{\omega_{{\bf k}}-sd_{{\bf k}}+i0^+}] =\delta(\omega_{{\bf k}}-sd_{{\bf k}}) \nonumber \\
&&B_s({{\bf k}},\omega)\equiv-{1\over \pi}
{{{\mathrm{Re}}\, }}[\frac{1}{\omega_{{\bf k}}-sd_{{\bf k}}+i0^+}], \nonumber\end{aligned}$$ we can write FT-LDOS (omitting a factor of $-\frac{1}{\pi}$), $$\begin{aligned}
\label{eq:RF-AA}
&&\delta n_{\alpha\beta}({{\bf p}},\omega)\nonumber \\
&=&{{\mathrm{Re}}\, }[P_{\alpha\beta}]\sum_{{{\bf k}}ss'} F^{ss'}_{\alpha\beta}[A_s({{{{\bf k}}+{{\bf p}}}},\omega)B_{s'}({{\bf k}},\omega)+A \leftrightarrow B]\nonumber \\
&+& {{\mathrm{Im}}\, }[P_{\alpha\beta}]\sum_{{{\bf k}}ss'} F^{ss'}_{\alpha\beta}[A_s({{{{\bf k}}+{{\bf p}}}},\omega)A_{s'}({{\bf k}},\omega)+A \leftrightarrow B]. \nonumber \\\end{aligned}$$
Since an autocorrelation of $AA$-type dominates typically a $BB$-type one in the second line of Eq. , we chose only the $AA$ term which corresponds to the joint density of states. As for an approximate yet faithful encoding of $\delta n_{\alpha\beta}({{\bf p}},\omega)$, we propose a so-called reduced response function $\mathcal{R}_{\alpha\beta}({{\bf p}},\omega)$ $$\label{def:RRF}
\mathcal{R}_{\alpha\beta}({{\bf p}},\omega) = \mathcal{S}_{\alpha\beta}({{\bf p}},\omega) + i \mathcal{J}_{\alpha\beta}({{\bf p}},\omega),$$ where $$\begin{aligned}
\label{eq:S&J}
\mathcal{S}_{\alpha\beta}({{\bf p}},\omega)&=&\sum_{{{\bf k}}ss'}F^{ss'}_{\alpha\beta}[A_s({{{{\bf k}}+{{\bf p}}}},\omega)B_{s'}({{\bf k}},\omega)+A \leftrightarrow B]\nonumber \\
\mathcal{J}_{\alpha\beta}({{\bf p}},\omega)&=& \sum_{{{\bf k}}ss'}F^{ss'}_{\alpha\beta}[A_s({{{{\bf k}}+{{\bf p}}}},\omega)A_{s'}({{\bf k}},\omega)].\end{aligned}$$ Note that $\mathcal{S}$ and $\mathcal{J}$ are related by the Hilbert transformation (or the Kramers-Kronig relation). The generalized joint density of states (GJDOS) $\mathcal{J}$ incorporates spin coherent factor into JDOS, and has been applied to analyze QPI for gapless topological systems in terms of spin-selective scattering probability [@beidenkopf_11; @batabyal_16]. The reduced response function contributes to FT-LDOS $\delta n_{\alpha\beta}({{\bf p}},\omega)$ by a projection onto the direction $P_{\alpha\beta}$ in the complex plane, namely $$n_{\alpha\beta}({{\bf p}},\omega)\simeq \frac{1}{2}(P_{\alpha\beta}^*\mathcal{R}_{\alpha\beta}({{\bf p}},\omega) +h.c.).$$
Spin coherent factor {#sec:analysis_F}
--------------------
Now we derive the expression of $P_{\alpha\beta}F^{ss'}_{\alpha\beta}$ and reveal the meaning of the spin coherent factor $F^{ss'}_{\alpha\beta}$. We exploit a representation of density matrix $\rho_{{{\bf k}},s}\equiv u_{{{\bf k}},s}u_{{{\bf k}},s}^+=\frac{1}{2}(1+s\hat{{{\bf d}}}_{{\bf k}}\cdot\boldsymbol{\sigma})$. Since ${{\rm Tr}}[abc]={{\rm Tr}}[bca]$, we have $$\begin{aligned}
&&P_{\alpha\beta}F^{ss'}_{\alpha\beta}({{{{\bf k}}+{{\bf p}}}},{{\bf k}},\omega)\nonumber\\
&=&
\nonumber V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}} {{\rm Tr}}[\sigma_\alpha\rho_{{{{{\bf k}}+{{\bf p}}}},s'}\sigma_\beta \rho_{{{\bf k}},s}]\\
&=&\frac{1}{4}V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}} \sum_{jl}{{\rm Tr}}[\sigma_\alpha\sigma_\beta+\sigma_j\sigma_\alpha\sigma_l\sigma_\beta\hat{d}^j_{{\bf k}}\hat{d}^l_{{{{\bf k}}+{{\bf p}}}}] \nonumber\end{aligned}$$ It should be reminded that $j,l=x,y,z$ while $\alpha,\beta=0,x,y,z$. Some interesting results can be immediately derived for the summation term denoted as $t^{ss'}_{\alpha\beta}({{{{\bf k}}+{{\bf p}}}},{{\bf k}})$. For $\alpha$ and $\beta$, there are three different cases:
1. $\alpha=\beta$. Then $t^{ss'}_{\alpha\alpha}=2(1+ss'\hat{{{\bf d}}}_{{{{\bf k}}+{{\bf p}}}}\cdot\hat R_\alpha\hat{{{\bf d}}}_{{\bf k}})$. Here $\hat{R}_\alpha$ is a mirror reflection with the $\alpha=x,y,z$ axis and $R_0$ means no operation.
2. $\alpha=0,\beta\neq 0$. Then $t^{ss'}_{0\beta}=2iss'L^\beta$, where $(L^x, L^y,L^z)=\hat{{{\bf d}}}_{{{{\bf k}}+{{\bf p}}}}\times\hat{{{\bf d}}}_{{\bf k}}$. By symmetry we have $t^{ss'}_{\beta0}=t^{ss'}_{0\beta}$.
3. $\alpha\neq 0,\beta\neq 0,\alpha$. Then $t^{ss'}_{\alpha\beta}=2ss'(\hat{d}^\alpha_{{\bf k}}\hat{d}^\beta_{{{{\bf k}}+{{\bf p}}}}+\hat{d}^\beta_{{\bf k}}\hat{d}^\alpha_{{{{\bf k}}+{{\bf p}}}})$.
Here the second case is remarkable. An factor of $i$ appears here. If $V^{\beta}_{{{{{\bf k}}+{{\bf p}}}},{{\bf k}}}$ is a real number, then we have $P_{0\beta}=i$, meaning that the GJDOS should account for quasiparticle interference and thus is expected to give sharp QPI features. However, those features may be depressed, for instance, in backscattering processes, where $t^{ss'}_{0\beta}(-{{\bf k}},{{\bf k}})=0$. Nevertheless, GJDOS can be used to explain QPI from scattering between quasiparticles locating at cusps of CCE for time-reversal breaking systems, such as d-wave superconductors [@capriotti_03; @wang_03].
Singular behaviors of the reduced repsonse function
---------------------------------------------------
The reduced response function can be written as two parts. The first part corresponds to terms of $AB+iAA$ type and the second part corresponds to $BA+iAA$ type, as written in Eq. and Eq. . In the following we give a derivation of the first part and the case of the second part can be obtained similarly. Our result shows that both the real and imaginary parts share the same singularities. To focus on the singular behavior, we omit the factor $F_{\alpha\beta}^{ss'}$ temporarily. The first part of RRF is $$\begin{aligned}
&&\sum_{ss'}r^{ss'}_{AB}({{\bf p}},\omega)=\nonumber \\
&&\sum_{{{\bf k}}ss'}\left[A_s({{{{\bf k}}+{{\bf p}}}},\omega)B_{s'}({{\bf k}},\omega)+A_s({{{{\bf k}}+{{\bf p}}}},\omega)A_{s'}({{\bf k}},\omega)\right] \nonumber\end{aligned}$$ Using $1/(x+0^+)=P(1/x)-i\pi\delta(x)$, the first part of RRF can be rewritten as $$\label{eq:cal_RF}
\begin{aligned}
\sum_{ss'}r^{ss'}_{AB}({{\bf p}},\omega)&=\sum_{ss'}\int \frac{1}{\omega-E^{s'}_{{{{\bf k}}+{{\bf p}}}}+i0^+} \delta(\omega-E^s_{{\bf k}}) dk_xdk_y\\
&=\sum_{ss'}\int_{E^s_\mathbf{k}=\omega} \frac{1}{\omega-E^{s'}_\mathbf{k+p}+i0^+}\frac{1}{|\nabla_\mathbf{k} E^s_\mathbf{k}|}dl.
\end{aligned}$$ The second part $\sum_{ss'}r^{ss'}_{BA}({{\bf p}},\omega)$ is obtained by a replacement ${{{{\bf k}}+{{\bf p}}}}\leftrightarrow{{\bf k}}$.
![Behavior of $\dot{E}_{{{\bf k}}_\omega(t_0)+{{\bf p}}}$. Blue and red lines stand for a small departure of scattered-in quasiparticle from ${{\bf k}}_\omega(t_0)$ but with the same ${{\bf p}}$. The left figure shows the general case while the right demonstrates the case of backscattering.[]{data-label="fig:backscattering_math"}](bc_math.jpg){width="0.8\columnwidth"}
The integral is along CCEs of $E^s_\mathbf{k}=\omega$. We may parametrize the $n$-th CCE as $\mathbf{k}^{n,s}_\omega(t)=(f^{n,s}_\omega(t),g^{n,s}_\omega(t))$ with $t\in [0,2\pi)$. For brevity we would omit the index $n$. After a tedious derivation we can show that the integral is singular for ${{\bf p}}$ under a given energy $\omega$ only if there exists $t_0$ that $$\begin{aligned}
\label{eq:singularity}
&&\omega-E^{s'}_{\mathbf{k}^s_\omega(t_0)+\mathbf{p}}=0 \nonumber\\
&&\dot{E}^{s'}_{\mathbf{k}^s_\omega(t_0)+\mathbf{p}}=\left.\frac{dE^{s'}_{\mathbf{k}^s_\omega(t)+\mathbf{p}}}{dt}\right|_{t=t_0}=0.\end{aligned}$$ The first condition promises that the scattered quasiparticle is on the CCE, and the second one points out that ${{\bf p}}$ should be specified. An illustration can be seen in Fig. to show that backscattering meets the above conditions. Moreover, Eq. servers as a mathematical foundation for exploring singular behavior analytically.
To study the singular behavior, we evaluate $r^{ss'}_{AB}(\mathbf{p}+\bm{\delta},\omega)$ with $|\bm{\delta}|\sim0$. An expression can be found as
$$\label{eq:sing_jcurvature}
r^{ss'}_{AB}(\mathbf{p}+\bm{\delta},\omega) \sim
- \frac{1}{|\mathbf{v^s}_{\mathbf{k}_0}||\mathbf{v^{s'}}_{\mathbf{k}_0+\mathbf{p}}|}
\frac{1}{|\bm{\delta}\cdot\bm{\varrho}^{ss'}_{\mathbf{k}_0+\mathbf{p},\mathbf{k}_0}|^{1/2}}
\left\{
\begin{aligned}
& \text{sgn}(\mathbf{n}_{\mathbf{k}_0+\mathbf{p}}\cdot\bm{\varrho}^{ss'}_{\mathbf{k}_0+\mathbf{p},\mathbf{k}_0}), & \bm{\delta}\cdot\bm{\varrho}^{ss'}_{\mathbf{k}_0+\mathbf{p},\mathbf{k}_0} > 0 & \\
& i, & \bm{\delta}\cdot\bm{\varrho}^{ss'}_{\mathbf{k}_0+\mathbf{p},\mathbf{k}_0} < 0 &
\end{aligned}
\right.
,$$
where we have defined the joint curvature, $$\label{j_curvature}
\bm{\varrho}^{ss'}_{\mathbf{k}',\mathbf{k}}=\kappa^{s'}_{\mathbf{k}'}\mathbf{n}^{s'}_{\mathbf{k}'}- \kappa^{s}_{\mathbf{k}}\mathbf{n}^{s}_{\mathbf{k}}.$$ Here $\kappa^s_\mathbf{k}$ is the curvature of the CCE $\omega=E^s_{{\bf k}}$ at $\mathbf{k}$, $\mathbf{n}^s_\mathbf{k}=\mathbf{v}^s_\mathbf{k}/|\mathbf{v}^s_\mathbf{k}|$ the direction vector of the group velocity, $\mathbf{k}_0$ the real solution of Eq. . We remark a similar expression can be found for $r^{ss'}_{AB}(\mathbf{p}+\bm{\delta},\omega)$ by an exchange of ${{\bf k}}_0+{{\bf p}}\leftrightarrow {{\bf k}}_0$.
The expression of singular behavior of Eq. in terms of the joint curvature provides a clear geometrical tool for analysis: (1) The real and imaginary parts of $R(\mathbf{p},\omega)$ share the same singularities and diverge complementarily in a inverse-square-root manner, unless the joint curvature is zero. (2) If the joint curvature of certain singularity is zero, the exponent of the power-law divergence is $-2/3$ and can be even higher. In Fig. , we illustrate singularities with nonzero and zero joint curvatures, and they gives divergence of $-\frac{1}{2}$ and at least $-\frac{2}{3}$, separately.
![CCE that changes sign of the curvature for a toy model. $\mathbf{p}_{1,2,3}$ are singularities of $R_{\alpha\beta}(\mathbf{p},\omega)$ because they connect points with their group velocities (anti)parallel. According to Eq. , only the joint curvature of $\mathbf{p}_3$ is zero which leads to the fact that the order of $\mathbf{p}_3$ is higher than those of $\mathbf{p}_{1,2}$.[]{data-label="fig:toy-jont-curvature"}](toy_jc.jpg){width="0.4\columnwidth"}
Stability of singularities for finite life-time quasiparticles
--------------------------------------------------------------
We consider the stability/robustness of singularities of $\mathcal{R}$ when quasiparticles have finite life-time. We argue that singularities are not stable for $\mathcal{S}$ when a length of CCE is approximate flat. We simply take $0^+\rightarrow \eta$, where $\eta$ is a positive finite number accounting for the finite life-time for quasiparticles.
A finite life-time of quasiparticle will smooth singularities. Note that $B({{\bf k}},\omega)$ and $A({{\bf k}},\omega)$ exhibit distinct behavior around $\omega=E_{{\bf k}}$(see Fig. (3a)). Remarkably, $B({{\bf k}},\omega)$ is zero at $\omega=E_{{\bf k}}$ and changes sign at two sides. The real part of RRF, $\mathcal{S}$, is an autocorrelation of $A({{\bf k}},\omega)$ and $A({{\bf k}},\omega)$, has a large weighting at $|\omega-E_{{\bf k}}|<\eta$. For a large-momentum backscattering in Fig. (3b), scattered-out quasiparticle always has energy $\omega>E_{{\bf k}}$, and thus gives a positive contribution to $\mathcal{S}$ . However, when scattering occurs on an approximated flat CCE (compared to $|{{\bf p}}|$), (see Fig. (3c)), then scattered-out quasiparticle of both $\omega>E_{{\bf k}}$ and $\omega<E_{{\bf k}}$ contribute to $\mathcal{S}$, and those contributions cancel out. The case for near-nesting can be analyzed similarly. In contrast, the joint density of states $\mathcal{J}$ is an autocorrlation of two $A({{\bf k}},\omega)$, which receive always positive contributions around and mainly from $\omega=E_{{\bf k}}$. Thus, when QPI pattern should be given by $\mathcal{S}$, $\mathcal{J}$ would give false features for scattering on an approximately flat band. For instance, a hot spot at ${{\bf p}}=0$ always presents in $\mathcal{J}$ but is absent in $\mathcal{S}$ for $\eta\neq 0$. This can clarify a discrepancy between FT-LDOS and GJDOS for explaining QPI [@derry_15].
![Stability of singularities for the real part of the reduced response function $\mathcal{S}$. (a) Amplitude of $A({{\bf k}},\omega)$ and $B({{\bf k}},\omega)$ with varying $\omega$ under positive finite $\eta$ that models a finite life-time of quasiparticle. (b). Stable singularity. Contributions to $\mathcal{S}$ are all positive (red arrows) under fixed ${{\bf p}}$ (green arrow). (c). Unstable singularity. Positive (red arrow) and negative (blue arrow) contributions to $\mathcal{S}$ under fixed ${{\bf p}}$. (d). Stable singularity for near-nesting scattering. (e). Unstable singularity for near-nesting scattering. []{data-label="fig:S_J_lifetime"}](S_lifetime2.jpg){width="0.8\columnwidth"}
Global patterns in QPI for ideal topological systems {#sec:global_patterns}
====================================================
In the above section, we have revealed that $\mathcal{S}$ and $\mathcal{J}$ share same singularities ideally. While in a real application (where quasiparticles have finite life-time), singularities of $\mathcal{S}$ disappear for approximately flat CCE, but are remained in $\mathcal{J}$, leading to a discrepancy. However, those discrepancies correspond to rather specified QPI patterns, such as hot spot or an asterisk-like pattern around ${{\bf p}}=0$ [@kohsaka_15]. Thus, we can take them as exceptions at first hand when using $\mathcal{J}$. After having justified the applicable of $\mathcal{J}$, we will derive a simple expression of $\mathcal{J}$ that can be used directly for analyzing QPI pattern in a geometrical fashion. We then use this expression to analyze global patterns in QPI for ideal topological Dirac points with varied topological numbers.
Analytical expressions of GJDOS
-------------------------------
Due to the properties of $\delta$-function， $J^{ss'}_{\alpha\beta}({{\bf p}},\omega)$ can be analytically evaluated, : $$\begin{aligned}
J_{\alpha\beta}({{\bf p}},\omega) &= \int F^{ss'}_{\alpha\beta}({{{{\bf k}}+{{\bf p}}}},{{\bf k}}) \frac{\delta(\omega-E^\prime) \delta(\omega-E)}{\left|\frac{\partial (E^{s'}_\mathbf{k+p}, E^s_\mathbf{k})}{\partial (k_x, k_y)}\right| } dEdE^\prime \\
&= \sum_{\mathbf{k}_0ss'} F^{ss'}_{\alpha\beta}({{{{\bf k}}+{{\bf p}}}},{{\bf k}})\left|\frac{\partial (E^{s'}_\mathbf{k+p}, E^s_\mathbf{k})}{\partial (k_x, k_y)} \right|^{-1}_{\mathbf{k}=\mathbf{k}_0}, \\
&= \sum_{\mathbf{k}_0ss'} \frac{F^{ss'}_{\alpha\beta}({{\bf k}}_0+{{\bf p}},{{\bf k}}_0)}{\left| \mathbf{v}^{s'}_{\mathbf{k}_0+\bf{p}}\times \mathbf{v}^s_{\mathbf{k}_0}\right|},
\end{aligned} \label{JDOS_exact}$$ where the sum is taken over all $\mathbf{k}_0$’s in the real solution set of the following two equations, $$\begin{aligned}
E^{s'}_{\mathbf{k}_0+\bf{p}}=\omega, \label{CCE1} \\
E^s_{\mathbf{k}_0}=\omega. \label{CCE2}\end{aligned}$$
The singularity is obtained by setting the dominator $$\label{jacobi}
\mathbf{v}^{s'}_{\mathbf{k}_0+\bf{p}} \times \mathbf{v}^s_{\mathbf{k}_0} = 0,$$ which could be seen as a generalization of Van Hove singularity with regards to joint density of states. The condition $\eqref{jacobi}$ is reached when $\mathbf{v}^s_{\mathbf{k}_0}=0$ or $\mathbf{v}^{s'}_{\mathbf{k}_0+\bf{p}}=0$, or $\mathbf{v}^s_{\mathbf{k}_0} \parallel \mathbf{v}^{s'}_{\mathbf{k}_0+\bf{p}}$. The latter, for example, explains backscattering in metals where quasiparticle of ${{\bf k}}$ is scattered to $-{{\bf k}}$. The condition of singularity also could be viewed as an envelope curve for one-parameter $\mathbf{p}$ curve family $E^{s'}_{\mathbf{k}+\bf{p}}=\omega$ and $E^s_{\mathbf{k}}=\omega$. This allows us to directly draw QPI pattern based on contours of constant energy. It should be pointed out that the condition Eq. can be related to the stationary phase approximation [@liu_12], which has been applied in the study of QPI for surface state of $\text{Bi}_2\text{Te}_3$.
The expression of Eq. , combined with geometrical representation of $F^{ss'}_{\alpha\beta}$ as discussed in Sec. , allows an analysis of QPI pattern from the band structure directly. In the following, we will first give examples for simple models of gapless topological systems.
Global patterns in QPI
----------------------
For a simple circle-like CCE, singularities come from backscattering from ${{\bf k}}_0$ to $-{{\bf k}}_0$. The topology of Dirac points manifests in their structures of spin-momentum binding, and will have effects on QPI patterns through the factor $F^{ss'}_{\alpha\beta}$ that may suppress the singularities. For illustration, we choose the Hamiltonian, $$\label{ham:ideal-dirac}
\mathcal{H}_{{{\bf k}}}=k^n(\cos{n\theta}~ \sigma_1+\sin{n\theta}~\sigma_2)$$ where $n$ is an arbitrary integer, as a model for a Dirac point with topological charge/number $Q=n$ [@zhao_13; @min_08]. Those systems have circle-like CCEs. Quasiparticle with eigenenergy $\pm|{{\bf k}}|^n$ has the wavefunction $u_{{{\bf k}},\pm}\text{exp}(i{{\bf k}}\cdot\mathbf{r})$, where $u_{{{\bf k}},\pm}=(1,\pm e^{-in\theta})^T$. We chose positive $\omega$, and indexes $s$ and $s'$ in the RRF framework are omitted.
We first consider the charge scattering and charge probe channel ($\alpha=\beta=0$) for systems with different topological numbers $Q$. In Fig. we demonstrate how to use GJDOS of Eq. to analyze QPI patterns. A distinct behavior can be identified between $Q=1$ and $Q=2$ that origins from their distinct structures of spin-momentum locking: QPI patterns are suppressed for $Q=1$ while restored for $Q=2$ for backscattering.
![Illustration of how spin direction(red arrows) and group velocity(blue arrows) along the CCE jointly determine QPI pattern. For backscattering group velocities are reversed and the denominator becomes divergent. However, spin directions are reversed for $Q=1$ but are the same for $Q=2$. Thus the divergence is kept only for $Q=2$. []{data-label="fig:jdos_00"}](jdos_00.jpg){width="0.8\columnwidth"}
We now consider $\alpha=\beta$ channel. Note that $F_{\alpha\alpha}=2(1+\hat{{{\bf d}}}_{{{{\bf k}}+{{\bf p}}}}\cdot\hat R_\alpha\hat{{{\bf d}}}_{{\bf k}})$. As the spin lies on $x-y$ plane, $R_z$ will not change $\hat{{{\bf d}}}_{{\bf k}}$. Interestingly, $F_{zz}$ is the same as that of $F_{00}$. Thus, although charge probe channel can not detect magnetic impurity $\sigma_z$ (the well-known prohibition of backscattering for Dirac fermion), a spin $\sigma_z$ resolved probe can [@guo_10]. On the other hand, $R_{x/y}$ will change the sign of $\hat{d}^{y/x}_{{\bf k}}$. As a result, for channel $\alpha=\beta=x/y$ suppressions of singularities will become direct dependent. In Fig. we show the case $\alpha=\beta=x$ for both $Q=1$ and $Q=2$, and QPI patterns have two and four separated hot arcs, separately. One can derive that the number of hot arcs turns to be $2n$ if the topological number $Q=n$, making it a global QPI pattern that can reveal topological number of the underlying system.
![Illustration of effective direction-selective prohibition of backscattering. The combination of scattering scattering channel $\alpha=x$ and probe channel $\beta=x$ effectively rotates the spin of quasiparticle around $x$ axis by $\pi$, leading to a twist of spin interference, and backscattering is prohibited along specified directions. For $Q=1$ and $Q=2$ QPI patterns have two and four hot arcs (brown color), respectively.[]{data-label="fig:jdos_xx"}](jdos_xx.jpg){width="0.8\columnwidth"}
Following the above examples we can exploit global QPI patterns from different combinations of scattering and probe channels as indicators for topological numbers of the underlying systems. We give some topological-number indicators as following:
1. *Odd-even indicator*. For scattering channel $\alpha=0$ and probe channel $\beta=0$, the spin coherent factor reads as $F_{00}\sim(1+\cos n(\theta_{-{{\bf k}}_0}-\theta_{{{\bf k}}_0}))$. As $\theta_{-{{\bf k}}_0}-\theta_{{{\bf k}}_0}=\pi$, then $\mathcal{J}_{00}\sim \lim\limits_{\theta{\rightarrow}\pi}\frac{F_{00}}{\sin\theta}\sim 0$ is zero only for odd $n$. This can be understood from the picture of spin-momentum locking, spins would be antiparallel/parallel for Dirac point with odd/even topological number at a pair $(-{{\bf k}}_0,{{\bf k}}_0)$. Thus, circle-like QPI patterns remain/disappear for even/odd topological numbers.
2. *Integer indicator*. For scattering channel $\alpha=x$ and probe channel $\beta=x$, the spin coherent factor reads as $F_{xx}\sim(1+\cos n(2\theta_{{{\bf k}}_0}+\pi))$. There is a $2n$-fold periodicity with angle $\theta$. Consequently, there are $2|n|$ disconnected hot arcs in the QPI pattern. For $\alpha=\beta=y$, the conclusion holds while hot arcs rotate $\frac{\pi}{2n}$ compared to the case $\alpha=\beta=x$.
3. *Positive-negative indicator.* Counting hot arcs alone can not tell whether topological charge is positive or negative, thus we need an additional indicator. This can be achieved by rotating the scattering channel and probe channel a little, e.g., rotating clockwise, and observing how QPI pattern would rotate. If it rotates clockwise (anti-clockwise), then topological charge is positive (negative).
Further, we can explore quasiparticle interference of two different Dirac points. For instance, intervalley scattering in graphene where two valleys have opposite topological numbers [@mallet_12]. Consider two Dirac points with topological charge $Q_1=n$ and $Q_2=m$, following the analysis of $F_{\alpha\beta}$, we find that the odd-even indicator($\alpha=\beta=0$) would tell odd or even for $n-m$, and integers indicator would give $n+m$ disconnected hot arcs. Those results provide further clues to identify topological charges of different Dirac points, which would be demonstrated on QPI patterns of the graphene family.
Applications to topological matters {#sec:qpi-materials}
===================================
We now apply the reduced response function approach for topological materials [@hasan_10; @qi_11; @zhao_13]. We chose three representative examples, including surface states of topological insulators $\text{Bi}_2\text{Te}_3$ [@fu_09; @zhang_09] and BiTeI [@crepaldi_12; @kohsaka_15], as well as the graphene family [@geim_07]. For those real materials there may be deviations of both dispersion and spin-momentum locking from ideal topological Dirac points described by ${\mathcal{H}}_n({{\bf k}})$ (Eq. ). Thus, they provides playgrounds for studying both local and global patterns of quasiparticle interference.
Surface states of topological insulator: $\text{Bi}_2\text{Te}_3$ {#subsection:qpi-ti}
-----------------------------------------------------------------
In the surface state of $\text{Bi}_2\text{Te}_3$, wrapping terms appears [@fu_09]. At the low energy limit, the CCE can be approximated as a circle, while at larger energy it becomes non-convex. Such distinct geometries of CCEs will lead to different patterns in quasiparticles interference at different energies [@zhou_09; @lee_09_TI]. On one hand, we expect remarkable new features in QPI; on the other hand, we can testify whether indicators of topological number are robust under complex CCEs when the energy increases.
To study the effect risen up by the complicated geometry of CCE at large energy, we consider the surface state of topological insulator $\text{Bi}_2\text{Te}_3$ which can be modeled as [@fu_09; @lee_09_TI], $$\label{ham:BiTe}
{\mathcal{H}}_{TI}({{\bf k}})=v(k_x\sigma_y-k_y\sigma_x)+\lambda k^3\cos{3\theta_{{\bf k}}}\sigma_z,$$ where $\theta_{{\bf k}}=\arctan{k_y/k_x}$. The system ${\mathcal{H}}_{TI}({{\bf k}})$ has time-reversal symmetry and belongs to AII class. It owns a $Z_2$ type topological number of $Q=1$ [@zhao_13]. Dispersions of two branches $s=\pm1$ are opposite and we take only the positive one ($s=1$) for consideration in RRF. At a low energy, the CCE is circle-like, and QPI patterns under different combinations of scattering and probe channels are consistent with those of ideal Dirac point with $Q=1$ (see Fig. (6) for details).
![QPI patterns for surface state of 3D topological insulator $\text{Bi}_2\text{Te}_3$ for circle-like contour of constant of energy at low energy $\omega=0.25$. Backscattering is prohibited for charge impurity and charge probe, as evidenced $\mathcal{S}_{00}$, while is restored for $\sigma_z$ impurity and $\sigma_z$-resolved probe. The QPI pattern shows two bright arcs for $\sigma_x$ impurity and $\sigma_x$-resolved probe. Those QPI patterns under different channels reveal a topological number $Q=1$.[]{data-label="fig:BiTe-QPI-small"}](BiTe-qpi-small.jpg){width="1.0\columnwidth"}
For a sufficient large energy, the wrapping term becomes important and CCE turns to be nonconvex and exhibit six cusps. A scattering between a pair of cusps can satisfy the condition of singularity Eq. but is not backscattering-type. Remarkably, ${{\bf p}}_1$ and ${{\bf p}}_4$ correspond to zero joint curvature. Moreover, ${{\bf p}}_3$ is a backscattering scattering between near-nesting arcs of CCE. Those vectors (${{\bf p}}_1,{{\bf p}}_3,{{\bf p}}_4$) thus have higher degrees of singularity than $\frac{1}{2}$, evidenced by hotter spots than ${{\bf p}}_4$ in joint density of states $J({{\bf p}},\omega)$ (See Fig. ). QPI under different combinations of scattering and probe channels exhibits more complex patterns as a consequence of both topological and geometrical aspects of the band structure. Numeral evaluation of $S_{00}({{\bf p}},\omega)$ shows hot spots locating at ${{\bf p}}_1, {{\bf p}}_4$ that are absent for small energy when CCE is convex. Due to the spin coherent factor, spot is darker for ${{\bf p}}_4$ than that of ${{\bf p}}_1$. Those results fit very well with the FT-STS experiment [@roushan_09; @beidenkopf_11]. We also present $J_{00}({{\bf p}},\omega)$ as a comparison. It can be seen that $J_{00}({{\bf p}},\omega)$ has some sharp features around the center. These features come from joint densities of states with the same arc of the CCE, and are not stable singularities. Thus, a direct application of JDOS should take those as false features, when QPI should be given by the real part of the RRF.
We then investigate QPI in the other situations, e.g, $S_{xx}({{\bf p}},\omega)$ and $S_{zz}({{\bf p}},\omega)$. Compared with QPI at low energy, we can see that while local features are different, the global patterns persist. For $S_{zz}({{\bf p}},\omega)$, the bright closed curve results from backscattering. For $S_{xx}({{\bf p}},\omega)$, QPI pattern is cut into two parts in the middle, showing a 2-fold pattern. Remarkably, ${{\bf p}}_4$ is enhanced while ${{\bf p}}_1$ is suppressed, due to the effective $R_x$ reflection of the spin of scattered-in quasiparticle. These distortions of local features plus persistences of global patterns suggest the robustness of indicators for topological numbers. To reveal these QPI patterns, magnetic impurity with specified direction as well as spin-resolved STS techniques should be required, which are expected to be fulfilled in future FT-STS experiments [@wiesendanger_09; @jeon_17; @cornils_17].
![QPI patterns for non-convex contour of constant energy for surface state of 3D topological insulator $\text{Bi}_2\text{Te}_3$. CCE at $\omega=1$ is given in the right-bottom (parameters in Eq. are $v=1$ and $\lambda=2$). Joint density of states $J$ evidences high degrees of singularities with hotter spots, e.g., ${{\bf p}}_1,{{\bf p}}_4$ due to zero joint curvatures, and ${{\bf p}}_3$ due to near-nesting. QPI patterns are given by $\mathcal{S}_{\alpha\alpha}$ for $\alpha=0,x,z$ scattering and probe channels. As a comparison with $\mathcal{S}_{00}$, $\mathcal{J}_{00}$ gives extra false features around the center.[]{data-label="fig:BeTe-RRF"}](BiTe-RRF2.jpg){width="1.0\columnwidth"}
Surface state of polar semiconductor: BiTeI {#subsection:BiTeI}
-------------------------------------------
We continue to study another topological state: surface state of polar semiconductor BiTeI [@crepaldi_12; @kohsaka_15; @kohsaka_17]. There are two concentric CCEs, due to a contribution from the bulk state. Those two CCEs correspond to two branches of $s=1,-1$, and thus are opposite in spin-momentum locking along the CCE (see Fig. ). Scattering between two concentric CCEs can not be ignored. Moreover, the outer CCE has a larger distortion from the circle. Such a system thus exhibits unconventional properties of the band structure that can lead to novel patterns of quasiparticle interference.
Following Ref. [@kohsaka_15], the surface state can be modeled as $$\begin{aligned}
H_0(k_x,k_y) &=\left(E_0 + \frac{k^2}{2m}E(k)\right)I + V(k)(k_x\sigma_y-k_y\sigma_x)
\nonumber\\ &+\Lambda(k)(3k_x^2-k_y^2)k_y\sigma_z,
\label{ham:BiTeI}\end{aligned}$$ where $I$ is the identity matrix, and $k=\sqrt{k_x^2+k_y^2}$. The last term of Eq. respects the C$_\mathrm{3v}$ symmetry of BiTeI [@ishizaka_11; @crepaldi_12]. Functions are $E(k)=1+\alpha_4k^2 + \alpha_6k^4$, $V(k)=v(1+\beta_3k^2+\beta_5k^4)$ and $\Lambda(k)=\lambda(1+\gamma_5k^2)$. Parameters as set as $m = 0.0168$ eV$^{-1}$Å$^{-2}$, $\alpha_4 = -2.03$ Å$^{-2}$, $\alpha_6 = 87.5$ Å$^{-4}$, $v = 3.13$ eVÅ$^{-1}$, $\beta_3 = -2.01$ Å$^{-2}$, $\beta_5 = 323$ Å$^{-4}$, $\lambda = -41.7$ eVÅ$^{-3}$, $\gamma_5 = 2.43$ Å$^{-2}$, and $E_0 = -0.352$ eV.
In the treatment of RRF, we should take all inter and intra CCEs scattering by a summation over $s=\pm1$ and $s'=\pm1$ in Eq. . Numeral simulations are given in Fig. for $\mathcal{S}_{00},\mathcal{S}_{zz},\mathcal{S}_{zx},\mathcal{S}_{xx}$. As we can see in $\mathcal{S}_{00}$, intra-CCE backscattering is prohibited for both the inner and the outer CCEs, due to a reversal of spin directions. However, inter-CCE scattering is allowed and a bright hexagon appears. QPI patterns in the case of $\mathcal{S}_{zz}$ is reversed: bright hexagon appears only for intra-CCE scattering. The most interesting case is spin $\sigma_x$ scattering and $\sigma_x$ probing channels: there are four disconnected bright arcs that respect a 2-fold symmetry. The outer arcs come from intra-CCE scattering: since $\hat{R}_x$ reflection will not change spin in $x$ direction, quasiparticle interference for ${{\bf p}}_3$ is still suppressed due to an approximately reversal of spins. However, hot arcs arises for wavevector ${{\bf p}}_2$, as $\hat{R}_x(\uparrow)=\downarrow$, thus the spin coherent factor now becomes nonzero. The inner arcs result from inter-CCE scattering: quasiparticle interference of wave vector ${{\bf p}}_1$ is enhanced while ${{\bf p}}_3$ is suppressed. The outer and inter hot arcs locate at horizon and vertical directions, respectively. Thus, they still respect the 2-fold symmetry. We also show $\mathcal{S}_{zx}$ that also respect the 2-fold symmetry. We remark that the $2$-fold symmetry can be considered as global patterns for indicting topological numbers $Q=1$, instead of merely counting the number of disconnected hot arcs.
![Quasiparticle interference for the surface state of BiTeI. In the middle is an inset of contours of constant energy at $\omega=-0.01$, which has two concentric CCEs. QPI for different combinations of scattering and probing channels are given by $\mathcal{S}_{00},\mathcal{S}_{zz},\mathcal{S}_{zx},\mathcal{S}_{xx}$, respectively. Inter-CCE (intra-CCE) scattering are favored (prohibited) in $\mathcal{S}_{00}$, while Intra-CCE (inter-CCE) scattering are favored (prohibited) in $\mathcal{S}_{zz}$. A global pattern of 2-fold symmetry appear in both $\mathcal{S}_{zx}$ and $\mathcal{S}_{xx}$. The four hot arcs in $\mathcal{S}_{xx}$ can be explained by direction selective spin coherent factor due to an effective spin reflection around the $x$ direction $\hat{R}_x$. []{data-label="fig:BiTeI-RRF"}](BiTeI-RRF2.jpg){width="0.8\columnwidth"}
The graphene family
-------------------
Graphene and its relatives of N-layer graphene host two topological Dirac points with opposite topological numbers [@min_08]. Moreover, the topological number can vary with the layer. For example, there are two gapless points with $Q=\pm 2$ for AB stacked bilayer graphene and $Q=\pm 3$ for ABC stacked trilayer graphene. Those make the graphene family an ideal platform for studying global patterns of quasiparticle interferences.
We consider N-layer graphene with ABC stacking. For the i-th layer two sublattices are denoted as $a_i$ and $b_i$, then there are isolated atoms $a_1$ and $b_N$ that are not directly coupled to other layers. Taking $a_1$ and $b_N$ as a pseudospin, we have an effective Hamiltonian $H_{G,N}({{\bf k}})=({{\mathrm{Re}}\, }{f({{\bf k}})})^N\sigma_x+({{\mathrm{Im}}\, }{f({{\bf k}})})^N\sigma_y$ to model ABC stacking N-layer graphene [@min_08]. Here $f({{\bf k}})=1+e^{i{{\bf k}}\cdot\bf{G}_1}+e^{i{{\bf k}}\cdot\bf{G}_2}$ with $\bf{G}_1,\bf{G}_2$ being primitive vectors for graphene. Topological Dirac points locate at $\mathbf{K}$ and $\mathbf{K}'$ with topological charges $Q=\pm N$ respectively. At low energy limit $H_{G,N}({{\bf k}})$ can be approximated by $H_{\pm N}({{\bf k}})$ with $f(k) \sim k^N$. It should be noted that the pseudospin here corresponds to sublattice, and the physical meaning of scattering and probing channel should be adjusted accordingly. For point-like impurity locating at site $a/b$ the scattering channel is $\tau_{a/b}=\frac{1}{2}(\sigma_0\pm\sigma_z)$ (here we have shortnoted $a/b$ for $a_1/b_N$). Similarly, LDOS is measured at site $a/b$ and probing channel is $\tau_{a/b}$. The reduced response functions $\mathcal{R}_{00}({{\bf p}},\omega)$ and $\mathcal{R}_{zz}({{\bf p}},\omega)$ can be obtained as follows, $\mathcal{R}_{00}({{\bf p}},\omega)=\frac{1}{2}(\mathcal{R}_{aa}({{\bf p}},\omega)+\mathcal{R}_{bb}({{\bf p}},\omega)+\mathcal{R}_{ab}({{\bf p}},\omega)+\mathcal{R}_{ba}({{\bf p}},\omega)+)$,$ \mathcal{R}_{zz}({{\bf p}},\omega)=\frac{1}{2}(\mathcal{R}_{aa}({{\bf p}},\omega)+\mathcal{R}_{bb}({{\bf p}},\omega)-\mathcal{R}_{ab}({{\bf p}},\omega)-\mathcal{R}_{ba}({{\bf p}},\omega))$, where $\mathcal{R}_{\mu\nu}$($\mu=a,b$ and $\nu=a,b$) correspond to probe channel $\tau_\mu$ and scattering channel $\tau_\nu$.
![QPI pattern for graphene family evaluated by $\mathcal{S}_{00}({{\bf p}},\omega)$ and $\mathcal{S}_{zz}({{\bf p}},\omega)$, as shown in (a) and (b), respectively. Here we chose $\omega=0.3,0.15,0.1$ for $N=1,2,3$, respectively. Inter-valley scattering leads to remarkable $2N$ hot arcs for $N=1,2,3$. CCEs and the joint density of states are shown in (c) for $N=1$. []{data-label="fig:qpi-graphene"}](graphene-rrf-3.jpg){width="1.0\columnwidth"}
As two valleys $\bf{K}$ and $\bf{K'}$ have the opposite topological charges, it is expected to see distinct QPI patterns between the center of ${{\bf p}}$-space due to intravalley scattering and those due to intervalley scattering away from the centre, such as around $\pm 2\mathbf{K}$, $\pm 2\mathbf{K}'$ and $\pm 2(\mathbf{K}-\mathbf{K}')$. We numeral evaluate $\mathbf{S}_{00}({{\bf p}},\omega)$ and $\mathbf{S}_{zz}({{\bf p}},\omega)$ for ABC stacked N-layer graphene for $N=1, 2, 3$ respectively, as seen in Fig. . For both channels, distinct $2N$ pieces of disconnected hot arcs appear for intravalley scattering and are rotated relatively. Those distinct features of QPI both for intervalley scattering indicate opposite topological charges for Dirac point located at the valleys. The simulations fit well with the FT-STS experiments [@brihuega_08; @mallet_12] for monolayer graphene, while better resolution is required for bilayer graphene. For intravalley scattering, a bright circle appears for channels $\alpha=\beta=z$ and is absent $\alpha=\beta=0$ for monolayer graphene. The persistent presence of bright circle around the center for bilayer and trilayer graphene may result from high density of states from dispersions of $k^N$ ($N>1$). We mention that joint density of states give false hot spots at ${{\bf p}}=0,\pm 2\mathbf{K},\pm 2\mathbf{K}$, and $\pm 2(\mathbf{K}-\mathbf{K}')$.
Conclusions
===========
We have developed a reduced response function approach for analyzing and simulating quasiparticle interference under different scattering and probing channels. The applicability of the generalized joint density of state has been clarified. We have analytically shown how singularities and spin coherent factor jointly determine QPI patterns. Remarkably, those local features and global patterns in QPI can be used to infer geometry details of dispersions and topology of band structures for the underlying system. We have also proposed indicators of topological numbers from global patterns of QPI for topological Dirac points. We have numerally simulated QPI of different topological materials, whose complicated geometrical features yet robust global patterns are evidenced and can be nicely captured in the RRF framework.
We remark that advances in spin-resolved STS technique [@wiesendanger_09; @jeon_17; @cornils_17] may experimentally visualize novel QPI patterns explored in this paper. Further investigations along this line include finding QPI that should be explained by the generalized joint density states, and an extension of the reduced response function approach for systems beyond two-band Hamiltonian description.
acknowledgement
===============
The work was supported by the GRF (No. HKU173057/17P) and CRF (No. C6005-17G) of Hong Kong, and the NKRDP of China (Grant No. 2016YFA0301800).
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---
author:
- 'Mangesh Mandlik,'
- and Somyadip Thakur
title: Stationary Solutions from the Large $D$ Membrane Paradigm
---
Introduction
============
The classical dynamics of $D$ dimensional spacetime is governed by Einstein’s theory in which matter influences spacetime via the celebrated Einstein’s equation $$R_{MN} - \frac{R}2 G_{MN} + \frac{D-1}\lambda G_{MN} = 8\pi T_{MN}$$ (in the natural units $c = G_N = e = 1$). The dynamical degrees of the spacetime are its metric $G_{MN}$. However even in the absence of matter ($T_{MN}=0$), it is one of the hardest equations to solve for a generic solution. This is because the curvature tensor $R_{MN}$ depends on the inverse of the metric tensor, which needs to be written in terms of metric components using the minors and the determinant, and ultimately we get $D(D+1)/2$ coupled differential equations of $D(D+1)/2$ functions of coordinates, which consist of rational functions of the metric components and their derivatives.\
The diffeomorphism gauge symmetry of this equation can be exploited to choose a clever coordinate system which enables us to obtain exact solutions in highly symmetric static or stationary situations. This equation can also be linearized to get weak gravitational solutions such as gravitational waves. However these techniques can not be generalized to interesting dynamical situations like, say, collision and merger of two black holes. For these, one needs to rely on solving this equation numerically, and the nonlinearity is quite demanding on the computational power.\
Recently the authors of [@Emparan:2013moa; @Emparan:2013xia; @Emparan:2013oza; @Emparan:2014aba] made an observation that in the limit where $D \gg 1$, the degrees of freedom of the black hole spacetime separate into light degrees with length scale $r_0$ and heavy degrees with length scale $\frac{r_0}D$, where $r_0$ is a characteristic length scale associated with black hole horizon. They also showed in [@Emparan:2014cia; @Emparan:2015rva] that the quasinormal modes about a static, spherically symmetric black hole in such a large number of dimensions show a similar separation of scales. So it should be possible to ‘integrate out’ the heavy modes and get an effective theory of light modes which are way fewer in number.[^1]\
In [@Bhattacharyya:2015dva] a systematic program of obtaining an effective theory of the light degrees of freedom of large $D$ black holes was initiated and named ‘a membrane paradigm at large $D$’. It was found out that $1/D$ becomes a new perturbative parameter in solving Einstein’s equation for black hole solutions which are symmetric in all but a finite number of directions. If we start with a particular very simple ansatz for a black hole metric that solves the Einstein’s equation without matter and cosmological constant at the leading order in $1/D$, and try solving for the perturbative corrections, it is observed that the black hole horizon acts like a codimension one ‘membrane’ embedded in a flat $D$ dimensional background and characterized by its shape as embedded in the background and a ‘velocity field’ defined on it. the dynamics of these shape and velocity is governed by the so called ‘membrane equations’. As the quasinormal modes about a static, spherically symmetric membrane match exactly with the light quasinormal modes obtained in [@Emparan:2014aba], this membrane paradigm is indeed an effective theory of the light modes.\
The perturbative expansion in $1/D$ indeed looks similar to the long wavelength expansion of Hydrodynamics, but there are many differences emerging from the nonlinearity of membrane equations at any given order. As a result, qualitative features of black hole like entropy production are visible even at the leading order. [@Dandekar:2016fvw] solve the next order in the perturbation and find the entropy production, indicating this membrane obeys the second law of thermodynamics like a black hole does. [@Bhattacharyya:2015fdk] addresses a membrane paradigm for charged asymptotically flat back holes, [@Bhattacharyya:2017hpj; @Bads2] deal with uncharged black hole membranes in spacetimes with cosmological constant while [@Bhattacharyya:2018ads] works with the charged black hole membranes in spacetimes with cosmological constant. Also, [@Bhattacharyya:2016nhn] has shown that the membrane equations that dictate the dynamics of the charged membranes in a flat background are actually a statement of conservation of a stress tensor and a charge current associated with the membrane. [^2] [^3]\
The equations that are obtained at any order in $1/D$ are still a system of nonlinear differential equations, but the advantage of this approach over solving the Einstein’s equations with brute force is that membrane equations don’t involve an inverse of some dynamical matrix, so the equations written in terms of the basic degrees of freedom of the membrane are ‘less nonlinear’, and hence will be less computationally taxing when this formalism is developed for highly dynamical processes.[^4]\
Though this membrane paradigm is potentially very useful for improving the efficiency of numerical black hole solutions, it would be very satisfying if it could give some analytic black hole solutions, at least perturbatively in $1/D$. An independent approach has led the authors of [@Suzuki:2015iha] to the conclusion that the stationary uncharged asymptotically flat or AdS black holes can be represented as membranes at the leading order in $1/D$ described by a very simple equation. So we can expect the leading order membrane equation presented in [@Bhattacharyya:2015fdk] or [@Bhattacharyya:2018ads] to reduce to the same equation when specialised to a stationary setting. The primary goal of this paper is to seek for the membrane equations for charged membrane in flat and AdS backgrounds specialised to general stationary configurations. These equations turn out to be very simple as expected, and reduce further to the equation in [@Bhattacharyya:2018ads] in the uncharged limit. These equations cannot be solved exactly in the simple examples we probe, namely a charged membrane rotating in a single plane, but can be solved perturbatively with ease. Also, these equations turn out to govern the thermodynamics of these membranes, which is quite expected because the equations we started from are the conservation of stress tensor and the charge current on the membrane.\
This paper is organized as follows. In section \[secstat\] we specialize the charged membrane equations for stationary configurations. In \[kill\] we establish which class of membranes corresponds to stationary configurations, and in \[subsecstat\] we find out the effective ‘stationary membrane equations’ that describe such membranes. In section \[thermo\] we describe the thermodynamics of these stationary membranes. The effective stress tensor and charge current on these membranes are proposed in \[EffcurrST\], thermodynamic data is extracted from them in \[flaw\] and this data is shown to satisfy the first law of thermodynamics, defining the temperature and chemical potential associated with the membrane in the process.\
Then in section \[rotmem\], we obtain the axially symmetric membrane solutions in flat and AdS backgrounds. We find exact solutions for uncharged membranes in both of these cases, while we obtain the charged counterparts perturbatively. In section \[rottherm\] we show how the thermodynamics of \[thermo\] is realized for the rotating membranes. The algebraic details relevant to this paper are given in the appendices in the end.
Membrane equations for stationary configurations {#secstat}
================================================
Before we begin, we give a quick overview of the notation used in this paper. Let’s denote the coordinates of the background spacetime by $\{X^M\}$ and the coordinates on the membrane world volume by $\{y^{\mu}\}$. The upper case Latin indices are reserved for the quantities that transform under the spacetime coordinate transformations, while the lower case Greek indices are for the quantities that transform under the membrane coordinate transformations. The membrane covariant quantities are written in terms of spacetime covariant quantities using the pullback factors $e^M_{\mu} \equiv \frac{\partial X^M}{\partial y^{\mu}}$ as
$$T_{\mu_1\cdots\mu_n} = T_{M_1\cdots M_n}e^{M_1}_{\mu_1}\cdots e^{M_n}_{\mu_m},$$
while the spacetime contravariant quantities are written in terms of membrane contravariant quantities as
$$T^{M_1\cdots M_n} = T^{\mu_1\cdots\mu_n}e^{M_1}_{\mu_1}\cdots e^{M_n}_{\mu_m}.$$
A large $D$ black hole membrane is characterized by:
- Its shape as embedded in the background spacetime, i.e. the shape function $\rho(X) = 1$. The quantities that appear here which encode this shape function are a unit (outward) normal $~n_M \equiv N\partial_M\rho~$ where $~N^{-2} = \partial_M\rho\partial^M\rho~$, extrinsic curvature tensor $K_{MN}$ (with trace $K$) and its derivatives. The lowering and raising are done by the background metric $G_{MN}$ and its inverse $G^{MN}$ respectively.
- A unit normalized velocity field $u^M$, $u^Mu_M = -1$ that lies in the membrane, i.e. $u^M n_M = 0$.
- A scalar charge field $Q(y)$.
The vector membrane equation has components entirely in the membrane. The constituents of both the vector and the scalar membrane equations such as the extrinsic curvature, the velocity and charge field also lie entirely in the membrane. Therefore in this paper we will work with quantities as a function of $\{y^\mu\}$ and they will transform under membrane coordinate transformations, i.e. they will have Greek lower case indices. Horizon Tunneling Revisited: The Case of Higher Dimensional Black Holes $$\begin{split}
g_{\mu\nu} &= G_{MN}e^M_{\mu}e^N_{\nu} ~~~~~~~~\text{(Membrane metric)},\\
K_{\mu\nu} &= K_{MN}e^M_{\mu}e^N_{\nu},\\
u_\mu &= u_Me^M_{\mu},\\
p_{\mu\nu} &= g_{\mu\nu} + u_\mu u_\nu~~~~~~~~\text{(Projector orthogonal to velocity)}.
\end{split}$$
Also, whenever the covariant derivatives and Riemann and Ricci curvature tensors are seen wearing hats, they should be assumed to be taken w.r.t. the membrane metric. Their unhatted counterparts are defined w.r.t. the background metric.
Large $D$ membrane equations {#memeq}
----------------------------
The charged membrane equations in presence of a cosmological constant are obtained in [@Bhattacharyya:2018ads], of which the vector equation is: $$\label{VeqAdS}
{\cal E}_\mu \equiv \left(\frac{\hat{\nabla}^2 u_\nu}{K}-(1-Q^2)\nabla_\nu \ln K + u\cdot K_\nu -(1+Q^2)u\cdot\hat{\nabla}u_\nu\right)p^\nu_\mu = 0,$$
while the scalar equation is: $$\label{SeqAdS}
{\cal E} \equiv \frac{\hat{\nabla}^2 Q}{QK}-u\cdot\nabla \ln Q - u\cdot\nabla\ln K +u\cdot K \cdot u+\frac{u\cdot R \cdot u}{K} = 0.$$
In the flat limit ($R_{MN} = 0$), the equations reduce to those in [@Bhattacharyya:2015fdk]. In uncharged limit ($Q = 0$) the vector equation reduces to the membrane equation in [@Bhattacharyya:2017hpj], while the scalar equation vanishes identically. This and the next section of this paper deal with stationary membranes in empty backgrounds with arbitrary cosmological constant (with magnitude ${\cal O}(D)$ or smaller) including zero, and any arbitrary charge (with $|Q| \sim {\cal O}(1)$ or smaller) including, again, zero. However in the section after that, \[rotmem\], where we specialize to the particular examples of rotating membrane solutions, we work with backgrounds of nonpositive cosmological constant, i.e. flat and AdS backgrounds, and we treat the uncharged and charged cases separately in each of them.
Time Like Killing Vectors and Stationary Membrane velocity configuration {#kill}
------------------------------------------------------------------------
As mentioned in the introduction, this paper deals with stationary solutions of the membrane equations and . But what defines the stationary solutions?\
The membrane equations that dictate the dynamics of the large $D$ black hole membrane are actually the equations of conservation of charge current and stress tensor defined on the membrane, as shown in [@Bhattacharyya:2016nhn] [^5]. This current and stress tensor can be seen as belonging to a fluid living on the membrane world volume [^6]. So if we look at the stress tensor of this ‘fluid’
$$\label{effST}
8\pi T_{\mu\nu}^{(eff)} =\left( \frac{1+Q^2}{2} \right) K u_\mu u_\nu + \left( \frac{1-Q^2}{2} \right)K_{\mu\nu}- \Sigma_{\mu\nu}
-Q\left( u_\mu{\cal V}_\nu + u_\nu{\cal V}_\mu\right),$$
where $$\label{visc}
\begin{split}
\Sigma_{\mu\nu} &= \tilde{\Sigma}_{\mu\nu} - \tilde{\Sigma}^\alpha_\alpha p_{\mu\nu},\\
\tilde{\Sigma}_{\mu\nu} &= p^{\alpha}_{\mu}\left(\frac{\hat\nabla_\alpha u_\beta + \hat \nabla_\beta u_\alpha}{2} \right)p^{\beta}_{\nu},
\end{split}$$ and $$\label{calv}
{\cal V}_\mu =\hat\nabla_\mu Q +Qu\cdot\hat{\nabla}u_{\mu},$$ we can make out the shear viscosity term $\Sigma_{\mu\nu}$. [^7] This suggests that the dynamics of the membrane is dissipative, and a general dynamical membrane will eventually settle down to a stationary configuration, and it will be characterized by disappearance of the dissipation term. Thus for the stationary configuration, $\Sigma_{\mu\nu} = 0$.\
Now consider the case in which the membrane has a time like Killing vector field $k^{\mu}$. Let us define a unit velocity field $u^{\mu}$ proportional to this time like Killing vector field as $$u^{\mu}=\gamma k^{\mu},$$ where $$\gamma^{-2} = -k_{\mu}k^{\mu}.$$ If we identify this velocity field with the velocity field that defines the membrane, we have then
$$\label{Sigstat}
\begin{split}
\tilde{\Sigma}_{\mu\nu} &= p_{\mu}^{\alpha}p_{\nu}^{\beta}\left(\hat{\nabla}_{\alpha}u_{\beta}+\hat{\nabla}_{\beta}u_{\alpha}\right),\\
&= p_{\mu}^{\alpha}p_{\nu}^{\beta}\left(\nabla_{\alpha}(\gamma k_{\beta})+\nabla_{\beta}(\gamma k_{\alpha})\right),\\
&= p_{\mu}^{\alpha}p_{\nu}^{\beta}\left(k_{\beta}\nabla_{\alpha}\gamma +k_{\alpha}\nabla_{\beta}\gamma \right),\\
&= 0.
\end{split}$$
Where going from second line to the third we have used the Killing equation and then we have used the fact that $p_{\mu\nu}$ is a projector orthogonal to $u^{\mu}$ ,and hence orthogonal to $k^{\mu}$. means $\Sigma_{\mu\nu} = 0$.\
Let’s appreciate the significance of this result for a moment. First we choose a timelike Killing vector $k(X)$ of the background, Since this generates an isometry of the empty background, it is also the symmetry of this background. Then let’s construct a membrane shape in this background that respects this symmetry. Now look at the restriction of $k(y)$ on the membrane. Since the membrane breaks any symmetry in the direction orthogonal to it, this restriction shouldn’t have a component normal to the membrane. $k(y)$ turns out to be the Killing vector of the metric induced on the membrane as well because $$\hat{\nabla}_\mu k_\nu+\hat{\nabla}_\nu k_\mu = e^M_\mu\left(\nabla_Mu_N+\nabla_Nu_M\right)e^N_\nu = 0.$$ Then we unit normalize $k(y)$ to get $u(y)$, and choose $u(y)$ to be the velocity field of the membrane.[^8] ensures that a membrane constructed in this way is stationary.[^9]\
Also, physically it makes sense that the charge field also should have the symmetry generated by $k$, making $k$ the generator of a symmetry of the membrane. However, this fixes neither the embedding of the membrane in the background spacetime nor the charge field as a function of the non-symmetry directions entirely, for that we have to solve the membrane equations.\
Stationary Membrane equations {#subsecstat}
-----------------------------
The previous subsection enables us to fix the velocity field on the membrane, so now there are only two unknown functions left, namely the shape function and the charge density function. On the other hand the number of independent equations to be solved is still $D-2$. For the system to be solvable, these equations must boil down to just two independent equations, and to respect diffeomorphism invariance, they must be scalar equations.\
Using for stationary configurations, the scalar equation reduces to
$$\label{Sstat}
\tilde{\nabla}^2\ln\frac{Q}{\gamma} = Ku\cdot\hat{\nabla}\ln(QK).$$
Using and , and rearranging terms we convert the vector equation into
$$\label{Vstat}
(1-Q^2)\tilde{\nabla}_\mu\ln K - (1+Q^2)\tilde{\nabla}_\mu\ln\gamma = ((1-Q^2)u\cdot\hat{\nabla}\ln K - (1+Q^2)u\cdot\hat{\nabla}\ln\gamma))u_\mu.$$
Since the membrane is symmetric along $u$, the RHS of both and vanish. From we get, assuming reasonable boundary conditions on the compact and extended directions of the membrane $$\label{Qgam}
Q = \alpha\gamma,$$ where $\alpha$ is a constant. Substituting in while setting the RHS of to zero, and an easy manipulation gives $$\tilde{\nabla}_\mu\ln\left(K\left(\frac{1}{\gamma}-\alpha^2\gamma\right)\right) = 0.$$ Which can be integrated over the membrane, and ultimately we get $$\label{Statsol}
\begin{split}
Q &= 2\sqrt{2\pi}\mu\gamma,\\
K &= \frac{4\pi T\gamma}{1-\alpha^2\gamma^2} = \frac{4\pi T\gamma}{1-Q^2}~~,
\end{split}$$ $T$ and $\mu$ are constant all over the membrane. The reason behind putting the seemingly arbitrary numerical factors in these equations will be clear when we look at the thermodynamic of the stationary membranes.\
In the absence of charge, the stationary membrane equations reduce to
$$\label{Statsolunch}
K = 4\pi T\gamma$$
Thermodynamics of stationary membranes {#thermo}
======================================
Effective stress tensor and charged current {#EffcurrST}
-------------------------------------------
The effective stress tensor and charge current belong to a family of stress tensors and charge currents the conservation of which gives the membrane equations at the leading order. We define those by simply covariantizing the ones defined in equations (6.19) and (6.34) in [@Bhattacharyya:2016nhn]. The actual stress tensor and charge current need to be found out by the procedure detailed in [@Bhattacharyya:2016nhn]. However for our purpose we only need the leading order stress tensor and charge current, and since the actual and effective quantities differ by subleading terms which don’t contribute even to the membrane equations, the leading parts can equally be read off from the effective quantities. The colour coding used in the equations in this section is explained in the appendix \[idens2\]
### The effective charge current and its conservation {#eccc}
We propose, as discussed above, the effective current on the membrane to be
$$\label{effcurr}
2\sqrt{2\pi} J_\mu^{(eff)}
=~Q {K} u_\mu -{\cal V}_\mu,$$
where ${\cal V}$ is defined in .
Therefore taking its divergence gives $$\label{effcurrcons}
\begin{split}
2\sqrt{2\pi} \hat{\nabla}^{\mu}J_\mu^{(eff)} =& u\cdot\nabla(KQ) \textcolor{blue}{+KQ\hat{\nabla}\cdot u}- \hat{\nabla}\cdot{\cal V}\\
=& ~u\cdot\nabla(KQ) - \hat{\nabla}^2Q - u\cdot R\cdot u - K~u\cdot K\cdot u + {\cal O}(1)\\
=& -KQ\left(\frac{\hat{\nabla}^2 Q}{QK}-u\cdot\nabla \ln Q - u\cdot\nabla\ln K +u\cdot K \cdot u+\frac{u\cdot R \cdot u}{K}\right)\\ &+ {\cal O}(1).
\end{split}$$ Here we have used . If we demand this current to be conserved, i,e, its divergence to vanish, implies the scalar membrane equation .
### The effective stress tensor and its conservation {#estc}
When we take the divergence of the effective stress tensor on the membrane, we get $$\label{effSTcons}
\begin{split}
8\pi \hat{\nabla}^\mu T_{\mu\nu}^{(eff)} =& \left( \frac{1+Q^2}{2} \right) Ku\cdot\hat{\nabla}u_{\nu} + \left( \frac{1+Q^2}{2} \right) (u\cdot\nabla K) u_{\nu}+KQ(u\cdot\nabla K) u_{\nu}\\
&+ \textcolor{blue}{\left( \frac{1+Q^2}{2} \right) K(\hat{\nabla}\cdot u)u_{\nu}}+\left( \frac{1-Q^2}{2} \right)\hat{\nabla}^\mu K_{\mu\nu} \textcolor{blue}{+Q(\nabla^\mu Q) K_{\mu\nu}}\\
&- \hat{\nabla}^{\mu}\Sigma_{\mu\nu} - Q(\hat{\nabla}\cdot{\cal V}) u_{\nu} \textcolor{blue}{-Q(\hat{\nabla}\cdot u){\cal V}_{\nu}-Qu\cdot\hat{\nabla}{\cal V}_{\nu}-Q{\cal V}\cdot\hat{\nabla}u_\nu}\\
&-\textcolor{blue}{(u\cdot\nabla Q){\cal V}_\nu - ({\cal V}\cdot\hat\nabla Q)u_\nu}\\
=& -\frac{K}2\left(\frac{\tilde{\nabla}^2 u_\mu}{K}-(1-Q^2)\nabla_\mu \ln K + u\cdot K_\mu -(1+Q^2)u\cdot\tilde{\nabla}u_\mu\right)\mathcal{P}^\mu_\nu\\
&- KQ^2\left(\frac{\hat{\nabla}^2 Q}{QK}-u\cdot\nabla \ln Q - u\cdot\nabla\ln K +u\cdot K \cdot u+\frac{u\cdot R \cdot u}{K}\right)u_{\nu}\\ &+{\cal O}(1).
\end{split}$$ Here we have used and . If we demand this stress tensor to be conserved, i,e, its divergence to vanish, the components of orthogonal to $u$ imply the vector membrane equation , whereas the component of along $u$ implies the scalar membrane equation .
### The uncharged limit {#unlim}
Setting $Q = 0$ in makes it identically vanish, while gives in this special case $$16\pi T_{\mu\nu}^{Q=0} = K u_\mu u_\nu + K_{\mu\nu}- 2\Sigma_{\mu\nu}
+ {\cal O}\left(\frac{1}{D}\right),$$ which matches with the stress tensor of the uncharged membrane proposed in [@Dandekar:2017aiv].\
In short, the uncharged limit and the flat limit of and agree with the known result, while their conservation gives the membrane equations. We take it as a sufficient evidence for these to be the AdS membrane effective charge current and stress tensor, and proceed to compute thermodynamic quantities using them. We will show that these quantities satisfy the first law of thermodynamics and thus further bolstering this claim.
Thermodynamic quantities from membrane data {#unch1stlaw}
-------------------------------------------
In this section we derive the thermodynamic quantities associated with a stationary membrane from the membrane data, i.e, the stress tensor, charge current and the entropy current. We show that these thermodynamic quantities obey the first law of black hole thermodynamics.\
We can define a conserved current from the leading order stress tensor as [^10] $$\label{concur}
J_{(E)}^\mu = -T^{\mu\nu}k_\nu = (1+Q^2)\frac{K}{16\pi\gamma}u^\mu.$$ Now let’s consider a spacelike slice of the membrane world volume with normal along $dt$. The conserved ‘total energy’ on this slice is given by
$$\label{memtoten}
\tilde{E} = \int d^{D-2}x \sqrt{-G} J_{(E)}^0 = \int d^{D-2}x \sqrt{-G} (1+Q^2)\frac{K}{16\pi},$$
since $u^0 = \gamma$. Note that this ‘total energy’ also includes the contributions from angular and linear momenta if $k$ has contributions from the rotation and spatial translation Killing vectors respectively.
A conserved charge is defined on such a slice from the leading order charge current $J^\mu = \frac{KQ}{2\sqrt{2\pi}}u^\mu$
$$\label{memcharge}
q = \int d^{D-2}x \sqrt{-G} J^0 = \int d^{D-2}x \sqrt{-G}\frac{KQ\gamma}{2\sqrt{2\pi}}.$$
And finally the entropy is simply
$$\label{mementropy}
S = \int d^{D-2}x \sqrt{-G}\frac{u^0}{4} = \int d^{D-2}x \sqrt{-G}\frac{\gamma}{4}.$$
All the integrations are taken over the aforementioned slice of the membrane worldvolume, $\{x\}$ are the $D-2$ coordinates on that slice.
First law of thermodynamics {#flaw}
---------------------------
Now let’s assume that the relative variations in $Q$, $\gamma$ and $K$ are of the same order, i.e. $$\frac{\delta Q}{Q} \sim \frac{\delta K}{K} \sim \frac{\delta \gamma}{\gamma}.$$ Let $l$ be a typical length scale of the membrane. Note that $K \sim \frac{D}{l}$ while $\sqrt{-G} \sim {l^{D-2}}$. Thus, $\frac{\delta\sqrt{-G}}{\sqrt{-G}} \sim -D\frac{\delta K}{K}$. Thus when the variations of thermodynamic quantities , and are taken, the variation of the volume factor dominates over the variation of the quantities that are getting integrated.
So $$\label{vartherm}
\begin{split}
\delta S &= \int_{\rho=1} d^{D-2}x~ \delta\sqrt{-G}~\frac{\gamma}{4},\\
\delta \tilde{E} &= \int_{\rho=1} d^{D-2}x~ \delta\sqrt{-G}~(1+Q^2)\frac{K}{16\pi},\\
\delta q &= \int_{\rho=1} d^{D-2}x~ \delta\sqrt{-G}~\frac{KQ\gamma}{2\sqrt{2\pi}}.
\end{split}$$
Now let’s simplify the combination $T\delta S + \mu\delta q$, where $T$ and $\mu$ are defined in . $$\label{thermoproof}
\begin{split}
T\delta S +\mu\delta q &= \int_{\rho=1} d^{D-2}x~ \delta\sqrt{-G}\left(T\frac{\gamma}{4} + \mu\frac{KQ\gamma}{2\sqrt{2\pi}}\right),\\
&= \int_{\rho=1} d^{D-2}x~ \delta\sqrt{-G}\left(\frac{(1-Q^2)K}{4\pi\gamma}\frac{\gamma}{4} + \frac{Q}{2\sqrt{2\pi}\gamma}\frac{KQ\gamma}{2\sqrt{2\pi}}\right),\\
&= \int_{\rho=1} d^{D-2}x~ \delta\sqrt{-G}~(1+Q^2)\frac{K}{16\pi},\\
&= \delta \tilde{E}.
\end{split}$$
Thus if we identify $T$ to be the membrane temperature and $\mu$ to be the membrane chemical potential, is the statement that a stationary membrane satisfies the first law of thermodynamics at the leading order in large $D$.
Stationary rotating membrane solutions {#rotmem}
======================================
In section \[secstat\] we have derived the stationary membrane equations which have solutions dual to black holes in large D for the flat space, $AdS$-spaces both in presence and absence of charge. In this section we would specialize in solving the membrane equations for some axially-symmetric configurations.
The effective membrane equation for Stationary ‘axially-symmetric’ configurations in flat space
-----------------------------------------------------------------------------------------------
In this subsection we will find axially-symmetric solutions to and in the ambient flat space both in presence and absence of charge which will be dual to asymptotically flat rotating black holes in presence and absence of charge respectively.\
A time like killing vector field of which generates the axially-symmetry of the flat background and hence the final membrane configuration is given by $$k=\frac{\partial}{\partial t}+\omega^{i}\frac{\partial}{\partial\theta_i}.$$ Hence, the unit time-like velocity field of the membrane that we consider is given by $$\label{gamma}
u^{\mu}=\gamma k^{\mu},\quad \text{where} \quad \gamma=\frac{1}{\sqrt{\left(1-\sum_{i}r_i^2 \omega_i^2\right)}}.$$ Now, the shape function of the surface that we consider should be such that its normal is orthogonal to the velocity field $u$. Also, we consider situation which preserves the $SO(D-2p-2)$ isometry of the metric . The surface equation that we consider is given by $$\label{shapeansatz2}
s^2=2g(\{r_i\}).$$ The unit normal to the above surface is given by $$n=2N\left(s ds-\frac{\partial g}{\partial r_i}\right), \quad \text{where,} \quad N=\frac{1}{2\sqrt{s^2+\sum_i\left(\frac{\partial g}{\partial r_i}\right)^2}}.$$ The determinant of the metric is given by $$\label{metdetflt}
\sqrt{-G}=s^{D-2p-2}\sqrt{\Omega_{D-2p-2}}\prod_ir_i.$$ Hence we can define the trace of the extrinsic curvature of the surface is given by $$K=\nabla_{\mu}n^{\mu}=\frac{1}{\sqrt{-G}}\partial_{\mu}\left(\sqrt{-G}n^\mu\right).$$ From it can be easily seen that the leading contribution to the $K$ comes from $s$ derivative acting on the determinant of the metric and hence the trace of the extrinsic curvature is given by $$\begin{split}\label{kappa}
K&=\frac{1}{\sqrt{-G}}n^s\partial_s(\sqrt{-G}),\\
K&=D\frac{n^s}{s}=D\frac{1}{\sqrt{s^2+\sum_i\left(\frac{\partial g}{\partial r_i}\right)^2}}.
\end{split}$$
Rotating uncharged membrane solutions in flat background
---------------------------------------------------------
The equation of the membrane in a stationary configuration with no charge($Q=0$) is given by $$\label{effleqn}
K=\frac{ \gamma}{\beta},$$ where we have defined $\beta=\frac{D}{4\pi T}$.\
Using the expressions of $K$ and $\gamma $ from (,) we get the effective equation for the surface defined on the $s^2=2g$ as $$\label{shapeflt}
2g+\sum_i(\partial_ig)^2
=\beta^2\left(1-\sum_ir_i^2\omega_i^2\right).$$ Let us redefine $g, r_i$ and $\omega_i$ such that we can rewrite in an intrinsic scale independent form. $$\label{redsur}
g=\beta^2 {{\cal G}},\qquad r_i=\beta x_i \qquad \text{and} \qquad\omega_i=\frac{{\Upsilon}_i}{\beta}.$$ In terms of these new variables we can write the equation of the membrane in the stationary configuration as $$\label{shapefltnw}
2{{\cal G}}+\sum_i(\partial_i{{\cal G}})^2
=\left(1-\sum_ix_i^2{\Upsilon}_i^2\right),$$ where the derivative is taken w.r.t, $x_i$.\
Let us now analyze the possible solutions of . One immediate observation is that if ${{\cal G}}$ is at most quadratic in $x_i$ then so is every term in the equation shape and it forms a closed set of linear algebraic equations. Moreover if the quadratic form in ${{\cal G}}$ is diagonal in $i$ space (i.e; there are no $x_i x_j$ terms) then the same is true of every term in . So it is natural to seek solutions to this equation of the form $${{\cal G}}(x_i)= \sum_i\left(\frac{A_i}{2} - \frac{C_i }{2}(x_i-B_i)^2\right).$$ For a non-trivial solution to we require that $C_i\neq0$ .\
Equating the coefficients of the constant, linear and quadratic terms respectively in yields following set of linear algebraic equations $$\label{quadsolflt} \begin{split}
&\sum_i\left(A_i+B_i^2C_i(C_i-1)\right) = 1,\\
&B_iC_i(C_i-1)=0,\\
&C_i(C_i-1)+ {\Upsilon}_i^2=0.\\
\end{split}$$ Let us now analyze the solutions to the set of equations in . We have two different sets of possible solutions which we list below,\
- For non-zero rotation $({\Upsilon}_i\neq 0)$ :\
Since $C_i\neq0$, the consistent set of solutions is $$\begin{split}
\sum_iA_i=1, \qquad B_i=0, \qquad \text{and}\qquad C_{i\pm}= \frac{1}{2} \left(1\pm \sqrt{1-4 {\Upsilon}_i ^2}\right).
\end{split}$$
- For static solutions $({\Upsilon}_i= 0)$ :\
Since $C_i\neq0$, the possible set of solutions is\
$$\begin{split}\label{sit2}
\sum_iA_i=1, \qquad B_i=\text{free parameter},\qquad C_i=1.
\end{split}$$
Let us now analyze each of these cases in some more detail.\
\
[ ****]{}\
In this case $A_i=1$, $B_i=0$ and $$\label{csol}
C_{i\pm}= \frac{1}{2} \left(1\pm \sqrt{1-4 {\Upsilon}_i ^2}\right).$$ The spacetime membrane is described by the equation $$\label{actsol}
s^2+ r_i^2 \frac{1 \pm \sqrt{1- 4 {\Upsilon}_i ^2}}{2} = \beta^2.$$ This is an ellipsoidal membrane which is fatter in the $i$ directions than all other directions (this excess bulge is a consequence of the centrifugal force). These configurations are dual to Myers Perry type black hole solutions. The above solution for the shape function is real and have a consistent solution for $$\label{Upsrange}
-1/2< {\Upsilon}_i <1/2.$$
Figure \[fig:flat1\] shows the surface for turning on rotation along one plane . The blue curve corresponds to the $-$ branch of and the red curve corresponds to the $+$ branch of .
![Uncharged Membrane surface with rotation along one direction for ${\Upsilon}=0.47$ for the $+$ and $-$ branch of []{data-label="fig:flat1"}](flat_space_pm.pdf){width="120mm"}
\
In the static configuration , i.e; ${\Upsilon}_i=0$, we have a special branch of solution where $C_i=1$ and $B_i$ is a free parameter. The membrane shape is given by the equation $$\label{surfmanif}
s^2+ \sum_i\left(r_i-\beta B_i\right)^2= \beta^2.$$ This does ${\it not}$ represent a spherical black hole with a shifted origin because the shift is in polar coordinates in a two plane. The surface above is non analytic at $r_i=0$ (and therefore unacceptable) for $B_i^2< A_i$. [^11] On the other hand if $B_i^2> A_i$ then the point $r_i=0$ does not lie on the manifold . In fact represents a smooth black ring solution with horizon topology $(S^1)^n \times S^{D-2n-2}$ where we have non-zero black ring parameter $B_i$ turned on along $n$ two planes and $n<\frac{D-2}{2}$. The surface may be thought of as $S^{(D-2n-2)}$ fibered over direct product of $n$ annuli [^12] . The inner radius of the annulus is $r_{i-}=\beta(B_i-\sqrt{A_i})$ and the outer radius is $r_{i+}=\beta(B_i+\sqrt{ A_i})$. The $S^{(D-2n-2)}$ shrinks to zero at both the inner and outer radius of the all the $n$ annuli $(r_{1\pm}, r_{2\pm},r_{3\pm}\cdots r_{n\pm})$.\
Notice that non rotating black rings exist only when their outer radius exceeds $2\beta\sqrt{ A_i}$. At fixed $\beta\sqrt{A_i}$ we thus have a single solution with outer radius $\beta\sqrt{ A_i}$, and then solutions (ring solutions) for all values of the outer radius greater than $2\beta\sqrt{ A_i}$.\
When $B_i\neq 0$ it describes a black ring solution as described above. However since $B_i$ is a free parameter we can have a possibility where it can be set to zero. When $B_i=0$ for all $i's$ the solution is spherical in all directions, and so represents the Schwarschild black hole. The spacetime membrane is described by the equation $$\label{actsschw}
s^2+ \sum_{i=0}^{p}r_i^2 = \beta^2.$$ In the static configuration we can also have an interesting solution where we can set $B_i=0$ in $(p-n)$ planes and in the rest $n$ planes it can be fixed non-zero constant parameter. The membrane shape is then given by $$\label{bhring}
s^2+ \sum_{i=0}^{n}\left(r_i-\beta B_i\right)^2+\sum_{j=0}^{p-n}r_j^2=\beta^2.$$ For the flat space the stationary configurations we can have solutions to the shape functions such that rotations are turned on in some planes and zero rotations in the rest of the planes. So the most general solution where rotations are turned on $k$ planes and zero rotations in $(p-k)$ planes is given by $$\label{genflt}
s^2+ \sum_{i=0}^{k}r_i^2 \frac{1 \pm \sqrt{1- 4 {\Upsilon}_i ^2}}{2}+\sum_{m=0}^{n}\left(r_{m}-\beta B_m\right)^2+\sum_{j=0}^{p-n-k}r_j^2 = \beta^2.$$ where we have rotating black hole solution in $k$ planes, black ring solutions in $n$ and non-rotating black hole in $p-n-k$ directions.
Rotating charged membrane solution in flat background
-----------------------------------------------------
The most general solutions of the membrane in presence of charge in a stationary configurations is given by $$\label{chflat}
\begin{split}
Q&=\alpha\gamma,\\
K&= \frac{\gamma}{\beta(1-Q^2)},
\end{split}$$ where we have used the definition $$\alpha=2\sqrt{2\pi} \mu$$ and $$\beta=\frac{D}{4\pi T}.$$
The extrinsic curvature of the membrane as embedded in flat space is given by and and the effective equation for the surface is given by $$\begin{split}
\label{efchflt}
& \left(2g+\sum_i(\partial_ig)^2\right)\left(1-\sum_ir_i^2\omega_i^2\right)\\
&=\beta^2\left(1-\sum_ir_i^2\omega_i^2-\alpha^2\right)^2.
\end{split}$$ With the redefinition we can rewrite in an intrinsically scale independent form as $$\begin{split}
\label{efchfltnw}
& \left(2{{\cal G}}+\sum_i(\partial_i{{\cal G}})^2\right)\left(1-\sum_ix_i^2{\Upsilon}_i^2\right)\\
&=\left(1-\sum_ix_i^2{\Upsilon}_i^2-\alpha^2\right)^2.
\end{split}$$ The static charged black hole and black ring solutions can be easily obtained by replacing $\beta$ with $\beta(1-\alpha^2)$ in the static uncharged case.[^13]\
The effective equations for the membrane in the stationary configuration is presence of charge is hard to solve non-perturbatively. However we can solve the equation perturbatively in two different scheme
- Perturbatively adding charge to neutral rotating membrane.
- Perturbatively adding rotation to static charged membrane.
[**[Perturbatively adding charge to neutral rotating membrane]{}**]{}\
\
We will solve the equation for the simplest case where only one rotation is turned on. Let $(r,\theta)$ be the polar co-ordinates in this plane and $x=\frac{r}{\beta}$. We will do a perturbative expansion in the charge about a neutral membrane rotating in one plane. Let us assume that the expansion of the shape function in the charge parameter is given by $${{\cal G}}= {{\cal G}}_0 + \alpha^2 {{\cal G}}_1 + \alpha^4 {{\cal G}}_2 + \alpha^6 {{\cal G}}_3 + \cdots$$
Substituting in and separating out powers of $\alpha$ we get following equations: $$\begin{split}
2 {{\cal G}}_0 + {{\cal G}}_0'^2 &= (1-{\Upsilon}^2 x^2),\\
2 {{\cal G}}_1 + 2 {{\cal G}}_1' {{\cal G}}_0' &= -2,\\
2 {{\cal G}}_2 + 2 {{\cal G}}_0' {{\cal G}}_2' + {{\cal G}}_1'^2 &= \frac{1}{1-{\Upsilon}^2 x^2},\\
2 {{\cal G}}_3 + 2 {{\cal G}}_0' {{\cal G}}_3' + 2 {{\cal G}}_1' {{\cal G}}_2' &= 0,\\
2 {{\cal G}}_4 + 2 {{\cal G}}_0' {{\cal G}}_4' + 2 {{\cal G}}_1' {{\cal G}}_3' + {{\cal G}}_2'^2 &= 0,\\
\dots
\end{split}$$ Where the primes denote the derivative w.r.t $x$.\
The zeroth order equation is the equation for the uncharged rotating case, so the solution for ${{\cal G}}_0$ should be the uncharged static black hole which we are perturbing. $$2{{\cal G}}_0 = 1 - C_{\pm} x^2,$$ where $$C_{\pm} = \frac{1\pm\sqrt{1-4{\Upsilon}^2}}{2}.$$ Which gives ${{\cal G}}_0'=-C_{\pm}~x$, and putting it in the next equation for ${{\cal G}}_1$ gives $$\label{cbhg1}
{{\cal G}}_1 = \kappa_1 x^{\frac{1}{C_{\pm}}} - 1.$$ Let us pause here to analyze the solution which gives two different conditions,\
\
$\bf{1}.$ If $1/C{\pm}$ is a not an even integer then after imposing regularity condition at $x=0$, we require $\kappa_1$ to vanish, giving ${{\cal G}}_1 = -1$. Hence ${{\cal G}}_1'=0$.\
\
$\bf{2}.$If $1/C{\pm}$ is an even integer then homogeneous part becomes analytic at $x=0$, hence we cannot set the $\kappa_1$ to zero.\
\
Here we will present only the solution when $1/C{\pm}$ is a not an even integer. The equation for ${{\cal G}}_2$ becomes, $${{\cal G}}_2 -C_{\pm}~x~{{\cal G}}_2' = \frac{\frac{1}{2}}{1-{\Upsilon}^2 x^2}.$$ Which gives $${{\cal G}}_2 = \kappa_2 x^{\frac{1}{C_{\pm}}} - \frac{1}{2C_{\pm}}({\Upsilon}x)^{\frac{1}{C_{\pm}}}\int_0^{{\Upsilon}x}\frac{y^{\frac{-1}{C_{\pm}}-1}}{1-y^2}dy.$$ The particular integral part of the solution is analytic by itself as $x\rightarrow0$. This demands $\kappa_2 = 0$. Hence $${{\cal G}}_2 = - \frac{1}{2C_{\pm}}({\Upsilon}x)^{\frac{1}{C_{\pm}}}\int_0^{{\Upsilon}x}\frac{y^{\frac{-1}{C_{\pm}}-1}}{1-y^2}dy$$\
The equation for ${{\cal G}}_3$ is $${{\cal G}}_3 -C_{\pm}~x~{{\cal G}}_3' = 0,$$ Which is a homogeneous equation, the solution of which we know is not analytic at $x\rightarrow 0$. So ${{\cal G}}_3 = 0$.\
And the equation for ${{\cal G}}_4$ is $$2{{\cal G}}_4 - 2 C_{\pm}~x~{{\cal G}}_4'= -{{\cal G}}_2'^2.$$ Here as well the particular integral turns out to be analytic, so we have to set the homogeneous solution to be zero.\
\
Hence the shape function for a charged rotating membrane can be written perturbatively upto second order in $\alpha^2$ as $$\label{chargefin}
2{{\cal G}}= (-C_{\pm}~x^2 + 1) - 2\alpha^2 + \left(- \frac{1}{2C_{\pm}}({\Upsilon}x)^{\frac{1}{C_{\pm}}}\int_0^{{\Upsilon}x}\frac{y^{\frac{-1}{C_{\pm}}-1}}{1-y^2}dy\right)\alpha^4+\cdots$$\
![The inner red surface corresponds to the solution for the charged membrane with $\alpha=0.2$ and ${\Upsilon}=0.47$ for the $-$ branch while the outer blue surface is for the uncharged membrane.[]{data-label="fig2"}](charge_flat_1.pdf){width="\textwidth"}
![The inner red surface corresponds to the solution for the charged membrane with $\alpha=0.2$ and ${\Upsilon}=0.47$ for the $-$ branch while the outer blue surface is for the uncharged membrane.[]{data-label="fig2"}](charge_flat_2.pdf){width="\textwidth"}
In figure \[fig1\] and \[fig2\] we show the surface comparison for the with the uncharged black hole solutions . The inner red surface corresponds to the solutions for the charged membrane with $\alpha=0.2$ and ${\Upsilon}=0.47$. The charge membrane is a deformed ellipsoid and it becomes flatter compared the uncharged rotating membrane for both the $+$ and $-$ branches.\
\
In this section we will solve the equation for the simplest case where only one rotation is turned on and do a perturbative expansion in ${\Upsilon}^2$ about a static charged membrane. Let us assume that the expansion of the shape function in ${\Upsilon}^2$ is given by $$\label{expomchft}
{{\cal G}}= {{\cal G}}_0 + {\Upsilon}^2 {{\cal G}}_1 + {\Upsilon}^4 {{\cal G}}_2 + {\Upsilon}^6 {{\cal G}}_3 + \dots$$
Substituting in and separating out powers of ${\Upsilon}^2$ we get following equations: $$\begin{split}
2{{\cal G}}_0 + {{\cal G}}_0'^2 &= (1-\alpha^2)^2,\\
2{{\cal G}}_1 + 2{{\cal G}}_1'g_0' &= x^2(\alpha^4-1),\\
2{{\cal G}}_2 + 2{{\cal G}}_0'{{\cal G}}_2' + {{\cal G}}_1'^2 &= \alpha^4x^4,\\
2{{\cal G}}_3 + 2{{\cal G}}_0'{{\cal G}}_3' + 2{{\cal G}}_1'{{\cal G}}_2' &= \alpha^4x^6,\\
2{{\cal G}}_4 + 2{{\cal G}}_0'{{\cal G}}_4' + 2{{\cal G}}_1'{{\cal G}}_3' +{{\cal G}}_2'^2 &= \alpha^4x^8,\\
\dots
\end{split}$$ Where the primes denote the derivative w.r.t $x$.\
The zeroth order equation is the equation for the charged static case, so the solution for ${{\cal G}}_0$ should be the charged static membrane which we are perturbing. $$\label{CG0}
2{{\cal G}}_0 = (1-\alpha^2)^2 - x^2.$$ Which gives ${{\cal G}}_0'=-x$.\
If we look at the equation at any order, we see that the homogeneous part of all the equations is the same, hence any equation can be written as $$2{{\cal G}}_n - 2{{\cal G}}_n' = f_n(x)$$ Which has a simple solution $${{\cal G}}_n = C_nx - \frac{x}{2}\int_0^x\frac{f_n(\rho)}{\rho^2}d\rho.$$ For the first order we have $f_1(x) = x^2(\alpha^4-1)$. So we get $${{\cal G}}_1 = C_1 x- \frac{1}{2} x^2(\alpha^4-1).$$ Since the solution has to be regular at $x = 0$, $C_1 = 0$. Now at any order by induction we can show that the particular integral is a polynomial in $x^2$. So the only odd order term is the homogeneous part which has to be set to zero at each order for the sake of regularity at $x = 0$. i.e. $C_n = 0$ for all $n$.\
For the second order $f_2(r) = \alpha^4 x^4 -(\alpha^4 - 1)^2x^2$. So we get $${{\cal G}}_2 = -\frac{1}{6}\alpha^4 x^4 + \frac{1}{2}(\alpha^4 - 1)^2x^2.$$ We can proceed in the similar way for higher orders in ${\Upsilon}$, $$\begin{split}
{{\cal G}}_3 =& -\frac{1}{10} x^6 \left(\alpha^ 4 \right)+\frac{2}{9} \alpha ^4 \left(\alpha ^4-1\right) x^4-\left(\alpha ^4-1\right)^3 x^2,\\
{{\cal G}}_4 =& -\frac{1}{14} x^8 \left(\alpha ^4 \right)+\frac{1}{225} \alpha ^4 \left(37 \alpha ^4-27\right) x^6-\frac{14}{27} x^4 \left(\alpha ^4 \left(\alpha ^4-1\right)^2 \right)\\ &+\frac{5}{2} \left(\alpha ^4-1\right)^4 x^2,\\
{{\cal G}}_5 =& -\frac{1}{18} x^{10} \left(\alpha ^4 \right)+\frac{2}{245} \alpha ^4 \left(17 \alpha ^4-10\right) x8-\frac{x^6 \left(\alpha ^4 \left(1471 \alpha ^8-2362 \alpha ^4+891\right) \right)}{3375}\\ &+\frac{116}{81} \alpha ^4 \left(\alpha ^4-1\right)^3 x^4-7 x^2 \left(\left(\alpha ^4-1\right)^5 \right),\\
\dots
\end{split}$$ Plugging back the solutions in we get the solution for charged rotating membrane at the leading order in $1/D$ expansion.[^14]
The effective membrane equation for Stationary ‘axially-symmetric’ configurations in $AdS$ {#rotmemAdS}
-------------------------------------------------------------------------------------------
In this subsection we will find axially-symmetric solutions to and in the ambient empty $AdS$ space both in presence and absence of charge which will be dual to rotating black hole in presence and absence of charge respectively.\
A time like killing vector field of which brings out the ‘axially-symmetric’ configuration of the final spacetime and hence the final membrane configuration is given by
$$k=\frac{\partial}{\partial t}+\omega^{i}\frac{\partial}{\partial\theta_i}.$$
Hence, the unit time-like velocity field of the membrane that we consider is given by $$\label{gammaads}
u^{\mu}=\gamma k^{\mu},\quad \text{where} \quad \gamma=\frac{1}{\sqrt{\left(1-\sum_{i}r_i^2 \omega_i^2\right)+\frac{1}{L^2}\sum_i (r_i^2+s^2)}}.$$
We consider the global $AdS$ metric with manifest $SO(D-p-2)$ isometry and . Since we view the membrane as propagating in an empty $AdS$ . The extrinsic curvature is given by $$\label{kappaads}
\begin{split}
K&=\frac{1}{\sqrt{-G}}n^s\partial_s(\sqrt{-G}),\\
K&=D\frac{n^s}{s}=D\frac{1+\frac{s^2}{L^2}-\frac{1}{L^2}\sum_i r_i \partial_i g}{\sqrt{s^2+\sum_i\left(\frac{\partial g}{\partial r_i}\right)^2+\frac{1}{L^2}\sum_i\left(s^2-r_i \partial_i g\right)^2}}\,\,.\\
\end{split}$$
Rotating uncharged membrane solutions in $AdS$ back ground {#rotsolAdS}
----------------------------------------------------------
The most general solutions of the membrane in without charge in a stationary configurations is given by $$\label{effleqnads}
K=\frac{ \gamma}{\beta},$$ where we have defined $\beta=\frac{D}{4\pi T}$.\
Using the expressions of $\mathcal{K}$ and $\gamma $ from (,) we get the effective equation for the surface defined on the $s^2=2g$ as $$\label{adsefeq1}
\begin{split}
& 2g+\sum_i(\partial_ig)^2+\frac{1}{L^2}\left(2g -\sum_{i}r_i\partial_ig\right)^2 \\
&=\beta^2 \left(1+2\frac{g}{L^2}-\frac{1}{L^2}\sum_ir_i\partial_ig\right)^2\left(1+\frac{2g}{L^2}-\sum_ir_i^2\left(\omega_i^2-\frac{1}{L^2}\right)\right).
\end{split}$$ Let us redefine $g, r_i$ and $\omega_i$ such that we can rewrite in an intrinsic scale independent form. $$\label{redsurds}
g=\beta^2 {{\cal G}},\qquad r_i=\beta x_i ,\qquad L= \beta {{\cal L}}\qquad \text{and} \qquad\omega_i=\frac{{\Upsilon}_i}{\beta},$$ with this new redefinition we can rewrite $$\begin{aligned}
\label{adsefeqnw}
&& 2{{\cal G}}+\sum_i(\partial_i{{\cal G}})^2+\frac{1}{{{\cal L}}^2}\left(2{{\cal G}}-\sum_{i}x_i\partial_i{{\cal G}}\right)^2\nonumber\\
&&= \left(1+2\frac{{{\cal G}}}{{{\cal L}}^2}-\frac{1}{{{\cal L}}^2}\sum_ix_i\partial_i{{\cal G}}\right)^2\left(1+\frac{2{{\cal G}}}{{{\cal L}}^2}-\sum_ix_i^2\left({\Upsilon}_i^2-\frac{1}{{{\cal L}}^2}\right)\right).\end{aligned}$$
The above differential equation is relatively hard to solve in general, however as has been explained in the situation of the flat space rotating solution we can solve this by a general quadratic ansatz as follows. Let us turn on rotation only along one plane and the general ansatz $${{\cal G}}(x)= \frac{A {{\cal L}}^2}{2}-\frac{C }{2}(x-B{{\cal L}})^2,$$ where $A,C$ and $B$ are functions of ${{\cal L}}$. For a non-trivial solution to we require that $C\neq0$.\
Plugging the quadratic ansatz in and equating the different powers of $x$ to zero we arrive at the following relations obeyed by the coefficients $A, B $ and $C$ are as follows $$\label{algeqns}
\begin{split}
&{{\cal L}}^2 \left(A-B^2 C\right) \left(A-B^2 C+1\right)-\left(A-B^2 C+1\right)^3+B^2 C^2 {{\cal L}}^2=0,\\
&\frac{4 B C \left(A-B^2 C+1\right)^2}{{{\cal L}}}-2 B C {{\cal L}}\left(A-B^2 C-C+1\right)=0,\\
&C \left(B^4 C (6 C-1)+B^2 \left(C \left({{\cal L}}^2-7\right)+2\right)+(C-1) {{\cal L}}^2+1\right)-1
+A^2 \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)\\&+A \left(C \left(B^2 \left(-7 C-2 {{\cal L}}^2 {\Upsilon}^2+2\right)+2\right)+2 {{\cal L}}^2 {\Upsilon}^2-2\right)+{{\cal L}}^2 {\Upsilon}^2 \left(B^2 C-1\right)^2=0,\\
&2 B C \left({{\cal L}}^2 {\Upsilon}^2 \left(A+B^2 (-C)+1\right)+A C-A-2 B^2 C^2+B^2 C+C-1\right)=0,\\
&B^2 C^2 \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)=0.
\end{split}$$ There are five equations with three undetermined variables, which makes the system of equations over-constrained. The last equation in gives two possible solutions $$\label{solbc}
\begin{split}
B&=0,\text{or}\\
C&=1-{{\cal L}}^2 {\Upsilon}^2.
\end{split}$$ Now if we consider the second solution in the system of equations in is still over-constrained and doesn’t have any solution, hence the only consistent solution is $B=0$. Setting $B=0$ gives us two equations in terms of two undetermined variables. Hence the ansatz which gives a consistent solution is $${{\cal G}}(x)= \frac{A {{\cal L}}^2}{2}-x^2\frac{C }{2}.$$ $$\label{algeqnscon}
\begin{split}
&{{\cal L}}^2 \left(A^2+A\right)-(A+1)^3=0,\\
&(C-1) \left((A+1)^2+C {{\cal L}}^2\right)+(A+1)^2 {{\cal L}}^2 {\Upsilon}^2=0.\\
\end{split}$$ The first equation on can be solved and we have the following set of solutions $$\label{solads1}
\begin{split}
A&= -1,\\
A_{\pm}&=\frac{1}{2} \left(\left({{\cal L}}^2\pm {{\cal L}}\sqrt{{{\cal L}}^2-4}\right)-2\right).\
\end{split}$$ Plugging back the solutions for $A_i$ in the second equation in we get a class of different solutions
- $$\label{sol1}\begin{split}
A&=-1,\\C&= 0 \qquad \text{or}\qquad 1.
\end{split}$$
- $$\label{sol2}\begin{split}
A_{+}&=\frac{1}{2} \left(\left({{\cal L}}^2+ {{\cal L}}\sqrt{{{\cal L}}^2-4}\right)-2\right),\\
C_{+\pm}&=-\frac{1}{2}\left(A_{+}-1\right)\pm\frac{1}{2}{{\cal L}}\sqrt{A_{+}(1-4{\Upsilon}^2)}.
\end{split}$$
- $$\label{sol3}\begin{split}
A_{-}&=\frac{1}{2} \left(\left({{\cal L}}^2- {{\cal L}}\sqrt{{{\cal L}}^2-4}\right)-2\right),\\
C_{-\pm}&=-\frac{1}{2}\left(A_{-}-1\right)\pm\frac{1}{2}{{\cal L}}\sqrt{A_{-}(1-4{\Upsilon}^2)}.
\end{split}$$
The solution gives a hyperbolic solution which is non-compact and hence an unphysical solution, hence we discard the solution. The only physical solutions to are and .
Let us now consider the validity regime of the solutions and , which gives us a compact and real solutions. For compactness of the solutions we require that [^15] $$\label{valsol}
\begin{split}
& A_{\pm}>0,\qquad C_{\pm\pm}>0,
\end{split}$$ for the solutions to be real we require [^16] $$\label{valsol2}
\begin{split}
& {{\cal L}}\geq 2,\qquad -\frac{1}{2}<{\Upsilon}\leq \frac{1}{2}.
\end{split}$$
In terms of the thermodynamic quantities we get a real and consistent rotating black hole (black brane) solutions in $AdS$ when the thermodynamic temperature $T$ and the rotation $\omega$ follows $$\begin{split}
T>\frac{D}{2\pi L} \quad\text{and} \qquad\frac{-2\pi T}{D}<\omega<\frac{2\pi T}{D}.
\end{split}$$\
${{\cal L}}\geq2$ also ensure that $A_{\pm}>0$ as required for compactness of the solution. The requirement of compactness of the solutions to gives the following conditions as listed in \[tab1\].
Parameter range allowed $A$ allowed $C$
------------------------------------------------------- ------------- -------------
$|\Upsilon| < \frac{1}{\mathcal{L}}$ $A_+$ $C_{++}$
$A_-$ $C_{-+}$
$\frac{1}{\mathcal{L}} < |\Upsilon| \leq \frac{1}{2}$ $A_-$ $C_{-+}$
$A_-$ $C_{--}$
: Allowed branches of solutions for different values of ${\Upsilon}$[]{data-label="tab1"}
\
$C_{+-} $ is always negative for any acceptable values of ${{\cal L}}$ and ${\Upsilon}$ as given in and hence is never give a compact solution .\
A detailed derivation of these limits is presented in \[valsolap\]. Hence only $C_{++}$, $C_{-+}$ and $C_{--}$ give a real and compact solution for the surface. The corresponding membrane are given by $$\label{adstsol}\begin{split}
s^2+ r^2 C_{(++)}= A_{+} L^2,\\
s^2+ r^2 C_{(-+)}= A_{-} L^2,\\
s^2+ r^2 C_{(--)}= A_{-} L^2.\\
\end{split}$$ These are ellipsoidal membrane which are fatter in the plane of rotation all other directions (this excess bulge is a consequence of the centrifugal force).\
Figure \[figads1\] shows the surface for turning on rotation in one plane for the first entry in \[tab1\] and figure \[figads2\] second entry in \[tab1\] respectively.
![ and with ${{\cal L}}= 2.3, {\Upsilon}=0.49$ []{data-label="figads2"}](s1s2_Ads.pdf){width="\textwidth"}
![ and with ${{\cal L}}= 2.3, {\Upsilon}=0.49$ []{data-label="figads2"}](s2s3_Ads.pdf){width="\textwidth"}
Rotating charged membrane solution in $AdS$
--------------------------------------------
The most general stationary charged membrane solutions in $AdS$ is given by $$\begin{split}
Q&=\alpha\gamma,\\
K&= \frac{\gamma}{\beta (1-Q^2)},
\end{split}$$
where we have used the definition $$\alpha=2\sqrt{2\pi} \mu$$ and $$\beta=\frac{D}{4\pi T}.$$
The extrinsic curvature is given by and and the effective equation for the surface is given by
$$\begin{split}
\label{adsefeqch}
& \left(2g+\sum_i(\partial_ig)^2+\frac{1}{L^2}\left(2g -\sum_{i}r_i\partial_ig\right)^2\right)\left(1+\frac{2g}{L^2}-\sum_ir_i^2\left(\omega_i^2-\frac{1}{L^2}\right)\right)\\
&=\beta^2 \left(1+2\frac{g}{L^2}-\frac{1}{L^2}\sum_ir_i\partial_ig\right)^2\left(1+\frac{2g}{L^2}-\sum_ir_i^2\left(\omega_i^2-\frac{1}{L^2}\right)-\alpha^2\right)^2.
\end{split}$$
With the redefinition we can rewrite in an intrinsically scale independent form as $$\begin{split}
\label{adsefeqchnw}
& \left(2{{\cal G}}+\sum_i(\partial_i{{\cal G}})^2+\frac{1}{{{\cal L}}^2}\left(2{{\cal G}}-\sum_{i}x_i\partial_i{{\cal G}}\right)^2\right)\left(1+\frac{2{{\cal G}}}{{{\cal L}}^2}-\sum_ix_i^2\left({\Upsilon}_i^2-\frac{1}{{{\cal L}}^2}\right)\right)\nonumber\\
&= \left(1+2\frac{{{\cal G}}}{{{\cal L}}^2}-\frac{1}{{{\cal L}}^2}\sum_ix_i\partial_i{{\cal G}}\right)^2\left(1+\frac{2{{\cal G}}}{{{\cal L}}^2}-\sum_ix_i^2\left({\Upsilon}_i^2-\frac{1}{{{\cal L}}^2}\right)-\alpha^2\right)^2.
\end{split}$$
The above equation is hard to solve non-perturbatively. However we can solve it perturbatively in two different scheme
- Perturbatively adding charge to neutral rotating membrane
- Perturbatively adding rotation to static charged membrane
\
We will consider the simplest case where only one rotation is turned on. Let us expand the shape function perturbatively in terms of $\alpha^2$ as $$\label{expchg}
{{\cal G}}= {{\cal G}}_0 + \alpha^2 {{\cal G}}_1 + \alpha^4 {{\cal G}}_2 + \alpha^6 {{\cal G}}_3 + \cdots$$
Substituting in and separating out powers of $\alpha$ we get the equations for the different orders in the expansions.\
The zeroth order equation is $$\begin{split}
&\frac{x^2 {{\cal G}}_0'(x)^2}{{{\cal L}}^2}-\frac{4 x {{\cal G}}_0(x) {{\cal G}}_0'(x)}{{{\cal L}}^2}+\frac{\left(-2 {{\cal G}}_0(x)+{{\cal L}}^2 \left(x^2 {\Upsilon}^2-1\right)-x^2\right) \left(-x {{\cal G}}_0'(x)+2 {{\cal G}}_0(x)+{{\cal L}}^2\right)^2}{{{\cal L}}^6}\\&+{{\cal G}}_0'(x)^2+\frac{4 {{\cal G}}_0(x)^2}{{{\cal L}}^2}+2 {{\cal G}}_0(x)=0,\\
\end{split}$$ Where the primes denote the derivative w.r.t $x$.\
The zeroth order equation is the equation for the uncharged rotating case, so the solution for ${{\cal G}}_0$ should be the uncharged rotating membrane . $$2{{\cal G}}_0 = A{{\cal L}}^2- C x^2,$$ where $A$ and $C$ are given in and . For each of the different allowed solutions, $A_{+}, C_{++}$ and $A_{-}, C_{-+},C_{--}$ we will have the different charged solutions in $AdS$. For simplicity we will denote the these by $A$ and $C$.\
Substituting the zeroth order solution in the equation for the first order in $\alpha^2$ is given by $$\label{eqg1}
\begin{split}
\left(a x+b x^3\right) {{\cal G}}_1'(x)+{{\cal G}}_1(x) \left(a_1+b_1 x^2\right)+c=0,
\end{split}$$ where the form of the coefficients are given by $$\begin{split}
&a=\frac{(A+1)^2}{{{\cal L}}^2}-A-C,\\
&b=-\frac{(A+1) \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)}{{{\cal L}}^4},\\
&c=(A+1)^2,\\
&a_1=-\frac{3 (A+1)^2}{{{\cal L}}^2}+2 A+1,\\
&b_1= \frac{2 (A+1) \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)}{{{\cal L}}^4}.
\end{split}$$ The solution to is given by $$\label{cbhadsg1}
{{\cal G}}_1(x)=\kappa x^{-\frac{a_1}{a}} \left(a+b x^2\right)^{\frac{1}{2} \left(\frac{a_1}{a}-\frac{b_1}{b}\right)}-\frac{c \, _2F_1\left(1,\frac{b_1}{2 b};\frac{a_1}{2 a}+1;-\frac{b x^2}{a}\right)}{a_1}.$$ After imposing regularity condition at $x=0$, we require $\kappa$ to vanish[^17], giving ${{\cal G}}_1 = \frac{c \, _2F_1\left(1,\frac{\text{b1}}{2 b};\frac{\text{a1}}{2 a}+1;-\frac{b x^2}{a}\right)}{\text{a1}}$.\
Substituting the values of the coefficients $a,b,a_1,b_1$ and $c$ we can rewrite the solution for the ${{\cal G}}_1(x)$ as $$\label{solg1}
\begin{split}
{{\cal G}}_1(x)&=\frac{x^2 \left(2 (A+1)^3 \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)\right)}{-(2 A+1) (2 C-1) {{\cal L}}^4-2 (A+1)^2 {{\cal L}}^2 (A-3 C+2)+3 (A+1)^4}\\&+\frac{(A+1)^2 {{\cal L}}^2}{3 (A+1)^2-(2 A+1) {{\cal L}}^2}.
\end{split}$$ Substituting the expression for the first order solution in , the equation for the order $\alpha^4$ is given by $$\begin{split}\label{eqg2}
{{\cal G}}_2'(x) \left(a_2 x^5+b_2 x^3+c_2 x\right)+{{\cal G}}_2(x) \left(a_3 x^4+b_3 x^2+c_3 \right)+\left(a_4 x^4+b_4 x^2+c_4\right)=0,
\end{split}$$ where the coefficients $a_2,a_3,a_4,b_2,b_3,b_4,c_2,c_3$ and $c_4$ are given in appendix \[coeffads\]
We can write in a simple form as $$\begin{split}
&{{\cal G}}_2'(x) +{{\cal G}}_2(x) P(x)+q(x)=0, \qquad\text{where}\\
&P(x)=\frac{\left(a_3 x^4+b_3 x^2+c_3 \right)}{\left(a_2 x^5+b_2 x^3+c_2 x\right)},\qquad q(x)=\frac{\left(a_4 x^4+b_4 x^2+c_4\right)}{\left(a_2 x^5+b_2 x^3+c_2 x\right)}.
\end{split}$$ The solution is of the form $$\begin{split}\label{g2sol}
{{\cal G}}_2(x)= -e^{-\int{P(x)dx}}\int_{0}^{x}\left(e^{\int{P(y)dy}}q(y)dy\right)+\kappa e^{-\int{P(x)dx}},
\end{split}$$ regularity of the solution forces $\kappa$ to vanish.\
Hence the shape function for a charged rotating black hole can be written perturbatively upto second order in $\alpha^2$ is given by $$2{{\cal G}}= (A {{\cal L}}^2 -Cx^2 \beta) + 2\alpha^2 {{\cal G}}_1 + 2{{\cal G}}_2\alpha^4+\cdots$$ where ${{\cal G}}_1$ and ${{\cal G}}_2$ are given in and respectively.\
[**[Perturbatively adding rotation to the charged static membrane]{}**]{}\
Let us expand the shape function of the charged rotating membrane perturbatively in terms ${\Upsilon}^2$ as $${{\cal G}}= {{\cal G}}_0 + {\Upsilon}^2 {{\cal G}}_1 + {\Upsilon}^4 {{\cal G}}_2 + {\Upsilon}^6 {{\cal G}}_3 + \dots$$
Substituting in and separating out powers of ${\Upsilon}$ , the zeroth order equation: $$\begin{split}
& \frac{x^2 {{\cal G}}_0'(x)^2}{{{\cal L}}^2}-\frac{4 x {{\cal G}}_0(x) {{\cal G}}_0'(x)}{{{\cal L}}^2}-\frac{\left(2 {{\cal G}}_0(x)-\left(\alpha ^2-1\right) {{\cal L}}^2+x^2\right)^2 \left(-x {{\cal G}}_0'(x)+2 {{\cal G}}_0(x)+{{\cal L}}^2\right)^2}{{{\cal L}}^6 \left(2 {{\cal G}}_0(x)+{{\cal L}}^2+x^2\right)}\\&+{{\cal G}}_0'(x)^2+\frac{4 {{\cal G}}_0(x)^2}{{{\cal L}}^2}+2 {{\cal G}}_0(x)=0.
\end{split}$$ Where the primes denote the derivative w.r.t. $x$.\
The solution to this equation is given by $$\label{g0sol}
2{{\cal G}}_0 = {\cal{A}}_{\pm} - x^2,$$ where ${\cal{A}}_{\pm}=\frac{1}{2} \left(\left(2 \alpha ^2\pm {{\cal L}}\sqrt{4 \alpha ^2+{{\cal L}}^2-4}\right)+{{\cal L}}^2-2\right).$ Which gives ${{\cal G}}_0'=-x$.\
The equation at any order can be written in the form
$$X(\alpha,{{\cal L}}){{\cal G}}_n - Y(\alpha,{{\cal L}}){{\cal G}}_n' =Z(\alpha,{{\cal L}}) f_n(x).$$
Which has a simple solution $${{\cal G}}_n =C_n e^{\frac{x X}{Y}}+e^{\frac{x X}{Y}} \int_1^x -\frac{Z f_n(r) e^{-\frac{r X}{Y}}}{Y} \, dr.$$ For the first order we get[^18] $$\begin{split}
{{\cal G}}_1 = &C_1 x^{2-\frac{{{\cal L}}^3+{{\cal L}}^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+2 \alpha ^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+4 \alpha ^2 {{\cal L}}}{2 {{\cal L}}}}\\+&\frac{x^2 \left({{\cal L}}^2 \left({{\cal L}}^3+{{\cal L}}^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+2 \alpha ^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+\left(4 \alpha ^2-2\right){{\cal L}}\right)\right)}{2 \left({{\cal L}}^3+{{\cal L}}^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+2 \alpha ^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+4 \alpha ^2 {{\cal L}}\right)}.
\end{split}$$ Since the solution has to be regular at $x = 0$, $C_1 = 0$. We have checked upto order ${\Upsilon}^8$ and we find that the particular integral is even order polynomial in $x$. So the only odd order term is the homogeneous part which has to be set to zero each order for the sake of regularity at $x = 0$. i.e. $C_n = 0$ for all $n$ for $n\leq4$.\
For the second order we get $$\label{solg2}
{{\cal G}}_2 = \zeta_1(\alpha,{{\cal L}}) x^2+\zeta_2 (\alpha,{{\cal L}})x^4,$$ where $\zeta_1$ and $\zeta_2$ are given in appendix \[coeffads\]
The higher order solutions are given below, where the constants $\zeta_i's$ are functions of $\alpha$ and ${{\cal L}}$ only, $$\begin{split}\label{higsol}
{{\cal G}}_3 =& x^6 \zeta_3+\zeta_4 x^4+\zeta_5 x^2,\\
{{\cal G}}_4 =& x^8 \zeta_6+\zeta_7 x^6 +\zeta_8 x^4+\zeta_9 x^2,\\
\dots
\end{split}$$ The coefficients $\zeta_i's$ are quite complicated functions of $\alpha$ and ${{\cal L}}$, but the general structure of the higher order solutions are given by , we don’t present the explicit form of the coefficients here.[^19]
Thermodynamics for the rotating membrane solutions {#rottherm}
==================================================
We have derived the thermodynamic relations and the first law of thermodynamics in \[unch1stlaw\] . In this section we specialize these relations for the rotating membrane configurations. We define a conserved current from the leading order stress tensor as $$\begin{split}
J_{(E)}^\mu &= -T^{\mu\nu}k_\nu \\
&=(1+Q^2)\frac{K}{16\pi\gamma}u^\mu\left(\gamma^2-\sum_ir_i^2\omega_i^2\gamma^2\right).\\
\end{split}$$ Now let’s consider a spacelike slice of the membrane world volume with normal along $dt$. The conserved ‘total energy’ on this slice is given by
$$\label{toten}
\tilde{ E} = \int d^{D-2}y \sqrt{-G} J_{(E)}^0 = \int d^{D-2}y \sqrt{-G} (1+Q^2)\frac{K}{16\pi}\left(\gamma^2-\sum_ir_i^2\omega_i^2\gamma^2\right),$$
since $u^0 = \gamma$. Note that this ‘total energy’ also includes the contribution from angular momentum since $k$ has contribution from the rotation Killing vectors. We define the intrinsic energy as
$$\label{inen}
E = \int d^{D-2}y \sqrt{-G} (1+Q^2)\frac{\gamma^2K}{16\pi}.$$
The angular momentum density is simply the $T^{0\theta_i}$ component of the leading order stress-energy tensor defined on a spacelike slice on the membrane world volume with normal along $dt$. $$T^{0\theta_i}=(1+Q^2)\frac{K}{16\pi} \delta(\rho-1)u^0u^{\theta_i},$$ angular momentum ${\cal J}_i$ in the $i'th$ plane is defined as $$\label{anmom}
{\cal J}_i = \int d^{D-2}y \sqrt{-G}g_{\theta_i\theta_j}T^{0\theta_j} = \int d^{D-2}y \sqrt{-G}\sum_i\omega_ir_i^2\frac{\gamma^2(1+Q^2) K}{16\pi}.$$ where $g_{\theta_i\theta_j}=r_i^2\delta_{ij}$ from and .\
Using and we can rewrite as $$\tilde{E}= E-\sum_i\omega_i{\cal J}_i.$$
Using , and $$\label{varthermrot}
\delta {\cal J}_i = \int d^{D-2}y \delta\sqrt{-G}\sum_i\omega^ir_i^2\frac{\gamma^2(1+Q^2) K}{16\pi},$$
we can simplify the combination $T\delta S ,\mu\delta q$ and $\delta J_i$ where $T$ and $\mu$ are defined in . $$\label{thermoproofrot}
\begin{split}
T\delta S +\mu\delta q +\sum_i\omega_i\delta {\cal J}_i &= \int d^{D-2}y~ \delta\sqrt{-G}\left(T\frac{\gamma}{4} + \mu\frac{KQ\gamma}{2\sqrt{2\pi}}\right),\\
&= \int d^{D-2}y~ \delta\sqrt{-G}\left(\frac{(1-Q^2)K}{4\pi\gamma}\frac{\gamma}{4} + \frac{Q}{2\sqrt{2\pi}\gamma}\frac{KQ\gamma}{2\sqrt{2\pi}}+\frac{1+Q^2}{16 \pi}K\gamma^2\right),\\
&= \int d^{D-2}y~ \delta\sqrt{-G}~(1+Q^2)\frac{K}{16\pi}\gamma^2,\\
&= \delta E.\\
T\delta S+\mu \delta q&=\delta E-\sum_i\omega_i \delta {\cal J}_i .
\end{split}$$
Which is the statement that rotating membrane configurations obey the first law of thermodynamics at the leading order in large $D$.
Explicit form of the thermodynamic quantities for rotating membranes
--------------------------------------------------------------------
Now let’s calculate at the leading order in $D$ the thermodynamic quantities associated with the membrane rotating in a single plane. From equations , , , and we notice that the densities of all the thermodynamic quantities on the membrane are functions of $r^2$. let $f(r^2)$ be the density for a quantity $F$. Therefore
$$\begin{split}
F &= \int dr d\theta d\Omega_{D-4} \sqrt{-G} f(r^2),\\
&= \int dr \left(\int d\theta d\Omega_{D-4} \sqrt{-G} \right) f(r^2),\\
&= 2\pi \Omega_{D-4} \int dr~r \left(2g(r^2)\right)^{\frac{D-4}2} f(r^2).
\end{split}$$
We have used the fact that for both the flat and AdS spaces in our choice of coordinates $$\int d\theta d\Omega_{D-4} \sqrt{-G} = 2\pi r \Omega_{D-4} s^{D-4} = 2\pi \Omega_{D-4} r \left(2g(r^2)\right)^{\frac{D-4}2}.$$
Due to a steep dependence of the integrand on $r$, we can utilize the saddle point approximation to do this integration. To obtain the saddle point we extremist the integrand w.r.t. $r$. Let the saddle point be $(r_*)^2 = \epsilon_0 + \frac{\epsilon_1}{D-4}$. Now we proceed to finding the saddle point
$$\left((D-4) (r_*)^2 \frac{2g'((r_*)^2)} {2g((r_*)^2)} + 1\right)f((r_*)^2) + (r_*)^2f'((r_*)^2) = 0.$$
Solving at the leading order in $\frac{1}{D-4}$, we get $\epsilon_0 = 0$ while at the subleading order we get $\epsilon_1 = -\frac{2g'(0)} {2g(0)}$ which is positive as $2g$ is a monotonically decreasing function of $r^2$. This means $$(r_*)^2 = -\frac{2g(0)} {2g'(0)(D-4)}.$$ A couple of things are easily noticeable. First, the saddle point at the leading order is ${\cal{O}}(\frac{1}{D})$, so it’s sufficient to calculate the quantities at the leading order in $r^2$ at the saddle point, because any higher order terms will be suppressed in $1/D$. And secondly, the leading order saddle point in independent of the density $f$ that is being integrated, so all the integrals here have the same saddle point.\
Now in its general form $2g(r^2) = A - Cr^2 + \mathcal{O}(r^4)$, so $(r_*)^2 = \frac{A}{C(D-4)}$. Thus $$2g((r_*)^2) = A\left(1 - \frac{1}{D-4}\right) + \mathcal{O}\left(\frac{1}{(D-4)^2}\right).$$
Hence $$2g((r_*)^2)^{\frac{D-4}2} = A^{\frac{D-4}2}\sqrt{e} \sim A^{\frac{D-4}2}.$$ (Since we have omitted other constant factors coming from the saddle point integration, we can also omit other constant multiplicative factors like $\sqrt{e}$ consistently.)\
Now we can write the expressions after the saddle point integration $$\label{thermsaddle}
\begin{split}
\tilde{E} &= V(1+\bar{Q}^2)\frac{\bar{K}}{16\pi},\\
E &= V(1+\bar{Q}^2)\frac{\bar{\gamma}^2\bar{K}}{16\pi},\\
\mathcal{J} &= V(1+\bar{Q}^2)\omega(r_*)^2\frac{\bar{\gamma}^2\bar{K}}{16\pi},\\
q &= V\frac{\bar{K}\bar{Q}\bar{\gamma}}{2\sqrt{2\pi}},\\
S &= V\frac{\bar{\gamma}}4,
\end{split}$$ where $V = 2\pi\Omega_{D-4}\frac{A^{\frac{D-3}2}}{\sqrt{(D-4)C}}$ multiplied by all the consistently omitted factors like $\sqrt{e}$ and the ones from the saddle point integration. The barred quantities are the quantities evaluated at $r = r_*$. As all of $K$, $Q$ and $\gamma$ are nonzero at $r = 0$, the barred quantities can be computed at $r = 0$ instead of $r_*$.\
Immediately we can see that $\mathcal{J}$ is subleading compared to $E$ because of the extra $(r_*)^2$ factor, which indicates that at the leading order in $D$ the contribution from the angular momentum to the total energy is negligible. Hence $\tilde{E} = E$ at the leading order. Also, $S$ doesn’t have an explicit factor of $K$ in contrast to other quantities, it looks to be subleading as well. However it enters thermodynamics with a factor of $T$ multiplying it which itself is order $D$, so the entropy contributes to the thermodynamics in this way.\
When the variations of these thermodynamic quantities are taken, the leading contribution comes from the variation of the ‘membrane volume’ $V \sim A^{\frac{D-3}2}$ as expected. Thus we can infer that these thermodynamic quantities satisfy the relation $$\label{thermrel}
\tilde{E} = \mu Q + T S.$$ Now we calculate the thermodynamic quantities explicitly for a flat space charged rotating membrane. The barred quantities in are calculated at $r = 0$. Also since $2g(r)$ is an even function of $r$, $g'(0) = 0$ and hence gives $$2g(0) = \beta^2(1-\alpha^2)^2,$$ where $\alpha = 2\sqrt{2\pi}\mu$ and $\beta = \frac{D}{4\pi T}$. Hence equations and give $$\begin{split}
\bar{K} &= K(0) = \frac{D}{\sqrt{2g(0)}} = \frac{D}{\beta(1-\alpha^2)},\\
\bar{\gamma} &= \gamma(0) = 1,
\end{split}$$ while leads to $$\bar{Q} = \alpha\gamma(0) = \alpha.$$ Plugging in , $$\label{thermsaddle2}
\begin{split}
\tilde{E} &= V\frac{D(1+\alpha^2)}{16\pi\beta(1-\alpha^2)},\\
q &= V\frac{D\alpha}{2\sqrt{2\pi}\beta(1-\alpha^2)},\\
S &= \frac{V}4,
\end{split}$$ which satisfy\
Notice that the thermodynamic quantities depend on $\omega$ only through $V$, because $V$ depends on $C$ which in turn depends on $\omega$. Also, the angular momentum doesn’t contribute as discussed earlier. So all the thermodynamic quantities associated with a charged rotating membrane are proportional to the corresponding quantities for the charged static spherical membrane at the same temperature. For a charged static spherical membrane, $C=1$ (see ) which means $\frac{V}{V_0} = \frac{1}{\sqrt{C}}$, where $V_0$ is the ‘membrane volume’ of the charged spherical static membrane. So now can be written as $$\tilde{E} = \frac{\tilde{E}_0}{\sqrt{C}},~~q = \frac{q_0}{\sqrt{C}},~~S = \frac{S_0}{\sqrt{C}},~~T = T_0,~~\mu = \mu_0,$$ where the quantities with subscript $0$ belong to the static spherical membrane. [^20]
Discussion
==========
The important results obtained in this paper can be summarized as follows:
- The membrane equations in [@Bhattacharyya:2018ads] are specialized to the stationary membrane configurations, which yield two scalar equations which we call the ‘stationary membrane equations’ (SMEs).
- The thermodynamic quantities namely energy, entropy and electric charge associated with a stationary membrane are read off from the effective stress tensor and charge current on that membrane, and it is shown that they satisfy the first law of thermodynamics if and only if the membrane satisfies the SMEs. The integration constants in the SMEs are identified with the temperature and chemical potential of the membrane.
- Exact solutions of the uncharged version of the SMEs are obtained in axially symmetric (rotating) case, and for the charged rotating stationary membranes perturbative solutions are obtained in both the flat background and the AdS background.
However in this paper we have not discussed the dynamical stability of these solutions. In many cases the rotating black holes are unstable and would change into a more stable configuration, see [@Dias:2010eu; @Emparan:2003sy]. This feature is likely to be seen in large number of dimensions as well, and [@Emparan:2014jca] discusses the stability of uncharged rotating black holes in large $D$. We would like to do a stability analysis by checking the quasinormal modes about the charged rotating solutions obtained in this paper, but we leave it for future work.\
The perturbative solution for a charged rotating membrane with charge added perturbatively to a rotating uncharged membrane as done in , has a subtlety. The perturbative corrections yield homogeneous solutions at every order, but they can be fixed by demanding the solutions to be regular at $r = 0$ except in the case where $\frac{1}{C}$ is an even integer. In such a case we have not been able to figure out the mechanism by which the homogeneous solutions can be fixed, but we guess that the solutions wouldn’t be finite in extent in $r$ unless these integration constants are set to zero. This may also be related to the perturbative instability of the rotating solutions in large $D$ reported in [@Tanabe:2016opw].\
In this paper we haven’t considered the rotating membranes in the background with positive cosmological constant, i.e. de Sitter spacetime. Naively one can work with quadratic ansatz for membrane solution in dS background as well, after the analytic continuation $L^2 \rightarrow -L^2$ in sections \[rotmemAdS\] and \[rotsolAdS\]. One obvious consequence is that there is an extra solution compared to AdS case, namely $A = -1, C = 1$ in equation which we throw away in the AdS case as doesn’t give a valid solution. This solution is peculiar as it is independent of $\omega$, However the dS case deserves an analysis more careful than just a naive analytic continuation in order to get more interesting solutions and to avoid potential subtleties.\
Finally, according to the particular membrane paradigm we are working in, the large $D$ membranes should have a one-to-one correspondence with black holes in large $D$. It would be interesting to obtain the metric and gauge field outside the black holes corresponding to the membrane solutions obtained in this paper by putting the explicit expressions for the shape function, charge and velocity fields in the expressions for metric and gauge field in [@Bhattacharyya:2018ads].
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to S. Minwalla for suggesting this problem to us and for being part of the original collaboration, and S. Bhattacharyya, S. Kundu and P. Nandi for providing us with the draft of [@Bhattacharyya:2018ads] for using the charged membrane equations in AdS which are pivotal in our work. We would especially like to thank S. Bhattacharyya, A. O’Bannon, Y. Dandekar, A.Saha and Song He for several extremely useful discussions over the course of this project. We would also like to thank S. Bhattacharyya, S. Minwalla and A.Saha and for comments on a preliminary draft of this manuscript. The work of ST is supported by ITP-CAS. The work of MM was supported in part by an ISF excellence center grant 1989/14, a BSF grant 2016324 and by the Israel Science Foundation under grant 504/13.
Identities for section \[subsecstat\] {#idens2}
=====================================
To make the equations easy to the eye, we introduce a colour coding to various terms in many equations in this paper, especially in the first two appendices and in section \[EffcurrST\]. We have maintained the leading order in all these equations to be ${\cal O}(D)$, and so whenever some manipulation gives subleading terms, they will be coloured as soon as they appear, and then will disappear in the nest step, and lumped together with other subleading terms in the form of the ${\cal O}(1)$ symbol at the end of the line.
The Gauss’ equation for a timelike hypersurface
$$\label{Gauss}
\begin{split}
\hat{R}_{\mu\nu\alpha\beta} &= R_{MNPQ}e^M_\mu e^N_\nu e^P_\alpha e^Q_\beta +\left(K_{\mu\alpha}K_{\nu\beta}- K_{\nu\alpha}K_{\mu\beta}\right),\\
\hat{R}_{\mu\nu} &= R_{MN} e^M_\mu e^N_\nu +\left(K~K_{\mu\nu}- K_\mu^\alpha K_{\nu\alpha}\right).
\end{split}$$
$$\label{gamp1}
(u\cdot\hat{\nabla})\gamma = \gamma^4k^{\mu}k^{\alpha}\hat{\nabla}_{\mu}k_{\alpha} =\frac{\gamma^4}2k^{\mu}k^{\alpha}(\hat{\nabla}_{\mu}k_{\alpha}+\hat{\nabla}_{\alpha}k_{\mu})= 0.$$
due to the Killing equation.\
$$\label{gamp2}
\begin{split}
u\cdot\hat{\nabla} u_\mu &= u\cdot\hat{\nabla}(\gamma k_\mu),\\
&= \gamma u\cdot\hat{\nabla}k_\mu,\\
&= \gamma^2 k^\nu\hat{\nabla}_\nu k_\mu,\\
&= -\gamma^2 k^\nu\hat{\nabla}_\mu k_\nu,\\
&= -\frac{\gamma^2}2 \hat{\nabla}_\mu(k_\nu k^\nu),\\
&= \frac{\gamma^2}2 \hat{\nabla}_\mu(\gamma^{-2}),\\
&= -\hat{\nabla}_\mu\ln\gamma.
\end{split}$$
$$\label{uKupuRu}
\begin{split}
K~u\cdot K\cdot u + u\cdot R\cdot u &= u\cdot \hat{R}\cdot u ~\textcolor{blue}{+~u\cdot K\cdot K\cdot u},\\
&= -u^\nu\left(\hat{\nabla}_\mu\hat{\nabla}_\nu u^\mu \textcolor{blue}{- \hat{\nabla}_\nu\hat{\nabla}_\mu u^\mu}\right)+\mathcal{O}(1),\\
&= \hat{\nabla}^\mu\left(u\cdot\hat{\nabla}u_\mu\right) \textcolor{blue}{-(\hat{\nabla}^\mu u^\nu)(\hat{\nabla}_\nu u_\mu)}+\mathcal{O}(1),\\
&= -\hat{\nabla}^2\ln\gamma +\mathcal{O}(1)
\end{split}$$
has been used here.
$$\label{tnabsqu}
\begin{split}
p^\nu_\mu\hat{\nabla}^2 u_\nu &= p^\nu_\mu\hat{\nabla}^\alpha\hat{\nabla}_\alpha u_\nu\\
&= p^\nu_\mu\left(\hat{\nabla}^\alpha(k_\alpha\hat{\nabla}_\nu\gamma+k_\nu\hat{\nabla}_\alpha\gamma)-\hat{\nabla}_\alpha\hat{\nabla}_\nu u^\alpha\right)\\
&= \textcolor{blue}{p^\nu_\mu\left((\hat{\nabla}\cdot k) \hat{\nabla}_\nu\gamma + (k\cdot\hat{\nabla})\hat{\nabla}_\nu\gamma + (\hat{\nabla}^\alpha k_\nu)\hat{\nabla}_\alpha\gamma\right)} - p^\nu_\mu\left(\textcolor{blue}{\hat{\nabla}_\nu(\hat{\nabla}\cdot k)} + \hat{R}_{\nu\alpha}u^{\alpha}\right),\\
&= -p^\nu_\mu\hat{R}_{\nu\alpha}u^{\alpha} + \mathcal{O}(1).
\end{split}$$
Therefore, $$\label{tnabsqupuK}
\begin{split}
p^\nu_\mu\left(\hat{\nabla}^2 u_\nu + K~u\cdot K_\nu\right) &= p^\nu_\mu\left(K~K_{\nu\alpha}-\hat{R}_{\nu\alpha}u^{\alpha}\right)+ \mathcal{O}(1),\\
&= -p^\nu_\mu e^M_\nu R_{MN} e^N_\alpha u^\alpha \textcolor{blue}{~+p^\nu_\mu (u\cdot K\cdot K)_\nu}+\mathcal{O}(1),\\
&= e^M_\mu\mathcal{P}_M^A R_{AN}u^N + \mathcal{O}(1),\\
&= \mathcal{O}(1)
\end{split}$$
$$\label{vmanip}
\begin{split}
\frac{(1+Q^2)}{(1-Q^2)}\hat\nabla_{\mu}\ln\gamma &= \frac{(1+\alpha^2\gamma^2)}{(1-\alpha^2\gamma^2)}\frac{\hat\nabla_{\mu}\gamma}\gamma\\
&= \frac{\frac{1}{\gamma^2}+\alpha^2}{\frac{1}{\gamma}-\alpha^2\gamma}\hat\nabla_{\mu}\gamma\\
&= -\hat\nabla_{\mu}\ln\left(\frac{1}{\gamma}-\alpha^2\gamma\right)
\end{split}$$
Identities for Section \[EffcurrST\] {#idens3}
====================================
$$\begin{split}
\hat{\nabla}^\mu \left(\frac{\hat{\nabla}^2u_\mu}{K}\right) &= \frac{\hat{\nabla}^\mu\hat{\nabla}^2u_\mu}{K} \textcolor{blue}{+\frac{(\hat{\nabla}^\mu K)\hat{\nabla}^2u_\mu}{K^2}},\\
&= \frac{\hat{\nabla}^\nu\hat{\nabla}_\mu\hat{\nabla}_\nu u^\mu}{K} + {\cal O}(1),\\
&= \textcolor{blue}{\frac{\hat{\nabla}^\nu\hat{\nabla}_\nu\hat{\nabla}_\mu u^\mu}{K}} + \frac{\hat{\nabla}^\nu(\hat{R}_{\mu\nu} u^\mu)}{K} + {\cal O}(1),\\
&= \textcolor{blue}{\frac{\hat{\nabla}^\nu(e^N_\nu (R\cdot u)_N)}{K}} + \frac{\hat{\nabla}^\nu(K~K_{\mu\nu}u^\mu)}{K} - \textcolor{blue}{\frac{\hat{\nabla}^\nu(K_\nu^\alpha K_{\alpha\mu} u^\mu)}{K}} + {\cal O}(1),\\
&= \hat{\nabla}^\nu(K_{\mu\nu}u^\mu) - \textcolor{blue}{\frac{(\hat{\nabla}^\nu K)K_{\mu\nu}u^\mu}{K}} + {\cal O}(1).
\end{split}$$
Hence $$\label{Bid1}
\hat{\nabla}^\mu \left(\frac{\hat{\nabla}^2u_\mu}{K} - (K\cdot u)_\mu\right) = {\cal O}(1).$$
$$\label{Bid2}
\begin{split}
\hat{\nabla}^\mu (u\cdot\hat{\nabla}u_\mu) &= u^\nu\hat{\nabla}_\mu\hat{\nabla}_\nu u^\mu + \textcolor{blue}{(\hat{\nabla}_\nu u^\mu)(\hat{\nabla}_\mu u^\nu)}\\
&= \textcolor{blue}{u^\nu\hat{\nabla}_\nu\hat{\nabla}_\mu u^\mu} + u^\nu\hat{R}_{\mu\nu} u^\mu + {\cal O}(1),\\
&= u\cdot R \cdot u + K~u\cdot K\cdot u - \textcolor{blue}{u\cdot K\cdot K\cdot u} + {\cal O}(1),\\
&= u\cdot R \cdot u + K~u\cdot K\cdot u + {\cal O}(1).
\end{split}$$
$$\tilde{\Sigma}^\alpha_\alpha = \textcolor{blue}{\hat{\nabla}\cdot u} = {\cal O} \left(\frac{1}D\right)$$
and $$\tilde{\Sigma}_{\mu\nu} = \frac{1}2\left(p^\alpha_\mu\hat{\nabla}_\alpha u_\nu + p^\alpha_\nu\hat{\nabla}_\alpha u_\mu\right)$$ also, $$\begin{split}
\hat{\nabla}^\mu p^\alpha_\mu &= \textcolor{blue}{(\hat{\nabla}\cdot u) u^\alpha + u\cdot \hat{\nabla}u^\alpha} = {\cal O}(1),\\
\hat{\nabla}^\mu p^\alpha_\nu &= \textcolor{blue}{(\hat{\nabla}_\mu u_\nu) u^\alpha + u_\nu \hat{\nabla}_\mu u^\alpha} = {\cal O}(1)
\end{split}$$ and $$\begin{split}
u^\mu\hat{\nabla}^2u_\mu &= \hat{\nabla}^\nu(u^\mu\hat\nabla_\nu u_\mu) \textcolor{blue}{-(\hat{\nabla}^\nu u^\mu)(\hat{\nabla}_\nu u_\mu)} \\
&= {\cal O}(1)
\end{split}$$ which means $$\hat{\nabla}^2u_\mu = p^\nu_\mu\hat{\nabla}^2u_\nu + {\cal O}(1)$$
Hence $$\Sigma_{\mu\nu} = \tilde{\Sigma}_{\mu\nu} + {\cal O} \left(\frac{1}D\right)$$ Thus $$\label{Bid3}
\begin{split}
\hat{\nabla}^\mu\Sigma_{\mu\nu} =& \hat{\nabla}^\mu\tilde{\Sigma}_{\mu\nu} + {\cal O} (1)\\
=& \frac{1}2\left(p^\alpha_\mu\hat{\nabla}^\mu \hat{\nabla}_\alpha u_\nu \textcolor{blue}{+(\hat{\nabla}^\mu p^\alpha_\mu)\hat{\nabla}_\alpha u_\nu}+ p^\alpha_\nu\hat{\nabla}^\mu\hat{\nabla}^\alpha u_\mu \textcolor{blue}{ + (\hat{\nabla}^\mu p^\alpha_\nu)\hat{\nabla}_\alpha u_\mu}\right)+ {\cal O} (1)\\
=& \frac{1}2\left(\hat{\nabla}^2u_\nu \textcolor{blue}{+u^\alpha u_\mu\hat{\nabla}^\mu \hat{\nabla}_\alpha u_\nu + p^\alpha_\nu\hat{\nabla}_\alpha\hat{\nabla}_\mu u^\mu}+ p^\alpha_\nu\hat{R}_{\mu\alpha} u^\mu \right)+{\cal O} (1)\\
=& \frac{1}2\left(\hat{\nabla}^2u_\alpha p^\alpha_\nu + K~p^\alpha_\nu K_{\mu\alpha} u^\mu \textcolor{blue}{-p^\alpha_\nu K_{\alpha}^{\beta} K_{\beta\mu} u^\mu} \right)+{\cal O} (1)\\
=& \frac{1}2\left(\hat{\nabla}^2u_\alpha + K~ (K\cdot u)_\alpha \right)p^\alpha_\nu+{\cal O} (1)
\end{split}$$
And finally $$\label{Vid}
\begin{split}
\hat{\nabla}\cdot{\cal V} =& \hat{\nabla}^\mu\left(\nabla_\mu Q + Qu\cdot\hat{\nabla}u_\mu\right)\\
=& \hat{\nabla}^2Q +Q\hat{\nabla}^\mu(u\cdot\hat{\nabla}u_\mu)\textcolor{blue}{+Q(\hat{\nabla}^\mu Q)(u\cdot\hat{\nabla}u_\mu)}\\
=& \hat{\nabla}^2Q + Qu\cdot R\cdot u + QK~u\cdot K\cdot u+{\cal O}(1)
\end{split}$$
Flat space metric split in two planes
=====================================
The flat space metric in $D$ dimension can be written as $$\label{fltmet1}
ds^2=-dt^2+dr^2+r^2d\Omega^2_{D-2}.$$
Now, we split the spacetime into $p$ two planes keeping a large $SO(D-2p-2)$ isometry intact by writing $$r^2=\sum_{i=1}^{2p}r_i^2+s^2.$$ $$\label{flsplit}
ds^2=-dt^2+\sum_i(dr_i^2+r_i^2 d\theta_i^2)+ds^2+s^2 d\Omega^2_{D-2p-2}.$$
The Global AdS metric with manifest $SO(D-2p-2)$ isometry
=========================================================
The metric of AdS written in global coordinates is given by $$\label{globalads1}
ds^2=-\left(1+\frac{r^2}{L^2}\right)dt^2+\frac{dr^2}{\left(1+\frac{r^2}{L^2}\right)}+r^2d\Omega^2_{D-2}.$$
Now, we split the spacetime into $p$ two planes keeping a large $SO(D-2p-2)$ isometry intact by writing $$r^2=\sum_{i=1}^{2p}r_i^2+s^2.$$ Doing this the global AdS metric becomes $$\label{globadssplit}
ds^2=-dt^2\left(1+\frac{\sum {r_i}^2+s^2}{L^2}\right)-\frac{\left(\sum r_i dr_i+s ds\right)^2}{\sum {r_k}^2+s^2+L^2}+\sum_i(dr_i^2+r_i^2 d\theta_i^2)+ds^2+s^2 d\Omega^2_{D-2p-2}.$$
All the uncharged rotating membrane solutions in flat and AdS backgrounds
=========================================================================
It is important to note that the equations and are actually the square of and respectively. So solving them may give some spurious solutions which we need to weed out carefully. In this appendix we will obtain the conditions for a solution not to be spurious.\
The large $D$ membrane paradigm is based on the following peculiarity of the large $D$ black holes that we consider. If we zoom into a patch around a point $y_0$ on the horizon of such a black hole, it looks like the spacetime near the horizon of a static black hole boosted with velocity $u(y_0)$, where $u$ is the velocity field of the membrane corresponding to that black hole. In other words, the membrane looks locally static at $y_0$ when seen from a frame of reference with a boost $u(y_0)$. Hence $u(y_0)$ at any $y_0$ on the membrane should be a physical velocity. This means $\gamma^2(y_0) > 0$ at all $y_0$ on the membrane.\
Since we are solving the equation $K^2 = \frac{\gamma^2}{\beta^2}$, it may give solutions with $K^2 <0$ and $\gamma^2 < 0$, which we have to watch out for. If a solution has $K^2 < 0$ then it also has $\gamma^2 <0$, so it would suffice to check the sign of $K^{-2}$.
Rotating membranes in flat background
-------------------------------------
For the flat space membranes, $$K^{-2} = D^{-2}(2g + (g'(r))^2),$$ which is positive definite, so it seems like any solution with $2g > 0$ will be acceptable. But if we look at $$\gamma^{-2} = 1 - \omega^2 r^2,$$ the non-compact solutions should be unacceptable as $\gamma^{-2}$ goes negative for arbitrarily large value of $r$ for nonzero $\omega$. However on solving the equation , it is evident that this equation has only compact solutions of the form $$2g = \beta^2 - Cr^2,$$ where $C$ is positive, and given by . Since $\gamma^{-2}$ given above is a monotonically decreasing function of $r$, its minimum value, which is at the maximum $r$ on the membrane, has to be positive. Since $r^2_{max} = C\beta^2$, $$\gamma^{-2}_{min} = 1 - C\omega^2 \beta^2 = 1 - C\Upsilon^2,$$ and $C<1$ from , so $$\gamma^{-2}_{min} > 1 - \Upsilon^2.$$ But from , $\Upsilon^2 < \frac{1}4$, and hence $\gamma^{-2}_{min} > 0$ for the solutions .
Rotating membranes in AdS background
------------------------------------
For the AdS space membranes, $$K^{-2} = \frac{1}{D^2\left(1+\frac{2g - rg'(r)}{L^2}\right)^2}\left(2g + (g'(r))^2 + \frac{(2g - rg'(r))^2}{L^2}\right),$$
The solutions we consider have the form $$2g = \beta^2A\mathcal{L}^2 - Cr^2 = A L^2 - Cr^2.$$
Now consider the solutions , i.e. $A = -1$, $C = 0,1$. Since $2g$ has to be positive somewhere, both these solutions are discarded.\
Now let’s consider the branch $$A\mathcal{L}^2 = (1+A)^2$$ Which means $A > 0$. For $2g > 0$, $K^{-2} > 0$ for any $A$ and $C$, and the solutions should be acceptable.\
Noncompact rotating membranes in AdS {#noncomp}
------------------------------------
In \[rotsolAdS\] we analyzed the compact membrane solutions in AdS, i.e. $C > 0$. However there are valid solutions for $C < 0$, i.e. the noncompact solutions which we discuss below.
$$2g = A L^2 - Cr^2$$ represents a noncompact solution that extends from $r = 0$ to $r = \infty$ for $C < 0$. The expressions for $A_\pm$ and $C_{\pm\pm}$ are given in and . These solutions are valid in the parameter ranges that are complementary to the ones in the table \[tab1\]; those are given in \[tab2\]. According to the above subsection, these solutions are valid in their respective parameter range.\
Parameter range allowed $A$ allowed $C$
------------------------------------------------------- ------------- -------------
$|\Upsilon| < \frac{1}{\mathcal{L}}$ $A_+$ $C_{+-}$
$A_-$ $C_{--}$
$\frac{1}{\mathcal{L}} < |\Upsilon| \leq \frac{1}{2}$ $A_+$ $C_{++}$
$A_+$ $C_{+-}$
: Allowed branches of solutions for different values of ${\Upsilon}$[]{data-label="tab2"}
Naively it looks like for these extended solutions as we go away from $r = 0$, the speed $v = r\omega$ grows larger and it may become superluminal at some point. But in the coordinate system we have chosen, the speed of light also grows larger as $r$ grows. So the membrane never attains superluminal speed.
Validity regimes of the solution for the rotating $AdS$ solutions {#valsolap}
=================================================================
In this appendix we give a derivation of the conditions presented in .\
For a real solution $\sqrt{1-4{\Upsilon}_i^2}>0$ which gives us the condition $$-\frac{1}{2}<{\Upsilon}_i<\frac{1}{2}.$$ The compactness and reality condition on $A_{i\pm}$ gives us the condition that $$\begin{split}
&{{\cal L}}^2-4>0,\\
&{{\cal L}}>2\qquad \text{or} \qquad{{\cal L}}<-2.
\end{split}$$ Since ${{\cal L}}$ is a positive quantity, ${{\cal L}}>2$. $$\label{amxmin}
A_{+,min}=1 \qquad A_{-,max}=1.$$ Hence $$A_{i+}-1>0,\qquad A_{i-}-1<0.$$
- For $C_{i++}>0$ $$\begin{split}
C_{++}&=\frac{1}{2}\left((1-A_{+}+{{\cal L}}\sqrt{A_{+}(1-4{\Upsilon}^2)})\right)>0,\\
&\Rightarrow{{\cal L}}\sqrt{A_{i+}(1-4{\Upsilon}_i^2)}>(A_{i+}-1),\\
&\Rightarrow A_{+}\left({{\cal L}}+\frac{1}{{\Upsilon}_i}\right)\left({{\cal L}}-\frac{1}{{\Upsilon}_i}\right)<0,\\
&\Rightarrow|{\Upsilon}|<\frac{1}{{{\cal L}}}.
\end{split}$$
- $C_{i+-}<0$ since both the terms are negative.
- $C_{i-+}>0$ since both the terms are positive definite.
- For $C_{i--}>0$ $$\begin{split}
C_{--}&=\frac{1}{2}\left((1-A_{-}-{{\cal L}}\sqrt{A_{+}(1-4{\Upsilon}^2)})\right)>0,\\
&\Rightarrow {{\cal L}}\sqrt{A_{i-}(1-4{\Upsilon}_i^2)}<(1-A_{i-}),\\
&\Rightarrow\left({{\cal L}}+\frac{1}{{\Upsilon}_i}\right)\left({{\cal L}}-\frac{1}{{\Upsilon}_i}\right)>0,\\
&\Rightarrow|{\Upsilon}|>\frac{1}{{{\cal L}}}.
\end{split}$$
Coefficients of the perturbative expansion for the charged $AdS$ solutions {#coeffads}
==========================================================================
The coefficients for the equation $$\begin{split}
a_2&=\frac{2 (A+1) \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)^2}{{{\cal L}}^2},\\
b_2&=-2 \left(2 (A+1)^2-A {{\cal L}}^2\right) \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right),\\
c_2&=-2 A (A+1) {{\cal L}}^4+2 (A+1)^3 {{\cal L}}^2-2 C,\\
a_3&=-\frac{4 (A+1) \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)^2}{{{\cal L}}^2},\\
b_3&=2 \left(5 (A+1)^2-(2 A+1) {{\cal L}}^2\right) \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right),\\
c_3&=2 (A+1) (2 A+1) {{\cal L}}^4-6 (A+1)^3 {{\cal L}}^2,\\
a_4&=\frac{\sigma_n}{\sigma_d},\\
\sigma_n&=4 (A+1)^4 \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)^2 (3 A^4+12 A^3+2 A^2 \left(-2 {{\cal L}}^4 {\Upsilon}^2+{{\cal L}}^2+9\right)\\&+4 A \left(-2 {{\cal L}}^4 {\Upsilon}^2+{{\cal L}}^2+3\right)-{{\cal L}}^4 \left((1-2 C)^2+4 {\Upsilon}^2\right)+2 {{\cal L}}^2+3),\\
\sigma_d&=\left((2 A+1) (2 C-1) {{\cal L}}^4+2 (A+1)^2 {{\cal L}}^2 (A-3 C+2)-3 (A+1)^4\right)^2,\\
b_4&= \frac{\xi_n}{\xi_d},\\
\xi_n&=4 (A+1)^3 {{\cal L}}^2 \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right) (-(A+1) (1-2 C)^2 {{\cal L}}^6+2 (A+1)^3 (2 C-1) {{\cal L}}^4\\&+4 (A+1)^4 {{\cal L}}^2 \left(C+{{\cal L}}^2 {\Upsilon}^2-1\right)-4 (A+1)^3 {{\cal L}}^2 \left((A+1)^2+(2 C-1) {{\cal L}}^2\right)\\&+2 \left((A+1) (2 C-1) {{\cal L}}^2+(A+1)^3\right)^2-2 \left((A+1)^2+(2 C-1) {{\cal L}}^2\right) \bigg((2 (A+1)^2\\&-(2 A+1) {{\cal L}}^2) ((A+1)^2+(2 C-1) {{\cal L}}^2)+2 (A+1)^4\bigg)+3 (A+1)^5 {{\cal L}}^2),\\
\xi_d&=\left((2 A+1) {{\cal L}}^2-3 (A+1)^2\right)^2 \left((A+1)^2+(2 C-1) {{\cal L}}^2\right)^2,\\
c_4&=\frac{-(A+1)^2 (2 A+1)^2 {{\cal L}}^8+2 (A+1)^4 {{\cal L}}^6+3 (A+1)^6 {{\cal L}}^4}{\left((2 A+1) {{\cal L}}^2-3 (A+1)^2\right)^2}.
\end{split}$$ The coefficients for $$\begin{split}
&\zeta_1={{\cal L}}^5 ({{\cal L}}^9-\left(\alpha ^4-10 \alpha ^2+4\right) {{\cal L}}^7+\left(-10 \alpha ^6+40 \alpha ^4-24 \alpha ^2+2\right) {{\cal L}}^5\\&+4 \alpha ^2 \left(-8 \alpha ^6+20 \alpha ^4-12 \alpha ^2+1\right) {{\cal L}}^3-8 \alpha ^8 \left(\alpha ^2-1\right) \sqrt{4 \alpha ^2+{{\cal L}}^2-4}\\&-2 \alpha ^4 \left(9 \alpha ^4-16 \alpha ^2+5\right) {{\cal L}}^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+{{\cal L}}^8 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}\\&-\left(\alpha ^4-8 \alpha ^2+2\right) {{\cal L}}^6 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}-8 \alpha ^2 \left(\alpha ^4-3 \alpha ^2+1\right) {{\cal L}}^4 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}\\&-32 \alpha ^6 \left(\alpha ^2-1\right)^2 {{\cal L}})\times\frac{1}{\chi_1},\\
&\chi_1 =2 \left(-4 \alpha ^6+4 \alpha ^4+{{\cal L}}^4-\left(\alpha ^2-4\right) \alpha ^2 {{\cal L}}^2\right) (8 \alpha ^6 \left(\alpha ^2-1\right)+{{\cal L}}^8+\left(10 \alpha ^2-3\right) {{\cal L}}^6\\&+3 \alpha ^2 \left(11 \alpha ^2-6\right) {{\cal L}}^4+\left(38 \alpha ^6-28 \alpha ^4\right) {{\cal L}}^2+4 \alpha ^4 \left(3 \alpha ^2-1\right) {{\cal L}}\sqrt{4 \alpha ^2+{{\cal L}}^2-4}\\&+{{\cal L}}^7 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+\left(8 \alpha ^2-1\right) {{\cal L}}^5 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+\alpha ^2 \left(19 \alpha ^2-4\right) {{\cal L}}^3 \sqrt{4 \alpha ^2+{{\cal L}}^2-4})\\
&\zeta_2=-\frac{\alpha ^4 {{\cal L}}^3 \left({{\cal L}}^3-{{\cal L}}^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}-2 \alpha ^2 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+4 \left(\alpha ^2+1\right) {{\cal L}}\right)}{\chi_2},\\
&\chi_2=2 \left(-4 \alpha ^6+4 \alpha ^4+3 {{\cal L}}^4+\left(-\alpha ^4+12 \alpha ^2+4\right) {{\cal L}}^2\right) (8 \alpha ^6 \left(\alpha ^2-1\right)+{{\cal L}}^8+\left(10 \alpha ^2-3\right) {{\cal L}}^6\\&+3 \alpha ^2 \left(11 \alpha ^2-6\right) {{\cal L}}^4+\left(38 \alpha ^6-28 \alpha ^4\right) {{\cal L}}^2+4 \alpha ^4 \left(3 \alpha ^2-1\right) {{\cal L}}\sqrt{4 \alpha ^2+{{\cal L}}^2-4}\\&+{{\cal L}}^7 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+\left(8 \alpha ^2-1\right) {{\cal L}}^5 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}+\alpha ^2 \left(19 \alpha ^2-4\right) {{\cal L}}^3 \sqrt{4 \alpha ^2+{{\cal L}}^2-4}).\end{split}$$
[10]{}
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[^1]: See [@Emparan:2014jca; @Emparan:2015hwa; @Emparan:2016sjk; @Tanabe:2016pjr; @Tanabe:2016opw] for related discussion.
[^2]: See [@Sadhu:2016ynd; @Dandekar:2016jrp; @Dandekar:2017aiv] for more work in this membrane paradigm.
[^3]: See [@Herzog:2016hob; @Rozali:2016yhw; @Chen:2016fuy; @Chen:2017wpf; @Chen:2017hwm; @Rozali:2017bll; @Chen:2017rxa; @Raman:2017rfv; @Herzog:2017qwp; @Emparan:2018bmi; @Chen:2018vbv; @Andrade:2018zeb] for other work that uses large $D$ expansion.
[^4]: This comparison of the large $D$ membrane paradigm is with a naive numerical solution of Einstein’s equations only. and not with the techniques used in the field of numerical relativity, for example at LIGO, which involve some very smart approximations that save a lot of computation.
[^5]: The result in [@Bhattacharyya:2016nhn] was obtained for a charged membrane in flat space, but the analysis can be extended to a charged membrane in AdS (or dS)
[^6]: This statement is not quite accurate. According to [@Dandekar:2017aiv] the membrane stress tensor is a fluid stress tensor plus the Brown York stress tensor of the membrane due to its embedding in the background. However the Brown York part is automatically conserved, plus it doesn’t contribute to the dissipative part of the stress tensor which is our focus in this argument.
[^7]: The bulk viscosity is subleading at the relevant order in $\frac{1}D$.
[^8]: Since $k$ doesn’t have a component normal to the membrane, it doesn’t matter it the normalization is done w.r.t. the background metric or the induced metric on the membrane.
[^9]: The association of a Killing vector with a stationary configuration has been already derived in [@Caldarelli:2008mv] in the context of a fluid (plasma), but the derivation here is much simpler and more relevant to the membrane picture.
[^10]: In we have defined the conserved current $J_{(E)}^{\mu} $ with a negative sign so that the energy density which is the $J_{(E)}^0$ component of the conserved current, positive.
[^11]: Where $A_i$ satisfies the constraint $\sum_iA_i=1$.
[^12]: See [@Armas:2014bia] for perturbative uncharged black rings in flat space for any value of D.
[^13]: See [@Armas:2014rva; @Armas:2015qsv] for perturbative calculations of charged black rings in Einstein-Maxwell-Dilaton theory in flat space AdS spacetime for arbitrary D.
[^14]: As we go higher order in perturbations, the solutions for ${{\cal G}}_n$ are polynomial of $x^{2n}$, which naively looks divergent at large values of $x$. However the radius of convergence of this perturbation series is $\frac{1}{|{\Upsilon}|}$ and perturbation series is well defined in this range.
[^15]: In this section and in the next where we obtain perturbative solutions for charged membranes, we consider only the compact solutions. There are valid noncompact solutions as well ,which are discussed in \[noncomp\]
[^16]: The other condition;i.e, ${{\cal L}}<-2$ is excluded from the physical requirement that ${{\cal L}}=\beta L$ and both the temperature and the $AdS$ radius scale are positive.
[^17]: Here we consider the case where $a_1/a$ is not an even integer
[^18]: Here we have used the positive branch of the solution in , $$2{{\cal G}}_0 = \frac{1}{2} \left(\left(2 \alpha ^2+{{\cal L}}\sqrt{4 \alpha ^2+{{\cal L}}^2-4}\right)+{{\cal L}}^2-2\right) - x^2.$$
[^19]: As we go higher order in perturbations, the solutions for ${{\cal G}}_n$ are polynomial of $x^{2n}$, which naively looks divergent at large values of $x$. However the radius of convergence of this perturbation series is $\frac{1}{|{\Upsilon}|}$ and perturbation series is well defined in this range.
[^20]: A similar analysis can be done for the AdS membrane and the thermodynamic quantities can be obtained, but from the physics point of view it brings nothing to the table other than what the analysis of the flat case illustrates. So we opt not to present those results here.
|
---
abstract: 'We propose that the energy source of the outburst of V838 Mon and similar objects is an accretion event, i.e., gravitational energy rather than thermonuclear runaway. We show that the merger of two main sequence stars, of masses $M_1 \simeq 1.5 M_\odot$, and $M_2 \simeq 0.1-0.5 M_\odot$ can account for the luminosity, large radius, and low effective temperture of V838 Mon and similar objects. Subsequent cooling and gravitational contraction lead such objects to move along the Hayashi limit, as observed. By varying the masses and types of the merging stars, and by considering slowly expanding, rather than hydrostatic, envelopes, this model can account for a large range in luminosities and radii of such outburst events.'
author:
- Noam Soker and Romuald Tylenda
title: 'Main-sequence stellar eruption model for V838 Mon'
---
Introduction
============
V838 Mon (Munari et al. 2002, hereafter [@mhk]; Kimeswenger et al. [@kls]), V4332 Sgr (Martini et al. [@mwt]), and most probably M31 RV (M31 Red Variable – Mould et al. [@mcg]), form a peculiar and mysterious group of erupting stellar objects. They do not fit to any known class of stellar outbursts. Just from the light curve they can be classified as slow novae but their spectral evolution clearly shows that they are not. After developing a K-type giant-supergiant spectrum near maximum they evolve to lower effective temperature and fade as very late M-type giants or supergiants. The bolometric fading at a very low effective temperature has undoubtly been observed for V4332 Sgr (Martini et al. [@mwt]) and M31 RV (Mould et al. [@mcg]). It seems that also V838 Mon is evolving in the same direction (e.g. Osiwa[ł]{}a et al. [@omt]). No hot phase, as e.g. nebular stage in classical novae, has been observed in any of the three objects. These observational facts almost certainly exclude any kind of thermonuclear event models, such as runaway models for classical novae (see e.g. that proposed for M31 RV by Iben & Tutukov [@it]) or a late post-AGB He-shell flash (born-again AGB), which are usually discussed while trying to interpret the above variables ([@mhk]; Kimesweneger et al. [@kls]). The point is that the thermonuclear models after relatively cool phase at (visual) maximum always evolve to higher and higher effective temperature before they star fading in luminosity. The reason is that the stellar envelope, inflated by the initial thermonuclear outburst, starts shrinking well before the nuclear reactions start declining.
As usually in discussions of nature of a stellar outburst the observational data on the pre-outburst object are very important. Unfortunantely they are very scant in our case. Photometric data for V838 Mon in pre-outburst are consistent with an F-type main sequence star ([@mhk]; Kimesweneger et al. [@kls]), althought observational uncertainities are important. For V4332 Sgr a K-type star is suggested as a progenitor (Martini et al., [@mwt]). There are no data on the pre-outburst state of M31 RV.
If a thermonuclear event is excluded as argued above, the next energy source that comes to mind is an accretion (or merging) event. We therefore investigate in this paper a scenario in which a $\sim$1.5 M$_\odot$ (F-type) main sequence star accretes a less massive objects. Given the energy liberated during the observed outbursts it is clear that accretion of at least a massive planet onto the main sequence star has to be involved.
While constraining our model we use the results from observations of V838 Mon, as this is the best observed object from the three (e.g. [@mhk], Kimesweneger et al. [@kls], Kolev et al. [@kmt], Osiwa[ł]{}a et al. [@omt]). This star erupted in the begining of 2002, first by $\sim$6 mag. (in V) and as a K-type star it reached a luminosity of $\sim 500(D/1{\rm kpc})^2\mbox{L}_\odot$, and a radius of $\sim 55(D/1{\rm kpc})\mbox{R}_\odot$, where $D$ is the distance to V838 Mon. Next, after a month of a slow decline and cooling it had another brightning by $\sim$4.5 mag. and reached a luminosity of $\sim 1.0 \times 10^4 (D/1 {\rm kpc})^2 \mbox{L}_\odot$, and a radius of $\sim 120 (D/1 {\rm kpc}) \mbox{R}_\odot$. Subsequent evolution can be described as a more or less horizontal evolution on the H-R diagram towards lower and lower effective temperatures somewhat interupted by a minor brightnening a month after the second outburst. The effective radius has countinuously been increasing and in April reached a value of $\sim 400 (D/1 {\rm kpc}) \mbox{R}_\odot$ ([@mhk]). Numerous spectral lines have been observed to show P-Cyg profiles during the outburst indicating an outflow with a velocity of 200-500 km${\rm s}^{-1}$, but no mass loss rate estimate is available. Lines of LiI and BaII have been observed, but no analysis of the element abundances has been done. In mid April the star started a rapid decline in V accompanied by the spectral evolution to late M-types (Osiwa[ł]{}a et al. [@omt]).
A scattered light echo have been detected around V838 Mon and it was used to drive its distance. While [@mhk] derive a distance of $D = 0.79$ kpc and Kimeswenger et al. ([@kls]) derive a distance of $D=0.66$ kpc, a calcualtion by Bond et al. ([@bps]; H.E. Bond, private communication) yields a much larger distance of $D \simeq 3$ kpc. Above, and in the rest of the paper, we scale the properties of V838 Mon with the distance, such that our model covers this entire distance scale. Indeed, one of the adavantages of our proposed model, is that it can account for a large range of luminosities, because the luminosity is fixed by the type and mass of the merging stars. Thus we can also account for M31 RV, which was significantly more luminous than the other two objects and reached $\sim 10^6 \mbox{L}_\odot$
Our paper is organized as follows. In Sect. 2 we estimate the general properties of a main sequence star, the primary, which accretes mass from a destructed main sequence companion. These results are used in Sect. 3 to proposed a scenario to the eruption of V838 Mon. Our discussion and summary are in Sect. 4.
Accreting main sequence stars
=============================
Accreting main sequence stars were found to expand to giant dimensions in several different kind of calculations. These include the merger of two main sequence stars, in studying the formation of blue straggler stars (e.g. Sills et al. [@slb]), mass accretion by a main sequence companion inside an envelope of a giant (Hjellming & Taam 1991, hereafter [@ht]), and an accreting low mass main sequence star (Prialnik & Livio 1985, hereafter [@pl]).
PL85 calculate the outcome of an accreting fully convective $0.2 M_\odot$ main sequence star. When a large fraction of the free fall energy of the accreted matter is retained, they find the star to expand. [@pl] parametrized the fraction of energy retain by a parameter $\alpha$, $$U= \alpha {G M_\ast M_{\rm acc}}/{R_\ast} ,$$ where $M_\ast$ and $R_\ast$ are the mass and redius of the accreting main sequence star, respectively, and $M_{\rm acc}$ is the accreted mass to the envelope. For an orbiting companion that is being destructed and accreted, the maximum available energy has $\alpha=0.5$. Not all this energy will be retain by the accreted matter which build the envelope in our model. Large fraction of the energy goes to radiaton and mass loss observed in the eruption of V838 Mon and the other objects. [@pl] calculate the accretion of mass up to $\Delta M = 2.5 \times 10^{-3} M_\odot$, and find that when $\alpha$ is not too small the star expands. A good fit to their results of $\alpha=0.1$ and an accretion rate of $\dot M_{acc} = 10^{-2} M_\odot \mathrm{yr^{-1}}$, and when the accreted mass is $M_{\rm acc} > 0.5 \times 10^{-3} M_\odot$, gives for the radius of the star $$\frac {R_\ast}{R_\odot} \simeq
2.2 \left(\frac {M_{\rm acc}}{10^{-3} M_\odot} \right) + 0.5.$$ Extrapolating this fit to much higher accreted mass and radii, we find that to obtain a radius of $R_\ast \simeq 50-150 R_\odot$ (characteristic fro the first brightening of V838 Mon), the accreted mass should be $M_{\rm acc} \simeq 0.02-0.07 M_\odot$. With higher accretion rate and higher values of $\alpha$, [@pl] find larger radii.
In the calculations of [@ht], a $1.25 M_\odot$ main sequence star accretes from a giant up to $M_{\rm acc} = 0.115 M_\odot$, at an increasing rate up to $\dot M_{\rm acc} \simeq 10^{-2} M_\odot \mathrm{yr^{-1}}$. The final radius of the accreting star depends on the entropy of the accreted gas, being lower for lower-entropy accreted gas. We take the higher entropy case; we later take the low entropy-high density gas to be accreted into the primary star. In this case ([@ht]’s model 2c), a good fit to figures 1a and 3a of [@ht] is $$\frac {R_{\ast}}{R_\odot} \simeq 1.4 \times
10^{17(M_{\rm acc}/M_\odot)}.$$ This is a much stronger dependence than that found by [@pl], and cannot be extrapolated to much larger accreted mass since then the radius becomes too large. However, this formula does demonstrate that a small accreted mass can cause a large expansion when the star is already a giant. By the last fit, to obtain a radius of $R \simeq 300-1500 R_\odot$ (characteristic for the second, major outburst of V838 Mon), the accreted mass should be $M_{\rm acc} \sim 0.14 -0.2M_\odot$. The strong dependance in the last equation, suggests that with more energy supplied, as in the cores merger we discussed later, less mass can be inflated to large radii. Despite the large differences between the two works cited above, and the very different fits to [@pl] and [@ht], the amount of accreted mass, required to inflate a giant to $R_\ast \simeq 300 R_\odot$ is similar, and amounts to $M_{\rm acc} \sim 0.15 M_\odot$. In the proposed scenario, two main sequence stars merge on a dynamical scale, following dynamical instability (Rasio & Shapiro [@rs]) and the accretion rate can be as high as $\dot M_{\rm acc} \sim 0.03 M_\odot / 1~{\rm day}=
10 M_\odot \mathrm{yr^{-1}}$. At such a high accretion rate, and with most energy deposited in the accreted mass, an accreted mass lower than $0.15 M_\odot$ can be inflated to giant dimensions. Therefore, for our scenario we take the inflated envelope mass to be $M_{\rm acc} \sim 0.05-0.3 M_\odot$.
In the calculations cited above the accreted mass was added to the outer layers of the inflated star. In our scenario, the mass is added at the base of the inflated envelope, at the surface of the original primary star. A somewhat different treatment of the energy budget is required. Let the destructed companion be split into low entropy-high density gas of mass $M_c$, i.e., the core of the companion, which is accreted and sinks into the core of the primary star, down to a radius $r_f<R_1$ (where $R_1$ is the initial radius of the accreting primary star) and a high entropy mass $M_{\rm acc}$ which goes into the inflated envelope. This assumption is based on the numerical calculations of Sandquist, Bolte & Hernquist ([@sbh]). The total energy of this structure is $$E_g \simeq - \frac {G M_1 M_c}{r_f} + E_{\rm env},$$ where $E_{\rm env}$ is the gravitational energy of the envelope. For an envelope having a density profile of $\rho = K r^{-2.5}$ from $R_1$ to $R_\ast$, where $K$ is a constant, which crudely fits the envelope found by [@pl] for the model cited above, the envelope energy has a simple approximated form for $R_\ast \gg R_1$, given by $$E_{\rm env} \simeq - \frac{ G (M_1+M_c)M_{\rm acc}}{R_1}
\left(\frac {R_1}{R_\ast} \right)^{1/2}.$$ The original energy of the binary system is $$E_0 \simeq - 0.5 \frac {G M_1 M_2}{a},$$ where $a$ is the orbital separation at destruction. In addition, an energy $E_2>0$ should be supplied to destruct the companion. The condition to inflate the envelope becomes $$E_0 > E_g + E_2.$$ If $M_c=0$, this condition can be met only if $a > 0.5 R_1 (R_\ast/R_1)^{1/2}$. For $R_\ast/R_1 \sim 500$, this condition reads $a>10 R_1$. However, we do expect the destruction to occur when $a <2 R_1$. We conclude that a large fraction of the destructed companion in our scenario must be accreted as a low entropy mass into the companion.
As an example, consider the case of $M_c = k M_2$, hence $M_{\rm acc}=(1-k) M_2$, $r_f \simeq 0.25 R_1$, as the size of the solar core, $R_\ast > 300 R_1$, and $a=2 R_1$. We also take the destruction energy of the companion to be $$E_2 = \beta \frac {G M_c M_{\rm acc}} {R_2}.$$ The envelope of the companion before destruction, of mass $M_{\rm acc}$, has a smaller average radius than $R_2$. On the other hand, there is thermal energy of the companion. Hence $\beta \sim 2-3$. Another energy source not considered here is the rotation energy of each of the two stars, which are likely to corotate in our scenario (see next section). Condition (7) becomes then $$1 < 16 k - 4 \beta k (1-k) \frac{R_1 M_2}{R_2 M_1}
+4 (1-k) \left(1+ k \frac{M_2}{M_1}\right)
\left(\frac {R_1}{R_\ast} \right)^{1/2} .$$ From this condition it is clearly seen that when $k=0$, the envelope can be inflated only to $R_\ast < 16 R_1$. For $k \ga 0.3$ and $R_\ast \gg R_1$, the last term, which is the energy of the inflated envelope can be neglected. Taking also $R_1 M_2 \simeq R_2 M_1$, the last condition is simplified to $$1 < 4 k [4 - \beta (1-k)]$$ for the assumed parameters here, the last condition is met by a large margin for $ k \ga 0.3$. This means that even by relaxing some of the assumptions used here, a giant may still be formed from such a process.
Not all the energy will be retain by the accreted matter which build the envelope in our model. Some fraction of the energy goes to the fast wind and radiation as observed in the eruptions. It is interesting to note that the radiation requires only small amount of mass to be accreted on the surface of the original primary star. Taking the energy of the effective-accreted mass that goes to radiation to be that of a Keplerian disk, we find this effective mass accretion rate to be $$\dot M_{\rm rad} \simeq 6 \times 10^{-4}
\left( \frac{L}{10^4 L_\odot} \right)
\left( \frac{R_1}{R_\odot} \right)
\left( \frac{M_\ast}{M_\odot} \right)^{-1} M_\odot \mathrm{yr^{-1}} .$$ For a distance to V838 Mon of $D =3$ kpc, the luminosity is $L \simeq 10^5 \mbox{L}_\odot$, and the accreted mass over several years is still very small.
The proposed scenario
=====================
Based on the previous section, in particular on our estimate that a mass of $M_{\rm acc} \sim 0.05-0.3$ is required to inflate the envelope to $\sim 500-1500 \mbox{R}_\odot$, we proposed the following general scenario for the eruption of V838 Mon. The progenitor was a close binary system composed of two main sequence stars: a primary with $M_1 \sim 1.5 M_\odot$, and a companion with a mass $0.1 \mbox{M}_\odot \la \mbox{M}_2 \la 0.5 \mbox{M}_\odot$. The small orbital separation implies that the spins of both stars were synchronized with the orbital period, and the orbit was circular. Due to evolution, mainly magnetic winds, possibly from both stars, the system lost angular momentum at a relatively fast rate, and the orbital separation decreased. One of the stars filled its Roche lobe, and transferred mass to the other. For main sequence stars, the increase of stellar radius with the stellar mass is steeper that the increase of the average Roche lobe size with mass (e.g., see formula by Eggleton [@egg]), and the more massive primary star is likely to fill its Roche lobe first. At about the same time as Roche lobe overflow occurs, a binary system of two low mass main sequence stars becomes dynamically unstable (Rasio & Shapiro [@rs]). This implies a very fast merger and mass transfer of the outer layers of the stars. It is possible that the mass transfer of the outer layer of the primary star onto the secondary caused the envelope to be first inflated around the [*secondary*]{} star. In any case, mass transfer of the outer layers of the primary or of both stars, we propose, caused the first burst in luminosity, when V838 Mon reached a radius of $\sim 50 R_\odot$. Then, the secondary was accreted onto the primary, in particular the core of the secondary merged with the core of the primary, a process that released much more energy. We proposed that this caused the second burst after about a month.
The merging process is likely to be complicated, with different epochs of variation in the luminosity, as parcells of gas being accreted and the merged stars, both envelope and the merging cores, are settled to a new equilibrium. As noted in equation (11), a very small amount of accreted mass can lead to a relatively strong burst. Such a variation could cause the third outburst, which occurred about a month after the second burst, and which required, if lasted one month, only $\sim 5 \times 10^{-5} M_\odot (D/1\ \mathrm{kpc})^2$ to be accreted on the initial surface of the primary star.
Discussion and Summary
======================
Our main goal is to point out that the energy source of the outburst of V838 Mon was, and still is, an accretion event, i.e., gravitational energy, rather than thermonuclear runaway. We considered the merger of two main sequence stars, of masses $M_1 \simeq 1.5 M_\odot$, and $M_2 \simeq 0.1-0.5 M_\odot$. In our proposed scenario, the first outburst, with a luminosity of $L_\ast \simeq 500 (D/1 {\rm kpc})^2 \mbox{L}_\odot$ and an envelope radius of $R_\ast \simeq 55 (D/1 {\rm kpc}) R_\odot$, was caused by the merger of the outer layers of the two stars, via dynamical instability. First mass was transfered from the more massive primary to the secondary; most likely the envelope was first inflated around the secondary. Then, after about a months, the cores of the two stars collided, releasing more energy, this led to the expansion of V838 Mon to $R_\ast \simeq 400 (D/1 {\rm kpc}) R_\odot$, and its luminosity to reache $L_\ast \simeq 1.0 \times 10^4 (D/1 {\rm kpc})^2 L_\odot$. Equation (11) shows that a small amount of accreted mass can lead to large luminoisty variations. As the two merged stars settled to equilibrium, the luminoisty varies; such a process caused the third outburst as well.
As mentioned in Sect. 1 a light echo have been detected around V838 Mon during outburst. [@mhk] and Kimeswenger et al. ([@kls]) interpret this as due to scattering on circumstellar dust. This obviously rises a question about the origin of the circumstellar dust. In our scenario the dust would reside in remains of a protostellar cloud from which the V838 Mon binary system has been formed. This would imply a relatively young age of V838 Mon which is consistent with its low distance ($\sim$13 pc) from the Galactic plane ([@mhk]). It is, however, also possible that the echo is due to interestellar dust, as suggested by Bond et al. ([@bps]), and then obviously it has nothing to do with the nature of V838 Mon.
In principle, mergers of other types of stars is possible. We note that the possible locations of the progenitor of V838 Mon on the HR-diagram ([@mhk]), intersects the location of a zero-age horizontal branch (HB) star which lost most of its envelope on the RGB (D’Cruz et al. [@cdr]). Merger of such an HB star with a low mass main sequence star may leed to an inflated envelope around the HB star, similar in many respects to the scenario proposed in the previous section. The HB star lost most of its envelope because of the interaction with the low mass main sequence companion via a common envelope evolution, which started when the HB progenitor was on the upper red ginat branch. In this scenario the dust-halo can be attributed to the high mass loss rate on the red giant branch.
Finally it is worth of noting that the effective photosphere of the three objects seems to increase more or less steadilly with time during outburst. In the case of M31 RV it expanded up to $8000 {\rm R}_\odot$ (Mould et al. [@mcg]). This would imply that the observed photospheric regions are not in hydrostatic equillibrium. The models of [@ht] and [@pl], on which our considerations in Sect. 2 are based, are in hydrostatic equillibrium. Slowly expanding envelopes would probably require less mass and thus lower $M_{\rm acc}$ than hydrostatic envelopes inflated to large radii. Thus accretion of an object of even lower mass then estimated in Sect. 2 could give origin to the observed outbursts.
This work has partly been supported by the Polish State Committee for Scientific Research through the grant No. 2.P03D.020.17. N. S. has also been supported by a grant from the USA-Israel Binational Science Foundation and the Israel Science Foundation.
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Osiwa[ł]{}a, J. P., Miko[ł]{}ajewski, M., Tomov, T., Ga[ł]{}an, & C., Nirski J. 2002 in Symbiotic Stars probing Stellar Evolution, eds. R.L.M. Corradi, J. Miko[ł]{}ajewska, & T.J. Mahoney, ASP Conf. Series, in press
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|
---
abstract: 'We study the production of $\Sigma^{\pm}\pi^{\mp}pK^+$ particle quartets in p+p reactions at 3.5 GeV kinetic beam energy. The data were taken with the HADES experiment at GSI. This report evaluates the contribution of resonances like $\Lambda(1405)$, $\Sigma(1385)^0$, $\Lambda(1520)$, $\Delta(1232)$, $N^*$ and $K^{*0}$ to the $\Sigma^{\pm} \pi^{\mp} p K^+$ final state. The resulting simulation model is compared to the experimental data in several angular distributions and it shows itself as suitable to evaluate the acceptance corrections properly.'
address:
- 'Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Sud, 95125 Catania, Italy'
- 'LIP-Laboratório de Instrumentação e Física Experimental de Partículas , 3004-516 Coimbra, Portugal'
- 'Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059 Kraków, Poland'
- 'GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany'
- 'Institut für Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany'
- 'Joint Institute of Nuclear Research, 141980 Dubna, Russia'
- 'Institut für Kernphysik, Goethe-Universität, 60438 Frankfurt, Germany'
- 'Excellence Cluster ’Origin and Structure of the Universe’ , 85748 Garching, Germany'
- 'Physik Department E12, Technische Universität München, 85748 Garching, Germany'
- 'II.Physikalisches Institut, Justus Liebig Universität Giessen, 35392 Giessen, Germany'
- 'Istituto Nazionale di Fisica Nucleare, Sezione di Milano, 20133 Milano, Italy'
- 'Institute for Nuclear Research, Russian Academy of Science, 117312 Moscow, Russia'
- 'Frederick University, 1036 Nicosia, Cyprus'
- 'Department of Physics, University of Cyprus, 1678 Nicosia, Cyprus'
- 'Institut de Physique Nucléaire (UMR 8608), CNRS/IN2P3 - Université Paris Sud, F-91406 Orsay Cedex, France'
- 'Nuclear Physics Institute, Academy of Sciences of Czech Republic, 25068 Rez, Czech Republic'
- 'LabCAF. Dpto. Física de Partículas, Univ. de Santiago de Compostela, 15706 Santiago de Compostela, Spain'
- 'Also at Lawrence Berkeley National Laboratory, Berkeley, USA'
- 'Also at ISEC Coimbra, Coimbra, Portugal'
- 'Also at ExtreMe Matter Institute EMMI, 64291 Darmstadt, Germany'
- 'Also at Technische Universität Darmstadt, Darmstadt, Germany'
- 'Also at Technische Universität Dresden, 01062 Dresden, Germany'
- 'Also at Dipartimento di Fisica, Università di Milano, 20133 Milano, Italy'
- 'Also at Dipartimento di Fisica Generale and INFN, Università di Torino, 10125 Torino, Italy'
author:
- 'G. Agakishiev'
- 'A. Balanda'
- 'D. Belver'
- 'A. Belyaev'
- 'J.C. Berger-Chen'
- 'A. Blanco'
- 'M. Böhmer'
- 'J. L. Boyard'
- 'P. Cabanelas'
- 'E. Castro'
- 'S. Chernenko'
- 'T. Christ'
- 'M. Destefanis'
- 'F. Dohrmann'
- 'A. Dybczak'
- 'E. Epple'
- 'L. Fabbietti'
- 'O. Fateev'
- 'P. Finocchiaro'
- 'P. Fonte'
- 'J. Friese'
- 'I. Fröhlich'
- 'T. Galatyuk'
- 'J. A. Garzón'
- 'R. Gernhäuser'
- 'C. Gilardi'
- 'M. Golubeva'
- 'D. González-Díaz'
- 'F. Guber'
- 'M. Gumberidze'
- 'T. Heinz'
- 'T. Hennino'
- 'R. Holzmann'
- 'A. Ierusalimov'
- 'I. Iori'
- 'A. Ivashkin'
- 'M. Jurkovic'
- 'B. Kämpfer'
- 'K. Kanaki'
- 'T. Karavicheva'
- 'I. Koenig'
- 'W. Koenig'
- 'B. W. Kolb'
- 'R. Kotte'
- 'A. Krása'
- 'F. Krizek'
- 'R. Krücken'
- 'H. Kuc'
- 'W. Kühn'
- 'A. Kugler'
- 'A. Kurepin'
- 'R. Lalik'
- 'S. Lang'
- 'J. S. Lange'
- 'K. Lapidus'
- 'T. Liu'
- 'L. Lopes'
- 'M. Lorenz'
- 'L. Maier'
- 'A. Mangiarotti'
- 'J. Markert'
- 'V. Metag'
- 'B. Michalska'
- 'J. Michel'
- 'E. Morinière'
- 'J. Mousa'
- 'C. Müntz'
- 'R. Münzer'
- 'L. Naumann'
- 'J. Otwinowski'
- 'Y. C. Pachmayer'
- 'M. Palka'
- 'Y. Parpottas'
- 'V. Pechenov'
- 'O. Pechenova'
- 'J. Pietraszko'
- 'W. Przygoda'
- 'B. Ramstein'
- 'A. Reshetin'
- 'A. Rustamov'
- 'A. Sadovsky'
- 'P. Salabura'
- 'A. Schmah'
- 'E. Schwab'
- 'J. Siebenson'
- 'Yu.G. Sobolev'
- 'S. Spataro'
- 'B. Spruck'
- 'H. Ströbele'
- 'J. Stroth'
- 'C. Sturm'
- 'A. Tarantola'
- 'K. Teilab'
- 'P. Tlusty'
- 'M. Traxler'
- 'R. Trebacz'
- 'H. Tsertos'
- 'V. Wagner'
- 'M. Weber'
- 'C. Wendisch'
- 'J. Wüstenfeld'
- 'S. Yurevich'
- 'Y. Zanevsky'
title: 'Production of $\Sigma^{\pm}\pi^{\mp}pK^+$ in p+p reactions at 3.5 GeV beam energy'
---
$\Lambda(1405)$ ,p+p collisions
Introduction
============
For already half a century the $\Lambda(1405)$ is a well known resonance with strangeness $S=-1$, Isospin $I=0$ and spin $\frac{1}{2}$. Even though its four star character suggests a good understanding of this baryon, its inner structure is still a topic of investigation. Indeed it is difficult to describe the $\Lambda(1405)$ as a three quark baryon, as it is lighter than its nucleon partner, the $N^*(1535)$. Also the large mass difference to the $\Lambda(1520)$ can not be understood in terms of spin-orbital coupling [@Hyodo:2011ur]. With the mass of the $\Lambda(1405)$ lying slightly below the $\bar{K}N$ threshold another picture of this particle was established. From the analysis of the $\bar{K}N$ scattering length Dalitz and Tuan predicted the $\Lambda(1405)$ in 1959 [@Dalitz1959dn; @Dalitz1959dq], already two years before its experimental discovery. Nowadays the $\Lambda(1405)$ is described in a coupled channel approach based on chiral dynamics [@Ikeda:2011dx]. Here this baryon is generated dynamically as an interference of two states, a $K^-p$ bound state and a $\Sigma\pi$ resonance. However, this two pole structure cannot be observed directly in the $\Sigma\pi$ invariant mass spectrum, as the $\Sigma\pi$ pole is located far in the imaginary part of the complex energy plane.\
With these predictions, the structure of the $\Lambda(1405)$ is interesting in terms of a deeper understanding of the kaon-nucleon interaction. Experimental data are available for $\pi^-$+p [@Chao:1973sa], $K^-$+p [@Hemingway:1984pz] and $\gamma$+p [@Moriya:2011af] reactions. First results on p+p data were reported in [@Zychor:2007gf]. But only the neutral decay channel ($\Lambda(1405)\rightarrow\Sigma^0\pi^0$) was investigated. We also study p+p reactions and concentrate on the charged decay channels ($\Lambda(1405)\rightarrow\Sigma^{\pm}\pi^{\mp}$). In order to extract precisely the spectral function of the $\Lambda(1405)$, all the reactions that contribute to the $\Sigma^{\pm}\pi^{\mp}pK^+$ particle quartet have to be considered. This includes the production of resonances like $K^{0*}$, $N^*$ and $\Delta^{++}(1232)$. A simplified model, assuming only an incoherent sum of these contributions is finally used to describe the experimental data and extract the acceptance corrections. This model reproduces the experimental data for many kinematical variables.\
The analyzed data were taken with the **H**igh **A**cceptance **D**i-**E**lectron **S**pectrometer (HADES) [@Agakishiev:2009am] at GSI in Darmstadt, Germany. In this beam time a proton beam of 3.5 GeV kinetic energy was incident on a liquid hydrogen target and a total statistic of about 1.2 billion events was collected.
Data analysis
=============
Evaluation of resonances contributions
--------------------------------------
The presented analysis concentrates on the production of the $\Lambda(1405)$ together with a proton and a $K^+$ followed by the decay of the $\Lambda(1405)$ into $\Sigma^{\pm}\pi^{\mp}$: $$\begin{aligned}
\label{rec:L1405}
p+p\rightarrow\Lambda(1405)+p+K^+\rightarrow(\Sigma^{\pm}\pi^{\mp})+p+K^+\rightarrow((n\pi^{\pm})\pi^{\mp})+p+K^+\end{aligned}$$ The general analysis steps to extract the $\Lambda(1405)$ signal are presented in detail in [@Siebenson:2010hh; @Siebenson:2010zz]. These steps consist in identifying the four charged final state particles ($p,K^+,\pi^+,\pi^-$) and the reconstruction of the neutron via the missing mass to the four particles. The neutron component can be enhanced by an appropriate cut on the corresponding missing mass. After this selection, the $\Sigma^+$ and $\Sigma^-$ hyperons are reconstructed via the missing mass of $p,K^+,\pi^-$ or $p,K^+,\pi^+$, respectively. By extracting the hyperon signals in the two spectra, the data sample is further purified, and at the same time it is divided into two subsamples. One subsample consists mainly of events with an intermediate $\Sigma^+$ hyperon, whereas the other subsample contains mainly events with an intermediate $\Sigma^-$ signal. For these two samples the missing mass spectrum of the proton and the $K^+$ ($MM(p,K^+)$), where the $\Lambda(1405)$ is expected to show up, is studied separately, see fig. \[fig:LA1405\_PlusNonRes\].
![(Color online) Missing mass distribution of proton and $K^+$ for the two different subsamples within the HADES acceptance. Panel a) for events showing an intermediate $\Sigma^+$ and panel b) for events showing an intermediate $\Sigma^-$. Experimental data (black dots) are compared to simulations. See text for details.[]{data-label="fig:LA1405_PlusNonRes"}](LA1405_PlusNonRes.eps){width="100.00000%"}
To show the pure physical signal, in both pictures the misidentification background is already subtracted. The treatment of this misidentification background is discussed extensively in [@Agakishiev:2011qw]. The data (black dots) are compared to a sum of simulations. The strengths of the different contributions were determined by a simultaneous fit to four different observables, namely the two missing masses in fig. \[fig:LA1405\_PlusNonRes\] and the two $p,K^+,\pi^{\mp}$ missing mass spectra where the missing mass of $\Sigma^+$ and $\Sigma^-$ are visible. Details about the fitting procedure to the four spectra can be found in [@Siebenson:2010hh]. To give a full description of the experimental data, several contributions have to be taken into account in the simulation. A list of considered channels with particles in the same final state as in reaction (\[rec:L1405\]) ($p,K^+,\pi^+,\pi^-$) is given in table \[tab:Tabel1\].
\[tab:Tabel1\] Channel $p+p\rightarrow$ Category
------------------------ ------------------------------------------------------------------------ ------------------------------
1 $\Lambda(1405)+p+K^+\rightarrow (\Sigma^{\pm}\pi^{\mp})+p+K^+$
2 $\Sigma(1385)^0+p+K^+\rightarrow(\Sigma^{\pm}\pi^{\mp})+p+K^+$ $\Sigma\pi$ resonant
3 $\Lambda(1520)+p+K^+\rightarrow(\Sigma^{\pm}\pi^{\mp})+p+K^+$
4 $\Sigma^++\pi^-+p+K^+$
5 $\Sigma^++K^++\Delta^0(1232)/N(1440)\rightarrow \Sigma^++K^++(p\pi^-)$ $\Sigma^+\pi^-$ non-resonant
6 $\Sigma^++p+K^{*0}\rightarrow \Sigma^++p+(K^+\pi^-)$
7 $\Sigma^-+\pi^++p+K^+$
8 $\Sigma^-+K^++\Delta^{++}(1232)\rightarrow \Sigma^-+K^++(p\pi^+)$ $\Sigma^-\pi^+$ non-resonant
9 $\Lambda+\pi^++n+K^+$
10 $K^{0}_{S}+p+n+K^+$
: Reactions taken into account for the analysis. The channels are classified into two main categories. See text for details.
The channels 9 and 10 can be rejected from the data sample, as demonstrated in [@Siebenson:2010hh]. The other channels, however, contain all the same final and intermediate state particles and can therefore contribute to the data in fig. \[fig:LA1405\_PlusNonRes\]. They are classified into two categories:
- “$\Sigma\pi$ resonant” are all channels, where the $\Sigma$ and the $\pi$ stem from the same mother particle. They should be visible as resonances in both $MM(p,K^+)$ spectra of fig. \[fig:LA1405\_PlusNonRes\].
- “$\Sigma^+\pi^-$ ($\Sigma^-\pi^+$) non resonant” are channels which have a $\Sigma^+\pi^-$ ($\Sigma^-\pi^+$) pair as an intermediate state, but the two particles are not stemming from a common mother particle. These channels give a broad, phase space like distribution in the spectra of fig. \[fig:LA1405\_PlusNonRes\]. The “$\Sigma^+\pi^-$ non resonant” channels can only contribute to fig. \[fig:LA1405\_PlusNonRes\] a), whereas the “$\Sigma^-\pi^+$ non resonant” channels can only give significant contribution to fig. \[fig:LA1405\_PlusNonRes\] b).
Indeed, clear peak structures around 1400 MeV/c$^2$ and 1500 MeV/c$^2$ can be observed in both spectra of fig. \[fig:LA1405\_PlusNonRes\]. They are attributed to the channels 1-3. The $\Lambda(1520)$ (green histograms) is well separated from the $\Lambda(1405)$ mass area and can therefore be isolated. However, the $\Sigma(1385)^0$ (violet histograms) overlaps completely with this area, and it is impossible to separate the $\Lambda(1405)$ and the $\Sigma(1385)^0$ in this data sample. Only in the neutral decay channels the two resonances show different properties ($\Lambda(1405)\rightarrow\Sigma^0\pi^0$, BR 33.3$\%$ and $\Sigma(1385)^0\rightarrow\Lambda\pi^0$, BR 88$\%$). This allows to analyze them independently. The obtained results for the p+p data at $E_{kin}=3.5$ GeV are reported in [@Epple:2011; @EppleFabbietti:2011] and yield a cross section ratio of $\sigma_{\Lambda(1405)}/\sigma_{\Sigma(1385)^0}\approx 1$. This value is used as an external constraint for the analysis presented here. It gives the contributions of $\Lambda(1405)$ and $\Sigma(1385)^0$ shown in fig. \[fig:LA1405\_PlusNonRes\]. For the simulation of the $\Lambda(1405)$ a Breit-Wigner distribution (black histograms) was used. However, for a good agreement between simulation and experiment, the Breit-Wigner had to be simulated with a pole mass of around 1385 MeV/c$^2$, which then results in the solid gray histograms.\
To describe the spectra in fig. \[fig:LA1405\_PlusNonRes\] completely, also phase space like distributions (red histograms), coming from the “$\Sigma\pi$ non-resonant” channels, are needed. A priori it is not clear to which extent the different channels in table \[tab:Tabel1\] contribute, as the spectra in fig. \[fig:LA1405\_PlusNonRes\] are not sensitive to this information. Therefore, other observables have to be studied.\
Fig. \[fig:InvMass\_p\_pim\_K\_pim\] concentrates on the subsample with an intermediate $\Sigma^+$ hyperon. It shows the same data set as in fig. \[fig:LA1405\_PlusNonRes\] a), but before subtracting the misidentification background (blue histograms). Fig. \[fig:InvMass\_p\_pim\_K\_pim\] a) displays the invariant mass distribution of the proton and the $\pi^-$ ($M(p,\pi^-)$), where possible $\Delta$ and $N^*$ resonances should appear. For extracting a possible contribution of a $K^{*0}$, the invariant mass distribution of the $K^+$ and the $\pi^-$ ($M(K^+,\pi^-)$) is shown in fig. \[fig:InvMass\_p\_pim\_K\_pim\] b).
![(Color online) a) Invariant mass of proton and $\pi^-$ and b) invariant mass of $K^+$ and $\pi^-$ for the subsample with an intermediate $\Sigma^+$ hyperon. The spectra show the results within the HADES acceptance.[]{data-label="fig:InvMass_p_pim_K_pim"}](InvMass_p_pim_K_pim.eps){width="100.00000%"}
Compared to the experimental data are simulations, where the scaling of the different channels is known from the simultaneous fit mentioned above. For the “$\Sigma^+\pi^-$ non-resonant” contribution only channel 4 is included. This assumption gives already a rather good description of the data. As indications neither of $\Delta$/$N^*$ nor of $K^{*0}$ resonances are visible, only this channel 4 is used in the further analysis. Indeed, also the “$\Sigma^+\pi^-$ non-resonant” part shown in fig. \[fig:LA1405\_PlusNonRes\] contains only this channel. However, possible contributions due to the channels 5 and 6 can not be excluded completely by this analysis. For example the production of a $K^{*0}(892)$ via channel 6 is only slightly above threshold and thus the cross section might be just too small to see a clear contribution to fig. \[fig:LA1405\_PlusNonRes\] b).\
To identify the different contributions to the “$\Sigma^-\pi^+$ non-resonant” part, the invariant mass of the proton and the $\pi^+$ ($M(p,\pi^+)$) is studied in fig. \[fig:InvMass\_p\_pip\_both\].
![(Color online) Invariant mass of the proton and the $\pi^+$ within the HADES acceptance. Panel a): only channel 7 is used for the $\Sigma^-\pi^+$ non-resonant part. Panel b): according to a $\chi^2$ minimization, only channel 8 is used for the $\Sigma^-\pi^+$ non-resonant part.[]{data-label="fig:InvMass_p_pip_both"}](InvMass_p_pip_both.eps){width="100.00000%"}
Only the data subsample of fig. \[fig:LA1405\_PlusNonRes\] b) with an intermediate $\Sigma^-$ hyperon is investigated. The misidentification background is not subtracted. Panel a) of fig. \[fig:InvMass\_p\_pip\_both\] shows the result where the “non resonant” simulations contain only channel 7. The scaling factors for the different channels are again known from the fitting procedure. The data show an enhanced structure, which can not be described by the simulations. As a comparison, fig. \[fig:InvMass\_p\_pip\_both\] b) shows exactly the same data, but now including channel 7 as well as channel 8 into the simulations. The relative contribution of these two channels is a free parameter, which is obtained by a $\chi^2$ fit to the experimental data points in fig. \[fig:InvMass\_p\_pip\_both\]. The fit results in a negligible contribution of channel 7. With the inclusion of the $\Delta^{++}(1232)$ the experimental data can be described rather well. Due to this result, it is concluded that the “$\Sigma^-\pi^+$ non-resonant” contribution is dominated by $\Delta^{++}(1232)$ production. Therefore, only the channel 8 is used in the simulation, which is already taken into account in fig. \[fig:LA1405\_PlusNonRes\] b).\
With the presented analysis a simulation model with several contributions is obtained, which gives reliable descriptions of the observables investigated so far. However, the goal of this analysis is to understand and to describe the experimental data within the full HADES acceptance. This asks for acceptance corrections. For this purpose it might be not sufficient to study only invariant mass distributions, as they are not very sensitive to e.g. angular distributions of the produced particles. Therefore, angular distributions in the Center-Mass System (CMS), Gottfried-Jackson system and helicity system are studied in the next part. Detailed information about the properties of these frames and the corresponding angular distributions can be found in [@Agakishiev:2011qw; @AbdelBary:2010pc].
Angular distributions
---------------------
Starting point is again reaction (\[rec:L1405\]). Here, three particles are produced in the entrance channel ($\Lambda(1405)$, $p$ and $K^+$). The momentum of the possible $\Lambda(1405)$ is reconstructed via the missing four-vector to the proton and the $K^+$ ($MV(p,K)$). It is clear from the results above that this hypothetical particle does not always refer to a $\Lambda(1405)$, but can also stem from all other channels of table \[tab:Tabel1\]. Fig. \[fig:Angular\_dist\_SigmaP\] and \[fig:Angular\_dist\_SigmaM\] show all angles between the three momenta in the three different frames for the subsample with an intermediate $\Sigma^+$ or $\Sigma^-$ hyperon, respectively.
![(Color online) Angular distributions within the HADES acceptance for events with an intermediate $\Sigma^+$ hyperon (top raw: distribution of $MV(p,K)$, $p$ and $K^+$ in the CMS, middle raw: helicity angles of $MV(p,K)$, $p$ and $K^+$, bottom raw: Gottfried-Jackson angles of $MV(p,K)$, $p$ and $K^+$.)[]{data-label="fig:Angular_dist_SigmaP"}](Angular_dist_SigmaP.eps){width="99.00000%"}
![(Color online) Angular distributions within the HADES acceptance for events with an intermediate $\Sigma^-$ hyperon (top raw: distribution of $MV(p,K)$, $p$ and $K^+$ in the CMS, middle raw: helicity angles of $MV(p,K)$, $p$ and $K^+$, bottom raw: Gottfried-Jackson angles of $MV(p,K)$, $p$ and $K^+$.)[]{data-label="fig:Angular_dist_SigmaM"}](Angular_dist_SigmaM.eps){width="99.00000%"}
The nomenclature is the following:
- $\theta^{A}_{CMS}$: angle between particle A and the beam(target) direction in the Center-Mass System.
- $\theta^{A-B}_{A-C}$: angle between particle A and particle B in the reference frame where particle A and particle C are going back to back and have equal momenta (helicity angle).
- $\theta^{A-bt}_{A-C}$: angle between particle A and the beam/target-type protons (bt) in the reference frame where particle A and particle C are going back to back and have equal momenta (Gottfried-Jackson angle).
By permutation of particles A,B and C nine different observables are obtained, where some of them are not kinematically independent. The different panels in fig. \[fig:Angular\_dist\_SigmaP\] and \[fig:Angular\_dist\_SigmaM\] show the comparison between experimental data and simulations. A reasonable agreement could not be obtained by using only phase space simulations for the different channels of tab. \[tab:Tabel1\]. The production of the $\Sigma(1385)^+$ in the CMS was found to be rather anisotropic [@Agakishiev:2011qw]. The production of the $\Sigma(1385)^0$ is assumed to show the same behavior. Therefore, the simulation of channel 2 was folded with an angular distribution in $cos\left(\theta^{MV(p,K)}_{CMS}\right)$. Additionally, the data sets in fig. \[fig:Angular\_dist\_SigmaP\],a) and \[fig:Angular\_dist\_SigmaM\],a) were subdivided into several angular regions of $\theta^{MV(p,K)}_{CMS}$. These subsamples were analyzed independently. In this way it was found that the $\Lambda(1405)$ and $\Lambda(1520)$ seem to be produced rather isotropically in $cos\left(\theta^{MV(p,K)}_{CMS}\right)$, whereas the “$\Sigma^+\pi^-$ ($\Sigma^-\pi^+$) non resonant” channels, namely channel 4 and 8, show an anisotropic behavior. This anisotropy is included by folding the simulation of these channels in $cos\left(\theta^{MV(p,K)}_{CMS}\right)$ with the corresponding distributions. The resulting distributions are included in fig. \[fig:Angular\_dist\_SigmaP\] and \[fig:Angular\_dist\_SigmaM\]. A reasonable agreement in all 18 pictures is obtained.
Summary
=======
We studied the production of the $\Sigma^{\pm}\pi^{\mp}pK^+$ particle quartets in p+p reactions. It was possible to describe the experimental missing mass distribution of the proton and $K^+$ by a sum of different simulations, including $\Lambda(1405)$, $\Sigma(1385)^0$, $\Lambda(1520)$ and “non-resonant $\Sigma\pi$” production. Studying several invariant mass spectra, the “non-resonant $\Sigma^+\pi^-$” production seems to stem mainly from the reaction $p+p\rightarrow\Sigma^++\pi^-+p+ K^+$ and no clear indication of intermediate resonances like $N^*/\Delta^0$ ($p+p\rightarrow N^*/\Delta^0+\Sigma^++K^+$) or $K^{0*}$ ($p+p\rightarrow\Sigma^++p+K^{*0}$) could be seen. However, the “non-resonant $\Sigma^-\pi^+$” production turned out to come exclusively from the reaction $p+p\rightarrow\Delta^{++}(1232)+\Sigma^-+K^+$. Furthermore, by including an anisotropic production of the $\Sigma(1385)^0$ channel as well as of the “non-resonant $\Sigma\pi$” channels, our simulations can describe the measured data for several angular distributions. The overall good agreement between experimental data and simulations is a necessary precondition to use the obtained simulation model for acceptance and efficiency corrections and to finally extract cross sections for the different channels.
Acknowledgements
================
The author gratefully acknowledge support from the TUM Graduate School.\
The following funding are acknowledged. LIP Coimbra, Coimbra (Portugal): PTDC/FIS/113339/2009, SIP JUC Cracow, Cracow (Poland): NN202286038, NN202198639, HZ Dresden-Rossendorf, Dresden (Germany): BMBF 06DR9059D, TU Muenchen, Garching (Germany) MLL Muenchen DFG EClust: 153 VH-NG-330, BMBF 06MT9156 TP5 TP6, GSI TMKrue 1012, GSI TMFABI 1012, NPI AS CR, Rez (Czech Republic): MSMT LC07050, GAASCR IAA100480803, USC - S. de Compostela, Santiago de Compostela (Spain): CPAN:CSD2007-00042, Goethe Univ. Frankfurt (Germany): HA216/EMMI, HIC for FAIR (LOEWE), BMBF06FY9100I, GSI F&E01, CNRS/IN2P3 (France).
[99]{} T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012). R.H. Dalitz and S. F. Tuan, Phys. Rev. Lett 2, 425 (1959) R.H. Dalitz and S. F. Tuan, Annal. Phys. 8, 100 (1959). Y. Ikeda et al., Prog. Theor. Phys. 125, 1205 (2011). Y.A Chao et al. Nucl. Phys. B 56, 46 (1973). R. J. Hemingway et al., Nucl. Phys. B 253, 742 (1985). K. Moriya and R. Schumacher, arXiv:1110.0469. I. Zychor et al. (ANKE Coll.), Phys. Lett. B 660, 167 (2008). G. Agakishiev et a. (HADES Coll.) Eur. Phys. J. A 41, 243 (2009). J. Siebenson, L. Fabbietti., A. Schmah et al., arXiv: 1009.0946. J. Siebenson et al. (HADES Coll.) AIP Conf. Proc. Vol. 1322, 389 (2010). G. Agakishiev et al. (HADES Coll.) accepted by PRC, arXiv:1109.6806. E. Epple et al. (HADES Coll.) Int. J. Mod. Phys. A 26, 616 (2011). L. Fabbietti and E. Epple (HADES Coll), arXiv:1202.0232. M. Abdel-Bary er al. (COSY-TOF Coll.) Eur. Phys. J. A 46, 27 (2010).
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abstract: 'We study the occurrence of negative differential conductance induced by resonance effects in a model for a multilayer heterostructure. In particular, we consider a system consisting of several correlated and non-correlated monoatomic layers, sandwiched between two metallic leads. The geometry confines electrons in wells within the heterostructures, which are connected to each other and to the leads by tunneling processes. The non-equilibrium situation is produced by applying a bias-voltage to the leads. Our results show that for specific values of the parameters resonance tunneling takes place. We investigate in detail its influence on the current-voltage characteristics. Our results are obtained via non-equilibrium real-space dynamical mean-field theory. As an impurity solver we use the so-called auxiliary master equation approach, which addresses the impurity problem within an auxiliary system consisting of a correlated impurity, a small number of uncorrelated bath sites, and two Markovian environments described by a generalized master equation.'
author:
- Irakli Titvinidze
- Antonius Dorda
- Wolfgang von der Linden
- Enrico Arrigoni
title: Resonance Effects in Correlated Multilayer Heterostructures
---
{width="\columnwidth"}
. \[schematicp\]
Introduction
============
Quantum mechanical resonance effects play an important role in physics and technology. A well known example is resonant tunneling through potential barriers. Tunneling through two barriers, which becomes resonant at a specific external bias voltage, underlies the functioning of resonant tunneling diodes. Their applications range from high-speed microwave systems to novel digital logic circuits. Resonant tunneling through potential barriers is interesting from the theoretical point of view as well. To investigate this effect, one usually considers double or multi-well structures made of semiconductor[@ch.es.74; @datta.ch2-4-5; @gu.li.96; @er.po.11] or hybrid superconductor-semiconductor[@gi.pi.01] materials, graphene[@ro.dr.16; @yo.ki.09; @la.wa.15; @fe.je.12] and graphene-boron[@gu.to.16; @fa.le.15; @br.lo.15; @ba.fe.15; @mi.tu.14; @brey.14; @ba.ga.14] heterostructures. Different approaches are used to theoretically investigate their properties. One can mention, for example, modified optical Bloch equations[@gu.li.96], self-consistent non-equilibrium Green’s functions,[@er.po.11; @go.sh.16u] the envelope wave-function formalism,[@go.sh.16u] adiabatic approximations[@pr.sj.96], combinations of quantum transport random matrix theory with Bogoliubov-de Gennes equations[@gi.pi.01], first-principle density functional theory[@br.lo.15], Bardeen transfer Hamiltonian approach[@fe.je.12], Wentzel-Kramers-Brillouin[@ro.dr.16], and Lorentzian approximation for the quasi-particle spectral function[@gu.to.16]. However, to our knowledge, effects of electron correlations on resonant tunneling have so far been either neglected or included in a perturbative or mean-field way only. Here, we present a first study which examines the effect of correlations on resonant tunneling in an accurate and non-perturbative manner.
Recent experimental progress makes it possible to fabricate correlated heterostructures[@an.ga.99; @is.og.01; @ga.ah.02; @oh.mu.02; @oh.hw.04; @zh.wa.12] with atomic resolution and in particular, growing atomically abrupt layers with different electronic structures[@is.og.01; @oh.mu.02; @ga.ah.02]. Here, we study a system which is composed of alternating strongly correlated and non-correlated metallic layers, as well as band insulator layers (see Fig. \[schematicp\]). The geometry of the system is such that electrons are confined in three wells connected by tunneling. The non-equilibrium situation is driven by applying a bias-voltage to the leads, which introduces a homogeneous electric field in the central region. Resonant tunneling is mainly induced by the particular geometry, rather than the specific values of the system parameters. Since our goal is to investigate the qualitative behavior of this effect, we mainly perform calculations for one representative set of model parameters. In addition, in order to address the effect of correlations on resonance tunneling, we also investigate the behavior of the resonance current as a function of the interaction $U$.
In contrast to the previous works mentioned above, we use dynamical mean-field theory (DMFT)[@ge.ko.96; @voll.10; @me.vo.89], which can treat electron-electron correlations accurately and is one of the most powerful methods to investigate high-dimensional correlated systems. Originally, DMFT was developed to treat equilibrium situations, and later extended[@ao.ts.14; @sc.mo.02u; @fr.tu.06; @free.08; @jo.fr.08; @ec.ko.09; @okam.07; @okam.08; @ar.kn.13; @ti.do.15; @do.ti.16] to the nonequilibrium case. This is formulated within the nonequilibrium Green’s function approach originating from the works of Kubo[@kubo.57], Schwinger[@schw.61], Kadanoff, Baym[@ba.ka.61; @kad.baym] and Keldysh[@keld.65].
DMFT is a comprehensive, thermodynamically consistent and non-perturbative scheme which becomes exact in infinite dimensions, but usually quite well describes two and three dimensional systems. The only approximation in DMFT is locality of the self-energy. The latter can be calculated by mapping the original problem onto a single impurity Anderson model (SIAM)[@ande.61], whose parameters are determined self-consistently. For homogeneous systems the self-energies are the same for each lattice site due to translational symmetry, and, therefore, one needs to solve only one SIAM problem. For systems with broken translational invariance, as in the present case, the self-energies depend on the layer index $z$. Therefore, it is necessary to generalize the formalism and take into account the spatial inhomogeneity of the system[@po.no.99.sm; @po.no.99.ms; @po.no.99.ldmft; @po.no.99.emfl; @free.06; @no.ma.10; @is.li.09; @no.ma.11; @okam11; @mi.fr.01; @free.04; @ok.mi.04.ldmft; @do.ko.97; @do.ko.98; @so.yu.08; @we.ha.11u; @he.co.08; @ko.hi.08; @ko.hi.09; @no.ka.09; @ko.ba.11; @bl.go.11; @ki.ki.11; @au.as.15; @sn.ti.08; @sn.ti.11; @ti.sc.12; @go.ti.10; @sc.ti.13; @au.ti.15; @ze.fr.09; @okam.07; @okam.08; @ec.we.13; @ec.we.14; @ha.fr.11], and, accordingly to solve several SIAM problems.
In the present work the nonequilibrium SIAM problem is treated by using a recently developed auxiliary master equation approach[@ar.kn.13; @do.nu.14; @ti.do.15], which treats the impurity problem within an auxiliary system consisting of a correlated impurity, a small number of uncorrelated bath sites and two Markovian environments described by a generalized master equation.
The paper is organized as follows: Sec. \[Model\] we introduce the Hamiltonian of the system. In Sec. \[Method\] we illustrate the application of real-space dynamical mean-field theory within the non-equilibrium steady-state Green’s function formalism for a system consisting of many layers. Afterwards, in Sec. \[Results\], we present our results. Our conclusions are presented in Sec. \[Conclusions\].
Model {#Model}
=====
The model, consisting of a central region ($c$) with $L=12$ infinite and translationally invariant layers sandwiched between two semi-infinite metallic leads ($\alpha=l,r$), is described by the Hamiltonian (see Fig. \[schematicp\]): $$\begin{aligned}
\label{Hamiltonian}
{\cal H}&=&
\hspace{-0.15cm}
-\hspace{-0.5cm}\sum_{z, \langle {\bf r}^{\phantom\dagger}_\perp,{\bf r}'_\perp\rangle_z, \sigma}\hspace{-0.5cm}t_{z} c_{z,{\bf r}_\perp,\sigma}^\dagger c_{z,{\bf r}'_\perp,\sigma}^{\phantom\dagger}
-\hspace{-0.45cm}\sum_{\langle z, z'\rangle, {\bf r}_\perp, \sigma}\hspace{-0.45cm} t_{zz'} c_{z,{\bf r}_\perp,\sigma}^\dagger c_{z',{\bf r}_\perp,\sigma}^{\phantom\dagger} \nonumber \\
&+&\hspace{-0.1cm}\sum_{z,{\bf r}_\perp}\hspace{-0.1cm} U_z n_{z,{\bf r}_\perp,\uparrow}n_{z,{\bf r}_\perp,\downarrow}
+\hspace{-0.1cm}\sum_{z,{\bf r}_\perp,\sigma}\hspace{-0.1cm}\varepsilon_z n_{z,{\bf r}_\perp,\sigma} \, ,\end{aligned}$$ with nearest-neighbor inter-layer (intra-layer) hopping $t_{zz'}$ ($t_{z}$), local onsite Hubbard interaction $U_z$ and local energy $\varepsilon_z$. $\langle z, z'\rangle$ stands for neighboring $z$ and $z'$ layers and $\langle {\bf r}^{\phantom\dagger}_\perp,{\bf r}'_\perp\rangle_z$ stands for neighboring ${\bf r}_\perp$ and ${\bf r}'_\perp$ sites of the $z$-th layer. $c_{z,{\bf r}_\perp,\sigma}^\dagger$ creates an electron at site ${\bf r}_\perp$ of layer $z$ with spin ${\sigma}$ and $n_{z, {\bf r}_\perp, \sigma}=c_{z, {\bf r}_\perp, \sigma}^\dagger c_{z, {\bf r}_\perp, \sigma}^{\phantom\dagger}$ denotes the corresponding occupation-number operator. $z=1, \ldots, 12$ describes the central layers, while $z<1$ and $z>12$ corresponds to the left and the right lead layers, respectively.
We assume isotropic nearest-neighbor hopping parameters within the central region ($t_{zz'}=t_{z}=t_c$) and within the leads ($t_{zz'}=t_{z}=t_{\alpha=l,r}$). The hybridization between the leads and central region is the same on both sides $t_{0,1}=v_{l}=t_{12,13}=v_{r}$.
Finally, the local energy and the chemical potential in the leads is determined by an applied voltage $\Phi$, i.e. and .
The leads are initially prepared in equilibrium and $T=0$ at the distant past (time $ \to -\infty$) when the hoppings between leads and layer are switched off. Then the hoppings are switched on and the system is allowed to evolve in time until steady state is reached. Notice that despite of the appearance of equilibrium Green’s functions in the expressions, there is no approximation of fixing the leads in equilibrium. In our approach, it is not necessary to solve explicitly for the transient time evolution, and we can directly address the steady state. Since the leads are infinite, they have equilibrium properties far away from the device, but near the device (within the healing length) there will be charge depletion or enhancement, i.e. charge reconstruction near the interfaces. In combination with the long-range part of the Coulomb interaction (LRCI) this could induce modifications in the singls-particle potential. LRCI could be included by a simultaneous solution of the Poisson and DMFT equation (see, e.g. [@free.06]), but this is beyond the scope of the present paper. Notice that this approximation is common in the framework of real-space DMFT calculations (see e.g. Refs. [@okam.07; @okam.08; @kn.li.11; @ne.ar.15; @ma.am.15; @ma.am.16; @am.we.12; @ri.an.16; @ze.fr.09; @ec.we.13; @ec.we.14; @ha.fr.11; @po.no.99.sm; @po.no.99.ms; @po.no.99.ldmft; @po.no.99.emfl]). Here, we approximate the effects of the LRCI, by introducing a linear behavior of the onsite energies (homogeneous electric field) in the central region as .
Real-space Dynamical Mean-Field theory {#Method}
=======================================
In order to investigate steady-state properties we use real-space Dynamical mean-field theory (R-DMFT), which is also known as inhomogeneous DMFT. Due to the finite number of layers translational invariance along the $z$ axes (perpendicular to the layers) is broken, but the system is still translationally invariant in the $xy$ plane. Therefore we can introduce a corresponding momentum ${{{\bf k} }}=(k_x,k_y)$. [@okam]
The Green’s function for the central region, can be expressed via Dyson’s equation $$\begin{aligned}
\label{GR}
&&[{\bf G^{-1}}]^{\gamma}(\omega, {{{\bf k} }}) =[{\bf g}_0^{-1}(\omega, {{{\bf k} }})]^\gamma - {\boldsymbol \Sigma}^\gamma(\omega)-{\boldsymbol \Delta}^\gamma(\omega, {{{\bf k} }}) \hspace{0.5cm} \;.\end{aligned}$$
Here, we use boldface symbols to indicate matrices in the indices $z=1, \ldots, 12$. Moreover, $\gamma\in\{R,A,K\}$ stands for retarded, advanced and Keldysh components, respectively, and ${\bf G}^{A}(\omega, {{{\bf k} }})=[{\bf G}^{R}(\omega, {{{\bf k} }})]^{\dagger}$.
The inverse of the non-interacting Green’s function for the isolated central region reads $$\begin{aligned}
[{\bf g^{-1}_0}]^R_{zz'} (\omega, {{{\bf k} }}) &=\left(\omega + i 0^{+}-E_z( {{{\bf k} }}) \right)\delta_{zz'} + t_{zz'} \, ,
\label{g0R}\\
[{\bf g^{-1}_0}]^K_{zz'} (\omega, {{{\bf k} }})&\simeq 0\;.
$$ with $E_z( {{{\bf k} }})=\varepsilon_z -2t_z (\cos k_x + \cos k_y)$. ${\boldsymbol \Delta}^\gamma(\omega, {{{\bf k} }})$ describes the hybridization with the leads and can be expressed as $$\label{Delta}
{\boldsymbol \Delta}^\gamma_{zz'}(\omega, {{{\bf k} }})=\delta_{z,z'}\left(\delta_{z,1}v_l^2 g_l^\gamma (\omega, {{{\bf k} }})+ \delta_{z,L}v_r^2 g_r^\gamma (\omega, {{{\bf k} }})\right) \, ,$$ where $g_l^\gamma (\omega, {{{\bf k} }})$ and $g_r^\gamma (\omega, {{{\bf k} }})$ describe the Green’s functions for the edge layers of the leads disconnected from the central region. Their retarded component can be expressed as[@po.no.99.sm; @po.no.99.ms; @hayd.80] $$\begin{aligned}
&&g_\alpha^R(\omega, {{{\bf k} }})= \frac{\omega -E_\alpha({{{\bf k} }})}{2t_\alpha^2}
- i\frac{\sqrt{4t_\alpha^2-(\omega-E_\alpha({{{\bf k} }}))^2}}{2t_\alpha^2} \, ,
\label{galphaR}\end{aligned}$$ with $E_\alpha( {{{\bf k} }})=\varepsilon_\alpha-2t_\alpha (\cos k_x + \cos k_y)$. The sign of the square-root for negative argument must be chosen such that the Green’s function has the correct $1/\omega$ behavior for $|\omega|\to \infty$. Since the disconnected leads are separately in equilibrium, we can obtain their Keldysh components from the retarded ones via the fluctuation dissipation theorem[@ha.ja] $$g_\alpha^K(\omega, {{{\bf k} }})= 2i(1-2f_\alpha(\omega))\;{{\rm Im\:}}g_{\alpha}^R(\omega,{{{\bf k} }}) \, .
\label{galphaK}$$ Here, $f_\alpha(\omega)$ is the Fermi distribution for chemical potential $\mu_\alpha$ and temperature $T_\alpha$.
Finally the self-energy ${\boldsymbol\Sigma}^\gamma_{zz'}(\omega)=\delta_{zz'} \Sigma^\gamma_z(\omega)$ is a diagonal and ${{{\bf k} }}$-independent matrix due to the DMFT approximation. To determine the self-energy for each correlated layer $z$ we solve a (non-equilibrium) quantum impurity model with Hubbard interaction $U_z$ and onsite energy $\varepsilon_z$, coupled to a self-consistently determined bath. The latter is specified by its hybridization function obtained as (see e.g. Ref. [@voll.10]) $$\begin{aligned}
&&\hspace{-0.5cm}\Delta_{{\rm bath},z}^R(\omega)=\omega + i 0^{+}-\varepsilon_z -\Sigma^R_z(\omega) -\frac{1}{G_{{\rm loc},z}^R(\omega)} \, , \\
\label{DeltaR}
&&\hspace{-0.5cm}\Delta_{{\rm bath},z}^K(\omega)=- \Sigma^K_z(\omega)+\frac{G_{{\rm loc},z}^K(\omega)}{|G_{{\rm loc},z}^R(\omega)|^2} \
\label{DeltaK} \end{aligned}$$ where the local Green’s function is defined as $$G_{{\rm loc},z}^\gamma(\omega)=\int\limits_{\rm BZ} \frac{d^2{{{{\bf k} }}}}{(2\pi)^2} {\bf G}_{zz}^\gamma(\omega,{{{\bf k} }}) \;.
\label{G_loc}$$ To calculate the diagonal elements of the matrices ${\bf G}^\gamma(\omega,{{{\bf k} }})$ one could invert the matrices in Eqs. . However, it is numerically more efficient to use the recursive Green’s function method[@th.ki.81; @le.ca.13; @ec.we.13], which we here generalize to Keldysh Green’s functions. For a given $z$ we decompose the system into three decoupled clusters by setting $t_{z-1,z}=t_{z,z+1}=0$ (for the first and the last layer into two decoupled cluster). The result is an isolated layer of the central region at position $z$ and the two remaining parts of the central region to the left and to the right of layer $z$. By $L_{z-1}^\gamma(\omega,{{{\bf k} }})$ ($R_{z+1}^\gamma(\omega,{{{\bf k} }})$) we denote the local Green’s function at layer $z-1$ ($z+1$) of the isolated cluster to the left (right) of layer $z$. In addition, we define ${g^{\gamma}_{z}}(\omega,{{{\bf k} }})$ as the full cluster Green’s function of layer $z$.[@inv_Keldish] For $z=2,\ldots,L-1$ it describes isolated layers, while for $z=1$ ($z=L$) it also contains the hybridization effects of the left (right) lead, which are covered by $\Delta^\gamma(\omega ,{{{\bf k} }})$. For the sake of better readability, we will suppress the argument $(\omega ,{{{\bf k} }})$ in the following equations. From and the ensuing definitions we readily see that the inverse cluster Green’s function $[g^{-1}_{z}]^{\gamma}$ is equal to diagonal elements of the inverse $[{\bf G^{-1}}]^\gamma_{zz}$ of the full Green’s function of the central region. The omitted hopping processes $t_{z-1,z}$ and $t_{z,z+1}$ can now be reintroduced by the Dyson equation, which is applicable due to the DMFT approximation of local self energies. We obtain $$\begin{aligned}
\label{GRzz}
&&{\bf G}_{zz}^R=\frac{1}{{[g^{-1}_{z}]^{R}} - t_{z-1,z}^2 L_{z-1}^R -t_{z,z+1}^2 R_{z+1}^R} \, ,\\
\label{GKzz}
&&{\bf G}_{zz}^K=-\frac{{[g^{-1}_{z}]^{K}} - t_{z-1,z}^2 L_{z-1}^K -t_{z,z+1}^2 R_{z+1}^K}{\bigl|{[g^{-1}_{z}]^{R}} - t_{z-1,z}^2 L_{z-1}^R -t_{z,z+1}^2 R_{z+1}^R \bigl|^2} \, .\end{aligned}$$ The Green’s functions $L_{z}^\gamma$ and $R_{z}^\gamma$ in turn are evaluated recursively as follows: $$\begin{aligned}
\label{LR}
&&L_{z}^R=\frac{1}{{[g^{-1}_{z}]^{R}}- t_{z-1,z}^2 L_{z-1}^R} \, ,\\
\label{LK}
&&L_{z}^K=-\frac{{[g^{-1}_{z}]^{K}}- t_{z-1,z}^2 L_{z-1}^K}{\bigl|{[g^{-1}_{z}]^{R}} - t_{z-1,z}^2 L_{z-1}^R \bigl|^2}\, ,\end{aligned}$$ for $z=2,3,\ldots L$ with initial values $$\begin{aligned}
\label{L_{1}}
L_{1}^R&=\frac{1}{{[g^{-1}_{1}]^{R}}} \, ,\qquad
L_{1}^K=-\frac{{[g^{-1}_{1}]^{K}}}{\bigl|{[g^{-1}_{1}]^{R}}\bigl|^2} \, , \end{aligned}$$ and $$\begin{aligned}
\label{RR}
&&R_{z}^R=\frac{1}{{[g^{-1}_{z}]^{R}}- t_{z,z+1}^2 R_{z+1}^R} \, ,\\
\label{RK}
&&R_{z}^K=-\frac{{[g^{-1}_{z}]^{K}}- t_{z,z+1}^2 R_{z+1}^K}{\bigl|{[g^{-1}_{z}]^{R}} - t_{z,z+1}^2 R_{z+1}^R \bigl|^2} \, ,\end{aligned}$$ for $z=L-1,L-2,\ldots 1$ with initial values $$\begin{aligned}
\label{R_L}
R_{L}^R&=\frac{1}{{[g^{-1}_{L}]^{R}}} \, ,\qquad
R_{L}^K=-\frac{{[g^{-1}_{L}]^{K}}}{\bigl|{[g^{-1}_{L}]^{R}}\bigl|^2} \, .\end{aligned}$$
In addition, the self-consistent DMFT loop works as follows: we start with an initial guess for the self-energies $\Sigma^\gamma_z(\omega)$, which typically was taken equal to zero, and based on Eqs. - we calculate the bath hybridization functions $\Delta_{{\rm bath},z}^R(\omega)$ and $\Delta_{{\rm bath},z}^K(\omega)$ for each correlated site. From them we solve the (non-equilibrium) quantum impurity models and calculate new self-energies as described below. We repeat this procedure until convergence is reached.[@convergance]
To address the impurity problem and evaluate self-energies, we adopt a recently developed auxiliary master equation approach (AMEA)[@ar.kn.13; @do.nu.14; @ti.do.15]. This method can be seen as a generalization of the equilibrium exact-diagonalization impurity solver to treat nonequilibrium steady-state situations. In AMEA dissipation, which is crucial in order to achieve a steady state, is included by additionally coupling the cluster to Markovian environments, which can be seen as particle sinks and reservoirs (for details see Refs. ). The accuracy of the impurity solver increases with increase of $N_b$ and becomes exponentially exact in the limit $N_b \rightarrow \infty$.
![(Color online) Current $J$ vs bias voltage $\Phi$. Solid, dashed and dotted lines are obtained by solving the impurity problem with $N_b=6$, $N_b=4$ and $N_b=2$, respectively (see text). Parameters are the same as in Fig. \[schematicp\] . []{data-label="Current_vs_Phi"}](Current_XOOOZOOZOOOX.pdf){width="0.75\columnwidth"}
\
![Current $J$ as a function of the Hubbard interaction $U$ at the resonance. On-site energies in the first and the last layers are fixed to $\varepsilon_1^{(0)}=\varepsilon_{12}^{(0)}=-4$. Results are obtained with $N_b=4$. Other parameters are the same as in Fig. \[schematicp\] .[]{data-label="fig:J_vs_U"}](J_vs_U.pdf){width="0.75\columnwidth"}
Results {#Results}
=======
Here, we presents results for the steady state properties of the system, displayed in Fig. \[schematicp\], consisting of twelve layers (central region) sandwiched between two semi-infinite metallic leads. Among these twelve central region layers only the first and the last layers are correlated, with Hubbard interactions $U_1=U_{12}=U=8$ and onsite energies $\varepsilon_1^{(0)}=\varepsilon_{12}^{(0)}=-U/2=-4$. The onsite energies of the fifth and the eight layers are $\varepsilon_8^{(0)}=-\varepsilon_5^{(0)}=4$ and $\varepsilon_z^{(0)}=0$ for all $z \neq 1, 5, 8, 12$. The hopping between nearest-neighbor central region sites $t_c=1$ is taken as unit of energy,[@initial_Sigma] while hopping between nearest-neighbor sites of the leads are $t_l=t_r=2$. Finally, the hybridizations between leads and central region are $v_l=v_r=1$. All calculations are performed for zero temperature in the leads ($T_l=T_r=0$).
The system is particle-hole symmetric. More specifically it is invariant under a simultaneous particle-hole transformation, a change of sign in the phase of one sublattice (as in the Hubbard model) together with a reflection of the $z$ axis. Therefore, properties of $z$-th and ${(L+1-z)}$-th layers are connected by particle-hole transformation. Consequently, we need to evaluate the self-energy for the $z=1$ layer only, and determine its value for $z=L$ layer based on the symmetry ($\Sigma_{12}^R(\omega)=-[\Sigma_{1}^R(-\omega)]^*+U$ and $\Sigma_{12}^K(\omega)=[\Sigma_{1}^K(-\omega)]^*$). All other layers are non interacting.
In Fig. \[Current\_vs\_Phi\] we plot the current-voltage characteristics of the system. Results are obtained with $N_b=2,4,6$ bath sites of the DMFT auxiliary impurity problem. We find that the difference between results obtained with $N_b=4$ and $N_b=6$ is small for all bias voltages. It indicates fast convergence of the current with respect to the bath sites $N_b$.
The current increases with increasing bias voltage and reaches a first maximum at $\Phi \simeq 2.5$. Further increasing $\Phi$ reduces the current until a minimum at $\Phi \simeq 4$ is reached. A second maximum occurs at $\Phi \simeq 5.25$. For larger bias voltages, the current again decreases due to the decreased overlap of the density of states.
For low bias, where there is a large overlap of the density of states of the left and the right leads, the conductivity is large and the system is in a high-conductivity regime. That is why results in this region are similar to the one of a single layer (see e.g. Refs. ). In contrast, for larger bias $\Phi \gtrsim 3$ we are in the tunneling regime and the behavior of the current-voltage characteristics is significantly different. As we discuss below, the results we are showing are due to the occurrence of resonant tunneling. To clarify this effect, we investigate the non-equilibrium spectral functions, which can be calculated from the corresponding Green’s functions via $A_z(\omega, \varepsilon_{{{{\bf k} }}})=-\frac{1}{\pi}{\rm Im}G_z(\omega, \varepsilon_{{{{\bf k} }}})$. Due to the geometry of the system (see fig. \[schematicp\]) three wells are formed in the intervals $2\le z \le 4$, $6\le z \le 7$, and $9\le z \le 11$, to which electrons are partially confined and form quasi-bound levels. This can be seen by the fact that all spectral functions $A_z(\omega, \varepsilon_{{{{\bf k} }}})$ within a given well display peaks for the same $(\omega, \varepsilon_{{{{\bf k} }}})$, corresponding to quantized quasi-stationary levels in this well. Electrons can leak from the one to the next well only by quantum tunneling.
In Fig. \[G\_omega\_epsilon\] we plot the steady state spectral functions $A_z(\omega, \varepsilon_{{{{\bf k} }}})$ as a function of $\omega-\varepsilon_{{{{\bf k} }}}$ for different $\varepsilon_{{{{\bf k} }}}$ and bias voltages $\Phi$. In particular, we show results for bias voltages that correspond to maxima ($\Phi \simeq 2.5$ and $\Phi \simeq 5.25$), to a minimum and for a value ($\Phi=1$) at half maximum of the first peak in Fig. \[G\_omega\_epsilon\].
The results have the correct property $A_{L+1-z}(\omega,\varepsilon_{{{{\bf k} }}})=A_{z}(-\omega,-\varepsilon_{{{{\bf k} }}})$, which is a consequence of the particle-hole symmetry of the Hamiltonian. Our calculations show that for each non-correlated layer ($1<z<12$) the position of the peaks of the spectral function $A_z(\omega, \varepsilon_{{{{\bf k} }}})$ depends only on the value of $\omega-\varepsilon_{{{{\bf k} }}}$ and not on $\omega$ and $\varepsilon_{{{{\bf k} }}}$ separately. This indicates that for the non-correlated layers one-dimensional physics dominates and $\varepsilon_{{{{\bf k} }}}$ only shifts the energy levels. Furthermore, peaks of the spectral functions $A_z(\omega, \varepsilon_{{{{\bf k} }}})$ for the non-correlated layers in the first ($z=2,3,4$) and the last ($z=9,10,11$) wells generate dips in the spectral functions $A_z(\omega, \varepsilon_{{{{\bf k} }}})$ of the first ($z=1$) and the last ($z=12$) correlated layers correspondingly. This can be qualitatively understood from Eq. , if one assumes that $[{\bf G^{-1}}]_{zz}^R$ is a smooth, function, while $-L_{z-1}^R$ or $-R_{z+1}^R$ (neighboring layer Green’s functions) have narrow peaks.
As central region (layers $1<z<12$) are non-interacting resonant tunneling occurs when quasistationary states, i.e. the peaks in the spectral function, of the first and the last well coincide for any $\varepsilon_{{{{\bf k} }}}$.[@er.po.11; @gu.to.16] This is the case for $\Phi \simeq 5.25$, as can be seen by the gray regions in Fig. \[Phi5.25\]. If these peaks are within the energetic transport window the current gets enhanced at the corresponding bias voltage. For all other bias voltages shown (see Figs. \[Phi1\]-\[Phi4\]), peaks of $A_z(\omega, \varepsilon_{{{{\bf k} }}})$ for different wells do not coincide, so no resonant tunneling takes place. The second maximum in the current-voltage characteristics (see Fig. \[Current\_vs\_Phi\]) can, therefore, be understood in terms of such a resonant tunneling effect. On the other hand, the first maximum is due to the finite bandwidth of the leads, similar to the one for a single layer case (see e.g. Refs. ). In contrast to the single layer case, in the current situation electrons tunnel through four layers ($z=1,5,8,12$) and therefore the current drops faster after the maximum.
In order to address the effect of electron correlation on the resonance, we investigate the behavior of the resonance current $J$ as a function of the interaction $U$. [@barrier] In Fig. \[fig:J\_vs\_U\] we plot the current $J$ as a function of the interaction $U$ at the corresponding resonance bias voltage. The figure clearly shows that correlation effects substantially enhance the resonance effect. However the current maximum is obtained at not too large values of $U\sim 5$. This enhancement behavior can be understood in terms of two competing effects occurring as a function of $U$: since the resonance takes place at relatively high bias, the [*one-dimensional*]{} density of states (DOS) of the two leads have a reduced overlap. This suppresses tunneling at small $U$ for which scattering (approximately) conserves the momentum parallel to the layers. Upon increasing $U$, scattering channels to different values of the in-plane [**k**]{} open, so that the [*three-dimensional*]{} DOS is available for scattering, thus enhancing the current. On the other hand, by increasing $U$ also backscattering is increased, which, in turns suppresses the current.
Conclusions {#Conclusions}
===========
Using non-equilibrium DMFT calculations we investigate steady state properties of a multilayer heterostructure consisting of correlated and non-correlated layers. Due to the fact that the system is inhomogeneous, no matter how many impurity problems have to be solved, “standard” DMFT is not applicable and one has to use the real-space generalization of it. As an impurity solver we used the recently introduced auxiliary master equation approach, which addresses the impurity problem within an auxiliary system consisting of a correlated impurity, a small number of uncorrelated bath sites and two Markovian environments described by a generalized master equation[@ar.kn.13; @do.nu.14; @ti.do.15].
In particular, our main goal was to investigate resonance effects in this system. For this purpose we chose an arrangement of layers such that electrons were confined in three different wells and transport through the central region was only possible by quantum tunneling. For a particular bias voltage ($\Phi \simeq 5.25$) we observed that quasi-stationary energy levels in the first and the last wells coincided and resonance tunneling between them takes place. At that bias voltage the current displays a maximum. According to our calculations the current has another maximum at $\Phi \simeq 2.5$. The latter is due to the finite bandwidth of the leads. We checked that these qualitative findings are robust up to some extent as a function of the model parameters.
Furthermore, we also investigate effect of the interaction strength on the current at the resonance. We obtain that correlation effects for weak up to strong interaction substantially enhance the resonance current.
We thank Andreas Weichselbaum, Jim Freericks, Elias Assmann, Max Sorantin for valuable discussions. This work was supported by the Austrian Science Fund (FWF): P26508, as well as SfB-ViCoM project F04103, and NaWi Graz. The calculations were partly performed on the D-Cluster Graz and on the VSC-3 cluster Vienna.
|
---
abstract: 'We consider an Information-Plus-Noise type matrix where the Information matrix is a spiked matrix. When some eigenvalues of the random matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the spikes. Note that, in an Appendix, we present alternative versions of the earlier results of [@BaiSilver] (“noeigenvalue outside the support of the deterministic equivalent measure”) and [@MC2014] (“exact separation phenomenon”) where we remove some technical assumptions that were difficult to handle.'
author:
- 'M. Capitaine[^1]'
title: 'Limiting eigenvectors of outliers for Spiked Information-Plus-Noise type matrices '
---
Introduction {#no}
============
In this paper, we consider the so-called Information-Plus-Noise type model $$\nonumber
M_N= \Sigma_N \Sigma_N^*
\mbox{~~where~~}
\Sigma_N =
\sigma \frac{X_N}{\sqrt{N}}+A_N,$$ defined as follows.
- $n=n(N)$, $n \leq N$, $c_N=n/ N\rightarrow_{N\rightarrow +\infty} c \in]0;1].$
- $\sigma \in ]0;+\infty[.$
- $X_N=[X_{ij}]_{1\leq i \leq n; 1\leq j \leq N}$ where $\{X_{ij}, i\in \mathbb{N}, j \in \mathbb{N}\}$ is an infinite set of complex random variables such that $\{\Re (X_{ij}), \Im (X_{ij}), i\in \mathbb{N}, j \in \mathbb{N}\}$ are independent centered random variables with variance $1/2$ and satisfy
1. There exists $K>0$ and a random variable $Z$ with finite fourth moment for which there exists $x_0>0$ and an integer number $n_0>0$ such that, for any $x >x_0$ and any integer numbers $n_1,n_2 >n_0$, we have $$\label{condition}\frac{1}{n_1n_2} \sum_{i\leq n_1,j\leq n_2}P\left( \vert X_{ij}\vert >x\right) \leq KP\left(\vert Z \vert>x\right).$$
2. $$\label{trois}\sup_{(i,j)\in \mathbb{N}^2}\mathbb{E}(\vert X_{ij}\vert^3)<+\infty.$$
- Let $\nu$ be a compactly supported probability measure on $\mathbb{R}$ whose support has a finite number of connected components. Let $\Theta=\{\theta _1; \ldots ; \theta _J\}$ where $\theta _1 > \ldots > \theta _J\geq 0$ are $J$ fixed real numbers independent of $N$ which are outside the support of $\nu $. Let $k_1,\ldots,k_J$ be fixed integer numbers independent of $N$ and $r=\sum_{j=1}^J k_j$. Let $\beta_j(N)\geq 0$, $r+1\leq j \leq n$, be such that $\frac{1}{n} \sum_{j=r+1}^{n} \delta _{\beta _j(N)}$ weakly converges to $\nu $ and $$\label{univconv}
\max _{r+1\leq j\leq n} {\rm dist}(\beta _j(N),{\rm supp}(\nu )){\mathop{\longrightarrow }}_{N \rightarrow \infty } 0$$ where ${\rm supp}(\nu )$ denotes the support of $\nu $.\
Let $\alpha_j(N), j=1,\ldots,J$, be real nonnegative numbers such that $$\lim_{N \rightarrow +\infty} \alpha_j(N)=\theta_j.$$ Let $A_N$ be a $n \times N $ deterministic matrix such that, for each $j=1,\ldots,J$, $\alpha_j(N) $ is an eigenvalue of $A_N A_N^*$ with multiplicity $k_j$, and the other eigenvalues of $A_N A_N^*$ are the $\beta_j(N)$, $r+1\leq j \leq n$. Note that the empirical spectral measure of ${A_N A_N^*} $ weakly converges to $\nu $.
Note that assumption such as appears in [@CSBD]. It obviously holds if the $X_{ij}$’s are identically distributed with finite fourth moment.
For any Hermitian $n\times n$ matrix $Y$, denote by $\rm{spect}(Y)$ its spectrum, by $$\lambda_1(Y) \geq \ldots \geq \lambda_n(Y)$$ the ordered eigenvalues of $Y$ and by $\mu_{Y}$ the empirical spectral measure of $Y$: $$\mu _{Y} := \frac{1}{n} \sum_{i=1}^n \delta_{\lambda _{i}(Y)}.$$ For a probability measure $\tau $ on ${\mathbb{R}}$, denote by $g_\tau $ its Stieltjes transform defined for $z \in {\mathbb{C}}\setminus {\mathbb{R}}$ by $$g_\tau (z) = \int_{\mathbb{R}}\frac{d\tau (x)}{z-x}.$$ When the $X_{ij}$’s are identically distributed, Dozier and Silverstein established in [@DozierSilver] that almost surely the empirical spectral measure $\mu _{M_N}$ of $M_N$ converges weakly towards a nonrandom distribution $\mu_{\sigma,\nu,c}$ which is characterized in terms of its Stieljes transform which satisfies the following equation: for any $z \in \mathbb{C}^+$, $$\label{TS} g_{\mu_{\sigma,\nu,c}}(z)=\int \frac{1}{(1-\sigma^2cg_{ \mu_{\sigma,\nu,c}}(z))z- \frac{ t}{1- \sigma^2 cg_{ \mu_{\sigma,\nu,c}}(z)} -\sigma^2 (1-c)}d\nu(t).$$ This result of convergence was extended to independent but non identically distributed random variables by Xie in [@Xi]. (Note that, in [@HLN] , the authors in- vestigated the case where $\sigma$ is replaced by a bounded sequence of real numbers.) In [@MC2014], the author carries on with the study of the support of the limiting spectral measure previously investigated in [@DozierSilver2] and later in [@VLM; @LV] and obtains that there is a one-to-one relationship between the complement of the limiting support and some subset in the complement of the support of $\nu$ which is defined in (\[cale\]) below.
\[caractfinale\] Define differentiable functions $\omega_{\sigma,\nu,c}$ and $\Phi_{\sigma,\nu,c}$ on respectively $ \mathbb{R}\setminus \mbox{supp}(\mu_{\sigma,\nu,c})$ and $ \mathbb{R}\setminus \mbox{supp}(\nu)$ by setting $$\label{defom}\omega_{\sigma,\nu,c} :\begin{array}{ll} \mathbb{R}\setminus \mbox{supp}(\mu_{\sigma,\nu,c}) \rightarrow \mathbb{R}\\
x \mapsto
x (1- \sigma^2 c g_{ \mu_{\sigma,\nu,c}}(x))^2 -\sigma^2 (1-c)(1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(x))\end{array}$$ and $$\Phi_{\sigma,\nu,c} :\begin{array}{ll} \mathbb{R}\setminus \mbox{supp}(\nu) \rightarrow \mathbb{R}\\
x \mapsto
x (1+c \sigma^2g_{ \nu}(x))^2 + \sigma^2 (1-c) (1+ c \sigma^2 g_\nu(x))\end{array}.$$ Set $$\label{cale} {\cal E}_{\sigma,\nu,c}:=\left\{ x \in \mathbb{R}\setminus \mbox{supp}(\nu), \Phi_{\sigma,\nu,c}^{'}(x) >0, g_\nu(x) >-\frac{1}{\sigma^2c}\right\}.$$ $\omega_{\sigma,\nu,c}$ is an increasing analytic diffeomorphism with positive derivative from $\mathbb{R}\setminus \mbox{supp}(\mu_{\sigma,\nu,c})$ to ${\cal E}_{\sigma,\nu,c}$, with inverse $\Phi_{\sigma,\nu,c}$.
Moreover, extending previous results in [@LV] and [@BGRao10] involving the Gaussian case and finite rank perturbations, [@MC2014] establishes a one-to-one correspondance between the $\theta_i$’s that belong to the set ${\cal E}_{\sigma,\nu,c}$ (counting multiplicity) and the outliers in the spectrum of $M_N$. More precisely, setting $$\label{defTheta}\Theta_{\sigma,\nu,c} = \left\{\theta \in \Theta, \Phi_{\sigma,\nu,c}^{'}(\theta) >0, g_\nu(\theta) >-\frac{1}{\sigma^2c}\right\},$$ and $$\label{defSsigma}{\cal S}=\mbox{~supp~} (\mu_{\sigma,\nu,c}) \cup \left\{ \Phi_{\sigma,\nu,c}({\theta}), \theta \in \Theta_{\sigma,\nu,c} \right\},$$ we have the following results.
\[inclusionth\][@MC2014] For any $\epsilon>0$, $$\mathbb P[\mbox{for all large N}, \rm{spect}(M_N) \subset \{x \in {\mathbb{R}}, \mbox{dist}(x,{\cal S})\leq \epsilon \}]=1.$$
[\[ThmASCV\]]{}[@MC2014] Let $\theta_j$ be in $ \Theta_{\sigma,\nu,c}$ and denote by $n_{j-1}+1, \ldots , n_{j-1}+k_j$ the descending ranks of $\alpha_j(N)$ among the eigenvalues of $A_NA_N^*$. Then the $k_j$ eigenvalues $(\lambda_{n_{j-1}+i}(M_N), \, 1 \leq i \leq k_j)$ converge almost surely outside the support of $\mu_{\sigma,\nu,c}$ towards $\rho _{\theta _j}:=\Phi_{\sigma,\nu,c}(\theta_j)$. Moreover, these eigenvalues asymptotically separate from the rest of the spectrum since (with the conventions that $\lambda_0(M_N)=+\infty$ and $\lambda_{N+1}(M_N)=-\infty$) there exists $ \delta_0 >0$ such that almost surely for all large N, $$\label{sepraj}\lambda_{n_{j-1}}(M_N) > \rho _{\theta _j} + \delta_0 \, \mbox{~and~} \, \lambda_{n_{j-1}+k_j +1}(M_N) < \rho _{\theta _j} - \delta_0
.$$
\[gege\] Note that Theorems \[inclusionth\] and \[ThmASCV\] were established in [@MC2014] for $A_N$ as below and with ${\cal S}\cup \{0\}$ instead of $ {\cal S}$ but they hold true as stated above and in the more general framework of this paper. Indeed, these extensions can be obtained sticking to the proof of the corresponding results in [@MC2014] but using the new versions of [@BaiSilver] and of the exact separation phenomenon of [@MC2014] which are presented in the Appendix A of the present paper.
The aim of this paper is to study how the eigenvectors corresponding to the outliers of $M_N$ project onto those corresponding to the spikes $\theta_i$’s. Note that there are some pionneering results investigating the eigenvectors corresponding to the outliers of finite rank perturbations of classical random matricial models: [@Paul] in the real Gaussian sample covariance matrix setting, and [@BGRao09; @BGRao10] dealing with finite rank additive or multiplicative perturbations of unitarily invariant matrices. For a general perturbation, dealing with sample covariance matrices, S. Péché and O. Ledoit [@PL] introduced a tool to study the average behaviour of the eigenvectors but it seems that this did not allow them to focus on the eigenvectors associated with the eigenvalues that separate from the bulk. It turns out that further studies [@MCJTP; @BBCF15] point out that the angle between the eigenvectors of the outliers of the deformed model and the eigenvectors associated to the corresponding original spikes is determined by Biane-Voiculescu’s subordination function. For the model investigated in this paper, such a free interpretation holds but we choose not to develop this free probabilistic point of view in this paper and we refer the reader to the paper [@CDM]. Here is the main result of the paper.
\[cvev1p1\] Let $\theta_j$ be in $\Theta_{\sigma,\nu,c}$ (defined in ) and denote by $n_{j-1}+1, \ldots , n_{j-1}+k_j$ the descending ranks of $\alpha_j(N)$ among the eigenvalues of $A_NA_N^*$. Let $\xi(j)$ be a normalized eigenvector of $M_N$ relative to one of the eigenvalues $(\lambda_{n_{j-1}+q}(M_N)$, $1\leq q\leq k_j)$. Denote by $\|\cdot \|_2$ the Euclidean norm on $\mathbb{C}^n$. Then, almost surely
- $\displaystyle{\lim_{N\rightarrow +\infty}\left\| P_{\mbox{Ker~}(\alpha_j(N) I_N-A_NA_N^*)}\xi(j)\right\|^2_2 = \tau(\theta_j)}$\
where $$\label{tau}\tau(\theta_j)= \frac{1-\sigma^2c g_{\mu_{\sigma,\nu,c}}(\rho_{\theta_j})}{\omega_{\sigma,\nu,c}'(\rho_{\theta_j})}=\frac{ \Phi_{\sigma,\nu,c}'({\theta_j})}{1+ \sigma^2 cg_\nu(\theta_j)}$$
- for any $\theta_i$ in $\Theta_{\sigma,\nu,c}\setminus\{\theta_j\}$, $$\displaystyle{\lim_{N\rightarrow +\infty}\left\| P_{ \mbox{Ker~}(\alpha_i(N) I_N-A_NA_N^*)}\xi(j)\right\|_2 = 0.}$$
The sketch of the proof of Theorem \[cvev1p1\] follows the analysis of [@MCJTP] as explained in Section 2. In Section 3, we prove a universal result allowing to reduce the study to estimating expectations of Gaussian resolvent entries carried on Section 4. In Section 5, we explain how to deduce Theorem \[cvev1p1\] from the previous Sections. In an Appendix A, we present alternative versions on the one hand of the result in [@BaiSilver] about the lack of eigenvalues outside the support of the deterministic equivalent measure, and, on the other hand, of the result in [@MC2014] about the exact separation phenomenon. These new versions deal with random variables whose imaginary and real parts are independent but remove the technical assumptions ((1.10) and “$b_1>0$” in Theorem 1.1 in [@BaiSilver] and “$\omega_{\sigma,\nu,c}(b)>0$” in Theorem 1.2 in [@MC2014]). This allows us to claim that Theorem \[ThmASCV\] holds in our context (see Remark \[gege\]). Finally, we present, in an Appendix B, some technical lemmas that are used throughout the paper.
Sketch of the proof
===================
Throughout the paper, for any $m\times p$ matrix $B$, $(m,p)\in {\mathbb{N}}^2$, we will denote by $\Vert B\Vert$ the largest singular value of $B$, and by $\Vert B\Vert_2=\{Tr (BB^*)\}^{\frac{1}{2}}$ its Hilbert-Schmidt norm.\
The proof of Theorem \[cvev1p1\] follows the analysis in two steps of [@MCJTP]. [**Step A.**]{} First, we shall prove that, for any orthonormal system $(\xi_1,\cdots,\xi_{k_j})$ of eigenvectors associated to the $k_j$ eigenvalues $\lambda_{n_{j-1}+q}(M_N)$, $1\leq q\leq k_j$, the following convergence holds almost surely: $\forall l =1,\ldots,J$, $$\label{5.4}
\sum_{p=1}^{k_j}\left\| P_{\ker(\alpha_l (N) I_N-A_NA_N^*)}\xi_p\right\|^2_2 \rightarrow_{N \rightarrow +\infty} \frac{k_j\delta_{jl} (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\rho_{\theta_j}))}{\omega_{\sigma,\nu,c}'(\rho_{\theta_j})}.$$ Note that for any smooth functions $h$ and $f$ on $\mathbb{R}$, if $v_1,\ldots, v_n$ are eigenvectors associated to $\lambda_1(A_NA_N^*), \ldots,\lambda_n(A_NA_N^*)$ and $w_1,
\ldots, w_n$ are eigenvectors associated to $\lambda_1(M_N), \ldots, \lambda_n(M_N)$, one can easily check that $$\label{equavec}
{\rm Tr} \left[h(M_N) f(A_NA_N^*)\right] =\sum_{m,p=1}^n h(\lambda_p(M_N)) f(\lambda_m(A_NA_N^*)) \vert
\langle v_m,w_p \rangle \vert^2.$$ Thus, since $\alpha_l(N)$ on one hand and the $k_j$ eigenvalues of $M_N$ in $(\rho_{\theta_j}
-\varepsilon,\rho_{\theta_j}+\varepsilon)$ (for $\epsilon$ small enough) on the other hand, asymptotically separate from the rest of the spectrum of respectively $A_NA_N^*$ and $M_N$, a fit choice of $h$ and $f$ will allow the study of the restrictive sum $\sum_{p=1}^{k_j}\left\| P_{\ker(\alpha_l(N) I_N-A_NA_N^*)} \xi_p\right\|^2_2$. Therefore proving (\[5.4\]) is reduced to the study of the asymptotic behaviour of ${\rm Tr}\left[h(M_N)f(A_NA_N^*)\right]$ for some functions $f$ and $h$ respectively concentrated on a neighborhood of $\theta_l$ and $\rho_{\theta_j}$.\
[**Step B:**]{} In the second, and final, step, we shall use a perturbation argument identical to the one used in [@MCJTP] to reduce the problem to the case of a spike with multiplicity one, case that follows trivially from Step A.\
Step B closely follows the lines of [@MCJTP] whereas Step A requires substantial work. We first reduce the investigations to the mean Gaussian case by proving the following.
\[compgaunogau\] Let $X_N$ as defined in Section \[no\]. Let ${\cal G}_N = [{\cal G}_{ij}]_{1\leq i\leq n, 1\leq j\leq N}$ be a $n\times N$ random matrix with i.i.d. standard complex normal entries. Let $h$ be a function in $\cal C^\infty ({\mathbb{R}}, {\mathbb{R}})$ with compact support, and $\Gamma_N$ be a $n\times n $ Hermitian matrix such that $$\sup_{n,N} \Vert \Gamma_N \Vert<\infty \text{~and~} \sup_{n,N} \rm{rank} (\Gamma_N) <\infty.$$ Then almost surely,\
${{\rm Tr}}\left(h\left(\left(\sigma \frac{X_N}{\sqrt{N}}+A_N\right)\left(\sigma \frac{X_N}{\sqrt{N}}+A_N\right)^*\right) \Gamma_N\right)$ $$-\mathbb{E}\left({{\rm Tr}}\left[h\left(\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)^*\right) \Gamma_N\right] \right)\rightarrow_{N \rightarrow +\infty} 0.$$
The asymptotic behaviour of $\mathbb{E}\left({{\rm Tr}}\left[h\left(\left(\sigma\frac{{\cal G}_N}{\sqrt{N}}+A_N\right)\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)^*\right) f(A_NA_N^*)\right] \right)$ can be deduced, by using the bi-unitarily invariance of the distribution of $ {\cal G}_N$, from the following Proposition \[estimfonda\] and Lemma \[approxpoisson\].
\[estimfonda\] Let ${\cal G}_N = [{\cal G}_{ij}]_{1\leq i \leq n, 1\leq j\leq N}$ be a $n\times N$ random matrix with i.i.d. complex standard normal entries. Assume that $A_N$ is such that $$\label{diagonale} A_N=\begin{pmatrix} d_1(N) ~~~~~~~~~~~~~~~~~~~~(0)\\
~~~(0)\\ ~~~~~~~~~~\ddots~~~~~~~~~~~~~( 0)\\ ~(0)~~~~~~~~~~~~~~~~~~~~\\ ~~~~~~~~~~~~~~~~~d_{n}(N)~~~ ( 0 ) \end{pmatrix}$$ where $n=n(N)$, $n \leq N$, $c_N=n/ N\rightarrow_{N\rightarrow +\infty} c \in]0;1]$, for $i=1,\ldots,n$, $d_i(N) \in \mathbb{C}$, $\sup_N \max_{i=1,\ldots,n} \vert d_i(N)\vert <+\infty$ and $\frac{1}{n} \sum_{i=1}^n \delta_{\vert d_i(N)\vert^2}$ weakly converges to a compactly supported probability measure $\nu$ on $\mathbb{R}$ when $N$ goes to infinity. Define for all $z\in \mathbb{C}\setminus \mathbb{R}$, $$G^{\cal G}_N(z) =\left (zI - \left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)^*\right)^{-1}.$$ Define for any $q=1,\ldots,n$, $$\label{qq}\gamma_q(N) =(A_NA_N^*)_{qq} =\vert d_q(N)\vert^2.$$ There is a polynomial $P$ with nonnegative coefficients, a sequence $(u_N)_N$ of nonnegative real numbers converging to zero when $N$ goes to infinity and some nonnegative real number $l$, such that for any $(p,q)$ in $\{1, \ldots,n\}^2$, for all $z\in \mathbb{C}\setminus \mathbb{R}$, $$\label{premier}
\mathbb{E} \left(\left( G^{\cal G}_N(z)\right)_{pq}\right) = \frac{1- \sigma^2 cg_{\mu_{\sigma,\nu,c}}(z)}{\omega_{\sigma, \nu, c}(z) -\gamma_q(N)} \delta_{pq}
+\Delta_{p,q,N}(z),$$ with $$\left| \Delta_{p,q,N} (z)\right| \leq (1+\vert z\vert)^l P(\vert \Im z \vert^{-1})u_N.$$
Proof of Proposition \[compgaunogau\]
=====================================
In the following, we will denote by $o_{C}(1)$ any deterministic sequence of positive real numbers depending on the parameter $C$ and converging for each fixed $C$ to zero when $N$ goes to infinity. The aim of this section is to prove Proposition \[compgaunogau\].\
Define for any $C>0$, $$\begin{aligned}
Y_{ij}^C &=&\Re X_{ij}{1\!\!{\sf I}}_{\vert \Re X_{ij} \vert \leq C} - \mathbb{E}\left( \Re X_{ij}{1\!\!{\sf I}}_{\vert \Re X_{ij} \vert \leq C} \right) \nonumber \\&&+ \sqrt{-1} \left\{
\Im X_{ij}{1\!\!{\sf I}}_{\vert \Im X_{ij} \vert \leq C} - \mathbb{E}\left( \Im X_{ij}{1\!\!{\sf I}}_{\vert \Im X_{ij} \vert \leq C} \right) \right\}.\label{ycdef}\end{aligned}$$
Set $$\theta^*=\sup_{(i,j)\in \mathbb{N}^2}\mathbb{E}(\vert X_{ij}\vert^3)<+\infty.$$ We have $$\begin{aligned}
\mathbb{E} \left( \vert X_{ij}-Y_{ij}^C\vert^2 \right)&=&\mathbb{E} \left( \vert \Re X_{ij}\vert^2 {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert > C} \right)
+ \mathbb{E} \left( \vert \Im X_{ij}\vert^2 {1\!\!{\sf I}}_{\vert \Im X_{ij} \vert > C} \right) \\&&
-\left\{ \mathbb{E} \left( \Re X_{ij} {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert > C} \right)\right\}^2-\left\{ \mathbb{E} \left( \Im X_{ij} {1\!\!{\sf I}}_{\vert \Im X_{ij} \vert > C} \right)\right\}^2\\& \leq & \frac{\mathbb{E} \left(\vert \Re X_{ij} \vert^3\right)+\mathbb{E} \left(\vert \Im X_{ij} \vert^3\right)}{C}\end{aligned}$$ so that $$\sup_{i\geq 1,j\geq 1}\mathbb{E} \left( \vert X_{ij}-Y_{ij}^C\vert^2 \right) \leq \frac{2\theta^*}{C}.$$
Note that $$\begin{aligned}
1 - 2 \mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right) &=& 1-2 \mathbb{E} \left\{\left(\Re X_{ij} {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert \leq C} - \mathbb{E} \left( \Re X_{ij} {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert \leq C}\right)\right)^2\right\}
\\&=& 2\left[ \frac{1}{2} -\mathbb{E} \left( \vert \Re X_{ij}\vert^2 {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert \leq C} \right)\right]
+2\left\{ \mathbb{E} \left( \Re X_{ij} {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert \leq C} \right)\right\}^2\\&=&
2 \mathbb{E} \left( \vert \Re X_{ij}\vert^2 {1\!\!{\sf I}}_{\vert\Re X_{ij} \vert > C} \right)+ 2\left\{\mathbb{E} \left( \Re X_{ij} {1\!\!{\sf I}}_{\vert \Re X_{ij} \vert > C}\right) \right\}^2, \end{aligned}$$ so that $$\sup_{i\geq 1, j\geq 1}\vert 1 - 2\mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right) \vert \leq \frac{4\theta^*}{C}.$$ Similarly $$\sup_{i\geq 1,j\geq 1}\vert 1 - 2\mathbb{E} \left( \vert \Im Y_{ij}^C\vert^2 \right) \vert \leq \frac{4\theta^*}{C}.$$ Let us assume that $C>8\theta^*.$ Then, we have $$\mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right)> \frac{1}{4} \; \mbox{and}\; \mathbb{E} \left( \vert \Im Y_{ij}^C\vert^2 \right)> \frac{1}{4}.$$ Define for any $C> 8\theta^*$, $X^C=(X^C_{ij})_{1\leq i \leq n; 1\leq j \leq N},$ where for any $1\leq i \leq n, 1\leq j \leq N$, $$\label{defxc}{X}_{ij}^C =\frac{\Re Y_{ij}^C}{\sqrt{2\mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right)}} +\sqrt{-1} \frac{\Im Y_{ij}^C}{\sqrt{2\mathbb{E} \left( \vert \Im Y_{ij}^C\vert^2 \right)}}.$$
Let ${\cal G} = [{\cal G}_{ij}]_{1\leq i\leq n, 1\leq j \leq N}$ be a $n \times N$ random matrix with i.i.d. standard complex normal entries, independent from $X_N$, and define for any $\alpha>0$, $$X^{\alpha,C}= \frac{ X^C +\alpha {\cal G}}{\sqrt{1+\alpha^2}}.$$ Now, for any $n\times N$ matrix $B$, let us introduce the $(N+n) \times (N+n)$ matrix $${\cal M}_{N+n}(B) =\left( \begin{array}{ll} 0_{n\times n}~~~ B+A_N\\ B^* +A_N^*~~~ 0_{N\times N} \end{array} \right).$$ Define for any $z\in \mathbb{C}\setminus \mathbb{R}$, $$\tilde G(z) = \left( z I_{N+n} - {\cal M}_{N+n}\left(\sigma \frac{X_N}{\sqrt{N}}\right)\right)^{-1},$$ and $$\tilde G^{\alpha,C}(z) = \left( z I_{N+n} - {\cal M}_{N+n}\left(\sigma \frac{X^{\alpha,C}}{\sqrt{N}}\right)\right)^{-1} .$$ Denote by $\mathbb{U}(n+N)$ the set of unitary $(n+N)\times (n+N)$ matrices. We first establish the following approximation result.
\[approximation\] There exist some positive deterministic functions $u$ and $v$ on $[0,+\infty[$ such that $\lim_{C\rightarrow +\infty} u(C) =0$ and $\lim_{\alpha \rightarrow 0} v(\alpha)=0,$ and a polynomial $P$ with nonnegative coefficients such that for any $\alpha$ and $ C>8\theta^*$, we have that\
$\bullet$ almost surely, for all large N,\
$\displaystyle{ \sup_{U\in \mathbb{U}(n+N)}\sup_{(i,j)\in \{1,\ldots,n+N\}^2}\sup_{z\in \mathbb{C}\setminus \mathbb{R} } |\Im z |^{2} \left| (U^*\tilde{G}^{\alpha,C}(z)U)_{ij}- (U^*\tilde{{G}}(z)U)_{ij}\right|}$ $$\label{EGC} \leq u(C)+v(\alpha),
$$ $\bullet$ for all large $N$,\
$\displaystyle{ \sup_{U\in \mathbb{U}(n+N)} \sup_{(i,j)\in \{1,\ldots,n+N\}^2} \sup_{z\in \mathbb{C}\setminus \mathbb{R} } \frac{1}{ P(|\Im z |^{-1}) }\left| \mathbb{E} \left( (U^*\tilde{G}^{\alpha,C}(z)U)_{ij}- (U^*\tilde{{G}}(z)U)_{ij}\right)\right|}$$$\label{EGCalpha} \leq u(C)+v(\alpha)+ o_{C}(1).$$
Note that $$\begin{aligned}
{X}_{ij}^C-Y_{ij}^C&=& \Re X_{ij}^C \left( 1-\sqrt{2} \mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right)^{1/2}\right)
+\sqrt{-1} \Im X_{ij}^C \left( 1-\sqrt{2} \mathbb{E} \left( \vert \Im Y_{ij}^C\vert^2 \right)^{1/2}\right)
\\&=&\Re X_{ij}^C\frac{1 - 2\mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right) }{1 + \sqrt{2}\mathbb{E} \left( \vert \Re Y_{ij}^C\vert^2 \right)^{1/2}}+
\sqrt{-1} \Im X_{ij}^C\frac{1 - 2\mathbb{E} \left( \vert \Im Y_{ij}^C\vert^2 \right) }{1 + \sqrt{2}\mathbb{E} \left( \vert \Im Y_{ij}^C\vert^2 \right)^{1/2}}.\end{aligned}$$ Then, $$\left\{\sup_{(i,j)\in \mathbb{N}^2}\mathbb{E} \left( \vert X^C_{ij}-Y_{ij}^C\vert^2 \right)\right\}^{1/2} \leq \frac{4\theta^*}{C},
\mbox{~and~}\sup_{(i,j)\in \mathbb{N}^2}\mathbb{E} \left( \vert X_{ij}^C-Y_{ij}^C\vert^3 \right) <\infty.$$ It is straightfoward to see, using Lemma \[lem0\], that for any unitary $(n+N)\times (n+N)$ matrix $U$,\
$
\left| (U^*\tilde{G}^{\alpha,C}(z)U)_{ij}- (U^*\tilde{{G}}(z)U)_{ij}\right|$ $$\begin{aligned}
&\leq & \frac{\sigma}{\vert \Im z \vert^2} \left\| \frac{X_N-{X}^{\alpha,C}}{\sqrt{N}} \right\| \nonumber \\
&\leq & \frac{\sigma}{\vert \Im z \vert^2} \left\{ \left\| \frac{X_N-Y^C}{\sqrt{N}} \right\| + \left\| \frac{{X}^C-Y^C}{\sqrt{N}} \right\|\right. \nonumber\\&& \left. + \left(1- \frac{1}{\sqrt{1+\alpha^2}}\right) \left\| \frac{ X^C}{\sqrt{N}} \right\| +\alpha \left\| \frac{{\cal G}}{\sqrt{N}} \right\| \right\} \label{inegu}.\end{aligned}$$ From Bai-Yin’s theorem (Theorem 5.8 in [@BaiSil06]) , we have $$\left\| \frac{{\cal G}}{\sqrt{N}} \right\|=2+o(1).$$ Applying Remark \[2.1\] to the $(n+N)\times (n+N)$ matrix $\tilde B=
\left( \begin{array}{ll} 0_{n\times n}~~~ B\\ B^*~~~ 0_{N\times N} \end{array} \right)$ for $B\in \{X_N-Y^C, X^C-Y^C,X^C\}$ (see also Appendix B of [@CSBD]), we have that almost surely $$\limsup_{N\rightarrow +\infty}\left\| \frac{{X^C}}{\sqrt{N}} \right\|\leq 2 \sqrt{2} ,~\limsup_{N\rightarrow +\infty}\left\| \frac{{X}^C-Y^C}{\sqrt{N}} \right\|\leq \frac{8\sqrt{2}\theta^*}{C},\;$$ and $$\limsup_{N\rightarrow +\infty}\left\| \frac{{X_N}-Y^C}{\sqrt{N}} \right\|\leq 4 \sqrt{ \frac{\theta^*}{C}}.$$ Then, (\[EGC\]) readily follows.
Let us introduce $$\Omega_{N,C}=\left\{ \left\| \frac{{\cal G}}{\sqrt{N}} \right\| \leq 4, \left\| \frac{X^C}{\sqrt{N}} \right\| \leq 4, \left\| \frac{X_N-Y^C}{\sqrt{N}} \right\| \leq 8 \sqrt{ \frac{\theta^*}{C}}, \left\| \frac{{X}^C-Y^C}{\sqrt{N}} \right\|\leq \frac{16\theta^*}{C} \right\}.$$ Using , we have\
$\left| \mathbb{E} \left( (U^*\tilde{{G}}^{\alpha,C}(z)U)_{ij}-(U^*\tilde{G}(z)U)_{ij}\right)\right|$ $$\begin{aligned}
&\leq &
\frac{4\sigma }{\vert \Im z \vert^2} \left[ 2 \sqrt{ \frac{\theta^*}{C}}+ \frac{4\theta^*}{C}+
\alpha +\left( 1-\frac{1}{\sqrt{1+\alpha^2}}\right) \right] \\& &+ \frac{2}{\vert \Im z \vert} \mathbb{P}(\Omega_{N,C}^c).\end{aligned}$$ Thus (\[EGCalpha\]) follows.
Now, Lemma \[approxpoisson\], Lemma \[approximation\] and Lemma \[HT\] readily yields the following approximation lemma.
\[approxCalpha\] Let $h$ be in $\cal C^\infty ({\mathbb{R}}, {\mathbb{R}})$ with compact support and $\tilde \Gamma_N$ be a $(n+N)\times (n+N)$ Hermitian matrix such that such that $$\sup_{n,N} \Vert \tilde \Gamma_N \Vert<\infty \text{~and~} \sup_{n,N} \rm{rank} (\tilde \Gamma_N) <\infty.$$ Then, there exist some deterministic functions $u$ and $v$ on $[0,+\infty[$ such that $\lim_{C\rightarrow +\infty} u(C) =0$ and $\lim_{\alpha \rightarrow 0} v(\alpha)=0,$ such that for all $C>0$, $\alpha>0$, we have almost surely for all large N, $$\label{diff} \left|{{\rm Tr}}\left[ h\left(({\cal M}_{N+n}\left(\frac{X^{\alpha,C}}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right]
- {{\rm Tr}}\left[ h\left(({\cal M}_{N+n}\left(\frac{X_N}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right]\right| \leq a^{(1)}_{C,\alpha},$$ and for all large $N$, $$\label{Ediff} \left|\mathbb{E}{{\rm Tr}}\left[ h\left(({\cal M}_{N+n}\left(\frac{X^{\alpha,C}}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right]
- \mathbb{E}{{\rm Tr}}\left[ h\left(({\cal M}_{N+n}\left(\frac{X_N}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right]\right| \leq a^{(2)}_{C,\alpha,N},$$ where $$a^{(1)}_{C,\alpha}= u(C)+v(\alpha),\; a^{(2)}_{C,\alpha,N}= u(C)+v(\alpha) + o_{C}(1).$$
Note that the distributions of the independent random variables $\Re (X_{ij}^{\alpha,C})$, $\Im (X_{ij}^{\alpha,C})$ are all a convolution of a centred Gaussian distribution with some variance $v_\alpha $, with some law with bounded support in a ball of some radius $R_{C,\alpha}$; thus, according to Lemma \[zitt\], they satisfy a Poincaré inequality with some common constant $ C_{PI}(C,\alpha)$ and therefore so does their product (see the Appendix B). An important consequence of the Poincaré inequality is the following concentration result.
\[Herbst\][ Lemma 4.4.3 and Exercise 4.4.5 in [@AGZ09] or Chapter 3 in [@Ledoux01]. ]{} There exists $K_1>0$ and $K_2>0$ such that for any probability measure $\mathbb{P}$ on $\mathbb{R^M}$ which satisfies a Poincaré inequality with constant $C_{PI}$, and for any Lipschitz function $F$ on $\mathbb{R}^M$ with Lipschitz constant $\vert F \vert_{Lip}$, we have $$\forall \epsilon> 0, \, \mathbb{P}\left( \vert F-{\mathbb{E}}_{\mathbb{P}}(F) \vert > \epsilon \right) \leq K_1 \exp\left(-\frac{\epsilon}{K_2 \sqrt{C_{PI}} \vert F \vert_{Lip}}\right).$$
In order to apply Lemma \[Herbst\], we need the following preliminary lemmas.
\[extlipschitz\](see Lemma 8.2 [@MCJTP]) Let $f$ be a real $C_{\cal L}$-Lipschitz function on $\mathbb{R}$. Then its extension on the $N\times N$ Hermitian matrices is $C_{\cal L}$-Lipschitz with respect to the Hilbert-Schmidt norm.
\[Lipschitz\] Let $\tilde \Gamma_N $ be a $(n+N)\times ( n+N)$ matrix and $h$ be a real Lipschitz function on $\mathbb{R}$. For any $n \times N$ matrix $B$, $$\left\{\left(\Re B(i,j),~ \Im B(i,j)\right)_{1\leq i \leq n, 1 \leq j \leq N}\right\} \mapsto Tr \left[ h\left(({\cal M}_{N+n}\left(B\right)\right) \tilde \Gamma_N\right]$$ is Lipschitz with constant bounded by $\sqrt{2} \left\| \tilde \Gamma_N \right\|_2 \Vert h \Vert_{Lip}$.
$ \left| {{\rm Tr}}\left[h ({\cal M}_{N+p}(B)) \tilde \Gamma_N\right]- {{\rm Tr}}\left[ h ({\cal M}_{N+p}(B^{'}))\tilde \Gamma_N\right]\right|$ $$\begin{aligned}
\nonumber\\ &\leq &
\left\| \tilde \Gamma_N \right\|_2 \left\| h ({\cal M}_{N+p}(B))-
h ({\cal M}_{N+p}(B^{'}))\right\|_2 \nonumber\\
&\leq & \left\| \tilde \Gamma_N \right\|_2 \left\|h \right\|_{Lip} \left\| {\cal M}_{N+p}(B)-{\cal M}_{N+p}(B^{'}) \right\|_2.\label{lip1}
\end{aligned}$$ where we used Lemma \[extlipschitz\] in the last line. Now, $$\left\| {\cal M}_{N+p}(B)-{\cal M}_{N+p}(B^{'}) \right\|^2_2= 2 \left\| B-B^{'}\right\|^2_2.\label{lip2}$$
Lemma \[Lipschitz\] readily follows from (\[lip1\]) and (\[lip2\]).
\[inegconc\] Let $\tilde \Gamma_N $ be a $(n+N)\times ( n+N)$ matrix such that $ \sup_{N,n} \left\| \tilde \Gamma_N \right\|_2 \leq K$. Let $h$ be a real Lipschitz function on $\mathbb{R}$. The random variable $F_N={\rm Tr} \left[ h\left( {\cal M}_{N+p}\left( \frac{X^{\alpha,C}}{\sqrt{N}}\right) \right) \tilde \Gamma_N\right]$ satisfies the following concentration inequality $$\forall \epsilon> 0, \, \mathbb{P}\left( \vert F_N-{\mathbb{E}}(F_N) \vert > \epsilon \right) \leq K_1 \exp\left(-\frac{\epsilon \sqrt{ N}}{K_2(\alpha,C) K \Vert h \Vert_{Lip}}\right),$$ for some postive real numbers $K_1$ and $K_2(\alpha,C)$.
Lemma \[inegconc\] follows from Lemmas \[Lipschitz\] and \[Herbst\] and basic facts on Poincaré inequality recalled at the end of the Appendix $B$.
By Borel-Cantelli’s Lemma, we readily deduce from the above Lemma the following
\[concentration\] Let $\tilde \Gamma_N $ be a $(n+N)\times ( n+N)$ matrix such that $ \sup_{N,n} \left\| \tilde \Gamma_N \right\|_2 \leq K$. Let $h$ be a real ${\cal C}^1$- function with compact support on $\mathbb{R}$.\
${\rm Tr} \left[ h\left( {\cal M}_{N+p}\left( \sigma \frac{X^{\alpha,C}}{\sqrt{N}}\right) \right)\tilde \Gamma_N\right]- \mathbb{E}\left[ {\rm Tr} \left[ h\left( {\cal M}_{N+p}\left( \sigma \frac{X^{\alpha,C}}{\sqrt{N}}\right) \right) \tilde \Gamma_N\right]\right]$ $$\label{concentre}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\stackrel{a.s}{\longrightarrow}_{N\rightarrow +\infty}0.$$
Now, we will establish a comparison result with the Gaussian case for the mean values by using the following lemma (which is an extension of Lemma \[gaussian\] below to the non-Gaussian case) as initiated by [@KKP96] in Random Matrix Theory.
\[IPP\] Let $\xi$ be a real-valued random variable such that ${\mathbb{E}}(\vert \xi \vert ^{p+2}) < \infty $. Let $\phi $ be a function from ${\mathbb{R}}$ to ${\mathbb{C}}$ such that the first $p+1$ derivatives are continuous and bounded. Then, $$\label{IPP2}
{\mathbb{E}}(\xi \phi (\xi )) = \sum_{a=0}^p \frac{\kappa _{a+1}}{a!}{\mathbb{E}}(\phi ^{(a)}(\xi )) + \epsilon ,$$ where $\kappa _{a}$ are the cumulants of $\xi $, $\vert \epsilon \vert \leq K \sup_t \vert \phi ^{(p+1)}(t)\vert {\mathbb{E}}(\vert \xi \vert ^{p+2})$, $K$ only depends on $p$.
\[Chatterjee\] Let ${\cal G}_N = [{\cal G}_{ij}]_{1\leq i\leq n, 1\leq j\leq N}$ be a $n\times N$ random matrix with i.i.d. complex $N(0,1)$ Gaussian entries. Define $$\tilde G^{{\tiny {\cal G}}}(z)= \left( zI_{N+n}- {\cal M}_{N+n}\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}\right) \right)^{-1}$$ for any $z\in \mathbb{C}\setminus \mathbb{R}.$ There exists a polynomial $P$ with nonnegative coefficients such that for all large $N$, for any $(i,j)\in \{1,\ldots,n+N\}^2$, for any $z\in \mathbb{C}\setminus \mathbb{R}$, for any unitary $(n+N)\times (n+N)$ matrix $U$, $$\label{gau} \left| \mathbb{E} \left[(U^*\tilde G^{{\tiny {\cal G}}}(z)U)_{ij}\right]- \mathbb{E}\left[(U^*\tilde G(z)U)_{ij}\right]\right| \leq \frac{1}{\sqrt{N}}P(\left|\Im z \right|^{-1}).$$ Moreover, for any $(N+n)\times (N+n)$ matrix $\tilde \Gamma_N$ such that $$\label{gauG} \sup_{n,N} \Vert \tilde \Gamma_N \Vert<\infty \text{~and~} \sup_{n,N} \rm{rank} (\tilde \Gamma_N) <\infty,$$ and any function $h$ in $\cal C^\infty ({\mathbb{R}}, {\mathbb{R}})$ with compact support, there exists some constant $K>0$ such that, for any large N,\
$ \left| \mathbb{E}\left[ {\rm Tr} \left[ h\left( {\cal M}_{N+n}\left( \sigma \frac{X_N}{\sqrt{N}}\right) \right)\tilde \Gamma_N\right]\right]-
\mathbb{E}\left[ {\rm Tr} \left[ h\left( {\cal M}_{N+n}\left( \sigma \frac{{\cal G}_N}{\sqrt{N}}\right) \right)\tilde \Gamma_N\right]\right] \right|
$ $$\label{comparaison}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\leq \frac{K}{\sqrt{N}}.$$
We follow the approach of [@PS] chapters 18 and 19 consisting in introducing an interpolation matrix $X_N(\alpha)= \cos \alpha X_N + \sin \alpha {\cal G}_N$ for any $\alpha$ in $[0;\frac{\pi}{2}]$ and the corresponding resolvent matrix $\tilde G(\alpha,z)= \left( zI_{N+n}- {\cal M}_{N+n}\left(\sigma \frac{X_N(\alpha)}{\sqrt{N}}\right) \right)^{-1}$ for any $z\in \mathbb{C}\setminus \mathbb{R}.$ We have, for any $(s,t)\in \{1,\ldots,n+N\}^2$, $$\mathbb{E}\tilde G^{{\tiny {\cal G}}}_{st}(z)- \mathbb{E} \tilde G_{st}(z)=\int_0^{\frac{\pi}{2}} \mathbb{E} \left( \frac{ \partial }{\partial \alpha} \tilde G_{st}(\alpha,z)\right) d\alpha$$ with $$\begin{aligned}
\frac{ \partial }{\partial \alpha} \tilde G_{st}(\alpha,z)&= &\frac{\sigma}{2\sqrt{N}}\sum_{l=1}^n \sum_{k=n+1}^{n+N }\left\{
\left[ \tilde G_{sl} (\alpha,z) \tilde G_{kt} (\alpha,z)+ \tilde G_{sk}(\alpha,z) \tilde G_{l t}(\alpha,z) \right]\right.\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~\times \left[ -\sin \alpha \Re X_{l(k-n)}+\cos \alpha \Re {\cal G}_{l(k-n)}\right] \\
&&\left.~~~~~~~~~~~~~~~~~~~~~~+i
\left[ \tilde G_{sl}(\alpha,z) \tilde G_{kt} (\alpha,z)- \tilde G_{sk} (\alpha,z)\tilde G_{l t}(\alpha,z) \right]\right.\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left.\times \left[-\sin \alpha \Im X_{l(k-n)} +\cos \alpha \Im {\cal G}_{l(k-n)} \right]\right\}.\end{aligned}$$ Now, for any $ l=1,\ldots,n $ and $k=n+1, \ldots, n+N$, using Lemma \[IPP\] for $p=1$ and for each random variable $\xi$ in the set $\left\{\Re X_{l(k-n)}, \Re {\cal G}_{l(k-n)},\Im X_{l(k-n)}, \Im {\cal G}_{l(k-n)} \right\} $, and for each $\phi$ in the set $$\left\{ (U^*\tilde G (\alpha ,z))_{ip}(\tilde G(\alpha,z)U)_{q j} ;
(p,q)=(l,k) \mbox{~or~}(k,l), (i,j)\in \{1,\ldots,n+N\}^2\right\},$$ one can easily see that there exists some constant $K>0$ such that $$\left| \mathbb{E} (U^* \tilde G^{{\tiny {\cal G}}}(z)U)_{ij}- \mathbb{E}(U^*\tilde G(z)U)_{ij} \right| \leq \frac{ K}{N^{3/2}}\sup_{Y \in {\cal H}_{n+N}(\mathbb{C})} \sup_{V\in \mathbb{U}(n+N)} S_V(Y)$$ where ${\cal H}_{n+N}(\mathbb{C})$ denotes the set of $(n+N) \times (n+N)$ Hermitian matrices and $S_V(Y)$ is a sum of a finite number independent of $N$ and $n$ of terms of the form $$\label{termes}\sum_{l=1}^n \sum_{k=n+1}^{n+N } \left|\left(U^*R(Y)\right)_{ip_1}\left(R(Y)\right)_{p_2p_3}\left(R(Y)\right)_{p_4p_5}\left(R(Y)U\right)_{p_6j} \right|$$ with $R(Y)=\left(zI_{N+n}- Y\right)^{-1}$ and $\{p_1, \ldots,p_6\}$ contains exactly three $k$ and three $l$.\
When $(p_1, p_6)=(k,l)$ or $(l,k)$, then, using Lemma \[lem0\],\
$ \sum_{l=1}^n \sum_{k=n+1}^{n+N }\left| \left(U^*R(Y)\right)_{ip_1}\left(R(Y)\right)_{p_2p_3}\left(R(Y)\right)_{p_4p_5}\left(R(Y)U\right)_{p_6j}\right|$ $$\begin{aligned}
&\leq& \frac{1}{\vert \Im z \vert^2} \sum_{k,l =1}^{n+N} \left|\left(U^*R(Y)\right)_{il}\left(R(Y)U\right)_{kj}\right| \\ &\leq& \frac{(N+n)}{\vert \Im z \vert^2}
\left(\sum_{l =1}^{n+N} \left|\left(U^*R(Y)\right)_{il}\right|^2 \right)^{1/2}\left(\sum_{k =1}^{n+N} \left|\left(R(Y)U\right)_{kj}\right|^2 \right)^{1/2}\\&=&\frac{(N+n)}{\vert \Im z \vert^2}
\left( \left(U^*R(Y)R(Y)^*U\right)_{ii} \right)^{1/2}\left( \left(U^*R(Y)^*R(Y)U\right)_{jj} \right)^{1/2}\\&\leq & \frac{(N+n)}{\vert \Im z \vert^4} \end{aligned}$$ When $p_1=p_6=k$ or $ l$, then, using Lemma \[lem0\],\
$\sum_{l=1}^n \sum_{k=n+1}^{n+N }\left| \left(U^*R(Y)\right)_{ip_1}\left(R(Y)\right)_{p_2p_3}\left(R(Y)\right)_{p_4p_5}\left(R(Y)U\right)_{p_6j}\right|$ $$\begin{aligned}
&\leq& \frac{N+n}{\vert \Im z \vert^2} \sum_{l =1}^{n+N} \left|\left(U^*R(Y)\right)_{il}\left(R(Y)U\right)_{lj}\right| \\ &\leq& \frac{(N+n)}{\vert \Im z \vert^2}
\left(\sum_{l =1}^{n+N} \left|\left(U^*R(Y)\right)_{il}\right|^2 \right)^{1/2}\left(\sum_{l =1}^{n+N} \left|\left(R(Y)U\right)_{lj}\right|^2 \right)^{1/2}\\&=&\frac{(N+n)}{\vert \Im z \vert^2}
\left( \left(U^*R(Y)R(Y)^*U\right)_{ii} \right)^{1/2}\left( \left(U^*R(Y)^*R(Y)U\right)_{jj} \right)^{1/2}\\&\leq & \frac{(N+n)}{\vert \Im z \vert^4} \end{aligned}$$ (\[gau\]) readily follows.
Then by Lemma \[HT\], there exists some constant $K>0$ such that, for any $N$ and $n$, for any $(i,j)\in \{1,\ldots,n+N\}^2$, any unitary $(n+N)\times (n+N)$ matrix $U$, $$\label{HT4E}\limsup_{y \rightarrow 0^+} \left| \int \left[ \mathbb{E}(U^*\tilde G(t+iy)U)_{ij}- \mathbb{E}(U^* {\tilde G^{{\tiny {\cal G}}}}(t+iy)U)_{ij}\right] h(t) dt \right| \leq \frac{ K}{\sqrt{N}}.$$ Thus, using (\[avecE\]) and (\[gauG\]), we can deduce (\[comparaison\]) from (\[HT4E\]).
The above comparison lemmas allow us to establish the following convergence result.
\[comptilde\] Let $h$ be a function in $\cal C^\infty ({\mathbb{R}}, {\mathbb{R}})$ with compact support and let $\tilde \Gamma_N $ be a $(n+N)\times ( n+N)$ matrix such that $ \sup_{n,N} \rm{rank} (\tilde \Gamma_N) <\infty$ and $ \sup_{n,N} \Vert \tilde \Gamma_N \Vert<\infty$. Then we have that almost surely\
$
Tr \left[ h\left(({\cal M}_{N+n}\left(\sigma \frac{X_N}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right]- \mathbb{E}\left[Tr \left[ h\left(({\cal M}_{N+n}\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right] \right]$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
{\longrightarrow}_{N\rightarrow +\infty}0.$$
Lemmas \[approxCalpha\], \[concentration\] and \[Chatterjee\] readily yield that there exist some positive deterministic functions $u$ and $v$ on $[0,+\infty[$ with $\lim_{C\rightarrow +\infty} u(C) =0$ and $\lim_{\alpha \rightarrow 0} v(\alpha)=0,$ such that for any $C>0$ and any $\alpha >0$, almost surely $$\limsup_{N \rightarrow +\infty} \left| Tr \left[ h\left(({\cal M}_{N+n}\left(\sigma \frac{X_N}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right]- \mathbb{E}\left[Tr \left[ h\left(({\cal M}_{N+n}\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}\right)\right) \tilde \Gamma_N\right] \right] \right|$$ $$\leq
u(C) +v(\alpha).$$ The result follows by letting $\alpha$ go to zero and $C$ go to infinity.
Now, note that, for any $N \times n$ matrix $B$, for any continuous real function function $h$ on ${\mathbb{R}}$, and any $n\times n $ Hermitian matrix $\Gamma_N$, we have $$Tr \left(h\left((B+A_N)(B+A_N)^*\right) \Gamma_N\right)=Tr \left[\tilde h\left({\cal M}_{N+n}\left(B\right)\right) \tilde \Gamma_N\right]$$ where $\tilde h(x)=h(x^2)$ and $ \tilde \Gamma_N= \begin{pmatrix} \Gamma_N & (0)\\ (0) & (0) \end{pmatrix}$. Thus, Proposition \[comptilde\] readily yields Proposition \[compgaunogau\].
Proof of Proposition \[estimfonda\]
===================================
The aim of this section is to prove Proposition \[estimfonda\] which deals with Gaussian random variables.Therefore we assume here that $A_N$ is as and set $\gamma_q(N)=(A_N A_N^*)_{qq}$. In this section, we let $X$ stand for ${\cal G}_N$, $A$ stands for $A_N$, $G$ denotes the resolvent of $M_N= \Sigma \Sigma^*$ where $\Sigma=\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N$ and $g_N$ denotes the mean of the Stieltjes transform of the spectral measure of $M_N$, that is $$g_N(z) = {\mathbb{E}}\left(\frac{1}{n} Tr G(z)\right), \, z \in {\mathbb{C}}\setminus {\mathbb{R}}.$$
Matricial master equation
-------------------------
To obtain the equation (\[mean\]) below, we will use many ideas from [@DHLN]. The following Gaussian integration by part formula is the key tool in our approach.
\[gaussian\]\[Lemma 2.4.5 [@AGZ09]\] Let $\xi$ be a real centered Gaussian random variable with variance $1$. Let $ \Phi$ be a differentiable function with polynomial growth of $\Phi$ and $\Phi'$. Then, $$\mathbb{E} \left( \xi \Phi(\xi) \right) = \mathbb{E} \left( \Phi^{'}(\xi) \right).$$
\[intbypart\] Let $z$ be in $\mathbb{C}\setminus \mathbb{R}$. We have for any $(p,q)$ in $\{1,\ldots,n\}^2$,\
$\mathbb{E}\left( G_{pq}(z)\right) \left\{ z(1-\sigma^2 c_N g_N(z)) -\frac{\gamma_q(N)}{ 1-\sigma^2 c_N g_N(z)} -\sigma^2 (1-c_N) +\frac{\sigma^2}{N}\sum_{p=1}^n \nabla_{pp}(z) \right\}$ $$\label{mean} \hspace*{2cm}=\delta_{pq}
+\nabla_{pq}(z),$$ where $$\label{nablapq} \nabla_{pq} = \frac{1}{1- \sigma^2 c_N g_N}\left\{
\frac{\sigma^2}{N} \frac{ \mathbb{E}\left( G_{pq}\right) }{1-\sigma^2 c_N g_N} \Delta_3 + \Delta_2(p,q) + \Delta_1(p,q).\right\},$$ $$\label{delta1pq}\Delta_{1}(p,q) = {\sigma^2}\mathbb{E}\left\{ \left[\frac{1}{N} Tr G - \mathbb{E}\left(\frac{1}{N} Tr G \right)\right] (G\Sigma \Sigma^*)_{pq} \right\} ,$$ $$\label{delta2pq}\Delta_{2}(p,q) = \frac{\sigma^2}{N}\mathbb{E}\left\{ Tr(GA\Sigma^* ) \left[G_{pq} - \mathbb{E}\left(G_{pq}\right) \right] \right\} ,$$ $$\label{delta3}\Delta_{3} = {\sigma^2}\mathbb{E}\left\{ \left[\frac{1}{N} Tr G - \mathbb{E}\left(\frac{1}{N} Tr G \right)\right] Tr (\Sigma^*GA ) \right\} .$$
Using Lemma \[gaussian\] with $\xi=\Re X_{ij} $ or $\xi= \Im X_{ij}$ and $\Phi= G_{pi} \overline{\Sigma_{qj}}$, we obtain that for any $j,q,p$, $$\begin{aligned}
\mathbb{E}\left[ \left( G \frac{\sigma X}{\sqrt{N}}\right)_{pj} \overline{\Sigma_{qj}} \right] &=& \sum_{i=1}^n \mathbb{E}\left[ G_{pi} \frac{\sigma X_{ij} }{\sqrt{N}}\overline{\Sigma_{qj}} \right]\\&=&\frac{\sigma^2}{N} \sum_{i=1}^n \mathbb{E}\left[ \left( G \Sigma\right)_{pj} G_{ii} \overline{\Sigma_{qj}} \right]
+ \frac{\sigma^2}{N} \mathbb{E}(G_{pq})\\&=& \frac{\sigma^2}{N} \mathbb{E}\left[ \left( Tr G\right) \left(G \Sigma\right)_{pj} \overline{\Sigma_{qj}} \right]
+ \frac{\sigma^2}{N} \mathbb{E}(G_{pq}). \label{1}\end{aligned}$$ On the other hand, we have $$\begin{aligned}
\mathbb{E}\left[ \left( G A \right)_{pj} \overline{\Sigma_{qj}} \right] &=&\mathbb{E}\left[ \left( G A \right)_{pj} \overline{A_{qj}} \right]
+\sum_{i=1}^n \mathbb{E}\left[ G_{pi} A_{ij} \frac{ \sigma \overline{X_{qj}}}{\sqrt{N}} \right]\\&=& \mathbb{E}\left[ \left( G A \right)_{pj} \overline{A_{qj}} \right] +\frac{\sigma^2}{N} \mathbb{E}\left[ G_{pq} \left( \Sigma^* G A \right)_{jj} \right] \label{2}\end{aligned}$$ where we applied Lemma \[gaussian\] with $\xi=\Re X_{qj} $ or $\xi= \Im X_{qj}$ and $\Psi= G_{pi}A_{ij}$. Summing (\[1\]) and (\[2\]) yields $$\begin{aligned}
\mathbb{E}\left[ \left( G \Sigma \right)_{pj} \overline{\Sigma_{qj}} \right] &=&\frac{\sigma^2}{N} \mathbb{E}(G_{pq})+
\frac{\sigma^2}{N} \mathbb{E}\left[ \left(Tr G\right) \left(G \Sigma\right)_{pj} \overline{\Sigma_{qj}} \right]\\&&+\frac{\sigma^2}{N} \mathbb{E}\left[ G_{pq} \left( \Sigma^* G A \right)_{jj} \right] + \mathbb{E}\left[ \left( G A \right)_{pj} \overline{A_{qj}} \right]. \label{delta1}\end{aligned}$$ Define $$\Delta_1(j)= \frac{\sigma^2}{N} \mathbb{E}\left[ \left(Tr G\right) \left(G \Sigma\right)_{pj} \overline{\Sigma_{qj}} \right]- \frac{\sigma^2}{N} \mathbb{E}\left[ Tr G \right] \mathbb{E}\left[ \left(G \Sigma\right)_{pj} \overline{\Sigma_{qj}} \right].$$ From (\[delta1\]), we can deduce that $$\begin{aligned}
\mathbb{E}\left[ \left( G \Sigma \right)_{pj} \overline{\Sigma_{qj}} \right] &=&\frac{1}{1-\sigma^2 c_N g_N} \left\{
\frac{\sigma^2}{N} \mathbb{E}(G_{pq})+
\frac{\sigma^2}{N} \mathbb{E}\left[ G_{pq} \left( \Sigma^* G A \right)_{jj} \right] \right.\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left.+ \mathbb{E}\left[ \left( G A \right)_{pj} \overline{A_{qj}} \right] + \Delta_1(j)\right\}.\end{aligned}$$ Then, summing over $j$, we obtain that\
$
\mathbb{E}\left[ \left( G \Sigma \Sigma^*\right)_{pq} \right] =\frac{1}{1-\sigma^2 c_N g_N} \left\{
{\sigma^2} \mathbb{E}(G_{pq})+
\frac{\sigma^2}{N} \mathbb{E}\left[ G_{pq} Tr\left( \Sigma^* G A \right) \right] \right.$ $$\label{71}\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ \mathbb{E}\left[ \left( G A A^*\right)_{pq} \right] + \Delta_1 (p,q)\right\},$$ where $\Delta_{1}(p,q)$ is defined by (\[delta1pq\]). Applying Lemma \[gaussian\] with $\xi=\Re X_{ij} $ or $ \Im X_{ij}$ and $\Psi= (GA)_{ij}$, we obtain that $$\mathbb{E} \left[Tr\left( \frac{\sigma X^*}{\sqrt{N}} G A \right) \right] =\frac{\sigma^2}{N} \mathbb{E}\left[ Tr G ~Tr\left( \Sigma^* G A \right) \right].$$ Thus, $$\mathbb{E}\left[ Tr\left( \Sigma^* G A \right) \right]=\mathbb{E}\left[ Tr\left( A^* G A \right) \right]+ \sigma^2 c_N g_N \mathbb{E}\left[ Tr\left( \Sigma^* G A \right) \right] + \Delta_3,$$ where $\Delta_3$ is defined by (\[delta3\]) and then $$\label{R3} \mathbb{E}\left[ Tr\left( \Sigma^* G A \right) \right]=\frac{1}{1-\sigma^2 c_Ng_N} \left\{ \mathbb{E}\left[ Tr\left( G A A^*\right) \right] + \Delta_3 \right\}.$$ (\[R3\]) and (\[delta2pq\]) imply that $$\label{ok}\frac{\sigma^2}{N} \mathbb{E}\left[ G_{pq}Tr\left( \Sigma^* G A \right) \right]= \frac{\sigma^2}{N}\frac{\mathbb{E}(G_{pq})}{1-\sigma^2 c_N g_N} \left\{ \mathbb{E}\left[ Tr\left( G A A^*\right) \right] + \Delta_3 \right\} + \Delta_2(p,q),$$ where $\Delta_2(p,q)$ is defined by (\[delta2pq\]). We can deduce from (\[71\]) and (\[ok\]) that\
$ \mathbb{E}\left[ \left( G \Sigma \Sigma^*\right)_{pq} \right] $ $$=\frac{1}{1-\sigma^2 c_Ng_N} \left\{
{\sigma^2} \mathbb{E}(G_{pq})+ \mathbb{E}\left[ \left( G A A^*\right)_{pq} \right]
+\frac{\sigma^2}{N} \frac{\mathbb{E}\left[ G_{pq}\right] }{1-\sigma^2 c_N g_N}\mathbb{E}\left[ Tr\left( G AA^* \right) \right]\right.$$ $$\label{73} \left.\hspace*{4cm}+\frac{\sigma^2}{N}\frac{\mathbb{E}(G_{pq})}{1-\sigma^2 c_N g_N} \Delta_3
+ \Delta_1 (p,q)+ \Delta_2 (p,q) \right\}.$$ Using the resolvent identity and (\[73\]), we obtain that $$\begin{aligned}
z\mathbb{E}\left( G_{pq} \right)& =&\frac{1}{1-\sigma^2 c_Ng_N} \left\{
{\sigma^2} \mathbb{E}(G_{pq})+ \mathbb{E}\left[ \left( G A A^*\right)_{pq} \right] \right.
\nonumber \\
&& \left.~~~~~~~~~~~~+\frac{\sigma^2}{N} \frac{\mathbb{E}\left[ G_{pq}\right] }{1-\sigma^2 c_Ng_N} \mathbb{E}\left[ Tr\left( G AA^* \right)\right] \right\}+ \delta_{pq}+\nabla_{pq}. \label{115}\end{aligned}$$ where $\nabla_{pq}$ is defined by (\[nablapq\]). Taking $p=q$ in (\[115\]), summing over $p$ and dividing by $n$, we obtain that $$\begin{aligned}
z g_N &=& \frac{\sigma^2 g_N}{1-\sigma^2 c_N g_N} + \frac{ Tr \left[ \mathbb{E} (G) AA^*\right]}{n(1-\sigma^2 c_N g_N)} \\
&&+ \frac{\sigma^2 g_N Tr \left[ \mathbb{E} (G) AA^*\right]}{N(1-\sigma^2 c_N g_N)^2} +1 + \frac{1}{n} \sum_{p=1}^n \nabla_{pp}\end{aligned}$$ It readily follows that $$\label{terme}\frac{ Tr \left[ \mathbb{E} (G) AA^*\right]}{n(1-\sigma^2 c_Ng_N)} \left( \frac{ \sigma^2 c_N g_N}{(1-\sigma^2 c_N g_N)} +1 \right)
=\left( z - \frac{ \sigma^2 }{(1-\sigma^2 c_N g_N)} \right) g_N -1 - \frac{1}{n} \sum_{p=1}^n \nabla_{pp}.$$
Therefore $$\frac{ Tr \left[ \mathbb{E} (G) AA^*\right]}{n(1-\sigma^2 c_N g_N)}=zg_N (1-\sigma^2 c_N g_N) - (1-\sigma^2 c_N g_N) -\sigma^2 g_N
- (1-\sigma^2 c_N g_N) \frac{1}{n} \sum_{p=1}^n \nabla_{pp}.$$ (\[terme\]) and (\[115\]) yield $$\mathbb{E}(G_{pq}) \times \left\{ z (1-\sigma^2 c_N g_N) -\frac{\gamma_q}{1-\sigma^2 c_N g_N} -\sigma^2(1-c_N) + \frac{\sigma^2}{N} \sum_{p=1}^n \nabla_{pp}\right\} =\delta_{pq} + \nabla_{pq}.$$ Proposition \[intbypart\] follows.
Variance estimates
------------------
In this section, when we state that some quantity $\Delta _N(z)$, $z \in {\mathbb{C}}\setminus {\mathbb{R}}$, is equal to $O(\frac{1}{N^p})$, this means precisely that there exist some polynomial $P$ with nonnegative coefficients and some positive real number $l$ which are all independent of $N$ such that for any $z \in {\mathbb{C}}\setminus {\mathbb{R}}$, $$\vert \Delta _N(z)\vert \leq \frac{(\vert z\vert+1)^l P( | \Im z |^{-1}) }{N^p}.$$ We present now the different estimates on the variance. They rely on the following Gaussian Poincaré inequality (see the Appendix B). Let $Z_1,\ldots,Z_q$ be $q$ real independent centered Gaussian variables with variance $\sigma^2$. For any ${\cal C}^1$ function $f: {\mathbb{R}}^q \rightarrow {\mathbb{C}}$ such that $f$ and ${\rm grad} f$ are in $L^2({\cal N}(0, \sigma^2I_q))$, we have $$\label{Poincare}
\mathbf{V}\left\{f(Z_1,\ldots ,Z_q)\right\}\leq \sigma^2 \mathbb{E} \left(\Vert ({\rm grad}f) (Z_1,\ldots,Z_q)\Vert_2^2\right) ,$$ denoting for any random variable $a$ by $\mathbf{V}(a) $ its variance $ \mathbb{E}(\vert a-\mathbb{E}(a)\vert^2)$. Thus, $(Z_1,\ldots,Z_q)$ satisfies a Poincaré inequality with constant $C_{PI}=\sigma^2$.
The following preliminary result will be useful to these estimates.
\[bornelambda1\] There exists $K>0$ such for all $N$, $$\mathbb{E} \left( \lambda_1\left( \frac{XX^*}{N} \right) \right) \leq K.$$
According to Lemma 7.2 in [@HT], we have for any $t \in ]0;N/2]$, $$\mathbb{E} \left[{{\rm Tr}}\left(\exp t \frac{XX^*}{N} \right) \right]\leq n \exp \left( (\sqrt{c_N} +1)^2 t +\frac{1}{N} (c_N +1) t^2 \right).$$ By the Chebychev’s inequality, we have $$\begin{aligned}
\exp \left(t \mathbb{E} \left( \lambda_1\left( \frac{XX^*}{N} \right) \right) \right)& \leq& \mathbb{E} \left( \exp t \lambda_1\left( \frac{XX^*}{N} \right)\right)\\ &\leq &\mathbb{E} \left[{{\rm Tr}}\left(\exp t \frac{XX^*}{N} \right) \right]\\&\leq& n \exp \left( (\sqrt{c_N} +1)^2 t +\frac{1}{N} (c_N +1) t^2 \right).\end{aligned}$$ It follows that $$\mathbb{E}\left(\lambda_1\left( \frac{XX^*}{N}\right)\right) \leq \frac{1}{t} \log n+ (\sqrt{c_N} +1)^2 + \frac{1}{N} (c_N +1) t.$$ The result follows by optimizing in $t$.
\[variance\] There exists $C>0$ such that for all large $N$, for all $z \in \mathbb{C}\setminus \mathbb{R}$, $$\label{esttrace} \mathbb{E}\left( \left| \frac{1}{n}{{\rm Tr}}G - \mathbb{E}(\frac{1}{n}{{\rm Tr}}G)\right|^2 \right)\leq \frac{C}{N^2 \vert \Im z \vert^4},$$ $$\label{Opq}\forall (p,q)\in \{1,\ldots,n\}^2, \;\mathbb{E}\left( \vert G_{pq} - \mathbb{E}(G_{pq})\vert^2 \right)\leq \frac{C}{N \vert \Im z \vert^4},$$ $$\label{V}\mathbb{E}\left( \vert {{\rm Tr}}\Sigma^*GA - \mathbb{E}({{\rm Tr}}\Sigma^* GA)\vert^2 \right)\leq \frac{C(1+\vert z \vert)^2}{ \vert \Im z \vert^4}.$$
Let us define $\Psi: \mathbb{R}^{2(n\times N)} \rightarrow { M }_{n\times N}(\mathbb{C})$ by $$\Psi:~~\{x_{ij},y_{ij},i=1,\ldots,n,j=1,\ldots,N\}\rightarrow \sum_{i=1,\ldots,n}\sum_{j=1,\ldots,N} \left( x_{ij} +\sqrt{-1} y_{ij} \right) e_{ij},$$ where $e_{ij}$ stands for the $n\times N$ matrix such that for any $(p,q)$ in $\{1,\ldots,n\} \times \{1,\ldots, N\}$, $(e_{ij})_{pq}=\delta_{ip}\delta_{jq}.$ Let $F$ be a smooth complex function on ${ M }_{n\times N}(\mathbb{C})$ and define the complex function $f$ on $\mathbb{R}^{2(n\times N)}$ by setting $f=F\circ \Psi$. Then,$$\Vert {\rm grad} f(u)\Vert_2 = \sup_{V\in { M }_{n\times N}(\mathbb{C}), Tr VV^*=1}
\left| \frac{d}{dt} F(\Psi(u)+tV)_{\vert_{t=0}}\right|.$$ Now, $X=\Psi(\Re(X_{ij}),
\Im(X_{ij}),1\leq i\leq n,1\leq j\leq N)$ where the distribution of $\{\Re(X_{ij}),
\Im(X_{ij}),1\leq i\leq n,1\leq j\leq N\}$ is ${\cal N}(0, \frac{1}{2}I_{2nN})$.\
Hence consider $F:~H \rightarrow \frac{1}{n} {{\rm Tr}}\left(zI_n -\left(\sigma\frac{ H}{\sqrt{N}} +A \right)\left(\sigma\frac{H}{\sqrt{N}} +A \right)^*\right)^{-1}$.\
Let $ V\in {M }_{n\times N}(\mathbb{C})$ such that $ Tr VV^*=1$.\
$\frac{d}{dt} F(X+tV)\vert_{t=0}$ $$=\frac{1}{n} \left\{{{\rm Tr}}\left(G\sigma\frac{V}{\sqrt{N}} \left(\sigma\frac{X}{\sqrt{N}} +A \right)^* G\right) + {{\rm Tr}}\left(G \left(\sigma\frac{X}{\sqrt{N}} +A \right)\sigma \frac{V^*}{\sqrt{N}} G\right)\right\}.$$ Moreover using Cauchy-Schwartz’s inequality and Lemma \[lem0\], we have\
$\left| \frac{1}{n} {{\rm Tr}}\left(G\sigma\frac{V}{\sqrt{N}} \left(\sigma\frac{X}{\sqrt{N}} +A \right)^* G\right)\right|
$ $$\begin{aligned}
&\leq &\frac{\sigma}{n} (TrVV^* )^{\frac{1}{2}}\left[\frac{1}{N}Tr (\left(\sigma\frac{X}{\sqrt{N}} +A \right)\left(\sigma\frac{X}{\sqrt{N}} +A \right)^*G^2(G^*)^2)\right]^{\frac{1}{2}}\\
&\leq & \frac{\sigma}{\sqrt{N}\sqrt{n}\vert \Im z\vert^2}\left[\lambda_1\left(\left(\sigma\frac{X}{\sqrt{N}} +A \right)\left(\sigma\frac{X}{\sqrt{N}} +A \right)^*\right)\right]^{\frac{1}{2}}.\end{aligned}$$ We get obviously the same bound for $\vert \frac{1}{n} {{\rm Tr}}\left(G \left(\sigma\frac{X}{\sqrt{N}} +A \right) \sigma \frac{V^*}{\sqrt{N}} G\right)\vert$. Thus\
$\mathbb{E}\left(\Vert {\rm grad}f\left(\Re(X_{ij}),
\Im(X_{ij}),1\leq i\leq n,1\leq j\leq N\right)\Vert_2^2 \right)$ $$\label{grad} \leq \frac{4 \sigma^2}{\vert \Im z\vert^4 Nn}\mathbb{E}\left[\lambda_1\left(\left(\sigma\frac{X}{\sqrt{N}} +A \right)\left(\sigma\frac{X}{\sqrt{N}} +A \right)^*\right)\right].$$ (\[esttrace\]) readily follows from (\[Poincare\]), (\[grad\]), Theorem A.8 in [@BaiSil06], Lemma \[bornelambda1\] and the fact that $\Vert A_N \Vert$ is uniformly bounded. Similarly, considering $$F:~H \rightarrow {{\rm Tr}}\left[\left(zI_N -\left(\sigma \frac{H}{\sqrt{N}} +A \right)\left(\sigma \frac{H}{\sqrt{N}} +A \right)^*\right)^{-1}E_{qp}\right],$$ where $E_{qp}$ is the $n\times n$ matrix such that $(E_{qp})_{ij}=\delta_{qi}\delta_{pj}$, we can obtain that, for any $V\in { M }_{n\times N}(\mathbb{C})$ such that $ {{\rm Tr}}VV^*=1$,\
$\left| \frac{d}{dt} F(X+tV)_{\vert_{t=0}} \right|$ $$\leq \frac{\sigma}{\sqrt{N}} \left\{\left(\left(GG^*\right)_{pp} \left(G^*\Sigma \Sigma^*G\right)_{qq}\right)^{1/2}
+ \left( \left(G^*G\right)_{qq} \left(G\Sigma \Sigma^*G^*\right)_{pp}\right)^{1/2}\right\}.$$Thus, one can get in the same way. Finally, considering $$F:~H \rightarrow {{\rm Tr}}\left[\left(\sigma \frac{H}{\sqrt{N}} +A \right)^*\left(zI_N -\left(\sigma \frac{H}{\sqrt{N}} +A \right)\left(\sigma \frac{H}{\sqrt{N}} +A \right)^*\right)^{-1}A\right],$$ we can obtain that, for any $V\in { M }_{n\times N}(\mathbb{C})$ such that $ {{\rm Tr}}VV^*=1$, $$\begin{aligned}
\left| \frac{d}{dt} F(X+tV)_{\vert_{t=0} }\right|& \leq&\sigma\left\{
\left(\frac{1}{N} {{\rm Tr}}\Sigma^* G A \Sigma^* GG^* \Sigma A^* G^*\Sigma \right)^{1/2} \right. \\&&\left.+
\left(\frac{1}{N} {{\rm Tr}}GA \Sigma^* G \Sigma \Sigma^* G^* \Sigma A^* G^* \right)^{1/2} \right. \\&&\left.+
\left(\frac{1}{N} {{\rm Tr}}GA A^* G^* \right)^{1/2}\right\} \end{aligned}$$
Using Lemma \[lem0\] (i), Theorem A.8 in [@BaiSil06], Lemma \[bornelambda1\], the identity $\Sigma \Sigma^*G=G\Sigma \Sigma^* = -I + z G,$ and the fact that $\Vert A_N \Vert$ is uniformly bounded, the same analysis allows to prove .
\[estimeesvariance\] Let $\Delta_1(p,q)$, $\Delta_2(p,q)$, $(p,q)\in \{1,\ldots,n\}^2$, and $\Delta_3$ be as defined in Proposition \[intbypart\]. Then there exist a polynomial $P$ with nonnegative coefficients and a nonnegative real number $l$ such that, for all large $N$, for any $z\in \mathbb{C}\setminus \mathbb{R}$, $$\label{estdelta3} \Delta_3(z)\leq \frac{P(\vert \Im z\vert^{-1} (1+\vert z \vert )^l}{N},$$ and for all $ (p,q)\in \{1,\ldots,n\}^2$, $$\label{estdelta1} \Delta_1(p,q)(z)\leq \frac{P(\vert \Im z\vert^{-1} (1+\vert z \vert )^l}{N},$$ $$\label{estdelta2} \Delta_2(p,q)(z)\leq \frac{P(\vert \Im z\vert^{-1} (1+\vert z \vert )^l}{N\sqrt{N}}.$$
Using the identity $$GM_N = -I + z G,$$ (\[estdelta1\]) readily follows from Cauchy-Schwartz inequality, Lemma \[lem0\] and (\[esttrace\]). (\[estdelta2\]) and (\[estdelta3\]) readily follows from Cauchy-Schwartz inequality and Lemma \[variance\]
Estimates of Resolvent entries
------------------------------
In order to deduce Proposition \[estimfonda\] from Proposition \[intbypart\] and Corollary \[estimeesvariance\], we need the two following Lemma \[majpre\] and Lemma \[gnmoinsgmu\].
\[majpre\] For all $z\in \mathbb{C}\setminus \mathbb{R}$, $$\label{in1}\frac{1}{\left|1- \sigma^2 c_N g_N(z)\right|} \leq \frac{\vert z\vert }{\vert \Im z \vert},$$ $$\label{in2}\frac{1}{\left|1- \sigma^2 c g_{\mu_{\sigma,\nu,c}}(z)\right|} \leq \frac{\vert z\vert }{\vert \Im z \vert}.$$
Since $\mu_{M_N}$ is supported by $[0,+\infty[$, readily follows from\
$\frac{1}{\left|1- \sigma^2 c_N g_N(z)\right|} =\frac{\vert z\vert }{\left|z- \sigma^2 c_N zg_N(z)\right|} $ $$\hspace*{1,7
cm}\leq \frac{\vert z\vert }{\left|\Im (z- \sigma^2 c_N zg_N(z))\right|}= \frac{\vert z\vert }{|\Im z |\left( 1+\sigma^2 c_N \mathbb{E} \int \frac{t}{|z-t|^2} d\mu_{M_N}(t)\right)}.$$ may be proved similarly.
Corollary \[estimeesvariance\] and Lemma \[majpre\] yields that, there is a polynomial $Q$ with nonnegative coefficients, a sequence $b_N$ of nonnegative real numbers converging to zero when $N$ goes to infinity and some nonnegative integer number $l$, such that for any $p,q$ in $\{1, \ldots,n\} $, for all $z\in \mathbb{C}\setminus \mathbb{R}$, $$\label{nablast}\nabla_{pq} \leq (1+\vert z\vert)^l Q(\vert \Im z \vert^{-1})b_N,$$ where $\nabla_{pq}$ was defined by (\[nablapq\]).\
\[gnmoinsgmu\] There is a sequence $v_N$ of nonnegative real numbers converging to zero when $N$ goes to infinity such that for all $z\in \mathbb{C}\setminus \mathbb{R}$,
$$\label{gnmoinsg} \left| g_N(z)-g_{\mu_{\sigma,\nu,c}}(z)\right| \leq \left\{\frac{\vert z\vert^2 +2}{\vert \Im z \vert^{2}}+ \frac{1}{\vert \Im z \vert}\right\}v_N.$$
First note that it is sufficient to prove (\[gnmoinsg\]) for $z\in \mathbb{C}^+:=\{z \in \mathbb{C}; \Im z >0\}$ since $ g_N(\bar z)-g_{\mu_{\sigma,\nu,c}} (\bar z)= \overline{g_N(z)-g_{\mu_{\sigma,\nu,c}}(z)}$. Fix $\epsilon>0.$ According to Theorem A.8 and Theorem 5.11 in [@BaiSil06], and the assumption on $A_N$, we can choose $K> \max\{ 2/\varepsilon; x, x \in \rm{supp}( \mu_{\sigma,\nu,c})\}$ large enough such that $\mathbb{P}\left( \left\|M_N\right\| >K\right)$ goes to zero as $N$ goes to infinity. Let us write $$\label{reecriture}g_N(z)=\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right) +\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert > K} \right).$$ For any $z \in \mathbb{C}^+$ such that $\vert z \vert > 2K$, we have $$\left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)\right| \leq \frac{1}{K} \leq \frac{\epsilon}{2} \mbox{~~and~~}\left|g_{\mu_{\sigma,\nu,c}}(z)\right| \leq \frac{1}{K} \leq \frac{\epsilon}{2}.$$ Thus, $\forall z \in \mathbb{C}^+,$ such that $ \vert z \vert > 2K$, we can deduce that\
$ \left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}}(z)\right|\frac{(\Im z)^2}{\vert z \vert^2 +2}$ $$\begin{aligned}
&\leq & \left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}} (z)\right| \nonumber \\&\leq& {\varepsilon} \label{horscompact}.\end{aligned}$$ Now, it is clear that $\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)$ is a sequence of locally bounded holomorphic functions on $\mathbb{C}^+$ which converges towards $g_{\mu_{\sigma,\nu,c}}$. Hence, by Vitali’s Theorem, $\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)$ converges uniformly towards $g_{\mu_{\sigma,\nu,c}}$ on each compact subset of $\mathbb{C}^+$. Thus, there exists $N(\epsilon)>0$, such that for any $N \geq N(\epsilon)$, for any $z\in \mathbb{C}^+$, such that $ \vert z \vert \leq 2K$ and $\Im z \geq {\varepsilon}$,\
$
\left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}}(z)\right|\frac{(\Im z)^2}{\vert z \vert^2 +2}$ $$\begin{aligned}
&\leq & \left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}}(z)\right| \nonumber \\&\leq&{ \varepsilon} \label{compactaudessus}.\end{aligned}$$ Finally, for any $z\in \mathbb{C}^+$, such that $\Im z \in ]0; {\varepsilon}[$, we have $$\label{compactaudessous}
\left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}}(z)\right|\frac{(\Im z)^2}{\vert z \vert^2 +2} \leq \frac{2}{\Im z} \frac{(\Im z)^2}{\vert z \vert^2 +2} \leq \Im z \leq {\varepsilon}.$$ It readily follows from (\[horscompact\]), (\[compactaudessus\]) and (\[compactaudessous\]) that for $N \geq N(\epsilon)$, $$\sup_{z\in \mathbb{C}^+}\left\{ \left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}}(z)\right|\frac{(\Im z)^2}{\vert z \vert^2 +2} \right\}\leq {\varepsilon}$$ Moreover, for $N \geq N'(\epsilon)\geq N(\epsilon)$, $\mathbb{P}\left( \left\|M_N\right\| >K\right) \leq \varepsilon.$ Therefore, for $N \geq N'(\epsilon)$, we have for any $z \in \mathbb{C}^+$,\
$\left| g_N(z)-g_{\mu_{\sigma,\nu,c}}(z)\right|$ $$\begin{aligned}
& \leq &\frac{\vert z\vert^2 +2}{\vert \Im z \vert^{2}} \sup_{z\in \mathbb{C}^+} \left\{\left|\mathbb{E}\left( \frac{1}{n} {{\rm Tr}}G_N(z) {{1\!\!{\sf I}}}_{\Vert M_N \Vert \leq K} \right)-g_{\mu_{\sigma,\nu,c}}(z)\right|\frac{(\Im z)^2}{\vert z \vert^2 +2}\right\} \nonumber\\&& + \frac{1}{\Im z} \mathbb{P}\left( \left\|M_N\right\| >K\right)\nonumber\\&\leq &
\varepsilon \left\{\frac{\vert z\vert^2 +2}{\vert \Im z \vert^{2}}+ \frac{1}{\Im z}\right\}\end{aligned}$$ Thus, the proof is complete by setting $$v_N= \sup_{z\in \mathbb{C}^+} \left\{\left|g_N(z)-g_{\mu_{\sigma,\nu,c}}(z) \right| \left(\frac{\vert z\vert^2 +2}{\vert \Im z \vert^{2}}+ \frac{1}{\Im z}\right)^{-1}\right\}.$$
Now set $$\tau_N= {(1-\sigma^2c_Ng_{ N}(z))z- \frac{ \gamma_q(N)}{1- \sigma^2 c_Ng_{N}(z)} -\sigma^2 (1-c_N)}$$ and $$\label{tau'} \tilde \tau_N={(1-\sigma^2cg_{ \mu_{\sigma,\nu,c}}(z))z- \frac{ \gamma_q(N)}{1- \sigma^2 cg_{ \mu_{\sigma,\nu,c}}(z)} -\sigma^2 (1-c)}.$$ Lemmas \[majpre\] and \[gnmoinsgmu\] yield that there is a polynomial $R$ with nonnegative coefficients, a sequence $w_N$ of nonnegative real numbers converging to zero when $N$ goes to infinity and some nonnegative real number $l$, such that for all $z\in \mathbb{C}\setminus \mathbb{R}$, $$\label{tau}\left|\tau_N - \tilde \tau_N\right| \leq (1+\vert z\vert)^l R(\vert \Im z \vert^{-1})w_N.$$ Now, one can easily see that, $$\label{ima2} \left|\Im \left\{(1-\sigma^2cg_{ \mu_{\sigma,\nu,c}}(z))z- \frac{ \gamma_q(N)}{1- \sigma^2 cg_{ \mu_{\sigma,\nu,c}}(z)} -\sigma^2 (1-c)\right\}\right| \geq \vert \Im z \vert,$$ so that $$\label{moinsun} \left| \frac{1}{\tilde \tau_N}\right| \leq \frac{1}{\vert\Im z \vert}.$$ Note that $$\label{re}\frac{1}{ \tilde \tau_N} =\frac{( {1- \sigma^2c g_{\mu_{\sigma,\nu,c}}(z)})}{\omega_{\sigma, \nu, c}(z) -\gamma_q(N)}.$$
Then, (\[premier\]) readily follows from Proposition \[intbypart\], (\[nablast\]), (\[tau\]), (\[moinsun\]), (\[re\]), and (ii) Lemma \[lem0\]. The proof of Proposition \[estimfonda\] is complete.
Proof of Theorem \[cvev1p1\]
============================
We follow the two steps presented in Section 2.\
[**Step A.**]{} We first prove (\[5.4\]).
Let $\eta>0$ small enough and $N$ large enough such that for any $l=1,\ldots, J$, $\alpha_l(N)\in [\theta_l-\eta,\theta_l+\eta]$ and $[\theta_l-2\eta,\theta_l+2\eta]$ contains no other element of the spectrum of $A_NA_N^*$ than $\alpha_l(N)$. For any $l=1,\ldots,J$, choose $f_{\eta,l}$ in $\mathcal{C}^\infty (\mathbb{R}, \mathbb{R})$ with support in $[\theta_l
-2\eta,\theta_l+2\eta]$ such that $f_{\eta,l}(x)=1$ for any $x \in [\theta_l
-\eta,\theta_l+\eta]$ and $0 \leq f_{\eta,l} \leq 1$. Let $0< \epsilon <\delta_0$ where $\delta_0$ is introduced in Theorem \[ThmASCV\]. Choose $h_{\varepsilon,j}$ in $\mathcal{ C}^\infty (\mathbb{R}, \mathbb{R})$ with support in $[\rho_{\theta_j} -\varepsilon
,\rho_{\theta_j}+\varepsilon
]$ such that $h_{\varepsilon,j} \equiv 1$ on $[\rho_{\theta_j} -\varepsilon/2
,\rho_{\theta_j}+\varepsilon/2
]$ and $0 \leq h_{\varepsilon
,j}\leq 1$.\
Almost surely for all large $N$, $M_N$ has $k_j$ eigenvalues in $]\rho_{\theta_j} -\varepsilon/2
,\rho_{\theta_j}+\varepsilon/2[$. According to Theorem \[ThmASCV\], denoting by $(\xi_1,\cdots,\xi_{k_j})$ an orthonormal system of eigenvectors associated to the $k_j$ eigenvalues of $M_N$ in $( \rho_{\theta_j} -\varepsilon/2, \rho_{\theta_j}+\varepsilon/2)$, it readily follows from (\[equavec\]) that almost surely for all large $N$, $$\sum_{n=1}^{k_j}\left\| P_{\ker(\alpha_l(N) I_n-A_NA_N^*)}\xi_n \right\|^2= {\rm Tr} \left[ h_{\varepsilon
,j}(M_N) f_{\eta,l}(A_NA_N^*)\right].$$ Applying Proposition \[compgaunogau\] with $\Gamma_N= f_{\eta,l}(A_NA_N^*)$ and $K=k_l$, the problem of establishing (\[5.4\]) is reduced to prove that\
$ \mathbb{E}\left({{\rm Tr}}\left[h_{\varepsilon,j} \left(\left(\sigma\frac{{\cal G}_N}{\sqrt{N}}+A_N\right)\left(\sigma\frac{{\cal G}_N}{\sqrt{N}}+A_N\right)^*\right) f_{\eta,l}(A_NA_N^*)\right] \right)$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\rightarrow_{N \rightarrow +\infty} \frac{k_j\delta_{jl} (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\rho_{\theta_j}))}{\omega_{\sigma,\nu,c}'(\rho_{\theta_j})}.$$ Using a Singular Value Decomposition of $A_N$ and the biunitarily invariance of the distribution of ${\cal G}_N$, we can assume that $A_N$ is as and such that for any $j=1,\ldots, J,$ $$(A_NA_N^*)_{ii}=\alpha_j(N) \mbox{~~ for $i=k_1+\ldots+k_{j-1}+l$, $l=1,\ldots,k_j$}.$$ Now, according to Lemma \[approxpoisson\],\
$ \mathbb{E}\left({{\rm Tr}}\left[h_{\varepsilon,j} \left(\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)\left(\sigma \frac{{\cal G}_N}{\sqrt{N}}+A_N\right)^*\right) f_{\eta,l}(A_NA_N^*)\right] \right)$ $$= - \lim_{y\rightarrow 0^{+}}\frac{1}{\pi} \int \Im \mathbb{E}{{\rm Tr}}\left[G^{\cal G}_N(t+iy) f_{\eta,l}(A_NA_N^*)\right] h_{\varepsilon,j}(t) dt,$$ with, for all large $N$, $$\begin{aligned}
\mathbb{E}{{\rm Tr}}\left[G^{\cal G}_N(t+iy) f_{\eta,l}(A_NA_N^*)\right]
&=&\sum_{k=k_1+\cdot+k_{l-1}+1}^{k_1+\cdot+k_{l}}
f_{\eta,l} (\alpha_l(N))\mathbb{E}[G^{\cal G}_N(t+iy)]_{kk} \\&=& \sum_{k=k_1+\cdot+k_{l-1}+1}^{k_1+\cdot+k_{l}}
\mathbb{E}[G^{\cal G}_N(t+iy)]_{kk}. \end{aligned}$$ Now, by considering $$\tau'={(1-\sigma^2cg_{ \mu_{\sigma,\nu,c}}(z))z- \frac{ \theta_l}{1- \sigma^2 cg_{ \mu_{\sigma,\nu,c}}(z)} -\sigma^2 (1-c)}$$ instead of dealing with $\tilde \tau_N$ defined in (\[tau’\]) at the end of the proof of Proposition \[estimfonda\], one can prove that there is a polynomial $P$ with nonnegative coefficients, a sequence $(u_N)_N$ of nonnegative real numbers converging to zero when $N$ goes to infinity and some nonnegative real number $s$, such that for any $k$ in $\{k_1+\ldots+k_{l-1}+1, \ldots,k_1+\ldots+k_l\}$, for all $z\in \mathbb{C}\setminus \mathbb{R}$, $$\label{deuxieme}
\mathbb{E} \left(\left( G^{\cal G}_N(z)\right)_{kk}\right) = \frac{1- \sigma^2 cg_{\mu_{\sigma,\nu,c}}(z)}{\omega_{\sigma, \nu, c}(z) -\theta_l}
+\Delta_{k,N}(z),$$ with $$\left| \Delta_{k,N} (z)\right| \leq (1+\vert z\vert)^s P(\vert \Im z \vert^{-1})u_N.$$ Thus, $$\mathbb{E}{{\rm Tr}}\left[G^{\cal G}_N(t+iy) f_{\eta,l}(A_NA_N^*)\right]
= k_l \frac{1- \sigma^2 cg_{\mu,\sigma,\nu}(t+iy)}{\omega_{\sigma, \nu, c}(z) -\theta_l}
+ \Delta_N(t+iy),$$ where for all $z \in \mathbb{C} \setminus \mathbb{R}$, $\Delta_N(z)= \sum_{k=k_1+\cdot+k_{l-1}+1}^{k_1+\cdot+k_{l}} \Delta_{k,N}(z),$ and $\left| \Delta_{N} (z)\right| \leq k_l (1+\vert z\vert)^s P(\vert \Im z \vert^{-1})u_N.$
First let us compute $$\lim_{y\downarrow0}\frac{k_l}{\pi}\int_{\rho_{\theta_j}-\varepsilon}^{\rho_{\theta_j}+\varepsilon}
\Im\frac{h_{\varepsilon,j}(t) (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(t+iy))}{\theta_l-\omega_{\sigma,\nu,c}(t+iy)}\,dt.$$ The function $\omega_{\sigma,\nu,c}$ satisfies $\omega_{\sigma,\nu,c}(\overline{z})=\overline{\omega_{\sigma,\nu,c}(z)}$ and $g_{\mu_{\sigma,\nu,c}}(\overline{z})=\overline{g_{\mu_{\sigma,\nu,c}}(z)}$, so that $\Im\frac{ (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(t+iy))}{\theta_l-\omega_{\sigma,\nu,c}(t+iy)}=\frac{1}{2i}[\frac{ (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(t+iy))}{\theta_l-\omega_{\sigma,\nu,c}(t+iy)}-
\frac{ (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(t-iy))}{\theta_l-\omega_{\sigma,\nu,c}(t-iy)}]$. As in [@MCJTP], the above integral is split into three pieces, namely $\int_{\rho_{\theta_j}-\varepsilon}^{\rho_{\theta_j}-\varepsilon/2}+
\int_{\rho_{\theta_j}-\varepsilon/2}^{\rho_{\theta_j}+\varepsilon/2}+\int_{\rho_{\theta_j}+\varepsilon/2}^{\rho_{\theta_j}+\varepsilon}$. Each of the first and third integrals are easily seen to go to zero when $y\downarrow0$ by a direct application of the definition of the functions involved and of the (Riemann) integral. As $h_{\varepsilon,j}$ is constantly equal to one on $[\rho_{\theta_j}-\epsilon/2; \rho_{\theta_j}+\epsilon/2]$, the second (middle) term is simply the integral $$\frac{k_l}{2\pi i}\int_{\rho_{\theta_j}-\varepsilon/2}^{\rho_{\theta_j}+\varepsilon/2}\frac{1-\sigma^2cg_{\mu_{\sigma,\nu,c}}(t+iy)}{\theta_l-\omega_{\sigma,\nu,c}(t+iy)}-
\frac{1-\sigma^2cg_{\mu_{\sigma,\nu,c}}(t-iy)}{\theta_l-\omega_{\sigma,\nu,c}(t-iy)}\,dt.$$ Completing this to a contour integral on the rectangular with corners $\rho_{\theta_j}\pm\varepsilon/2\pm iy$ and noting that the integrals along the vertical lines tend to zero as $y\downarrow0$ allows a direct application of the residue theorem for the final result, if $l=j$, $$\frac{k_j (1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\rho_{\theta_j}))}{\omega_{\sigma,\nu,c}'(\rho_{\theta_j})}.$$ If we consider $\theta_l$ for some $l\neq j$, then $z\mapsto (1-\sigma^2 cg_{\mu_{\sigma,\nu,c}}(z))(\theta_l-
\omega_{\sigma,\nu,c}(z))^{-1}$ is analytic around $\rho_{\theta_j}$, so its residue at $\rho_{\theta_j}$ is zero, and the above argument provides zero as answer.
Now, according to Lemma \[HT\], we have $$\limsup_{y\rightarrow 0^+}~(u_N)^{-1}\left|\int h_{\varepsilon,j}
(t)\Delta_N(t+iy)dt\right| <+\infty$$ so that $$\label{reste}
\lim_{N\rightarrow + \infty}\limsup_{y \rightarrow 0^+} \left| \int h_{\varepsilon,j}
(t) \Delta_N(t+iy)dt
\right| =0.$$
This concludes the proof of (\[5.4\]).\
[**Step B:**]{} In the second, and final, step, we shall use a perturbation argument identical to the one used in [@MCJTP] to reduce the problem to the case of a spike with multiplicity one, case that follows trivially from Step A. A further property of eigenvectors of Hermitian matrices which are close to each other in the norm will be important in the analysis of the behaviour of the eigenvectors of our matrix models. Given a Hermitian matrix $M\in\ M_N(\mathbb C)$ and a Borel set $S\subseteq\mathbb R$, we denote by $E_M(S)$ the spectral projection of $M$ associated to $S$. In other words, the range of $E_M(S)$ is the vector space generated by the eigenvectors of $M$ corresponding to eigenvalues in $S$. The following lemma can be found in [@BBCF15].
\[eigenspaces\] Let $M$ and $M_0$ be $N \times N$ Hermitian matrices. Assume that $\alpha,\beta,\delta\in\mathbb R$ are such that $\alpha<\beta$, $\delta>0$, $M$ and $M_0$ has no eigenvalues in $[\alpha-\delta,\alpha]\cup
[\beta,\beta+\delta]$. Then, $$\|E_{M}((\alpha,\beta))-E_{M_0}((\alpha,\beta))\|<\frac{4(\beta-\alpha+2\delta)}{\pi\delta^2}\|M-M_0\|.$$ In particular, for any unit vector $\xi\in E_{M_0}((\alpha,\beta))(\mathbb C^N)$, $$\|(I_N-E_{M}((\alpha,\beta)))\xi\|_2<\frac{4(\beta-\alpha+2\delta)}{\pi\delta^2}\|M-M_0\|.$$
Assume that $\theta_i$ is in $\Theta_{\sigma,\nu,c}$ defined in (\[defTheta\]) and $k_i\neq 1$. Let us denote by $V_1(i),\ldots, V_{k_i}(i)$, an orthonormal system of eigenvectors of $A_NA_N^*$ associated with $\alpha_i(N)$. Consider a Singular Value Decomposition $A_N=U_ND_NV_N$ where $V_N$ is a $N\times N$ unitary matrix, $U_N$ is a $n\times n$ unitary matrix whose $k_i$ first columns are $ V_1(i),\ldots, V_{k_i}(i)$ and $D_N$ is as with the first $k_i$ diagonal elements equal to $\sqrt{\alpha_i(N)}$.
Let $ \delta_0$ be as in Theorem \[ThmASCV\]. Almost surely, for all $N$ large enough, there are $k_i$ eigenvalues of $M_N$ in $(\rho_{\theta_i}- \frac{\delta_0}{4}, \rho_{\theta_i}+ \frac{\delta_0}{4})$, namely $\lambda_{n_{i-1}+q}(M_N)$, $q=1,\ldots,k_i$ (where $n_{i-1}+1,\ldots,n_{i-1}+k_i $ are the descending ranks of $\alpha_i(N)$ among the eigenvalues of $A_NA_N^*$), which are moreover the only eigenvalues of $M_N$ in $(\rho_{\theta_i}-\delta_0,\rho_{\theta_i}+\delta_0)$. Thus, the spectrum of $M_N$ is split into three pieces: $$\{\lambda_1(M_N),\dots,\lambda_{n_{i-1}}(M_N)\}\subset (\rho_{\theta_i}+\delta_0,+\infty[,$$ $$\{\lambda_{n_{i-1}+1}(M_N),\dots,\lambda_{n_{i-1}+k_i}(M_N)\}
\subset(\rho_{\theta_i}- \frac{\delta_0}{4},\rho_{\theta_i}+ \frac{\delta_0}{4}),$$ $$\{\lambda_{n_{i-1}+k_i+1}(M_N),\dots,
\lambda_{N}(M_N)\}\newline\subset [0,
\rho_{\theta_i}-\delta_0).$$ The distance between any of these components is equal to $3\delta_0/4$. Let us fix $\epsilon_0$ such that $0\leq \theta_i ( 2 \epsilon_0 k_i +\epsilon_0^2k_i^2) < dist(\theta_i, \mbox{supp~}\nu \cup_{i\neq s}\theta_s )$ and such that $[\theta_i; \theta_i+\theta_i ( 2 \epsilon_0 k_i +\epsilon_0^2k_i^2)] \subset {\cal E}_{\sigma, \nu, c}$ defined by (\[cale\]). For any $0< \epsilon<\epsilon_0$, define the matrix $A_N(\epsilon)$ as $A_N(\epsilon)=U_ND_N(\epsilon) V_N$ where $$\left(D_N(\epsilon)\right)_{m, m}=\sqrt{\alpha_{i}(N)} [1 + \epsilon (k_i-m+1)],\text{~~for $m\in\{1,\ldots,k_i\}$},$$ and $\left(D_N(\epsilon)\right)_{pq}=\left(D_N\right)_{pq}$ for any $(p,q)\notin \{ (m,m), m\in\{1,\ldots,k_i\}\}$.
Set $$M_N(\epsilon)=\left(\sigma \frac{X_N}{\sqrt{N}} +A_N(\epsilon)\right)\left( \sigma \frac{X_N}{\sqrt{N}} +A_N(\epsilon)\right)^*.$$ For $N $ large enough, for each $m\in\{1,\ldots,k_i\}$, $ \alpha_{i}(N) [1 + \epsilon (k_i-m+1)]^2$ is an eigenvalue of $A_NA_N^*(\epsilon)$ with multiplicity one. Note that, since $\sup_N\Vert A_N \Vert <+\infty$, it is easy to see that there exist some constant $C$ such that for any $N$ and for any $0< \epsilon<\epsilon_0$, $$\left\|M_N(\epsilon)-M_N\right\|\leq
C\epsilon \left( \left\| \frac{X_N}{\sqrt{N}}\right\| +1\right) .$$ Applying Remark \[2.1\] to the $(n+N)\times (n+N)$ matrix $\tilde X_N=
\left( \begin{array}{ll} 0_{n\times n}~~~ X_N\\ X_N^*~~~ 0_{N\times N} \end{array} \right)$ (see also Appendix B of [@CSBD]), it readily follows that there exists some constant $C'$ such that a.s for all large N, for any $0< \epsilon<\epsilon_0$, $$\label{normediff} \left\| M_N(\epsilon)-M_N\right\| \leq C' \epsilon.$$ Therefore, for $\epsilon $ sufficiently small such that $C' \epsilon < \delta_0/4$, by Theorem A.46 [@BaiSil06], there are precisely $n_{i-1}$ eigenvalues of $M_N(\epsilon)$ in $[0,\rho_{\theta_i}-3\delta_0/4)$, precisely $k_i$ in $(\rho_{\theta_i}-\delta_0/2,\rho_{\theta_i}+\delta_0/2)$ and precisely $N-(n_{i-1}+k_i)$ in $(\rho_{\theta_i}+3\delta_0/4,+ \infty[$. All these intervals are again at strictly positive distance from each other, in this case $\delta_0/4$.
Let $\xi$ be a normalized eigenvector of $M_N$ relative to $\lambda_{n_{i-1}+q}(M_N)$ for some $q\in\{1,\ldots, k_i\}$. As proved in Lemma \[eigenspaces\], if $E(\epsilon)$ denotes the subspace spanned by the eigenvectors associated to $\{\lambda_{n_{i-1}+1}(M_N(\epsilon)),\dots,\lambda_{n_{i-1}+k_i}
(M_N(\epsilon))\}$ in $\mathbb C^N$, then there exists some constant $C$ (which depends on $\delta_0$) such that for $\epsilon$ small enough, almost surely for large $N$, $$\label{least2}
\left\|P_{ E(\epsilon)^{\bot}}\xi\right\|_2\leq C \epsilon
.$$ According to Theorem \[ThmASCV\], for $j\in\{1,\ldots,k_i\}$, for large enough $N$, $\lambda_{n_{i-1}+j}(M_N(\epsilon))$ separates from the rest of the spectrum and belongs to a neighborhood of $\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon))$ where $$\theta_i^{(j)}(\epsilon)=\theta_i\left( 1+\epsilon (k_i -j+1) \right)^2.$$ If $\xi_{j}(\epsilon,i
)$ denotes a normalized eigenvector associated to $\lambda_{n_{i-1}+j}(M_N(\epsilon))$, Step A above implies that almost surely for any $p\in \{1,\ldots, k_i\}$, for any $\gamma>0$, for all large $N$, $$\label{5.14}
\left|\left| \langle V_{p}(i),\xi_{j}(\epsilon,i)\rangle \right|^2 - \frac{\delta_{jp}\left(1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon)))\right)}{ \omega_{\sigma,\nu,c}'\left(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon))
\right)}\right|<\gamma.$$ The eigenvector $\xi$ decomposes uniquely in the orthonormal basis of eigenvectors of $M_N(
\epsilon)$ as $\xi=\sum_{j=1}^{k_i}c_j(\epsilon)\xi_j(\epsilon,i)+\xi(\epsilon)^\perp$, where $c_j(\epsilon)=\langle\xi|\xi_j(\epsilon,i)\rangle$ and $\xi(\epsilon)^\perp=P_{ E(\epsilon)^{\bot}}\xi$; necessarily $\sum_{j=1}^{k_i}|c_j(\epsilon)|^2+\|\xi(\epsilon)^\perp\|_2^2=1$. Moreover, as indicated in relation , $\|\xi(\epsilon)^\perp\|_2\leq C \epsilon.$ We have $$\begin{aligned}
P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi&=&\sum_{j=1}^{k_i}c_j(\epsilon)
P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi_j(\epsilon,i)\\&&+P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi(\epsilon)^\perp\\
&=&\sum_{j=1}^{k_i}c_j(\epsilon)
\sum_{l=1}^{k_i}\langle \xi_j(\epsilon,i) | V_{l}(i)\rangle
V_{l}(i)\\
& & \mbox{}+P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi(\epsilon)^\perp.\end{aligned}$$ Take in the above the scalar product with $\xi=\sum_{j=1}^{k_i}c_j(\epsilon)\xi_j(\epsilon,i)+\xi(\epsilon)^\perp$ to get $$\begin{aligned}
\lefteqn{\langle P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi|\xi\rangle =}\\
& & \mbox{}
\sum_{j,l,s=1}^{k_i}c_j(\epsilon)\langle \xi_j(\epsilon,
i) |V_{l}(i) \rangle\overline{c_s(\epsilon)}\langle V_{l}(i)|\xi_s(\epsilon,
i)\rangle\\
& & \mbox{}+\sum_{j=1}^{k_i}c_j(\epsilon)
\sum_{l=1}^{k_i}\langle \xi_j(\epsilon,i)| V_{l}(i) \rangle
\langle V_{l}(i)|\xi(\epsilon)^\perp\rangle\\
& & \mbox{}+\langle P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi(\epsilon)^\perp|\xi\rangle.\end{aligned}$$ Relation indicates that $$\begin{aligned}
\lefteqn{ \hspace*{-2cm}\sum_{j,l,s=1}^{k_i}c_j(\epsilon)\langle \xi_j(\epsilon,
i) | V_{l}(i) \rangle\overline{c_s(\epsilon)}\langle V_{l}(i)|\xi_s(\epsilon,
i)\rangle}\\
&= & \mbox{} \sum_{j=1}^{k_i}|c_j(\epsilon)|^2|\langle V_{j}(i)|\xi_j(\epsilon,
i)\rangle|^2+\Delta_1\\
& =& \mbox{} \sum_{j=1}^{k_i}|c_j(\epsilon)|^2 \frac{\left(1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon)))\right)}{ \omega_{\sigma,\nu,c}'\left(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon))
\right)}+\Delta_1 + \Delta_2,\end{aligned}$$ where for all large $N$, $\vert \Delta_1\vert \leq \sqrt{ \gamma} k_i^3$ and $\vert \Delta_2\vert \leq \gamma $. Since $\|\xi(\epsilon)^\perp\|_2\leq C\epsilon$,\
$
\left|\sum_{j=1}^{k_i}c_j(\epsilon)
\sum_{l=1}^{k_i}\langle \xi_j(\epsilon,i) | V_{l}(i) \rangle
\langle V_{l}(i)|\xi(\epsilon)^\perp\rangle \right.$\
$\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\langle P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi(\epsilon)^\perp|\xi\rangle\right|
\leq\left(k_i^2+1\right){C\epsilon}.
$\
Thus, we conclude that almost surely for any $\gamma>0$, for all large $N$,$$\left|\langle P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi|\xi\rangle-
\sum_{j=1}^{k_i}\frac{|c_j(\epsilon)|^2 \left(1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon)))\right)}{ \omega_{\sigma,\nu,c}'\left(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon))
\right)}
\right|$$ $$\label{5.15}~~~~~~~~~~~~~~~~~~~~~~\leq(k_i^2+1)C\epsilon+ \sqrt{\gamma}k_i^3+\gamma .$$ Since we have the identity $$\langle P_{\ker(\alpha_i(N)I_N-A_NA_N^*)}\xi|\xi\rangle=\|P_{\ker( \alpha_i(N)I_N-A_NA_N^*)}\xi\|_2^2$$ and the three obvious convergences $\lim_{\epsilon\to0}\omega_{\sigma,\nu,c}'\left(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon))
\right)=\omega_{\sigma,\nu,c}'(\rho_{\theta_i})$, $\lim_{\epsilon\to0}g_{\mu_{\sigma,\nu,c}}\left(\Phi_{\sigma,\nu,c} (\theta_i^{(j)}(\epsilon))
\right)=g_{\mu_{\sigma,\nu,c}}(\rho_{\theta_i})$ and $\lim_{\epsilon\to0}\sum_{j=1}^{k_i}|c_j(\epsilon)|^2=1$, relation concludes Step B and the proof of Theorem \[cvev1p1\]. (Note that we use (2.9) of [@MC2014] which is true for any $x\in \mathbb{C}\setminus \mathbb{R}$ to deduce that $1-\sigma^2 c g_{\mu_{\sigma,\nu,c}}(\Phi_{\sigma,\nu,c}(\theta_i))=
\frac{ 1}{1+ \sigma^2 cg_\nu(\theta_i)}$ by letting $x$ goes to $\Phi_{\sigma,\nu,c}(\theta_i)$).
Appendix A {#appendix-a .unnumbered}
==========
We present alternative versions on the one hand of the result in [@BaiSilver] about the lack of eigenvalues outside the support of the deterministic equivalent measure, and on the other hand of the result in [@MC2014] about the exact separation phenomenon. These new versions (Theorems \[noeigenvalue\] and \[sep\] below) deal with random variables whose imaginary and real parts are independent, but remove the technical assumptions ((1.10) and “$b_1>0$” in Theorem 1.1 in [@BaiSilver] and “$\omega_{\sigma,\nu,c}(b)>0$” in Theorem 1.2 in [@MC2014]). The proof of Theorem \[noeigenvalue\] is based on the results of [@BC]. The arguments of the proof of Theorem 1.2 in [@MC2014] and Theorem \[noeigenvalue\] lead to the proof of Theorem \[sep\].
\[pasde\] Consider $$\label{modele}M_N=( \sigma \frac{ X_N}{\sqrt{N}}+A_N)(\sigma \frac{ X_N}{\sqrt{N}}+A_N)^*,$$ and assume that
1. $X_N = [X_{ij}]_{1\leq i\leq n, 1\leq j\leq N}$ is a $n\times N$ random matrix such that $ [X_{ij}]_{i\geq1,j\geq 1}$ is an infinite array of random variables which satisfy and and such that $ \Re(X_{ij})$, $ \Im(X_{ij})$, $(i,j)\in \mathbb{N}^2$, are independent, centered with variance $1/2$.
2. $A_N$ is an $n\times N$ nonrandom matrix such that $\Vert A_N \Vert$ is uniformly bounded.
3. $n\leq N$ and, as $N$ tends to infinity, $c_N=n/N \rightarrow c\in ]0,1]$.
4. $[x,y] $, $x<y$, is such that there exists $\delta>0$ such that for all large $N$, $ ]x-\delta; y+\delta[ \subset \mathbb{R}\setminus \rm{supp} (\mu_{\sigma,\mu _{A_N A_N^*},c_N})$ where $\mu_{\sigma,\mu _{A_N A_N^*},c_N}$ is the nonrandom distribution which is characterized in terms of its Stieltjes transform which satisfies the equation where we replace $c$ by $c_N$ and $\nu$ by $\mu _{A_N A_N^*}.$
Then, we have $$\mathbb P[\mbox{for all large N}, \rm{spect}(M_N) \subset {\mathbb{R}}\setminus [x,y] ]=1.$$
Since, in the proof of Theorem \[pasde\], we will use tools from free probability theory, for the reader’s convenience, we recall the following basic definitions from free probability theory. For a thorough introduction to free probability theory, we refer to [@VDN].
- A ${\cal C}^*$-probability space is a pair $\left({\cal A}, \tau\right)$ consisting of a unital $ {\cal C}^*$-algebra ${\cal A}$ and a state $\tau$ on ${\cal A}$ i.e a linear map $\tau: {\cal A}\rightarrow \mathbb{C}$ such that $\tau(1_{\cal A})=1$ and $\tau(aa^*)\geq 0$ for all $a \in {\cal A}$. $\tau$ is a trace if it satisfies $\tau(ab)=\tau(ba)$ for every $(a,b)\in {\cal A}^2$. A trace is said to be faithful if $\tau(aa^*)>0$ whenever $a\neq 0$. An element of ${\cal A}$ is called a noncommutative random variable.
- The noncommutative $\star$-distribution of a family $a=(a_1,\ldots,a_k)$ of noncommutative random variables in a ${\cal C}^*$-probability space $\left({\cal A}, \tau\right)$ is defined as the linear functional $\mu_a:P\mapsto \tau(P(a,a^*))$ defined on the set of polynomials in $2k$ noncommutative indeterminates, where $(a,a^*)$ denotes the $2k$-uple $(a_1,\ldots,a_k,a_1^*,\ldots,a_k^*)$. For any selfadjoint element $a_1$ in ${\cal A}$, there exists a probability measure $\nu_{a_1}$ on $\mathbb{R}$ such that, for every polynomial P, we have $$\mu_{a_1}(P)=\int P(t) \mathrm{d}\nu_{a_1}(t).$$ Then we identify $\mu_{a_1}$ and $\nu_{a_1}$. If $\tau$ is faithful then the support of $\nu_{a_1}$ is the spectrum of $a_1$ and thus $\|a_1\| = \sup\{|z|, z\in \rm{support} (\nu_{a_1})\}$.
- A family of elements $(a_i)_{i\in I}$ in a ${\cal C}^*$-probability space $\left({\cal A}, \tau\right)$ is free if for all $k\in \mathbb{N}$ and all polynomials $p_1,\ldots,p_k$ in two noncommutative indeterminates, one has $$\label{freeness}
\tau(p_1(a_{i_1},a_{i_1}^*)\cdots p_k (a_{i_k},a_{i_k}^*))=0$$ whenever $i_1\neq i_2, i_2\neq i_3, \ldots, i_{k-1}\neq i_k$, $(i_1,\ldots i_k)\in I^k$, and $\tau(p_l(a_{i_l},a_{i_l}^*))=0$ for $l=1,\ldots,k$.
- A noncommutative random variable $x$ in a ${\cal C}^*$-probability space $\left({\cal A}, \tau\right)$ is a standard semicircular random variable if $x=x^*$ and for any $k\in \mathbb{N}$, $$\tau(x^k)= \int t^k d\mu_{sc}(t)$$ where $d\mu_{sc}(t)=
\frac{1}{2\pi} \sqrt{4-t^2}{{1\!\!{\sf I}}}_{[-2;2]}(t) dt$ is the semicircular standard distribution.
- Let $k$ be a nonnull integer number. Denote by ${\cal P}$ the set of polynomials in $2k $ noncommutative indeterminates. A sequence of families of variables $ (a_n)_{n\geq 1} =
(a_1(n),\ldots, a_k(n))_{n\geq 1}$ in $C^* $-probability spaces $\left({\cal A}_n, \tau_n\right)$ converges in $\star$-distribution, when n goes to infinity, to some $k$-tuple of noncommutative random variables $a=(a_1,\ldots,a_k)$ in a ${\cal C}^*$-probability space $\left({\cal A}, \tau\right)$ if the map $P\in {\cal P} \mapsto
\tau_n(
P(a_n,a_n^*))$ converges pointwise towards $P\in {\cal P} \mapsto\tau(
P(a,a^*))$.
- $k$ noncommutative random variables $a_1(n),\ldots, a_k(n)$, in $C^* $-probability spaces $\left({\cal A}_n, \tau_n\right)$, ${n\geq 1}$, are said asymptotically free if $(a_1(n),\ldots, a_k(n))$ converges in $\star$-distribution, as n goes to infinity, to some noncommutative random variables $(a_1,\ldots,a_k)$ in a ${\cal C}^*$-probability space $\left({\cal A}, \tau\right)$ where $a_1, \ldots,a_k$ are free.
We will also use the following well known result on asymptotic freeness of random matrices. Let ${\cal A}_n$ be the algebra of $n\times n$ matrices with complex entries and endow this algebra with the normalized trace defined for any $M\in {\cal A}_n$ by $\tau_n(M) =\frac{1}{n}{{\rm Tr}}(M)$. Let us consider a $n\times n$ so-called standard G.U.E matrix, i.e a random Hermitian matrix ${\cal G}_n = [{\cal G}_{jk}]_{j,k=1}^n$, where ${\cal G}_{ii}$, $\sqrt{2} \Re e({\cal G}_{ij})$, $\sqrt{2} \Im m({\cal G}_{ij})$, ${i<j}$ are independent centered Gaussian random variables with variance $1$. For a fixed real number $t$ independent from n, let $H_n^{(1)}, \ldots, H_n^{(t)}$ be deterministic $n\times n$ Hermitian matrices such that $\max_{i=1}^t\sup_n \Vert H_n^{(i)} \Vert < +\infty$ and $(H_n^{(1)}, \ldots, H_n^{(t)})$, as a t-tuple of noncommutative random variables in $({\cal A}_n, \tau_n)$, converges in distribution when n goes to infinity. Then, according to Theorem 5.4.5 in [@AGZ09], $ \frac{{\cal G}_n}{\sqrt{n}}$ and $(H_n^{(1)}, \ldots, H_n^{(t)})$ are almost surely asymptotically free i.e almost surely, for any polynomial P in t+1 noncommutative indeterminates, $$\label{sansD}\tau_n\left\{ P\left({ H_n^{(1)}},\ldots,{ H_n^{(t)}},\frac{{\cal G}_n}{\sqrt{n}}\right)\right\} \rightarrow_{n\rightarrow +\infty} \tau \left( P(h_1,\ldots,h_t,s)\right)$$ where $h_1,\ldots,h_r$ and $s$ are noncommutative random variables in some ${\cal C}^*$-probability space $({\cal A}, \tau)$ such that $(h_1,\ldots,h_r)$ and $s$ are free, $s$ is a standard semi-circular noncommutative random variable and the distribution of $(h_1,\ldots,h_t)$ is the limiting distribution of $(H_n^{(1)}, \ldots, H_n^{(t)})$.\
Finally, the proof of Theorem \[pasde\] is based on the following result which can be established by following the proof of Theorem 1.1 in [@BC]. First, note that the algebra of polynomials in non-commuting indeterminates $X_1,\ldots, X_k$, becomes a $\star$-algebra by anti-linear extension of $(X_{i_1}X_{i_2}\ldots X_{i_m})^*=X_{i_m}\ldots X_{i_2}X_{i_1}$.
\[noeigenvalue\] Let us consider three independent infinite arrays of random variables, $ [W^{(1)}_{ij}]_{i\geq1,j\geq 1}$, $ [W^{(2)}_{ij}]_{i\geq1,j\geq 1}$ and $ [X_{ij}]_{i\geq1,j\geq 1}$ where
- for $l=1,2$, $W^{(l)}_{ii}$, $\sqrt{2}Re(W^{(l)}_{ij})$, $\sqrt{2} Im(W^{(l)}_{ij}), i<j$, are i.i.d centered and bounded random variables with variance 1 and $W^{(l)}_{ji}=\overline{W^{(l)}_{ij}}$,
- $\{\Re (X_{ij}), \Im (X_{ij}), i\in \mathbb{N}, j \in \mathbb{N}\}$ are independent centered random variables with variance $1/2$ and satisfy and .
For any $(N,n)\in \mathbb{N}^2$, define the $(n+N)\times (n+N) $ matrix: $$\label{Wigner}W_{n+N}=\begin{pmatrix} W_n^{(1)} & X_N \\ X_N^* & W_N^{(2)} \end{pmatrix}$$ where $X_N=[X_{ij}]_{\tiny \begin{array}{ll}1\leq i\leq n\\1\leq j\leq N\end{array}}, \; W^{(1)}_n= [W^{(1)}_{ij}]_{1\leq i,j\leq n},\;
W^{(2)}_N= [W^{(2)}_{ij}]_{1\leq i,j\leq N}$.\
Assume that $n=n(N)$ and $\lim_{N\rightarrow +\infty}\frac{n}{N}=c \in ]0,1].$\
Let $t$ be a fixed integer number and $P$ be a selfadjoint polynomial in $t+1$ noncommutative indeterminates.\
For any $N \in \mathbb{N}^2$, let $(B_{n+N}^{(1)},\ldots,B_{n+N}^{(t)})$ be a $t-$tuple of $(n+N)\times (n+N)$ deterministic Hermitian matrices such that for any $u=1,\ldots,t$, $ \sup_{N} \Vert B_{n+N}^{(u)} \Vert< \infty$. Let $({\cal A}, \tau)$ be a $C^*$-probability space equipped with a faithful tracial state and $s$ be a standard semi-circular noncommutative random variable in $({\cal A}, \tau)$. Let $b_{n+N}=(b_{n+N}^{(1)},\ldots,b_{n+N}^{(t)})$ be a t-tuple of noncommutative selfadjoint random variables which is free from $s$ in $({\cal A},\tau)$ and such that the distribution of $b_{n+N}$ in $({\cal A},\tau)$ coincides with the distribution of $(B_{n+N}^{(1)},\ldots, B_{n+N}^{(t)})$ in $({ M}_{n+N}(\mathbb{C}), \frac{1}{n+N}{{\rm Tr}})$.\
Let $[x,y]$ be a real interval such that there exists $\delta>0$ such that, for any large $N$, $[x-\delta,y+\delta]$ lies outside the support of the distribution of the noncommutative random variable $ P\left(s, b_{n+N}^{(1)},\ldots,b_{n+N}^{(t)}\right)$ in $({\cal A},\tau)$. Then, almost surely, for all large N, $$\rm{spect}P\left(\frac{{ W}_{n+N}}{\sqrt{n+N}}, B_{n+N}^{(1)},\ldots,B_{n+N}^{(t)})\right) \subset {\mathbb{R}}\setminus [x,y].$$
We start by checking that a truncation and Gaussian convolution procedure as in Section 2 of [@BC] can be handled for such a matrix as defined by , to reduce the problem to a fit framework where,
- for any $N$, $(W_{n+N})_{ii}$, $\sqrt{2}Re((W_{n+N})_{ij})$, $\sqrt{2} Im((W_{n+N})_{ij}), i<j, i \leq n+N,\; j\leq n+N$, are independent, centered random variables with variance 1, which satisfy a Poincaré inequality with common fixed constant $C_{PI}$.
Note that, according to Corollary 3.2 in [@Ledoux01], (H) implies that for any $p\in \mathbb{N}$, $$\label{moments} \sup_{N\geq 1} \sup_{1\leq i,j\leq n+N} \mathbb{E}\left(\vert (W_{n+N})_{ij}\vert^p\right) <+\infty.$$
\[2.1\] Following the proof of Lemma 2.1 in [@BC], one can establish that, if $(V_{ij})_{i\geq 1, j\geq 1}$ is an infinite array of random variables such that $\{\Re (V_{ij}), \Im (V_{ij}), i\in \mathbb{N}, j \in \mathbb{N}\}$ are independent centered random variables which satisfy and , then almost surely we have $$\limsup_{N\rightarrow +\infty} \left\| \frac{Z_{n+N}}{\sqrt{N+n}}\right\| \leq 2\sigma^*$$ where $$Z_{n+N}=\begin{pmatrix} (0) & V_N \\ V_N^* & (0) \end{pmatrix} \; \mbox{ with}\; V_N=[V_{ij}]_{\tiny \begin{array}{ll}1\leq i\leq n\\1\leq j\leq N\end{array}} \mbox{and}\; \sigma^*=\left\{\sup_{(i,j)\in \mathbb{N}^2}\mathbb{E}(\vert V_{ij}\vert^2)\right\}^{1/2}.$$
Then, following the rest of the proof of Section 2 in [@BC], one can prove that for any polynomial $P$ in $1+t$ noncommutative variables, there exists some constant $L>0$ such that the following holds. Set $\theta^*=\sup_{i,j}\mathbb{E}\left(\left|X_{ij}\right|^3\right)$. For any $0<\epsilon<1$, there exist $C_\epsilon>8\theta^* $ (such that $C_\epsilon >\max_{l=1,2} \vert W^{(l)}_{11}\vert $ a.s.) and $\delta_\epsilon>0$ such that almost surely for all large $N$, $$\label{fit} \left\| P\left(\frac{W_{n+N}}{\sqrt{n+N}},B_{n+N}^{(1)},\ldots,B_{n+N}^{(t)}\right)- P\left(\frac{\tilde W_{n+N}^{C_\epsilon,\delta_\epsilon}}{\sqrt{n+N}}, B_{n+N}^{(1)},\ldots,B_{n+N}^{(t)}\right)\right\|\leq L \epsilon,$$ where, for any $C>8 \theta^*$ such that $C>\max_{l=1,2} \vert W^{(l)}_{11}\vert $ a.s., and for any $\delta>0$, $\tilde W_{N+n}^{C,\delta}$ is a $(n+N)\times (n+N) $ matrix which is defined as follows. Let $({\cal G}_{ij})_{i\geq 1, j\geq 1}$ be an infinite array which is independent of $\{X_{ij}, W^{(1)}_{ij}, W^{(2)}_{ij}, (i,j)\in \mathbb{N}^2\}$ and such that $\sqrt{2} \Re e{\cal G}_{ij}$, $ \sqrt{2} \Im m{\cal G}_{ij}$, $i<j$, ${\cal G}_{ii}$, are independent centred standard real gaussian variables and ${\cal G}_{ij}=\overline{\cal G}_{ji}$. Set ${\cal G}_{n+N}= [{\cal G}_{ij}]_{1\leq i,j \leq n+N }$ and define $X_N^C=[X_{ij}^C]_{\tiny \begin{array}{ll}1\leq i\leq n\\1\leq j\leq N\end{array}}$ as in . Set $$\tilde W_{n+N}^C=\begin{pmatrix} W_n^{(1)} & X_N^C \\ (X_N^C)^* & W_N^{(2)} \end{pmatrix}\; \mbox{and} \; \tilde W_{N+n}^{C,\delta}= \frac{ \tilde W_{n+N}^C +\delta {\cal G}_{n+N}}{\sqrt{1+\delta^2}}.$$ $\tilde W_{N+n}^{C,\delta}$ satisfies (H) (see the end of Section 2 in [@BC]). readily yields that it is sufficient to prove Theorem \[noeigenvalue\] for $\tilde W_{N+n}^{C,\delta}$.\
Therefore, assume now that $W_{N+n}$ satisfies (H). As explained in Section 6.2 in [@BC], to establish Theorem \[noeigenvalue\], it is sufficient to prove that for all $m \in \mathbb{N}$, all self-adjoint matrices $\gamma, \alpha, \beta_1, \ldots, \beta_t$ of size $m\times m$ and all $\epsilon >0$, almost surely, for all large $N$, we have\
$
spect(\gamma \otimes I_{n+N} + \alpha\otimes \frac{W_{n+N}}{\sqrt{n+N}}+ \sum_{u=1}^t \beta_u \otimes B_{n+N}^{(u)})$ $$\label{spectre3} \subset
spect(\gamma \otimes 1_{\cal A} + \alpha \otimes s+ \sum_{u=1}^t \beta_u \otimes b_{n+N}^{(u)}) + ]-\epsilon, \epsilon[.$$ ( is the analog of Lemma 1.3 for $r=1$ in [@BC]). Finally, one can prove by following Section 5 in [@BC].
We will need the following lemma in the proof of Theorem \[pasde\].
\[interpretation\] Let $A_N$ and $c_N$ be defined as in Theorem \[pasde\]. Define the following $(n+N)\times (n+N) $ matrices: $P=\begin{pmatrix} I_n & (0) \\ (0) & (0) \end{pmatrix}$ and $Q=\begin{pmatrix} (0) & (0) \\ (0) & I_N\end{pmatrix}$ and ${\bf A}=\begin{pmatrix} (0) & A_N \\ (0) & (0) \end{pmatrix}$. Let $s,p_N,q_N, {\bf a}_N$ be noncommutative random variables in some $\mathcal{C}^*$-probability space $\left( {\cal A}, \tau\right)$ such that $s$ is a standard semi-circular variable which is free with $(p_N,q_N, {\bf a}_N)$ and the $\star$-distribution of $({\bf A},P,Q)$ in $\left(M_{N+n}(\mathbb{C}),\frac{1}{N+n} {{\rm Tr}}\right)$ coincides with the $\star$-distribution of $({\bf a}_N, p_N,q_N)$ in $\left( {\cal A}, \tau\right). $ Then, for any $\epsilon \geq 0$, the distribution of $ ({\sqrt{1+c_N}}\sigma p_N s q_N+ {\sqrt{1+c_N}}\sigma q_N s p_N + {\bf a}_N+ {\bf a}_N^*)^2 +\epsilon p_N$ is $\frac{n}{N+n} T_\epsilon \star \mu_{\sigma, \mu_{A_NA_N^*}, c_N} +\frac{n}{N+n} \mu_{\sigma, \mu_{A_NA_N^*}, c_N}+\frac{N-n}{N+n} \delta_{0}$ where ${T_\epsilon} {\star} \mu_{\sigma, \mu_{A_NA_N^*}, c_N}$ is the pushforward of $ \mu_{\sigma, \mu_{A_NA_N^*}, c_N}$ by the map $z\mapsto z+\epsilon$.
[*Here $N$ and $n$ are fixed*]{}. Let $k\geq 1$ and $C_k$ be the $k\times k$ matrix defined by $$C_k= \begin{pmatrix} \hspace*{-0.4cm} (0)&{1} \\ \hspace*{-0.1cm} \; \; \; \; \; \; \;\mbox{ \reflectbox{$\ddots$}} \\ \hspace*{-0.4cm}{1} & (0)\end{pmatrix}.$$ Define the $k(n+N)\times k(n+N)$ matrices $$\hat A_k= C_k\otimes {\bf A},\; \hat P_k=I_k\otimes P, \; \hat Q_k= I_k\otimes Q.$$ For any $k \geq 1$, the $\star$-distributions of $(\hat A_k, \hat P_k, \hat Q_k)$ in $( M_{k(N+n)}(\mathbb{C}), \frac{1}{k(N+n)}{{\rm Tr}})$ and $({\bf A}, P, Q)$ in $( M_{(N+n)}(\mathbb{C}), \frac{1}{(N+n)}{{\rm Tr}})$ respectively, coincide. Indeed, let ${\cal K}$ be a noncommutative monomial in $\mathbb{C}\langle X_1,X_2,X_3,X_4\rangle$ and denote by $q$ the total number of occurrences of $X_3$ and $X_4$ in ${\cal K}$. We have $${\cal K}(\hat P_k, \hat Q_k, \hat A_k, \hat A_k^*)=C_k^q \otimes {\cal K}(P,Q,{\bf A}, {\bf A}^*),$$ so that $$\frac{1}{k(n+N)} {{\rm Tr}}\left[{\cal K}(\hat P_k, \hat Q_k, \hat A_k, \hat A_k^*)\right]= \frac{1}{k}{{\rm Tr}}(C_k^q) \frac{1}{(n+N)}{{\rm Tr}}\left[{\cal K}(P,Q,{\bf A}, {\bf A}^*)\right].$$ Note that if $q$ is even then $C_k^q=I_k$ so that $$\label{egalite}\frac{1}{k(n+N)} {{\rm Tr}}\left[{\cal K}(\hat P_k, \hat Q_k, \hat A_k, \hat A_k^*)\right]=\frac{1}{(n+N)}{{\rm Tr}}\left[ {\cal K}(P,Q,{\bf A}, {\bf A}^*)\right].$$ Now, assume that $q$ is odd. Note that $PQ=QP=0, \;{\bf A}Q={\bf A}, \; Q{\bf A}=0, \; {\bf A}P=0$ and $P{\bf A}={\bf A}$ (and then $Q{\bf A}^*={\bf A}^*, \; {\bf A}^*Q=0,\, P{\bf A}^*=0$ and ${\bf A}^* P={\bf A}^*$). Therefore, if at least one of the terms $X_1 X_2$, $X_2 X_1$, $X_2 X_3$, $ X_3 X_1$, $X_4 X_2$ or $X_1 X_4$ appears in the noncommutative product in ${\cal K}$, then $ {\cal K}(P,Q,{\bf A}, {\bf A}^*)=0,$ so that still holds. Now, if none of the terms $X_1 X_2$, $X_2 X_1$, $X_2 X_3$, $ X_3 X_1$, $X_4 X_2$ or $X_1 X_4$ appears in the noncommutative product in ${\cal K}$, then we have ${\cal K}(P,Q,{\bf A}, {\bf A}^*)=\tilde {\cal K}({\bf A}, {\bf A}^*)$ for some noncommutative monomial $\tilde {\cal K}\in \mathbb{C} \langle X,Y\rangle $ with degree $q$. Either the noncommutative product in $\tilde {\cal K}$ contains a term such as $X^p$ or $Y^p$ for some $p \geq 2$ and then, since ${\bf A}^2= ({\bf A}^*)^2=0$, we have $\tilde {\cal K}({\bf A}, {\bf A}^*)=0$, or $\tilde {\cal K}(X,Y)$ is one of the monomials $ (XY)^{\frac{q-1}{2}}X$ or $Y(XY)^{\frac{q-1}{2}}$. In both cases, we have ${{\rm Tr}}\tilde {\cal K}({\bf A}, {\bf A}^*)=0$ and still holds.\
Now, define the $k(N+n)\times k(N+n)$ matrices $$\tilde P_k=\begin{pmatrix} I_{kn} & (0) \\ (0) & (0) \end{pmatrix}, \; \;\tilde Q_k= \begin{pmatrix} (0) & (0) \\ (0) & I_{kN}\end{pmatrix},\;
$$ A\_k=
\(0) &\
(0) & (0)
$$ where $\check{ A}$ is the $kn\times kN$ matrix defined by $$\check{ A}=\begin{pmatrix} (0)& A_N \\ \; \; \; \; \; \; \;\mbox{ \reflectbox{$\ddots$}} \\ A_N & (0)\end{pmatrix}.$$ It is clear that there exists a real orthogonal $k(N+n)\times k(N+n)$ matrix $O$ such that $\tilde P_k=O\hat P_k O^*$, $\tilde Q_k=O\hat Q_k O^*$ and $\tilde A_k=O\hat A_k O^*$. This readily yields that the noncommutative $\star$-distributions of $(\hat A_k, \hat P_k, \hat Q_k)$ and $({\tilde A}_k, \tilde P_k, \tilde Q_k)$ in $( M_{k(N+n)}(\mathbb{C}), \frac{1}{k(N+n)}{{\rm Tr}})$ coincide. Hence, for any $k\geq 1$, the distribution of $({\tilde A}_k, \tilde P_k, \tilde Q_k)$ in $( M_{k(N+n)}(\mathbb{C}), \frac{1}{k(N+n)}{{\rm Tr}})$ coincides with the distribution of $({\bf a}_N,p_N,q_N)$ in $\left( {\cal A}, \tau\right). $ By Theorem 5.4.5 in [@AGZ09], it readily follows that the distribution of $({\sqrt{1+c_N}}\sigma p_N s q_N+ {\sqrt{1+c_N}}\sigma q_N s p_N + {\bf a}_N+ {\bf a}_N^*)^2 +\epsilon p_N$ is the almost sure limiting distribution, when $k$ goes to infinity, of $({\sqrt{1+c_N}}\sigma \tilde P_k\frac{{\cal G} }{\sqrt{k(N+n)}}\tilde Q_k+ {\sqrt{1+c_N}} \sigma \tilde Q_k \frac{{\cal G} }{\sqrt{k(N+n)}}\tilde P_k+\tilde A_k+\tilde A_k^*)^2+\epsilon \tilde P_k$ in $( M_{k(N+n)}(\mathbb{C}), \frac{1}{k(N+n)}{{\rm Tr}})$, where ${\cal G}$ is a $k(N+n)\times k(N+n)$ GUE matrix with entries with variance 1. Now, note that $$\left[{\sqrt{1+c_N}}\sigma\left\{\tilde P_k \frac{{\cal G} }{\sqrt{k(N+n)}} \tilde Q_k
+\tilde Q_k \frac{{\cal G} }{\sqrt{k(N+n)}}\tilde P_k\right\} +\tilde A_k +\tilde A_k^*\right]^2 +\epsilon \tilde P_k$$ $$=
\begin{pmatrix} (\sigma \frac{{\cal G}_{kn\times kN}}{\sqrt{kN}}+\check{ A})(\sigma \frac{{\cal G}_{kn\times kN}}{\sqrt{kN}}+\check{ A})^*+\epsilon I_{kn} &(0)\\ (0)& (\sigma \frac{{\cal G}_{kn\times kN}}{\sqrt{kN}}+\check{A})^*(\sigma \frac{{\cal G}_{kn\times kN}}{\sqrt{kN}}+\check{A})
\end{pmatrix}$$ where ${\cal G}_{kn\times kN}$ is the upper right $kn\times kN$ corner of $ {\cal G}$. Thus, noticing that $\mu_{\check{ A}\check{ A}*}=\mu_{A_NA_N^*}$, the lemma follows from [@DozierSilver].
[*Proof of Theorem \[pasde\]*]{}. Let $W$ be a $(n+N)\times (n+N) $ matrix as defined by in Theorem \[noeigenvalue\]. Note that, with the notations of Lemma \[interpretation\], for any $\epsilon\geq 0$,\
$
\begin{pmatrix} (\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac{X_N}{\sqrt{N}}+A_N)^*+\epsilon I_n &(0)\\ (0)& (\sigma \frac{X_N}{\sqrt{N}}+A_N)^*(\sigma \frac{X_N}{\sqrt{N}}+A_N)
\end{pmatrix}$ $$\begin{aligned}
&=& \begin{pmatrix} (0)&(\sigma \frac{X_N}{\sqrt{N}}+A_N) \\(\sigma \frac{X_N}{\sqrt{N}}+A_N)^* &(0)
\end{pmatrix}^2 +\epsilon P
\\
&=& \left({\sqrt{1+c_N}}P\frac{\sigma W}{\sqrt{N+n}}Q+ {\sqrt{1+c_N}}Q\frac{\sigma W}{\sqrt{N+n}}P+{\bf A}+{\bf A}^*\right)^2+\epsilon P.\end{aligned}$$ Thus, for any $\epsilon \geq 0$,\
$ \rm{spect}\left\{(\sigma \frac{X_N}{\sqrt{N}}+A)(\sigma \frac{X_N}{\sqrt{N}}+A)^*+\epsilon I_n\right\}$ $$\label{inclusion}\subset \rm{spect}\left\{\left({\sqrt{1+c_N}}P\frac{\sigma W}{\sqrt{N+n}}Q+ {\sqrt{1+c_N}}Q\frac{\sigma W}{\sqrt{N+n}}P+{\bf A}+{\bf A}^*\right)^2+\epsilon P\right\}.$$ Let $[x,y] $ be such that there exists $\delta>0$ such that for all large $N$, $ ]x-\delta; y+\delta[ \subset \mathbb{R}\setminus \rm{supp} (\mu_{\sigma,\mu _{A_N A_N^*},c_N})$.
- Assume $x>0.$ Then, according to Lemma \[interpretation\] with $\epsilon=0$, there exists $\delta'>0$ such that for all large $n$, $ ]x-\delta'; y+\delta'[$ is outside the support of the distribution of $ ({\sqrt{1+c_N}}\sigma p_N s q_N+ {\sqrt{1+c_N}}\sigma q_N s p_N + {\bf a}_N+ {\bf a}_N^*)^2 $. We readily deduce that almost surely for all large N, according to Theorem \[noeigenvalue\], there is no eigenvalue of $({\sqrt{1+c_N}}P\frac{\sigma W}{\sqrt{N+n}}Q+ {\sqrt{1+c_N}}Q\frac{\sigma W}{\sqrt{N+n}}P+{\bf A}+{\bf A}^*)^2 $ in $[x,y]$. Hence, by with $\epsilon=0$, almost surely for all large N, there is no eigenvalue of $M_N$ in $[x,y].$
- Assume $x= 0$ and $y > 0$. There exists $0<\delta'<y$ such that $[0,3\delta']$ is for all large $N$ outside the support of $\mu_{\sigma, \mu_{A_NA_N^*}, c_N}$. Hence, according to Lemma \[interpretation\], $[\delta'/2,3\delta']$ is outside the support of the distribution of $ ({\sqrt{1+c_N}}\sigma p_N s q_N+ {\sqrt{1+c_N}}\sigma q_N s p_N + {\bf a}_N+ {\bf a}_N^*)^2 +\delta' p_N$. Then, almost surely for all large N, according to Theorem \[noeigenvalue\], there is no eigenvalue of $({\sqrt{1+c_N}}P\frac{\sigma W}{\sqrt{N+n}}Q+ {\sqrt{1+c_N}}Q\frac{\sigma W}{\sqrt{N+n}}P+{\bf A}+{\bf A}^*)^2 +\delta' P $ in $[\delta',2\delta']$ and thus, by , no eigenvalue of $ (\sigma \frac{X}{\sqrt{N}}+A_N)(\sigma \frac{X_N}{\sqrt{N}}+A_N)^*+\delta' I_n$ in $[\delta',2\delta']$. It readily follows that, almost surely for all large N, there is no eigenvalue of $ (\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac{X_N}{\sqrt{N}}+A_N)^*$ in $[0,\delta']$. Since moreover, according to (i), almost surely for all large N, there is no eigenvalue of $ (\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac{X_N}{\sqrt{N}}+A_N)^*$ in $[\delta',y]$, we can conclude that there is no eigenvalue of $M_N$ in $[x,y]$.
The proof of Theorem \[pasde\] is now complete. $\Box$\
We are now in a position to establish the following exact separation phenomenon.
\[sep\] Let $M_n$ as in with assumptions \[1-4\] of Theorem \[pasde\]. Assume moreover that the empirical spectral measure $\mu_{A_NA_N^*}$ of $A_NA_N^*$ converges weakly to some probability measure $\nu$. Then for $N$ large enough, $$\label{lemme31}\omega_{{\sigma,\nu,c}}([x,y])=[\omega_{{\sigma,\nu,c}}(x);\omega_{{\sigma,\nu,c}}(y)] \subset \mathbb{R} \setminus \mbox{supp}(\mu _{A_N A_N^*}),$$ where $\omega_{\sigma,\nu,c}$ is defined in . With the convention that $\lambda _0(M_N)=\lambda _0(A_NA_N^*)=+\infty $ and $\lambda _{n+1}(M_N)=\lambda _{n+1}(A_NA_N^*)=-\infty $, for $N$ large enough, let $i_N\in \{0,\ldots,n\}$ be such that $$\label{iN2}\lambda_{i_N+1}(A_N A_N^*) <\omega_{{\sigma,\nu,c}}(x) \mbox{~~ and ~~} \lambda_{i_N}(A_N A_N^*) > \omega_{{\sigma,\nu ,c}}(y).$$ Then $$\label{sepeq}P[\mbox{for all large N}, \lambda_{i_N+1}(M_N) <x\mbox{~and} ~ \lambda_{i_N}(M_N)>
y] = 1.$$
\[incl\] Since $\mu_{\sigma,\mu _{A_N A_N^*},c_N}$ converges weakly towards $\mu_{\sigma,\nu,c}$ assumption 4. implies that $\forall 0< \tau< \delta$, $[x-\tau; y+\tau] \subset \mathbb{R} \setminus \rm{supp}~ \mu_{\sigma,\nu,c}$.
is proved in Lemma 3.1 in [@MC2014].
- If $\omega_{{\sigma,\nu,c}}(x)< 0$, then $i_N=n $ in and moreover we have, for all large N, $\omega_{{\sigma,\mu _{A_N A_N^*},c_N}}(x)<0$. According to Lemma 2.7 in [@MC2014], we can deduce that, for all large $N$, $[x,y] $ is on the left hand side of the support of $\mu_{\sigma,\mu _{A_N A_N^*},c_N}$ so that $]-\infty; y+\delta]$ is on the left hand side of the support of $\mu_{\sigma,\mu _{A_N A_N^*},c_N}$. Since $[-\vert y \vert -1,y]$ satifies the assumptions of Theorem \[pasde\], we readily deduce that almost surely, for all large $N$, $\lambda_{n}(M_N)> y.$ Hence holds true.
- If $\omega_{{\sigma,\nu,c}}(x)\geq 0$, we first explain why it is sufficient to prove for $x$ such that $\omega_{{\sigma,\nu,c}}(x)>0.$ Indeed, assume for a while that is true whenever $\omega_{{\sigma,\nu,c}}(x)>0$. Let us consider any interval $[x,y]$ satisfying condition 4. of Theorem \[pasde\] and such that $\omega_{{\sigma,\nu,c}}(x)= 0$; then $i_N=n $ in . According to Proposition \[caractfinale\], $\omega_{{\sigma,\nu,c}}(\frac{x+y}{2})> 0$ and then almost surely for all large N, $\lambda_n (M_N)>y.$ Finally, sticking to the proof of Theorem 1.2 in [@MC2014] leads to for $x$ such that $\omega_{{\sigma,\nu,c}}(x)>0.$
Appendix B {#appendix-b .unnumbered}
==========
We first recall some basic properties of the resolvent (see [@KKP96], [@CD07]).
\[lem0\] For a $N \times N$ Hermitian matrix $M$, for any $z \in {\mathbb{C}}\setminus {\rm spect}(M)$, we denote by $G(z) := (zI_N-M)^{-1}$ the resolvent of $M$.\
Let $z \in {\mathbb{C}}\setminus {\mathbb{R}}$,
- $\Vert G(z) \Vert \leq |\Im z|^{-1}$.
- $\vert G(z)_{ij} \vert \leq |\Im z|^{-1}$ for all $i,j = 1, \ldots , N$.
- $G(z)M=MG(z) =-I_N +zG(z)$.
Moreover, for any $N \times N$ Hermitian matrices $M_1$ and $ M_2$, $$(zI_N-M_1)^{-1}-(zI_N-M_2)^{-1}=(zI_N-M_1)^{-1}(M_1-M_2)(zI_N-M_2)^{-1}.$$
The following technical lemmas are fundamental in the approach of the present paper.
\[approxpoisson\]\[Lemma 4.4 in [@BBCF15]\] Let $h: \mathbb{R}\rightarrow \mathbb{R}$ be a continuous function with compact support. Let $B_N$ be a $N\times N $ Hermitian matrix and $C_N$ be a $N\times N $ matrix . Then $$\label{sansE}{{\rm Tr}}\left[h(B_N) C_N\right]= - \lim_{y\rightarrow 0^{+}}\frac{1}{\pi} \int \Im {{\rm Tr}}\left[(t+iy-B_N)^{-1}C_N\right] h(t) dt.$$ Moreover, if $B_N$ is random, we also have $$\label{avecE}\mathbb{E}{{\rm Tr}}\left[h(B_N) C_N\right]= - \lim_{y\rightarrow 0^{+}}\frac{1}{\pi} \int \Im \mathbb{E}{{\rm Tr}}\left[(t+iy-B_N)^{-1}C_N\right] h(t) dt.$$
\[HT\] Let $f$ be an analytic function on ${\mathbb{C}}\setminus {\mathbb{R}}$ such that there exist some polynomial $P$ with nonnegative coefficients, and some positive real number $\alpha$ such that $$\label{nestimgdif}
\forall z \in \mathbb{C}\setminus \mathbb{R},~~\vert f(z)\vert \leq (\vert z\vert +1)^\alpha P(\vert \Im z\vert ^{-1}).$$ Then, for any $h$ in $\cal C^\infty ({\mathbb{R}}, {\mathbb{R}})$ with compact support, there exists some constant $\tau$ depending only on $h$, $ \alpha$ and $P$ such that $$\limsup _{y\rightarrow 0^+}\vert \int _{\mathbb{R}}h (x)f(x+iy)dx\vert < \tau.$$
We refer the reader to the Appendix of [@CD07] where it is proved using the ideas of [@HaaThor05].\
Finally, we recall some facts on Poincaré inequality. A probability measure $\mu$ on $\mathbb{R}$ is said to satisfy the Poincar' e inequality with constant $C_{PI}$ if for any ${\cal C}^1$ function $f: {\mathbb{R}}\rightarrow {\mathbb{C}}$ such that $f$ and $f' $ are in $L^2(\mu)$, $$\mathbf{V}(f)\leq C_{PI}\int \vert f' \vert^2 d\mu ,$$ with $\mathbf{V}(f) = \int \vert
f-\int f d\mu \vert^2 d\mu$.\
We refer the reader to [@BobGot99] for a characterization of the measures on $\mathbb{R}$ which satisfy a Poincaré inequality.
If the law of a random variable $X$ satisfies the Poincaré inequality with constant $C_{PI}$ then, for any fixed $\alpha \neq 0$, the law of $\alpha X$ satisfies the Poincaré inequality with constant $\alpha^2 C_{PI}$.\
Assume that probability measures $\mu_1,\ldots,\mu_M$ on $\mathbb{R}$ satisfy the Poincaré inequality with constant $C_{PI}(1),\ldots,C_{PI}(M)$ respectively. Then the product measure $\mu_1\otimes \cdots \otimes \mu_M$ on $\mathbb{R}^M$ satisfies the Poincaré inequality with constant $\displaystyle{C_{PI}^*=\max_{i\in\{1,\ldots,M\}}C_{PI}(i)}$ in the sense that for any differentiable function $f$ such that $f$ and its gradient ${\rm grad} f$ are in $L^2(\mu_1\otimes \cdots \otimes \mu_M)$, $$\mathbf{V}(f)\leq C_{PI}^* \int \Vert {\rm grad} f \Vert_2 ^2 d\mu_1\otimes \cdots \otimes \mu_M$$ with $\mathbf{V}(f) = \int \vert
f-\int f d\mu_1\otimes \cdots \otimes \mu_M \vert^2 d\mu_1\otimes \cdots \otimes \mu_M$ (see Theorem 2.5 in [@GuZe03]) .
\[zitt\]\[Theorem 1.2 in [@BGMZ]\] Assume that the distribution of a random variable $X$ is supported in $[-C;C]$ for some constant $C>0$. Let $g$ be an independent standard real Gaussian random variable. Then $X+\delta g$ satisfies a Poincaré inequality with constant $C_{PI}\leq \delta^2 \exp \left( 4C^2/\delta^2\right)$.
[**Acknowlegements.**]{} The author is very grateful to Charles Bordenave and Serban Belinschi for several fruitful discussions and thanks Serban Belinschi for pointing out Lemma \[gnmoinsgmu\]. The author also wants to thank an anonymous referee who provided a much simpler proof of Lemma \[majpre\] and encouraged the author to establish the results for non diagonal perturbations, which led to an overall improvement of the paper.
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abstract: 'The effects of the string cloud on higher-dimensional holographic superconductors in Einstein-Gauss-Bonnet gravity are investigated in the probe limit. The critical temperature is analytically obtained using Sturm-Liouville eigenvalue method. It is observed that the critical temperature only exists in an allowed region of the parameter space. Also, the presence of the string cloud with the sufficiently large density should prevent the the existence of the critical temperature. As increasing the string cloud density parameter, the critical temperature decreases in the region of the sufficiently low charge density but increases in the region of the sufficiently high charge density. Whereas, the presence of Gauss-Bonnet terms always makes the critical temperature decreasing. In addition, the expression of the condensation operator and the critical exponent are computed analytically.'
author:
- 'Cao H. Nam'
title: 'Effects of string cloud on Gauss-Bonnet holographic superconductors'
---
Introduction
============
The AdS/CFT correspondence [@Maldacena], which relates a weakly coupling gravity theory in AdS spacetime to a strongly coupling field theory on the boundary of AdS spacetime, has attracted consideration attention because of its computational power. According this correspondence, one can calculate the quantities of the strongly coupling field theory by using the gravitational dual in one higher dimension. In recent years, the AdS/CFT correspondence has been used to study various phenomena in condensed matter physics [@Hartnoll2010]. In particular, much attention have been dedicated to apply the holographic method for superconductors because high temperature superconductors are in the strong coupling regime and thus can not be described by the BCS theory. The dual gravitational description of $s$-wave superconductors was first proposed by Hartnoll et al. at which a phase transition from a black hole with no hair to one with scalar hair is interpreted as the (normal) conductor/superconducting phase transition in the dual field theory [@Horowitz2008a; @Horowitz2008b]. Following this work, the investigations of holographic superconductors in the framework of Einstein gravity have attracted a lot of attention [@Rodriguez-Gomez2010; @Jing2010; @Therrien2010; @Yang2010; @Pan2011; @Zong2011; @Zhang2011; @GeLeng2012; @Gangopadhyay2012; @Jing2013; @Erdmenger2013; @Lai2015; @Ghorai2016; @Liu2016; @Zangeneh2016; @Sheykhi2016; @Salahi2016; @Ghazanfari2018; @Asl2018].
At short distances, it is expected that Einstein gravity would obtain the higher-order curvature corrections. Einstein-Gauss-Bonnet gravity is one of the natural modifications for Einstein gravity at short distances by including Gauss-Bonnet term which arises naturally from the low-energy limit of heterotic string theory [@Zwiebach1985; @Witten1986; @Gross1987]. Importantly, the presence of Gauss-Bonnet term does not lead to more than second derivatives of the metric in the corresponding field equations and thus the theory is ghost-free. Also, Gauss-Bonnet term only plays the role in spacetime with the dimension $d>4$. Whereas, for $d\leq4$, Gauss-Bonnet term is a topological invariant and thus it does not contribute to the field equations. Studies of holographic superconductors have been generalized in the framework of Einstein-Gauss-Bonnet gravity [@Gregory2009; @Pavan2010; @Wang2010; @Barclay2010; @Cai2011; @Kanno2010; @Kanno2011; @Pan-Chen2011; @Barclay2011; @Li-Zhang2011; @Wang2011; @Jing-Chen2012; @Cai-Zhang2013; @Cui-Xue2013].
In the string theory, the fundamental building blocks are the one-dimensional strings. Inspired by this, the concept of string cloud was introduced as the one-dimensional analogous of a dust cloud. The black hole solution with the source of the string cloud was first found by Letelier [@Letelier1979]. Later, various black hole solutions in the presence of string cloud and their thermodynamic properties have been studied in the literature [@Herscovich2010; @Baboolal2014].
Inspired by these ideals, in this paper we study holographic superconductors with involving simultaneously the higher-order curvature corrections and string cloud. In Sect. \[HDM\], we set up a gravitational dual model. The starting point is to construct a $d$-dimensional black brane solution sourced by a string cloud and negative cosmological constant in Einstein-Gauss-Bonnet gravity, which is the background geometry for studying holographic superconductors in the probe limit. The matter action and the corresponding equations of motion are also provided. In Sect. \[CTCD\], we obtain analytically the expression of the critical temperature in terms of the charge density, string cloud density parameter, Gauss-Bonnet coupling, and spacetime dimension. Furthermore, we obtain the expression of the condensation operator or order parameter near the critical temperature in Sect. \[CVCE\].
\[HDM\] Holographic dual model
==============================
In this section, we will build a holographic dual model which is used for the subsequent computation of critical phenomena. First let us introduce the action describing $d$-dimensional Einstein-Gauss-Bonnet gravity coupled to a electromagnetic field and scalar field surrounded by a string cloud in the AdS spacetime background as $$S=\frac{1}{2}\int
d^dx\sqrt{-g}\left[R+\frac{(d-1)(d-2)}{l^2}+\alpha\mathcal{L}_{GB}+\mathcal{L}_m\right]+I_{\text{string-cloud}},\label{EGB-ED-adS}$$ where $R$ is the scalar curvature of the spacetime, $l$ is the curvature radius of the AdS spacetime, and $\mathcal{L}_{GB}$ is the Gauss-Bonnet term given as $$\mathcal{L}_{GB}=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\lambda}R^{\mu\nu\rho\lambda},$$ and $\alpha$ is the Gauss-Bonnet coupling parameter. The matter term is given as $$\mathcal{L}_m=-\frac{F_{\mu\nu}F^{\mu\nu}}{4}-|(\nabla_\mu-iqA_\mu)\psi|^2-m^2|\psi|^2,$$ where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is the strength tensor of the electromagnetic field $A_\mu$, and $\psi$ refers to the scalar field of the charge $q$ and the mass $m$. $I_{\text{string-cloud}}$ is the action for the string cloud.
Black brane solution with source of string cloud
------------------------------------------------
First let us review briefly the model of the string cloud [@Letelier1979]. For a classical relativistic string moving in the spacetime, its dynamics is described by the Nambu-Goto action $I_{\text{NG}}$ given by $$I_{\text{NG}}=-T_p\int_{\Sigma}\sqrt{-\gamma}d\sigma^0d\sigma^1,$$ where $T_p$ is the tension of the string and $(\sigma^0,\sigma^1)$ are the coordinates parameterizing the worldsheet $\Sigma$, $\gamma$ is the determinant of the induced metric $\gamma_{ab}=g_{\mu\nu}\frac{\partial x^\mu}{\partial\sigma^a}\frac{\partial x^\nu}{\partial\sigma^b}$. The worldsheet $\Sigma$ can be described by a bivector of the form $$\Sigma^{\mu\nu}=\epsilon^{ab}\frac{\partial x^\mu}{\partial\sigma^a}\frac{\partial x^\nu}{\partial\sigma^b},$$ where $\epsilon^{ab}$ is the two-dimensional Levi-Civita tensor, $\epsilon^{00}=\epsilon^{11}=0$ and $\epsilon^{01}=-\epsilon^{10}=1$. This bivector satisfies the identities, $\Sigma^{\mu[\rho}\Sigma^{\lambda\nu]}=0$ and $\nabla_\mu\Sigma^{\mu[\rho}\Sigma^{\lambda\nu]}=0$ (where the square brackets refer to antisymmetrization in the closed indices), which by the Frobenius’s theorem it leads to a parameterized surface. Also, the bivector $\Sigma^{\mu\nu}$ satisfies another identity, $\Sigma^{\mu\rho}\Sigma_{\rho\lambda}\Sigma^{\lambda\nu}=\gamma\Sigma^{\mu\nu}$. In this description, the Lagrangian of the string is given by $$\mathcal{L}_{\text{str}}=-T_p\left(-\frac{\Sigma^{\mu\nu}\Sigma_{\mu\nu}}{2}\right)^{1/2}.$$ Since we can obtain the energy-momentum tensor for the string as $T^{\mu\nu}=-2\frac{\partial\mathcal{L}_{\text{str}}}{\partial g_{\mu\nu}}=T_p(-\gamma)^{-1/2}\Sigma^{\mu\sigma}{\Sigma_\sigma}^\nu$.
In this work, we consider the string cloud with the energy-momentum tensor given by $$T^{\mu\nu}=\rho\frac{\Sigma^{\mu\sigma}{\Sigma_\sigma}^\nu}{\sqrt{-\gamma}},$$ where $\rho$ is the density of the string cloud. For the spherically symmetric and static string cloud, the ansatz of the bivector $\Sigma^{\mu\nu}$ is given as $$\Sigma^{\mu\nu}=B(r)\left({\delta}^\mu_t{\delta}^\nu_r-{\delta}^\nu_t{\delta}^\mu_r\right).$$ As a result, one can find the non-zero components of the energy-momentum tensor for the string cloud as $${T^t}_t={T^r}_r=-\rho|B(r)|.$$ By using the relation $\partial_\mu\left(\sqrt{-g}\rho\Sigma^{\mu\nu}\right)=0$, we can obtain explicitly ${T^t}_t$ and ${T^r}_r$ as $${T^t}_t={T^r}_r=-\frac{a}{r^{d-2}},$$ where $a$ is a positive real constant.
Varying the action (\[EGB-ED-adS\]) with respect to the spacetime metric $g_{\mu\nu}$, we derive the equations of motion for $g_{\mu\nu}$ as $$\begin{aligned}
{G^\mu}_\nu+\alpha{H^\mu}_\nu-\frac{(d-1)(d-2)}{2l^2}{\delta^\mu}_\nu&=&{T^\mu}_\nu(\text{matter})+{T^\mu}_\nu(\text{string-cloud}),\label{Eeq}\end{aligned}$$ where $${H^\mu}_\nu=2\left(R{R^\mu}_\nu-2R^{\mu\sigma}R_{\sigma\nu}-2R^{\sigma\rho}{R^\mu}_{\sigma\nu\rho}+{R^\mu}_{\rho\sigma\lambda}{R_\nu}^{\rho\sigma\lambda}\right)-\frac{1}{2}{\delta^\mu}_\nu\mathcal{L}_{GB}.$$ In this work, we would like to investigate the role of the string cloud on holographic superconductors in Einstein-Gauss-Bonnet gravity. Thus, we shall consider the backreaction of the string cloud on the spacetime geometry. Whereas, the matter fields (the gauge field $A_\mu$ and the scalar field $\psi$) are considered in the probe limit which means that their backreaction or the matter term $\mathcal{L}_m$ on the spacetime geometry is ignored. In this sense, the spacetime geometry in Einstein-Gauss-Bonnet gravity is sourced by the negative cosmological constant and the string cloud. Whereas, the matter fields decouple to gravity.
Now, we find a planar Schwarzschild-AdS black hole solution given by ansatz as $$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2h_{ij}dx^idx^j,$$ where $h_{ij}dx^idx^j=dx^2_1+dx^2_2+...+dx^2_{d-2}$ is the line element of the $(d-2)$-dimensional planar hypersurface. The $(t,t)$ component of Einstein’s field equations leads to the equation for the function $f(r)$ as $$\frac{d-2}{2r^4}\left\{r^2(d-3)f+r^3f'-\widetilde{\alpha}f\left[(d-5)f+2rf'\right]\right\}-\frac{(d-1)(d-2)}{2l^2}=-\frac{a}{r^{d-2}},$$ where $\widetilde{\alpha}=\alpha(d-3)(d-4)$. By solving this equation, we can obtain the $$f(r)=\frac{r^2}{2\widetilde{\alpha}}\left(1-\sqrt{1-\frac{4\widetilde{\alpha}}{l^2}+\frac{4\widetilde{\alpha}m}{r^{d-1}}+\frac{8a\widetilde{\alpha}}{(d-2)r^{d-2}}}\right),$$ where the reduced mass $m$ is related the total mass $M$ of the black hole as $$M=\frac{(d-2)\omega_{d-2}}{16\pi}m,$$ with $\omega_{d-2}$ to be the surface area of the unit $(d-2)$-sphere. The function $f(r)$ can be expressed in another form as $$f(r)=\frac{r^2}{2\widetilde{\alpha}}\left(1-\sqrt{1-\frac{4\widetilde{\alpha}}{l^2}\left(1-\frac{r^{d-1}_+}{r^{d-1}}\right)+\frac{8a\widetilde{\alpha}}{(d-2)r^{d-2}}\left(1-\frac{r_+}{r}\right)}\right),$$ where $r_+$ is the event horizon radius. The asymptotic behavior of the function $f(r)$, corresponding to $r\rightarrow\infty$, is given by $$f(r)=\frac{r^2}{l^2_{\text{eff}}},$$ where $$l^2_{\text{eff}}=\frac{2\widetilde{\alpha}}{1-\sqrt{1-\frac{4\widetilde{\alpha}}{l^2}}},$$ is the effective AdS radius. Note that, for the well-defined theory, the condition $\widetilde{\alpha}\leq l^2/4$ must be satisfied. The Hawking temperature of the black hole is given by $$T_H=\frac{1}{4\pi}\left[\frac{(d-1)r_+}{l^2}-\frac{2a}{(d-2)r^{d-3}_+}\right],\label{bhtemp}$$ which is interpreted as the temperature of the dual field theory.
Basic setup for matter fields
-----------------------------
Varying the action (\[EGB-ED-adS\]) with respect to the electromagnetic field $A_\mu$ and the scalar field $\phi$, we obtain $$\begin{aligned}
\nabla^\nu F_{\mu\nu}+iq\left[\psi^*(\nabla_\mu-iqA_\mu)\psi-\psi(\nabla_\mu+iqA_\mu)\psi^*\right]&=&0,\nonumber\\
\left(\nabla_\mu-iqA_\mu\right)\left(\nabla^\mu-iqA^\mu\right)\psi-m^2\psi &=&0,\label{MFeq}\end{aligned}$$ In order to solve these equations, we adopt spherically-symmetric and static ansatz for $A_\mu$ and $\psi$ as $$A_\mu=\phi(r)\delta^t_\mu,\ \ \ \ \psi=\psi(r).$$ By substituting this ansatz into Eq. (\[MFeq\]), it leads to the equation for the functions $\phi(r)$ and $\psi(r)$ as $$\begin{aligned}
\phi''(r)+\frac{d-2}{r}\phi'(r)-\frac{2q^2\phi(r)}{f(r)}\psi^2(r)&=&0,\label{r-phi-Eq}\\
\psi''(r)+\left[\frac{f'(r)}{f(r)}+\frac{d-2}{r}\right]\psi'(r)+\left[\frac{q^2\phi^2(r)}{f^2(r)}-\frac{m^2}{f(r)}\right]\psi(r)&=&0,\label{r-psi-Eq}\end{aligned}$$ where the prime is denoted the derivative with respect to the coordinate $r$. The regularity requirement for the fields $A_\mu$ and $\psi$ at the event horizon leads to the boundary conditions, $\phi(r_+)=0$ and $\psi(r_+)=\frac{f'(r_+)\psi'(r_+)}{m^2}$. Also, the behavior of $\phi(r)$ and $\psi(r)$ near the AdS boundary ($r\rightarrow\infty$) is given by $$\begin{aligned}
\phi(r)&=&\mu-\frac{\rho}{r^{d-3}},\label{phi-asy-beh}\\
\psi(r)&=&\frac{\langle\mathcal{O}_-\rangle}{r^{\Delta_-}}+\frac{\langle\mathcal{O}_+\rangle}{r^{\Delta_+}},\end{aligned}$$ where $\mu$ and $\rho$ are the chemical potential and the charge density, respectively, and $\Delta_\pm=\left[(d-1)\pm\sqrt{(d-1)^2+4m^2l^2_{\text{eff}}}\right]/2$ are the conformal dimensions of $\langle\mathcal{O}_\pm\rangle$, respectively.[^1] According to AdS/CFT correspondence, either $\langle\mathcal{O}_-\rangle$ or $\langle\mathcal{O}_+\rangle$ is interpreted as the source and the other is interpreted as the expectation value of the condensation operator in the boundary field theory. In this work, we consider $\langle\mathcal{O}_+\rangle$ as the expectation value of the condensation operator. Whereas $\langle\mathcal{O}_-\rangle$ is considered as the source which is set to be zero because the $\mathrm{U}(1)$ symmetry is broken spontaneously.
Interestingly, Eqs. (\[r-phi-Eq\]) and (\[r-psi-Eq\]) possess the following scaling symmetries
1. $\phi\rightarrow\lambda_1\phi,\ \ \psi\rightarrow\lambda_1\psi,\ \ q\rightarrow\lambda^{-1}_1q$.
2. $\phi\rightarrow\lambda_2\phi,\ \ \psi\rightarrow\lambda^{1/2}_2\psi,\ \ a\rightarrow\lambda_2a,\ \ \widetilde{\alpha}\rightarrow\lambda^{-1}_2\widetilde{\alpha},\ \ l\rightarrow\lambda^{-1/2}_2l,\ \ m\rightarrow\lambda^{1/2}_2m$,
which can be used to set $q=l=1$.
With new coordinate $z=\frac{r_+}{r}$, Eqs. (\[r-phi-Eq\]) and (\[r-psi-Eq\]) are rewritten as $$\begin{aligned}
\phi''(z)+\frac{4-d}{z}\phi'(z)-\frac{2 r^2_+\phi(z)}{z^4f(z)}\psi^2(z)&=&0,\label{z-phi-Eq}\\
\psi''(z)+\left(\frac{f'(z)}{f(z)}+\frac{4-d}{z}\right)\psi'(z)+\frac{r^2_+}{z^4}\left(\frac{\phi^2(z)}{f^2(z)}-\frac{m^2}{f(z)}\right)\psi(z)&=&0,\label{z-psi-Eq}\end{aligned}$$ where the prime is denoted the derivative with respect to the coordinate $z$, and $$f(z)=\frac{r^2_+}{2\widetilde{\alpha}z^2}\left(1-\sqrt{1-4\widetilde{\alpha}\left(1-z^{d-1}\right)+\frac{8a\widetilde{\alpha}z^{d-2}}{(d-2)r^{d-2}_+}(1-z)}\right).$$
\[CTCD\] Critical temperature versus charge density
===================================================
In this section, we will obtain the expression for the critical temperature $T_c$ and investigate how it behaves under the change of the string cloud density parameter $a$, the spacetime dimension $d$ as well as the Gauss-Bonnet coupling $\alpha$. The analytical calculation is based on the Sturm-Liouville eigenvalue method.
The condensation is zero at the critical temperature $T_c$, and thus the scalar field vanishes, $\psi=0$. Consequently, Eq. (\[z-phi-Eq\]) yields the following equation $$\phi''(z)+\frac{4-d}{z}\phi'(z)=0.$$ The general solution of this equation is given as $$\phi(z)=C_1+C_2z^{d-3}.\label{Tc-z-phi-sol1}$$ By using the boundary condition at the horizon $\phi(1)=0$ and the asymtotic behavior at Eq. (\[phi-asy-beh\]), we obtain $$\phi(z)=\frac{\rho}{r^{d-3}_{+c}}\left(1-z^{d-3}\right),\label{Tc-z-phi-sol2}$$ where $r_{+c}$ is denoted the event horizon radius of the black hole with the temperature $T_c$. The function $\phi(z)$ is expressed as follows $$\phi(z)=\lambda r_{+c}\xi(z),\label{near-Ads-phi}$$ where $\lambda=\frac{\rho}{r^{d-2}_{+c}}$ and $\xi(z)=1-z^{d-3}$. In the limit $T\rightarrow T_c$, we express $\psi(z)$ by $$\psi(z)=\langle\mathcal{O}_+\rangle\frac{z^{\Delta_+}}{r^{\Delta_+}_+}F(z),\label{near-Ads-psi}$$ where $F(z)$ is the trial function satisfying the boundary condition $F(0)=1$ and $F'(0)=0$. By substituting this expression of $\psi(z)$ and the solution of $\phi(z)$ at the temperature $T_c$ into Eq. (\[z-psi-Eq\]), we obtain $$F''(z)+p(z)F'(z)+q(z)F(z)+\lambda^2w(z)\xi^2(z)F(z)=0,$$ where $$\begin{aligned}
p(z)&=&\frac{4-d+2\Delta_+}{z}+\frac{g'(z)}{g(z)},\nonumber\\
q(z)&=&\frac{\Delta_+}{z}\left[\frac{3-d+\Delta_+}{z}+\frac{g'(z)}{g(z)}\right]-\frac{2\widetilde{\alpha}m^2}{z^4g(z)},\nonumber\\
w(z)&=&\frac{4\widetilde{\alpha}^2}{z^4g^2(z)},\nonumber\\
g(z)&=&\frac{1}{z^2}\left(1-\sqrt{1-4\widetilde{\alpha}\left(1-z^{d-1}\right)+\frac{8a\widetilde{\alpha}z^{d-2}(1-z)}{(d-2)r^{d-2}_+}}\right).\end{aligned}$$ Interestingly, this equation can be written in the form of the Sturm-Liouville equation as $$\left[T(z)F'(z)\right]'-Q(z)F(z)+\lambda^2P(z)F(z)=0, \label{SL-eq}$$ where $$\begin{aligned}
Q(z)&=&-T(z)q(z),\nonumber\\
P(z)&=&T(z)w(z)\xi^2(z),\nonumber\\
T(z)&=&e^{\int p(z)dz}=z^{2-d+2\Delta_+}\left[1-z^{d-1}-2b(1-z)z^{d-2}\right]\left\{
1-\left[2b(1-z)+z\right]z^{d-2}\widetilde{\alpha}\right.\nonumber\\
&&\left.-\left[2b(1-z)+z\right]\left(4bz^{d+1}-4bz^d-2z^{d+1}+3z^2\right)z^{d-4}\widetilde{\alpha}^2+\mathcal{O}\left(\widetilde{\alpha}^3\right)\right\},\end{aligned}$$ where $b\equiv\frac{a}{(d-2)r^{d-2}_+}$. It is known from the Sturm-Liouville eigenvalue problem that the eigenvalues of Eq. (\[SL-eq\]) are obtained by minimizing the following expression $$\lambda^2=\frac{\int^1_0T(z)F'^2(z)dz+\int^1_0Q(z)F^2(z)dz}{\int^1_0P(z)F^2(z)dz},\label{sqlam-func}$$ where the trial function $F(z)$ is chosen as $F(z)=1-\beta z^2$ [@Therrien2010]. With this result, from Eq. (\[bhtemp\]), we derive the critical temperature $T_c$ which depends on the charge density $\rho$, string cloud density parameter $a$, Gauss-Bonnet coupling $\alpha$ and spacetime dimension $d$ as $$T_c=\frac{1}{4\pi}\left[(d-1)\left(\frac{\rho}{\lambda_{\text{min}}}\right)^{\frac{1}{d-2}}-\frac{2a}{(d-2)}\left(\frac{\lambda_{\text{min}}}{\rho}\right)^{\frac{d-3}{d-2}}\right],\label{ctemp-chden}$$ where $\lambda_{\text{min}}$ is determined by minimizing the function $\lambda^2$ given in Eq. (\[sqlam-func\]). This is one of the main results in this present work. Clearly, the effect of the string cloud on the critical temperature $T_c$ enters through both the eigenvalue $\lambda_{\text{min}}$ and the expression for $T_c$. Whereas, Gauss-Bonnet term affects indirectly the critical temperature $T_c$ entering through the eigenvalue $\lambda_{\text{min}}$. It should be noted that, as the critical temperature is low enough, which means that $T_c$ goes to the zero, from Eq. (\[bhtemp\]) we have the following relation $$\begin{aligned}
r^{d-2}_+\simeq\frac{2a}{(d-1)(d-2)},\end{aligned}$$ by which the parameter $b$ in Eq. (39) becomes $b\simeq\frac{d-1}{2}$. This suggests that in this limit the change of the string cloud density parameter $a$ affects no longer $\lambda_{\text{min}}$. On the other hand, in this limit the change of the string cloud density parameter $a$ affects the critical temperature through that $a$ enters directly in the expression for $T_c$. With $a=0$ corresponding to the absence of the string cloud, we get the usual expression for the critical temperature of the holographic superconductors in Einstein-Gauss-Bonnet gravity as, $T_c=\frac{d-1}{4\pi}\left(\frac{\rho}{\lambda_{\text{min}}}\right)^{1/(d-2)}$. One can easily see that, from the expression (\[ctemp-chden\]), the critical temperature $T_c$ is possibly negative. As a result, the presence of the string cloud leads to a fact that the critical temperature only exists in an allowed region of the parameter space, given by $$\frac{\rho}{a\lambda_{\text{min}}}>\frac{2}{(d-1)(d-2)}.$$ For example, with $d=5$, $\alpha=0.02$, $a=1.5$, $r_+=1$ and $\Delta_+=3$, the critical temperature only exists for the charge density satisfying, $\rho\gtrsim0.975$. In addition, as the string cloud density parameter $a$ increases, $\lambda^2_{\text{min}}$ should decrease and become negative above a critical value $a_{\text{crt}}$. For example, the critical value of the string cloud density parameter is given by, $a_{\text{crt}}\approx5.55369$, with $d=5$, $\alpha=0.1$, $r_+=1$ and $\Delta_+=3$. Therefore, the presence of the string cloud with the sufficiently large density should prevent the existence of the critical temperature below which the superconducting phase will appear in the dual field theory.
In this way, in the presence of the string cloud, the critical temperature only exists if the string cloud density parameter $a$ is below a certain critical value. Note that, in order to form the superconducting state, it needs to have another condition which the chemical potential is above a critical value (for the charge density kept fixed). Thus, for the appearance of the superconducting state, the fact that the parameter $a$ is below a critical value can be considered as a necessary condition. Whereas, the fact that the chemical potential is above a critical value can be considered as a sufficient condition.
In order to see more clearly, we plot the critical temperature $T_c$ as a function of the charge density $\rho$, for various values of $\alpha$, $a$ and $d$, in Fig. \[Tc-rho\].
[cc]{}\
We observe that, for the region of the sufficiently low charge density, the critical temperature decreases as increasing the string cloud density parameter $a$. This suggests that the condensation becomes harder in the presence of the string cloud. Whereas, for the region of the sufficiently high charge density, the critical temperature increases when the string cloud density parameter $a$ increases. And, since the presence of the string cloud makes the condensation getting easier. In addition, for studying the behavior of the critical temperature $T_c$ under the change of the spacetime dimension $d$, we plot $T_c$ as a function of $d$, for various values of $a$, in Fig. \[Tc-dimension\].
[cc]{}
We can see from this figure that, in the presence of the string cloud, the critical temperature increases with the growth of the spacetime dimension. This is consistent with the previous works that showed superconductors of the high temperature can be achieved in the higher dimensional spacetime. Furthermore, we present a comparison between the results which are derived from the analytic and numerical calculations in Table. \[table-Tc-rho\], which shows the good agreement between two results.
----- ------------------------------------ ------------------------------------ ----- ------------------------------------ ------------------------------------
$b$ $\text{Analytical}$ $\text{Numerical}$ $b$ $\text{Analytical}$ $\text{Numerical}$
0 $0.195\rho^{1/3}$ $0.197\rho^{1/3}$ 0 $0.191\rho^{1/3}$ $0.193\rho^{1/3}$
0.3 $0.2\rho^{1/3}-0.04\rho^{-2/3}$ $0.201\rho^{1/3}-0.04\rho^{-2/3}$ 0.3 $0.196\rho^{1/3}-0.042\rho^{-2/3}$ $0.197\rho^{1/3}-0.041\rho^{-2/3}$
0.6 $0.205\rho^{1/3}-0.076\rho^{-2/3}$ $0.206\rho^{1/3}-0.076\rho^{-2/3}$ 0.6 $0.202\rho^{1/3}-0.079\rho^{-2/3}$ $0.202\rho^{1/3}-0.078\rho^{-2/3}$
0.9 $0.212\rho^{1/3}-0.108\rho^{-2/3}$ $0.212\rho^{1/3}-0.107\rho^{-2/3}$ 0.9 $0.209\rho^{1/3}-0.11\rho^{-2/3}$ $0.21\rho^{1/3}-0.11\rho^{-2/3}$
1.2 $0.221\rho^{1/3}-0.132\rho^{-2/3}$ $0.221\rho^{1/3}-0.132\rho^{-2/3}$ 1.2 $0.219\rho^{1/3}-0.134\rho^{-2/3}$ $0.22\rho^{1/3}-0.133\rho^{-2/3}$
----- ------------------------------------ ------------------------------------ ----- ------------------------------------ ------------------------------------
: Analytical and numerical values of the critical temperature $T_c$, for the different values of $\alpha$ and $b\equiv\frac{a}{(d-2)r^{d-2}_+}$, at $d=5$ and $m^2l^2_{\text{eff}}=-3$ (or $\Delta_+=3$).[]{data-label="table-Tc-rho"}
Also, from this table, it is easily to see that the critical temperature decreases as increasing the Gauss-Bonnet coupling as indicated in the literature.
\[CVCE\] Condensation value and critical exponent
=================================================
In this section, we would like to calculate the condensation value $\langle\mathcal{O}_+\rangle$ near the critical temperature $T_c$ and derive the corresponding critical exponent for Gauss-Bonnet holographic superconductors in the presence of the string cloud. In order to do this, we need to look at the behavior of the gauge field near $T_c$. It is important to note that near $T_c$ the condensation value $\langle\mathcal{O}_+\rangle$ is very small. This suggests that we can expand the function $\phi(z)$ in $\langle\mathcal{O}_+\rangle$ as $$\phi(z)=\lambda r_+\left(1-z^{d-3}\right)+\frac{\langle\mathcal{O}_+\rangle^2}{r^{2\Delta_+}_+}r_+\chi(z)+\cdots,\label{phi-Exp}$$ where the function $\chi(z)$ satisfies the boundary condition, $\chi(1)=\chi'(1)=0$. Then, substituting this expansion into Eq. (\[z-phi-Eq\]), we obtain $$\chi''(z)+\frac{4-d}{z}\chi'(z)-\frac{2\lambda r^2_+\left(1-z^{d-3}\right)}{f(z)}z^{-4+2\Delta_+}F^2(z)=0.\label{chi-Eq}$$ This equation can be written as $$\left[\frac{\chi'(z)}{z^{d-4}}\right]'=\frac{2\lambda r^2_+\left(1-z^{d-3}\right)}{f(z)}z^{-d+2\Delta_+}F^2(z).$$ Using the condition $\chi'(1)=0$, we can integrate two sides of the above equation, which leads to $$\begin{aligned}
\chi'(z)&=&z^{d-4}\int^{z}_1\frac{2\lambda r^2_+\left(1-\widetilde{z}^{d-3}\right)}{f(\widetilde{z})}\widetilde{z}^{-d+2\Delta_+}F^2(\widetilde{z})d\widetilde{z}.\label{Der-psi}\end{aligned}$$
By identifying the asymptotic behavior of $\phi(z)$ from Eqs. (\[phi-asy-beh\]) and (\[phi-Exp\]), we get the following relation $$\mu-\frac{\rho}{r^{d-3}_+}z^{d-3}=\lambda r_+\left(1-z^{d-3}\right)+\frac{\langle\mathcal{O}_+\rangle^2}{r^{2\Delta_+-1}_+}\left[\chi(0)+\chi'(0)z+\frac{\chi''(0)}{2}z^2+\cdots+\frac{\chi^{(d-3)}(0)}{(d-3)!}z^{d-3}+\cdots\right].$$ Comparing the coefficients of the terms relating to $z^{d-3}$ in the right-hand and left-hand sides of the above equation, we find $$\frac{\rho}{r^{d-2}_+}=\lambda-\frac{\langle\mathcal{O}_+\rangle^2}{r^{2\Delta_+}_+}\frac{\chi^{(d-3)}(0)}{(d-3)!}.\label{rel}$$ It should also note that we can determine $\chi'(0)=\chi''(0)=\cdots=\chi^{(d-4)}(0)=0$ which is consistent to the expression of $\chi'(z)$ given in Eq. (\[Der-psi\]). With the result (\[rel\]), we find the expression for the condensation value $\langle\mathcal{O}_+\rangle$ as $$\begin{aligned}
\langle\mathcal{O}_+\rangle=\frac{\Theta T^{\Delta_+}}{\sqrt{d-2}}\sqrt{\left(\frac{T_c}{T}\right)^{d-2}-1}\approx\Theta T^{\Delta_+}_c\sqrt{1-\frac{T}{T_c}},\end{aligned}$$ where $$\Theta=\left(\frac{4\pi}{d-1}\right)^{\Delta_+}\sqrt{-\frac{\lambda(d-2)!}{\chi^{(d-3)}(0)}}.$$ This expression shows clearly that the critical exponent is $1/2$, which is consistent with the mean-field value and is independent on the string cloud density parameter, Gauss-Bonnet coupling and spacetime dimension. In this sense, the value $1/2$ of the critical exponent seems to be a universal constant. Also, the presence of the string cloud and Gauss-Bonnet term do not affect the expression form of the condensation operator, which is a universal property for holographic superconductors. On the other hand, the presence of the string cloud and Gauss-Bonnet term only affect indirectly $\langle\mathcal{O}_+\rangle$ through modifying the value of $T_c$ and $\Theta$.
In order to calculate $\Theta$, we use a fact that in the limit $z\rightarrow0$ Eq. (\[chi-Eq\]) becomes $$\chi''(0)=\frac{d-4}{z}\chi'(z)\Big|_{z\rightarrow0}.$$ This leads to the following relation between the $(d-3)$-th and first-order derivatives of $\chi$ at $z=0$ $$\frac{\chi^{(d-3)}(0)}{(d-4)!}=\frac{\chi'(z)}{z^{d-4}}\Big|_{z\rightarrow0}.$$ Using this relation with $\chi'(z)$ given at Eq. (\[Der-psi\]), we obtain $$\Theta=\left(\frac{4\pi}{d-1}\right)^{\Delta_+}\sqrt{\frac{(d-2)(d-3)}{\mathcal{A}}},$$ where $\mathcal{A}$ is given by $$\begin{aligned}
\mathcal{A}&=&\int^{1}_0\frac{2r^2_+\left(1-z^{d-3}\right)}{f(z)}z^{-d+2\Delta_+}F^2(z)dz,\nonumber\\
&=&4\widetilde{\alpha}\int^{1}_0\frac{z^{-d+2+2\Delta_+}(1-z^{d-3})(1-\beta z^2)^2}{1-\sqrt{1-4\widetilde{\alpha}\left(1-z^{d-1}\right)+8b\widetilde{\alpha}z^{d-2}(1-z)}}dz,\nonumber\\
&=&2\int^{1}_0\frac{z^{-d+2+2\Delta_+}(1-z^{d-3})(1-\beta z^2)^2}{1-z^{d-1}-2bz^{d-2}(1-z)}dz+\widetilde{\alpha}\left(\frac{2}{d-3-2\Delta_+}+\frac{1}{\Delta_+}\right.\nonumber\\
&&\left.+\frac{\beta(d-3)}{(5-d+2\Delta_+)(1+\Delta_+}\right)-2\widetilde{\alpha}^2\left[\frac{\beta}{d-5-2\Delta_+}+\frac{1}{3-d+2\Delta_+}\right.\nonumber\\
&&\left.+\frac{2b}{d-2+2\Delta_+}+\frac{1-2b}{d-1+2\Delta_+}-\frac{2\beta b}{d+2\Delta_+}
+\frac{(2b-1)\beta}{d+1+2\Delta_+}-\frac{2b}{1+2\Delta_+}\right.\nonumber\\
&&\left.-\frac{2\beta b}{3+2\Delta_+}-\frac{1}{2}\left(\frac{1}{\Delta_+}+\frac{1-2b-\beta}{1+\Delta_+}+\frac{(2b-1)\beta}{2+\Delta_+}\right)\right]+\mathcal{O}\left(\widetilde{\alpha}^3\right),
\label{Afun}\end{aligned}$$ which can be, with the given parameters, integrated numerically.
With the expression (\[Afun\]), we now are in position to perform the explicit calculation for the value of the dimensionless condensation operator $\langle\mathcal{O}_+\rangle/T^{\Delta_+}_c$. In Fig. \[conds\], we plot $\langle\mathcal{O}_+\rangle/T^{\Delta_+}_c$ as a function in terms of $T/T_c$ for various values of $\alpha$, $a$ and $d$. We observe that, for $T/T_c$ kept fixed, as the string cloud density parameter $a$ increases, the value of the dimensionless condensation operator becomes smaller.
[cc]{}\
\[conclu\] Conclusion
=====================
In this paper, holographic superconductors have been investigated in the presence of a string cloud and Gauss-Bonnet term. In the probe limit at which the backreaction of the gauge field and scalar field on the background geometry is ignored all, we solve Einstein’s field equation to obtain the planar AdS Gauss-Bonnet black hole solution sourced by the string cloud and negative cosmological constant. By adopting the Sturm-Liouville eigenvalue method, the critical temperature has been obtained as a function of the charge density, string cloud density parameter, Gauss-Bonnet coupling, and spacetime dimension. The presence of the string cloud modifies the usual form of the critical temperature (which is proportional to $\rho^{1/(d-2)}$). As a result, the existence of the critical temperature is only allowed in a certain region of the parameter space. In particular, the critical temperature, below which the condensation will be formed, only exists as the string cloud density parameter is smaller than a certain critical value. Also, it is indicated that when the string cloud density parameter increases, the critical temperature decreases in the region of the sufficiently low charge density, whereas it increases in the region of the sufficiently high charge density. In addition, the critical temperature growths with increasing the spacetime dimension, which suggests that the superconductors of the high temperature can be achieved in the higher spacetime dimension. It is also observed that increasing Gauss-Bonnet coupling always makes the critical temperature decreasing. Finally, the condensation operator and the critical exponent of holographic superconductors are computed at which the critical exponent is always equal to $1/2$ independently on the properties of the system.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to express sincere gratitude to the referees for their constructive comments and suggestions which have helped us to improve the quality of the paper.
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---
abstract: 'Spectral imagers, the classic example being the color camera, are ubiquitous in everyday life. However, most such imagers rely on filter arrays that absorb light outside each spectral channel, yielding $\sim\!\!1/N$ efficiency for an $N$-channel imager. This is especially undesirable in thermal infrared (IR) wavelengths, where sensor detectivities are low, as well as in highly compact systems with small entrance pupils. Diffractive optics or interferometers can enable efficient spectral imagers, but such systems are too bulky for certain applications. We propose an efficient and compact thermal infrared spectral imager comprising a metasurface composed of sub-wavelength-spaced, differently-tuned slot antennas coupled to photosensitive elements. Here, we demonstrate this idea using graphene, which features a photoresponse up to thermal IR wavelengths. The combined antenna resonances yield broadband absorption in the graphene exceeding the $1/N$ efficiency limit. We establish a circuit model for the antennas’ optical properties and demonstrate consistency with full-wave simulations. We also theoretically demonstrate broadband $\sim\!36\%$ free space-to-graphene coupling efficiency for a six-spectral-channel metasurface. This research paves the way towards compact CMOS-integrable thermal IR spectral imagers.'
author:
- 'Jordan A. Goldstein'
- 'Dirk R. Englund'
bibliography:
- 'antennatheoryacsphot.bib'
title: 'Imaging Metasurfaces based on Graphene-Loaded Slot Antennas'
---
![a) Illustration of a broadband absorbing slot antenna metasurface consisting of six differently tuned slot antennas tiled with subwavelength periodicity. The graphene patches are color-coded by antenna length, and the diagram is drawn to proportion based on the device dimensions used to produce Figure \[antennas6\]. b) Depiction of a single graphene-coupled slot antenna-based photodetector based on the photothermoelectric effect.[]{data-label="figure1"}](figure1abwaxes){width="\columnwidth"}
###
We take spectral imaging for granted in daily life. Our eyes are spectral imagers, providing information about the composition of what we see. The infrared electromagnetic band covers many chemical absorption resonances and thus also reveals compositional information. In particular, infrared spectral imaging is applied in areas such as gas emission monitoring[@emissions; @gases; @etalonplume; @boat; @agent; @seepage; @hawaii], ecological monitoring[@soil; @trees; @pines], food quality control[@foodreview; @foodquality; @corn], waste sorting[@plastic], biological research[@dichroic] and oceanography[@ocean].
Spectral imaging aims to measure a “data cube” representing light intensity over two spatial dimensions $x$ and $y$ and one spectral dimension $\lambda$ with $N$ channels. Scanning spectral imagers sequentially measure different portions of the data cube over multiple exposures to form the full data cube. A common example is the pushbroom scanner which measures $x \times \lambda$ data cube slices while scanning $y$ and is thus typically associated with satellites and conveyor belts where either the camera or subject is gradually moving in one direction[@satellite; @emissions; @foodquality; @plastic]. The spectral axis may also be scanned such as in tunable filter-based imagers [@etalonplume; @liquidcrystal], which feature at most $1/N$ light utilization efficiency, or Fourier transform interferometer-based imagers which are bulky and have moving parts[@gases]. In contrast, snapshot spectral imagers (SSIs) capture a data cube with a single exposure[@snapshot]. This may be achieved using a color filter array similar to that of a color camera, thus limiting efficiency to $1/N$[@atwater; @nanowires; @snapshot]. Another category of SSIs uses dichroic or dispersive optics to break up incoming light by wavelength before arriving on a focal plane array (FPA). There are many variations of this approach[@bodkin; @coded; @ims; @gorman; @dichroic], but they all require increasing the etendue of the incoming light beam by a factor of $N$, leading to an unfavorable tradeoff between total FPA area, input acceptance angle and spectral resolution[@snapshot].
Compared to these technologies, the category of imagers based on inherently multispectral pixels is less explored. One such example is the Foveon RGB sensor, which extracts three different electrical signals from different depths in the optically active silicon region, as shorter wavelengths are absorbed closer to the sensor surface[@foveon]. Another approach uses nanoantennas, optical resonators of subwavelength dimensions which nevertheless feature absorption cross sections of order $\lambda^2$ if the antenna is conjugate impedance matched with its load; or, equivalently, if the antenna is critically coupled to the vacuum. Here, we propose a thermal IR multispectral imager where $N$ differently sized metallic slot antennas with infrared-sensitive loads targeting $N$ spectral channels are tiled to form a metasurface featuring efficient free space-to-load optical energy transfer. Figure \[figure1\]a shows such a metasurface for $N=6$. We model graphene as the photosensitive load because its broadband absorption in the mid-IR[@grcond] and processing flexibility[@grcmoscamera; @grcmoschem] make it suitable for this platform. Not only do the antennas sort incident light by spectral channel, but they also enhance the absorption of the graphene load, bridging the gap between the impedance of free space and graphene, which, when undoped, has an optical sheet resistance no lower than $16.1\,\text{k}\Omega$.[@grcond]. Figure \[figure1\]b shows in detail a single such antenna-coupled graphene photodetector. This detector is designed for a strong photothermoelectric response, in which absorbed light heats up the electron gas in the graphene, resulting in an electromotive force due to the Seebeck effect. The graphene channel is assumed to be isolated from the metasurface by a several-nanometer layer of dielectric, thin enough to not impact the optical properties of the system. The asymmetric position of the graphene channel with respect to the slot allows half of the graphene channel to be gated by metal underneath, yielding the asymmetric graphene Fermi level profile needed for a nonzero net photoresponse[@gabor; @jsong]. Note that while perfect absorption in this wavelength range has been demonstrated in heavily doped graphene accompanied with metal nanostructures[@seyoon], we limit our consideration to undoped graphene as the peak Seebeck coefficient occurs at very low doping levels[@gabor].
For optical absorption, slot antennas offer a few advantages over planar designs such as dipole or bowtie antennas. They have unidirectional radiation patterns, and thus an array of them can perfectly absorb an incident beam, whereas planar antennas require a quarter-wave back-reflector to do so [@alaee; @eggles]. The wavelength dependence of the back-reflector phase complicates design of broadband absorbing metasurfaces and exacerbates undesirable antenna-antenna coupling. Additionally, since planar antennas must be supported by transparent dielectric, they cannot be embedded in a CMOS process as the inter-layer dielectric strongly absorbs thermal infrared radiation[@kischkat; @lin], whereas for slot antennas the dielectric on top and in the perforations may be etched away without sacrificing the mechanical integrity of the antenna.
### Results and Discussion
![a) Graphene-loaded slot antenna with physical features corresponding to the components in circuit b) labelled. b) Circuit schematic of a graphene-coupled slot antenna. $V_A$ represents incoming light, $Z_A$ is the radiation impedance of the slot aperture, $Z_\text{gr}$ is the impedance of the graphene sheet and $Z_\text{wg}$ is that of the slot, effectively a short-circuit waveguide stub.[]{data-label="figure2"}](figure2abSmallText){width="\columnwidth"}
We model the slot antenna depicted in Figure \[figure2\]a as an aperture antenna fed by a rectangular waveguide terminated in a short circuit a distance $d$ behind the aperture. We represent the graphene sheet as a shunt impedance connected in parallel with the waveguide stub. Figure \[figure2\]b illustrates this circuit with a Thévenin equivalent radiation impedance $Z_A$ and source $V_A$ representing the aperture antenna[@stutzman]. The rectangular waveguide stub presents an impedance $$Z_\text{wg} = Z_0\,\frac{e^{jn_\text{eff}k_0d} + r e^{-jn_\text{eff}k_0d}}{e^{jn_\text{eff}k_0d} - r e^{-jn_\text{eff}k_0d}},$$ where $Z_0$ and $n_\text{eff}$ are the characteristic impedance and effective index of the $\text{TE}_{10}$ mode of the slot and $k_0$ is the vacuum wavenumber. $r$ is the Fresnel reflection coefficient between vacuum and metal for an *s*-polarized plane wave at an incident angle of $\arccos(n_\text{eff})$, which describes the $\text{TE}_{10}$ mode.
The graphene presents a mostly resistive impedance of $$Z_\text{gr}=\frac{\pi^2w}{8h\sigma_\text{gr}},
\label{zgr}$$ using power/current impedance normalization[@schelkunoff]. $w$ and $h$ are defined in Figure \[figure1\]b and $\sigma_\text{gr}$ is the sheet conductance of the graphene, modeled here as intrinsic. We calculate $Z_A$ using finite element simulations for various $h$ and $w$; we provide more details on these calculations in the Methods section. See Supplementary Figure 1 for an example of the frequency dependence of the impedances in the circuit.
Define $\eta_\text{gr}$ as the fraction of available power from Thévenin source $V_A$, $Z_A$ that is dissipated in $Z_\text{gr}$. Solving the above circuit, we arrive at $$\eta_\text{gr}=4\left|\frac{Z_\text{gr}\parallel Z_\text{wg}}{Z_\text{gr}}\right|^2\frac{\operatorname{Re}(Z_A) \operatorname{Re}(Z_\text{gr})}{\left|Z_A+(Z_\text{gr}\parallel Z_\text{wg})\right|^2},
\label{etaeqn}$$ where $\parallel$ represents reciprocal addition. Define $A_\text{gr}=\frac{P_\text{gr}}{I_\text{inc}}$ as the partial absorption cross section of light of intensity $I_\text{inc}$ coupled into the graphene and $P_\text{gr}$ as the power absorbed in the graphene. For a lossless, conjugate matched antenna, antenna theory predicts $A_\text{gr,max}=\frac{D\left(\theta,\phi\right)}{4\pi}\lambda^2$ where $D\left(\theta,\phi\right)$ is the antenna’s directivity at the given incident angle and polarization[@stutzman], which we calculate using finite element simulations. $\theta$ here is the polar angle and $\phi$ is the azimuthal angle. We omit the polarization angle in our notation, implicitly setting it to maximize $D$. Additional ohmic losses and impedance mismatch then reduce the actual absorption cross-section into graphene by a factor $\eta_\text{gr}$, i.e. $$A_\text{gr}=\eta_\text{gr}\frac{D\left(\theta,\phi\right)}{4\pi}\lambda^2.
\label{agr}$$ We obtain $A_\text{gr}$ from FDTD simulations of plane waves incident on the graphene-loaded antennas, which we then use to calculate $\eta_\text{gr}$ via Equation \[agr\].
![Comparison of simulated and modeled $\eta_\text{gr}$. Modeled antennas are $400\,\text{nm}$ wide. Dashed lines represent the fullwave FDTD absorption results, while the solid lines represent the impedance model results.[]{data-label="efficiencies"}](geosweepfinal){width="\columnwidth"}
Fig. \[efficiencies\] compares $\eta_\text{gr}$ between the model described by Equation \[etaeqn\] and FDTD results for antennas of various dimensions. The data show that the model is accurate to within 10% of the $\eta_\text{gr}$ peak amplitude and 2% of the resonance wavenumber. We attribute these discrepancies to aspects not captured by the quasi-analytical model, such as our assumption of a perfectly conducting outer antenna face and finite meshing. Despite these shortcomings, the present model allows us to predict slot antenna absorption properties to tolerances comparable to the uncertainty due to variations in metal quality.
![$\eta_\text{gr}$ versus wavenumber for various values of the load sheet conductance, where $\sigma_\text{gr}$ is the optical conductivity of intrinsic graphene. Dashed lines represent the fullwave FDTD absorption results, while the solid lines represent the model results. The antenna featured here has dimensions $d = 2.0\,{\textrm{\textmu m}}$, $h = 5.5\,{\textrm{\textmu m}}$, and $w = 0.4\,{\textrm{\textmu m}}$. []{data-label="grcondvar"}](scalingsweepfinal){width="\columnwidth"}
To further validate our model, we artificially scale the sheet conductance of the graphene load by factors ranging from $0.33$ to $5$ and compare the resulting $\eta_\text{gr}$ between the model and FDTD for a single such antenna. The results, shown in Figure \[grcondvar\], show that our model accurately predicts the sublinear scaling of $\eta_\text{gr}$ with respect to the load conductivity, with the $\eta_\text{gr}$ peak amplitude reaching $0.8$ for a load conductivity of $5\sigma_\text{gr}$.
Having modeled the individual components, we now discuss broadband absorbing metasurfaces incorporating differently tuned antennas tiled in a periodic array. We use a three-step process to design such metasurfaces. We first constrain $d$ and $w$ to be the same for all antennas in the metasurface, and choose their values to yield high peak $\eta_\text{gr}$ for antennas resonant in the targeted wavelength range. Secondly, we choose the values of $h$ for the antennas, following the heuristic that at the wavelength where one antenna’s $\eta_\text{gr}$ falls to half its peak amplitude, the next antenna’s $\eta_\text{gr}$ should have risen to half its peak amplitude. Finally, we choose the arrangement and pitch of the antennas to be as closely packed as possible while satisfying qualitative fabricability constraints. We also avoid juxtaposing antennas of adjacent wavelength channels to minimize antenna-antenna crosstalk.
The antenna pitch, more accurately described by the Bravais lattice vectors of the array, is a critical parameter in determining the potential absorption efficiency of the array. Light incident from a given direction can only be scattered by the two-dimensional diffraction orders of the lattice. The array can only perfectly absorb an incoming light beam if no nonzero diffraction orders fall within the light cone, barring the event that all higher diffraction orders overlap with nodes in the individual unit cell radiation pattern. By “light cone”, we refer here to the region in the Fourier transform space of the $xy$–plane for which radiation can occur, namely $k_x^2 + k_y^2 < k_0^2$. For a square lattice, if the lattice pitch $a<\lambda_\text{min}/2$, no higher diffraction orders are within the light cone for any incident angle. In practice, the limited numerical aperture of imaging systems relaxes this constraint.
![Graphene absorption efficiency of the six-antenna metasurface. The colored curves represent the contributions of each antenna to the overall graphene absorption of the metasurface, which is represented by the black dashed curve. The curve colors here match the antenna colors in Figure \[figure1\]a. Shorter antennas yield higher resonance wavenumbers.[]{data-label="antennas6"}](antennas6fig1acolors){width="\columnwidth"}
Targeting the $6-10\,{\textrm{\textmu m}}$ wavelength range, we follow the above methodology and arrive at an array of six antennas with $d=2.25\,{\textrm{\textmu m}}$, $w=400\,{\textrm{nm}}$, and $h={3.41, 3.81, 4.41, 5.04, 5.80, 7.36}\,{\textrm{\textmu m}}$, uniformly spaced and tiled as shown in Figure \[figure1\]a with a $7\,{\textrm{\textmu m}}$ by $12.667\,{\textrm{\textmu m}}$ unit cell. Figure \[antennas6\] shows the absorption efficiency of normally incident light into the graphene load of each antenna as well as their sum, simulated with FDTD. With an average efficiency of $36\%$ across the $1050\,{\textrm{cm}}^{-1}$ to $1600\,{\textrm{cm}}^{-1}$ band, this structure improves upon the $1/N$ limit of filter array-based imagers by roughly a factor of two. Note that the unit cell highlighted in Figure \[figure1\]a is not the primitive unit cell of the lattice, although it was used as the FDTD simulation region due to software constraints.
![a) Total graphene absorption efficiency versus wavenumber for metasurfaces with varied unit cell pitch. The legend indicates the width of the unit cell as illustrated in Figure \[figure1\]a. b) Mean absorption efficiency for each of the curves in a), averaged between $1050\,{\textrm{cm}}^{-1}$ and $1600\,{\textrm{cm}}^{-1}$. c) Unit cell dimensions used to obtain the results in a) and b) and the corresponding maximum wavenumbers for which normally incident light experiences no higher order diffraction. The unit cell x and y pitch were varied together to attain a roughly uniform antenna-antenna proximity.[]{data-label="pitchsweep"}](efficiencyPitchWInset){width="\columnwidth"}
To further understand the physics of these metasurfaces, we vary the pitch of the unit cell while keeping all other parameters constant. The resulting total absorption efficiency spectra for six different cases are shown in Figure \[pitchsweep\]a, and their mean efficiencies averaged between $1050\,{\textrm{cm}}^{-1}$ and $1600\,{\textrm{cm}}^{-1}$ are plotted in Figure \[pitchsweep\]b. We show the individual absorption contributions from each antenna for each metasurface pitch in Supplementary Figure 2, and we list the unit cell widths and heights in Figure \[pitchsweep\]c. The data show that the absorption efficiencies are roughly constant, and comparable with the peak efficiencies obtained in Figure \[efficiencies\], up to the $7\,{\textrm{\textmu m}}$–wide unit cell. For larger unit cells, the mean efficiency decreases with increasing unit cell size. This can be understood by analyzing the diffraction characteristics of the various metasurfaces. Sparser metasurfaces have tighter reciprocal lattices and thus more diffraction orders are available within the light cone. For normally incident illumination as we are using here, the minimum wavelength for which no higher-order diffraction occurs (i.e., completely specular reflection) is given by $\lambda_c=\max\!\left(\left(a_1^{-2}+a_2^{-2}\right)^{-1/2},\frac{a_1}{2},\frac{a_2}{2}\right)$, where $a_1$ and $a_2$ are the unit cell dimensions. The corresponding wavenumbers for the various unit cells used here are listed in Figure \[pitchsweep\]c. For the $9\,{\textrm{\textmu m}}$ and $11\,{\textrm{\textmu m}}$ width unit cells, the threshold wavenumber falls in the middle of the range of interest, resulting in reduced efficiency as diffracted light cannot participate in destructive interference with specularly scattered light. This is especially apparent for the $9\,{\textrm{\textmu m}}$ unit cell, which exhibits a clear spectral transition between high and low efficiency at the diffraction threshold. Note that choosing an excessively tight antenna spacing is also detrimental, not only due to fabrication difficulty, but also because graphene detectors feature minimum Johnson noise-dominated noise-equivalent power for channel lengths comparable to the hot carrier cooling length, which can range from $100\,{\textrm{nm}}$ to over $1\,{\textrm{\textmu m}}$ depending on the substrate and graphene quality [@cosmi; @jsong; @gabor]. Exceedingly tight antenna spacings may not provide room for such long graphene channels.
For a spectrally sensitive metasurface to be practical, not only must it maintain a high absorption efficiency for a reasonable range of incoming light directions, but also the absorption spectra of the individual antennas must not shift or distort too strongly as the incoming light angle varies. The directional dependence of our metasurface arises from two factors: The directivity profile $D\left(\theta,\phi\right)$ of the individual antennas, and array effects resulting from interference and antenna-antenna coupling. Although full angle-dependent simulation results for our gold metasurfaces are outside the scope of this paper due to the extreme computational overhead of off-angle periodic structure simulations[@bfast], we can still provide insight by elaborating on the aformentioned factors, and we also perform off-angle simulations of a simplified metasurface constructed of Perfect Electrical Conductor (PEC) which permits a much larger mesh size. Supplementary Figures 3a, b and c display the directivity profile of a $6.5\,{\textrm{\textmu m}}\times 400\,{\textrm{nm}}$ antenna on resonance. This profile is similar to those of the other antenna lengths used in the metasurface. Intuitively, the directivity decreases as the incident angle approaches the long axis of the antenna, and we thus expect a similar trend in the directional dependence of the array. In Supplementary Figure 3d, we plot for three wavenumbers the sets of incident light directions, represented as components $k_x$, $k_y$ of the incident wavevector $\mathbf{k}$, for which only specular reflection from the metasurface occurs. For $\lambda^{-1}=1050\,\text{cm}^{-1}$, all light incident within $45^\circ$ of normal is only specularly reflected. This range decreases with increasing wavenumber until normally incident light is pinched off at $1579\,\text{cm}^{-1}$. As in Figure \[pitchsweep\], we expect efficiency to suffer when the specular reflection-only condition is not met. Supplementary Figure 4 shows the results of off-angle simulations of the simplified PEC metasurface. The data show that for light incident off-angle but perpendicular to the antennas’ long axes, antenna resonances falling outside the specular reflection-only region are subject to decreased absorption efficiency as well as blue-shifting. For light incident off-angle and perpendicular to the antennas’ short axes, we obtain similar results, except that the peaks red-shift instead of blue-shift, and we observe an overall reduction of the absorption at steep incident angles due to the reduced directivity.
We also investigate metasurfaces comprising numbers of spectral channels besides $N=6$. Supplementary Figure 5 shows the geometric details and simulation results for metasurfaces with $N=3$, 4, 5, and 8 as well as the default value of 6. We achieve good results with uniformly high absorption efficiency for $N=5$ and $6$. For lower values of $N$, the wide frequency spacing between the individual resonances yields deep troughs in the overall absorption efficiency curve, whereas for higher values of $N$, excessive overlap between the antenna resonances causes the overall metasurface efficiency to suffer.
![a) Cross-section of slot antenna with narrowed input slit, as well as dimension definitions. b) Illustration of how such an antenna could be implemented in the wiring layers of a CMOS chip. c) $\eta_\text{gr}$ versus wavenumber for various antenna geometries and gold optical models. “Open” refers to the the normal slot antenna design in Figure \[figure1\]b), while “slitted” refers to the slitted design in a). “Evap” refers to evaporated gold as used throughout the paper[@palik], whereas “SC” refers to single crystal gold [@crc]. For the open antennas plotted here, $d=2.75\,{\textrm{\textmu m}}$. For the slitted inlet antennas, $d=1.75\,{\textrm{\textmu m}}$, $w_\text{i}=0.15\,{\textrm{\textmu m}}$, and $d_\text{i}=0.2\,{\textrm{\textmu m}}$. These simulations use solid, not perforated sidewalls.[]{data-label="improvedgeometries"}](improvedAntennaGeometries4){width="\columnwidth"}
With realistic metal, the efficiency of these devices is ultimately limited by ohmic losses. However, more advanced antenna designs can be used to improve $\eta_\text{gr}$. As it turns out, not only can the copper layers in the back end of a CMOS process be designed to incorporate slot antennas, but they also provide a convenient medium to realize those antenna designs that would be exceedingly difficult to fabricate in an academic setting[@gupta]. Figure \[improvedgeometries\]a introduces such a design in which the slot inlet is narrower than the internal cavity width. This design concentrates the electric field around the graphene, which reduces $Z_\text{gr}$ without increasing the $\text{TE}_{10}$ mode loss. Figure \[improvedgeometries\]b shows a CMOS adaptation of this design, in which the walls of the cavity are perforated to comply with via design rules. If the perforations are sub-cutoff for the resonant wavelength and sufficiently deep, they do not leak light. Additionally, it would be necessary to etch away the inter-layer dielectric to prevent light absorption.
Besides different antenna designs, material quality also affects efficiency considerably. We explore both of these variations in Figure \[improvedgeometries\]c, which plots $\eta_\text{gr}$ versus wavenumber for intrinsic graphene as simulated by FDTD for narrowed inlet (“slitted”) antennas and normal slot antennas, as well as with a single-crystal gold model [@crc] in addition to our default evaporated gold model. The data show that adopting a slitted antenna design increases the peak $\eta_\text{gr}$ from $0.4$ to $0.6$, and then to $0.9$ for single-crystal gold. We can thus hope to achieve $\eta_\text{gr}\approx0.6$ for CMOS-integrated antennas, as copper has been shown to exhibit slightly superior plasmonic properties to gold given suitable deposition conditions[@plasmonicfilms]. While one can achieve high efficiencies with clever antenna designs and more opaque loads than monolayer graphene, the Q is ultimately bounded by that of a sealed metal cavity which we simulate to be about 40 for the present gold model and antenna shapes. Silver has less mid-IR optical loss than copper or gold, but unlike copper it is not considered CMOS-compatible and thus may be difficult to integrate. Polar materials supporting optical phonons in the mid-IR are reported to have high Q plasmonic resonances and are thus worth investigating if higher Q is necessary, although plasmonic behavior only occurs in narrow wavelength bands, limiting applicability[@midirplasmonics].
We can apply data describing the resistivity of gated graphene to our model to estimate the room temperature detectivity of the spectral imager described by Figures \[figure1\] and \[antennas6\]. Using the methods described in Song et al.[@jsong] and the electrical properties of polycrystalline, non-annealed graphene achieved by de Fazio et al.[@defazio], we arrive at the values in Table \[sensitivity\] with and without accounting for graphene contact resistance for normally incident $x$-polarized light. Using the more advanced antenna designs shown in Figure \[improvedgeometries\], the detectivities would reach the $10^8\,\text{Jones}$ range. We discuss these calculations in the Methods section. For comparison, more conventional bolometer-based thermal IR FPAs operating at room temperature have been reported to achieve detectivities in the $\text{1--2}\times10^9\,\text{Jones}$ range, although such devices do not feature spectral resolution and are limited to millisecond-range response times[@foote; @nanovox].
[c | c c c]{} $R_\text{c}\;[\Omega\,{\textrm{\textmu m}}]$& $\mathcal{R}\;[\text{A}/\text{W}]$ & NEP \[$\text{pW}\,\text{Hz}^{-\frac{1}{2}}$\] & $D^*\,\text{[Jones]}$\
\
0 & $0.83$ & $9.6$ & $6.9\times10^7$\
1000 & $0.42$ & $12.$ & $5.5\times10^7$\
Finally, we would like to emphasize the general applicability of the antenna metasurface concept to other wavelength ranges, photosensitive elements and antenna designs. Indeed, Tamang et al. have proposed a similar concept applied to RGB imaging where silicon nanorods act as both the antenna and sensitive element [@tamang]. We propose that besides graphene and other 2D materials, III-V or HgCdTe semiconductor photosensitive elements could also be incorporated by a transfer printing heterointegration process[@transferprinting]. The slot antenna-based metasurface imager approach could also scale to terahertz, where Ohmic losses are much less than in the mid-IR[@ordal] and the antennas could be fabricated directly in a circuit board-like platform.
### Conclusion
In conclusion, we introduced a six-spectral-channel graphene-coupled slot antenna metasurface with $36\%$ efficiency functioning as a spectral imager in the thermal IR, as well as a model for estimating the optical properties of the individual antennas therein. This device is appropriate for integration in the wiring layers of a CMOS process with suitable post-processing to remove inter-layer dielectric within the cavity and transfer graphene. We have shown that more sophisticated antenna designs can improve the efficiency of optical energy transfer to the load to above $\eta_\text{gr}=0.6$. Further research on this concept may focus on experimental demonstration of the absorption enhancement functionality, or on optimizing the device design to meet certain engineering goals.
### Methods
*Simulation details*Unless otherwise specified, we use the evaporated gold optical model described in Palik et al. throughout the paper [@palik]. We model graphene as an infinitely thin conductive sheet using the optical conductivity model described in Hanson [@hanson]. As input parameters to the model, we use a temperature of $300\,\text{K}$, intrinsic graphene (zero Fermi level), and a scattering rate $\Gamma = 0.514\,\text{meV}$.
We use the Lumerical FDTD package for our FDTD simulations. For simulations of individual antennas, we use $x$-, $y$- and $z$-meshes of $8\,{\textrm{nm}}$ in the vicinities of the slot aperture and slot bottom, as well as a $z$-mesh of $40\,{\textrm{nm}}$ within the slot. As such, the finest meshes coincide with metallic surfaces and corners, allowing us to capture the nonzero skin depth of the metal, whereas the more gradual $z$-dependence of the fields inside the slot permits a coarser mesh. We find that a minimum mesh size of $8\,{\textrm{nm}}$ yields converged results for these simulations. We use PML boundary conditions on all sides except for the $-z$ side, where we use a metallic boundary condition as the light does not penetrate beyond the slots anyway. We also apply symmetry conditions across the $xz$ and $yz$ planes. For the metasurface simulations, we use the same meshing scheme, but with a fine mesh of $15\,{\textrm{nm}}$ and a $z$ mesh of $100\,{\textrm{nm}}$ within the slot due to computational resource availability limits. To illustrate the error associated with this choice of mesh, we plot mesh-dependent absorption efficiency curves for a $9\,{\textrm{\textmu m}}$ by $13.333\,{\textrm{\textmu m}}$ unit cell, $N=6$ metasurface in Supplementary Figure 6, with the mean efficiency averaged between $1050\,{\textrm{cm}}^{-1}$ and $1600\,{\textrm{cm}}^{-1}$ plotted in the inset. The results validate our qualitative conclusions and put an approximately $3\%$ relative error bound on the spectrally averaged efficiencies of the metasurfaces, although the coarse mesh does somewhat distort the actual spectra. For the metasurface simulations, we change the $x$- and $y$- boundary conditions to Bloch boundary conditions to reflect the periodic nature of the metasurface, and we apply symmetry across only the $xz$ plane.
*Impedance model details*We calculate $Z_0$ and $n_\text{eff}$ for our impedance model using the Lumerical MODE waveguide mode solver with an $8\,{\textrm{nm}}$ mesh. We use the graphene model from Falkovsky [@grcond], which gives almost identical results to the Hanson model used by Lumerical for the parameters we use. We use Ansys HFSS finite element software to calculate $Z_A$. These simulations excite the aperture from within by its $\text{TE}_{10}$ mode yielding the $S_{11}$ scattering coefficient of the internally reflected light, from which the software calculates $Z_A$. In the finite element simulations, we model the aperture and slot as perfect electrical conductors, as we expect the real part of the antenna impedance to be dominated by radiative loss (rather than ohmic loss) and the imaginary part by energy storage in the near field of the aperture (rather than plasmonically within the metal). From these same simulations we also extract the antenna directivity $D\left(\theta,\phi\right)$.
*Detectivity estimation*We base our estimation of the detectivity $D^*$ on the formulation put forth in Song et al.[@jsong]. To calculate the electronic temperature profile of graphene suspended over the slot, we solve the 2-dimensional partial differential equation: $$-\nabla\cdot\left(\kappa\Delta T_\text{el}\right) + \gamma C_\text{el}\Delta T_\text{el}=\alpha\epsilon_0\dot{N}-\mathbf{j}\cdot\nabla\Pi
\label{diffeq}$$ Here $\kappa$ represents the electronic planar thermal conductivity of the graphene; $\Delta T_\text{el}$ is the difference between the thermally excited electronic temperature $T_\text{el}$ and the lattice temperature $T_0=300\,\text{K}$, $\gamma$ represents the electron-phonon thermal decay rate, and $C_\text{el}$ represents the electronic heat capacity of graphene. $\alpha$ is the efficiency with which optical energy absorbed by the graphene is deposited into the electronic system on a sub-picosecond timescale, taken to be unity since the incident photon energy is below graphene’s optical phonon energy. $\epsilon_0\dot{N}$ represents the intensity profile of absorbed light. $\mathbf{j}$ is the electrical current density, and $\Pi$ refers to the Peltier coefficient. We choose an antenna with $w = 400\,{\textrm{nm}}$ and $h=5.5\,{\textrm{\textmu m}}$ in these calculations, extracting the spatial dependence of $\epsilon_0\dot{N}$ from Ansys HFSS finite element simulations. For the graphene’s conductivity $\sigma$ as a function of Fermi level, we use the data measured by de Fazio et al. for unannealed, polycrystalline graphene [@defazio]; this is then used to calculate $\kappa$ via the Wiedemann-Franz law and $\Pi$ as well as the Seebeck coefficient $S$ via the Mott formula. The value of $\gamma C_\text{el}$ is estimated by assuming a electronic thermal cooling length of $\sqrt{\kappa/\gamma C_\text{el}}=1\,{\textrm{\textmu m}}$, an empirical value[@cosmi]. As shown in Figure \[figure1\]b, the graphene is assumed to be terminated at the slot edge on one side, and is taken to extend $400 {\textrm{nm}}$ past the slot edge on the other side, where its Fermi level is gated through the metal to the n-type Seebeck coefficient peak. The graphene Fermi level in the suspended region is simply taken to be the zero-gate-voltage Fermi level from de Fazio et al. as it cannot be controlled. For simplicity we assume a sharp jump in the values of $\sigma$, $\kappa$, $\Pi$ and $S$ between the n- and p-doped sides of the device, neglecting fringing fields from the gate. The graphene channel is assumed to be short-circuited with graphene-metal contact resistances $R_\text{c}$ of either $0$ or $1000\,\Omega\,{\textrm{\textmu m}}$ per contact, the latter being consistent with 1-dimensional contacts to graphene near the Dirac point[@contacts]. The average $\Delta T_\text{el}$ at the graphene p-n junction and the Seebeck coefficients on either side determine the thermal electromotive force via $\mathcal{E}=-S\nabla T$[@grosso], which in turn determines $\mathbf{j}$ via the total device resistance. Thus, Equation \[diffeq\] including the Peltier term may be solved directly, as $\mathbf{j}$ is a linear functional of $\Delta T_\text{el}$. Solving for $\Delta T_\text{el}$ over two spatial dimensions, we find linear thermal decay profiles along the $+x$ and $-x$ directions away from the $\Delta T_\text{el}$ peak which indicates that the device is in the short-channel regime where carrier cooling is dominated by the $\Delta T_\text{el}=0$ boundary conditions, and $\sqrt{\kappa/\gamma C_\text{el}}$ is large enough for the electron-phonon interaction term to be inconsequential.
Having obtained the short-circuit responsivity $\mathcal{R}$ under zero bias as such, we calculate the noise-equivalent power of the device assuming Johnson noise at $300\,\text{K}$ as the dominant noise source, a reasonable assumption for an unbiased device[@graphenejohnsonnoise]. The detectivity for the array is calculated incorporating the antenna pitch, noting that there are two antennas per unit cell. To account for the decreased optical absorption efficiency of a metasurface loaded with graphene doped to the Seebeck coefficient peaks of roughly $\pm0.05\,\text{eV}$, we redo the simulation used to generate Figure \[antennas6\] with the graphene doped as such. We plot the resulting absorption spectra in Supplementary Figure 7, which shows a mean absorption efficiency of $33\%$ averaged between $1050\,{\textrm{cm}}^{-1}$ and $1600\,{\textrm{cm}}^{-1}$. We finally calculate the external values of responsitivity, NEP, and detectivity by scaling the internal values by this efficiency factor.
### Acknowledgement
The authors would like to thank Chris Panuski and Dr. Laura Kim of MIT as well as Sebastian Castilla of Institut de Ciències Fotòniques (ICFO) for helpful feedback and insight in the course of preparing this article. This research was funded in part by a grant from the Institute of Soldier Nanotechnologies of MIT. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1122374. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Supporting Information
----------------------
Additional plots to supplement the main text, including:
1. Example wavenumber dependence of the impedances used in Eqn. \[etaeqn\];
2. Graphene absorption efficiency spectra and individual antenna components for metasurfaces of various unit cell pitches;
3. Antenna radiation pattern and diagram of specular reflection-only condition for off-angle excitation;
4. Off-angle graphene absorption spectra for simplified PEC metasurface model;
5. Graphene absorption spectra for metasurfaces with varied numbers $N$ of different antenna sizes;
6. Graphene absorption spectra for a metasurface simulated with different FDTD mesh sizes;
7. Graphene absorption spectra for a metasurface loaded with graphene doped to $0.05\,\text{eV}$.
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---
abstract: 'We clarify a few conceptual problems of quantum field theory on the level of exactly solvable models with fermions. The ultimate goal of our study is to gain a deeper understanding of differences between the usual (“spacelike”) and light-front forms of relativistic dynamics. We show that by incorporating operator solutions of the field equations to the canonical formalism the spacelike and light front Hamiltonians of the derivative-coupling model acquire an equivalent structure. The same is true for the massive solvable theory, the Federbush model. In the conventional approach, physical predictions in the two schemes disagree. Moreover, the derivative-coupling model is found to be almost identical to a free theory, in contrast to the conventional canonical treatment. Physical vacuum state of the Thirring model is then obtained by a Bogoliubov transformation as a coherent state quadratic in composite boson operators. To perform the same task in the Federbush model, we derive a massive version of Klaiber’s bosonization and show that its light-front form is much simpler.'
author:
- 'L$\!\!$’ubomír Martinovic'
- Pierre Grangé
title: Hamiltonian formulation of exactly solvable models and their physical vacuum states
---
The usual “spacelike” (SL) and the lightfront (LF) [@Dir] forms of relativistic quantum field theory (QFT) are two independent representations of the same physical reality. There are however striking differences between both schemes already at the level of basic properties [@LKS; @LS]. This concerns the mathematical structure as well as some physical aspects (nature of field variables, division of the Poincaré generators into the kinematical and dynamical sets, status of the vacuum state, etc.) Exactly solvable models offer an opportunity to study the structure of the two theoretical frameworks and their relationship since in these models exact operator solutions of field equations are known. From the solutions, essential properties like correlation functions can be computed nonperturbatively and independently of conformal QFT methods [@Zam]. Note that not all solvable models belong to the conformal class. Investigations of their properties in a hamiltonian approach permits us to study directly the role of the vacuum state and of the operator structures in both forms of QFT. Recall that in the LF scheme, Fock vacuum is often the lowest-energy eigenstate of the [*full*]{} Hamiltonian. This feature is not present in the SL theory. In this letter, we give a brief survey of the hamiltonian study of the derivative-coupling model (DCM)[@Schr], the Thirring [@Thir] and the Federbush model (FM) [@Fed]. The unifying idea is to benefit from the knowledge of operator solutions of the field equations to re-express the corresponding Hamiltonians (both in the SL and LF versions) entirely in terms of true degrees of freedom which are the free fields. This previously overlooked aspect not only simplifies the overall physical picture but also removes structural differences between SL and LF Hamiltonians. For example, in the case of the simplest theory, the derivative-coupling model, the standard canonical procedure applied to the SL and LF Lagrangians leads to a striking result: the SL Hamiltonian contains an interaction term while its LF analog does not. Making use of exact solutions of the field equations, the SL version of the DCM Hamiltonian is found to have also the interaction-free form. Consequently, the physical SL vacuum coincides with the Fock vacuum in a full agreement with the LF result. However, for truly interacting models, the Fock vacuum is an eigenstate of the free SL Hamiltonians only. Interaction parts of the Hamiltonians are generally non-diagonal when expressed in terms of creation and annihilation operators. To find the true vacuum state, they have to be diagonalized. This is a complicated dynamical problem which however turns out to be tractable analytically for the Thirring and Federbush models. Our approach is to cast their Hamiltonians to the quadratic form by bosonization of the vector current and to diagonalize them by a Bogoliubov transformation, generating thereby the true ground state as a transformed Fock vacuum (a coherent state). We will show this explicitly for the Thirring model. As for the FM, the conventional approach yields a vanishing interaction Hamiltonian for the LF case and a nonvanishing one for the SL case. This discrepancy is removed when the solution of field equations is taken into account leading to interaction Hamiltonians of the same structure. Finally, note that the solvability of the (conformally-noninvariant) massive FM allows one also to test the methods of CFT where the mass term is treated as a perturbation [@Zam]. [**Derivative-coupling model.**]{} It is instructive to explain our main ideas at a very simple theory – a massive fermion field interacting with a massive scalar field via a gradient coupling. Its Lagrangian and field equations are &&=(\^ - m)+ ( - \^2 \^2) - gJ\^,\
&&i\^= m+ g\^,\
&&+ \^2 = gJ\^= 0. \[DCML\] Here $\gg^0=\gs^1,\gg^1=i\gs^2,~\ga^1=\gg^5=\gg^0\gg^1$ and $\gs^i$ are the Pauli matrices. $J^\mu(x)$ is the (normal-ordered) vector current. The original Schroer’s model [@Schr] had $\mu=0$. Using the notation $d\tilde{k}^1 = dk^1/\sqrt{4\gp E(k^1)}$, $\hat{k}.x = E(k^1)t - k^1x^1$, $E(k^1) = \sqrt{k_1^2 + \mu^2}$, the free scalar field, quantized by $[a(k^1),a^\dagger(l^1)]=\gd(k^1-l^1)$, is expanded as &&(x) = \^[+]{}\_[-]{} d\^1. It enters into the operator solution of the equation (\[DCML\]): (x) = :e\^[ig(x)]{}:(x), i\^(x) = m(x). \[opsol\] The Fock expansion of the free massive fermion field $\psi(x)$ (x) = \^[+]{}\_[-]{} [d\^1]{} \[mfexp\] contains the spinors $u^\dagger(p^1) = (\sqrt{p^-},\sqrt{p^+})$, $v^\dagger(p^1) = u^\dagger(p^1)\gg_5$, $p^\pm = E(p^1) \pm p^1$. The conjugate momenta \_= \_0(x) - gJ\^0, \_= \^, \_ = -lead to the Hamiltonian $H = H_{0B} + H^{'}$, where $H_{0B}$ corresponds to the free massive scalar field and &&H\^[’]{} = . Since the term $(i/2)\Psib\gg^\mu\delrlmu\Psi$ in the Lagrangian is conventionally taken in terms of the free field, the first term in $H^\prime$ becomes simply $-i\psid\ga^1\partial_1
\psi$. Setting $m=0$ for simplicity, the interaction Hamiltonian acquires the form && H\_g = -\_[-]{}\^[+]{} , \[wroh\] where the Klaiber’s bosonized representation [@Klaib] of the massless vector current was used (see Eq.(\[bocur\]) below). The Hamiltonian (\[wroh\]) is nondiagonal. Its diagonalization can be performed by a Bogoliubov transformation implemented by a unitary operator $U = \exp(iS)$ with iS() = (k\^1). As a result, the real ground state has nontrivial structure, $\vert \tilde{\gO}\rangle =
\exp\big(-iS(\gg_d)
\big)\vert 0 \rangle$ [@foot] (cf. Eqs.(\[cshf\]-\[vacsim\]) below). All this is true provided the starting Hamiltonian is the correct one. However, this is not the case. The point is that we did not use our knowledge of the operator solution (\[opsol\]) in the course of the derivation of the Hamiltonian (\[wroh\]). In the case of models which are not exactly solvable, one knows $\gg^\mu\delmu\Psi(x)$ from the corresponding Dirac equation. This expression should not be used in the Lagrangian since the latter would vanish (extremum of the action). In our case, we actually know more, namely $\delmu\Psi(x)$ from the solution (\[opsol\]) and this information (not the free Dirac equation, however!) [ *should*]{} be used in the starting Lagrangian (\[DCML\]). This is similar to elimination of a nondynamical field by using its constraint in the Lagrangian. Thus, the correct procedure in our case is to insert the solution for the interacting field $\Psi(x)$ into the kinetic term. We find that the interaction cancels completely in the Lagrangian (\[DCML\]) and we get $H=H_{0B}+H_{0F}$, H\_[0F]{} = . Although the full Hamiltonian is simply the sum of free Hamiltonians of the massive scalar and fermion fields and hence the ground state of the DCM is just the Fock vacuum, the model is not completely trivial: one generates the correct Heisenberg equations $i\partial_0\Psi(x) = -[H,\Psi(x)] = -i\ga^1\partial_1\Psi +
m\gg^0\Psi
- g\partial_0\phi\Psi
-g\ga^1\partial_1\phi\Psi$ with $H$. Correlation functions computed from the solution (\[opsol\]) will depend on the coupling constant, too. Note also that in the usual treatment the momentum operator contains interaction. This deffect is cured in our approach.
The same picture is obtained in the LF analysis. Our notation is $\psi^\dg =(\psild,\psihd)$, $x^\mu=(\xpl,\xmin)$, $\partial_{\pm} = \partial/\partial x^{\pm}$, $\hat{p}.x = 1/2(\hat{p}^-\xpl + p^+
\xmin)$, $\hat{p}^-=m^2/p^+$, where $x^+,p^+$ and $j^\mu = (j^+,j^-)$ are the LF time, momentum and current. Inserting now the solution (\[opsol\]) of the field equations &&2i\_2(x) = m\_1(x) +2g(x)\_2(x),\
&&2i\_1(x) = m\_2(x) +2g(x)\_1(x) \[lfeq\] into the LF Lagrangian &&\_[lf]{}=2 -\^2\^2 + i+ i\
&& - m(+ ) -gj\^+ - gj\^-, \[lfdcl\] we obtain the Lagrangian of the free LF massive fermion and boson fields with the corresponding LF Hamiltonian P\^- = . \[lfhdcm\] We recall that in the conventional treatment, one gets a controversial picture: the LF Hamiltonian still remains free while the SL Hamiltonian contains an interaction term (\[wroh\]) and its ground state is the coherent state $\vert \tilde{\gO}
\rangle$.
It is interesting that the analogous model with the axial vector current $j^\mu_{5}(x)$ is not solvable. One reason is that $j^\mu_5(x)$ is not conserved, and hence the (pseudo)scalar field is not free. More importantly, the naive generalization of the solution (\[opsol\]) to $\Psi(x) = :\exp\{-ig\gg^5\phi(x)\}:\psi(x)$ actually does not satisfy the corresponding Dirac equation On the other hand, the Rothe-Stamatescu model [@RS] ($m=0$) is exactly solvable but its Hamiltonian is actually a free one, contrary to the claim made in [@RS]. [**The Thirring model**]{} [@Thir] was extensively studied in the past. A detailed analysis of its operator solution has been made in [@Klaib]. A systematic hamiltonian treatment based on the model’s solvability was not given so far. The Lagrangian density of the massless Thirring model and the corresponding field equations read &&[L]{}=\^- gJ\_J\^,\
&&i\^\_(x) = gJ\^(x)\_(x),\
&&J\^=:\^:, \_J\^(x) = 0. \[TL\] The simplest classical solution is &&(x) = e\^[-i(g/)j(x)]{}(x), \^\_(x) = 0, \[tsol\] implying that the interacting current $J^\mu(x)$ coincides with the free current $j^\mu(x)$. The latter is obtained from the operator $j(x)$ as $\sqrt{\gp}j_\mu(x) = \partial_\mu j(x)$ (see below). The expansion of the massless spinor field is the $m =0$ limit of Eq.(\[mfexp\]). After the Fourier transformation, the current $j^\mu(x)=(\psid(x)
\psi(x),$$\psid(x)
\ga^1\psi(x))$ is represented in terms of boson operators: &&j\^(x) = {c(k\^1)e\^[-i.x]{} - c\^(k\^1)e\^[i.x]{}},\
&&c(k\^1) = dp\^1{ ( p\^1k\^1) +\
&& +(p\^1) (p\^1(k\^1-p\^1))d(k\^1-p\^1)b(p\^1)}, \[bocur\] with $c, c^\dagger$ obeying the canonical commutation relation, = (p\^1-q\^1), c(k\^1)0 = 0. The Hamiltonian of the model is obtained from the Lagrangian (\[TL\]) after inserting the solution (\[tsol\]) into it. The interaction is [*not*]{} canceled indicating a less trivial dynamics than found in the DC model. However, the contribution of the term $(i/2)\Psib\gg^\mu\delrlmu\Psi$ just reverses the sign of the interaction term leading to H = . \[THam\] The usual treatment gives $+1/2g$ in Eq.(\[THam\]) [@FI1; @Jap]. In the Fock representation, $H_0$ and $H_{g}$ have the form &&H\_0 = p\^1 ,\
&&H\_[g]{} = \_[-]{}\^ [+]{} dk\^1 k\^1 . \[thmh\] $H_g$ is not diagonal and thus $\vert 0 \rangle$ is not an eigenstate of $H=H_0+H_g$. To diagonalize $H$, we form the new Hamiltonians $\hat{H}_0 = H_0 - T,~~\hat{H}_{g} = H_{g} + T$ [@MaLi], where $T = \intek \vert k^1 \vert c^\dagger(k^1)c(k^1)$, and implement a Bogoliubov transformation by the unitary operator $U=e^{iS}$, &&iS = (p\^1) . \[U\] $\hat{H}_0$ is invariant with respect to $U$. $\hat{H}_{g}$ transforms non-trivially since $\big[S,c(k^1)\big] = i\gg(k^1)c^\dagger(-k^1)$. This, by $c(k^1) \rightarrow e^{iS}c(k^1)e^{-iS}$, implies &&c(k\^1) c(k\^1)(k\^1) - c\^(-k\^1)(k\^1). \[cshf\] The transformed Hamiltonian $e^{iS}\hat{H}_{g}e^{-iS}$ is diagonal, \^d\_[g]{}= k\^1 c\^(k\^1)c(k\^1), \[DH\] if we set $\gg(k^1) = \gg_d = \hlf{\rm artanh}\frac{g}
{\pi}$. Thus we have &&e\^[iS]{}e\^[-iS]{}0 = 0 = e\^[-iS]{}0 . \[ftv\] The new vacuum state $\vert \gO \rangle$ (where $\kappa =
\hlf\tanh\gg_d$), = N 0 , \[vacsim\] corresponds to a coherent state of pairs of composite bosons with zero total momentum, $P^1 \vert \gO \rangle = 0.$ The vacuum $\vert \gO \rangle$ is invariant under axial $U(1)$ transformations &&V()= , V() = e\^[iQ\_5]{},\
&&Q\_5 = (q\^1). \[QQ\] Thus, no chiral symmetry breaking occurs. This finding disagrees with the results [@FI1] where a BCS type of ansatz for the vacuum state was used. The true vacuum has to be an eigenstate of the full Hamiltonian, however. $\vert \gO \rangle$ is such a state while the BCS-like state is not.
Correlation functions can be calculated from the normal-ordered operator solution [@Klaib] (x)=e\^[(-ig/)j\^[(+)]{}(x)]{}(x)e\^[(-ig/) j\^[(-)]{}(x)]{}, using the vacuum $\vert \gO\rangle$. $j^{(\pm)}(x)$ are positive and negative-frequency parts of the integrated current, $j = j^{(+)} + j^{(-)}$, &&j\^[(-)]{}(x) = ( q\^1 -)e\^[-i.x]{}. \[jj\] $j^{(+)}(x) = j^{(-)\dagger}(x)$ and $\eta$ is the conventional infrared cutoff. Further details will be given elsewhere [@lupi]. [**The Federbush model**]{} is defined by the Lagrangian && = \^- m+ \^- -\
&&-g\_J\^H\^, \^ = -\^, \[FEL\] which desribes two species of coupled fermion fields with masses $m$ and $\mu$ [@Fed]. Both currents $J^\mu=
\Psib\gg^\mu\Psi, H^\mu=\Phib\gg^\mu\Phi$ are conserved. The coupled field equations &&i\^\_(x) = m(x) +g\_\^H\^(x)(x),\
&&i\^\_(x) = (x) -g\_\^J\^(x) (x) \[FEQ\] are exactly solvable even for non-zero masses: &&(x) = e\^[-i(g/)h(x)]{}(x), i\^\_(x) = m(x),\
&&(x) = e\^[i(g/)j(x)]{}(x), i\^\_(x) = (x). \[Fsolu\] In quantum theory, the above exponentials have to be regularized by the “triple-dot ordering” [@Wight]. The potentials $j(x), h(x)$ given in terms of the free currents as $\delmu j(x) = \sqrt{\gp}
\ge_{\mu\nu}j^\nu(x)$, $\delmu h(x) = \sqrt{\gp}\ge_{\mu\nu}h^\nu(x)$ enter into the solutions (\[Fsolu\]) in an “off-diagonal” way. After inserting the solutions into the Lagrangian (\[FEL\]), the interaction term changes its sign leading to the Hamiltonian H = H\_0 +g(j\^0h\^1 - j\^1h\^0). \[FEH\] where $H_0$ is the sum of two free fermion Hamiltonians. The LF field equations are also solved by (\[Fsolu\]) with the free LF fields $\psi(x)$, $\varphi(x)$; $j(x),h(x)$ are given by $2\delm j(x) = \sqrt{\gp}j^+(x)$, $2\delm h(x) = \hlf\sqrt{\gp}h^+(x)$. In the standard LF treatment, one would simply insert the solution of the fermionic constraint into ${\cal L}$. This yields however the free LF Hamiltonian! It is only after inserting the full solution like in the SL case that one obtains the four-fermion interaction term also in the LF case: P\^-\_g=g(j\^+ h\^- - j\^-h\^+). \[Flfh\] The interacting SL Hamitonian (\[FEH\]) contains terms composed solely from creation or annihilation operators, so the Fock vacuum is not its eigenstate. The diagonalization can be performed by a Bogoliubov transformation using a [*massive*]{} current bosonization. This is considerably more complicated than the massless case [@Klaib] but the generalized operators $c(k^1)$ can be derived [@lupi]. The analogous LF operators $A,A^\dagger$ are much simpler and have a structure similar to the massless SL case (\[bocur\]). The bosonized LF current is similar to the SL current (\[bocur\]): j\^[+]{}(x) = \_0\^ k\^+ A(k\^+,)e\^[-k\^+]{} + H.c. \[lfmasb\] Due to $[A(k^+),A^\dagger(l^+)] = \gd(k^+ - l^+)$, valid at $\xpl=\ypl$, the solution (\[Fsolu\]) can be easily normal-ordered: &&(x) = {-i \^(x) } {-i(x)}(x),\
&&(x) = \_0\^ A(k\^+,)e\^[k\^+]{}. \[lfnos\] Similar formulae hold for the solution $\Psi(x)$ built from the operators $B(k^+,\xpl), B^\dagger(k^+,\xpl)$ which are constructed from $h^+(x)$. The $j^-$ and $h^-$ currents contain the boson operators $C(k^+,\xpl), D(k^+,\xpl)$ and their conjugates, related to $A,A^\dg,B,B^\dg$ via the current conservation. In contrast to its SL analog, the interacting LF Hamiltonian is diagonal and therefore $ \vert 0 \rangle$ is its lowest-energy eigenstate: &&P\^-\_g = k\^+ . \[lfhf\] Diagonalization of the bosonized SL Hamiltonian yielding the true SL vacuum state $\vert \gO \rangle$ will be given in [@lupi].
The next step will be to compute the correlation functions in both schemes. This task is not simple since one needs to know the commutators of the composite boson operators at unequal times [@lupi]. This is the place where complexities of the usual triple-dot ordering technique [@ST] enter into our bosonization approach. The knowledge of exact correlation functions will allow us to get a deeper insight into the relation between the SL and LF forms of the Federbush model and of QFT in general.
[**Acknowledgements.**]{} This work was supported by the VEGA grant No.2/0070/2009, by the Slovak CERN Commission and by IN2P3 funding at the Université Montpellier II.
[99]{} P. A. M. Dirac, Rev. M. Phys. 21, 392 (1949). H. Leutwyler, J. R. Klauder and L. Streit, Nuovo Cim. A66, 536 (1970). H. Leutwyler and J. Stern, Ann. Phys. 112, 94 (1978). A. Belavin, A. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241, 333 (1984). B. Schroer, Fort. Physik 1, 1 (1962). W. E. Thirring, Ann. Phys. 3 (1958) 91. K. Federbush. Phys. Rev. 121, 1247 (1961). B. Klaiber, in [Lectures in Theoretical Physics]{}, Vol. Xa, New York (1968), p. 141. Our discussion here is a little heuristic for simplicity. Some operators have to be regularized by a cutoff to exist mathematically. A more rigorous treatment with the same physical conclusions based on fields as operator-valued distributions [@GW] will be given in [@lupi]. Also, our results remain unaltered when the products of the fermion fields are regularized by point-splitting. K. D. Rothe and I. O. Stamatescu, Ann. Phys. 95 (1975) 202. P. Grangé and E. Werner, Nucl. Phys. B (Proc. Suppl.) 161 75 (2006). L. Martinovic and P. Grangé, to published. M. Faber and A. N. Ivanov, Eur. Phys. J. C 20 (2001) 723, Phys. Lett. B563 (2003) 231. T. Fujita, M. Hiramoto T. Homma and H. Takahashi, J. Phys. Soc. Jap. 74 (2005) 1143, hep-th/0410221. D. C. Mattis and E. H. Lieb, J. M. Phys. 6 (1965) 304. A. Wightman, in [High Energy Interactions and Field Theory]{}, Cargese Lect. in Theor. Physics (1964), p.171. B. Schroer, T.Truong and P. Weisz, Ann. Phys. 102, 156 (1976).
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---
abstract: 'We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $W_8$ and $W_9$ singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes.'
address: |
Warsaw University of Technology\
Faculty of Mathematics and Information Science\
Plac Politechniki 1\
00-661 Warsaw\
Poland\
author:
- Żaneta Trȩbska
title: 'Symplectic $W_8$ and $W_9$ singularities'
---
Introduction
============
In this paper we study the symplectic classification of singular curves under the following equivalence:
\[symplecto\] Let $N_1, N_2$ be germs of subsets of symplectic space $(\mathbb{R}^{2n}, \omega)$. $N_1, N_2$ are **symplectically equivalent** if there exists a symplectomorphism-germ $\Phi:(\mathbb{R}^{2n}, \omega) \rightarrow(\mathbb{R}^{2n}, \omega)$ such that $\Phi(N_1)=N_2$.
We recall that $\omega$ is a symplectic form if $\omega$ is a smooth nondegenerate closed 2-form, and $\Phi:\mathbb{R}^{2n}
\rightarrow\mathbb{R}^{2n}$ is a symplectomorphism if $\Phi$ is diffeomorphism and $\Phi ^* \omega=\omega$.
Symplectic classification of curves was first studied by V. I. Arnold. In [@Ar1] and [@Ar2] the author studied singular curves in symplectic and contact spaces and introduced the local symplectic and contact algebra. In [@Ar2] V. I. Arnold discovered new symplectic invariants of singular curves. He proved that the $A_{2k}$ singularity of a planar curve (the orbit with respect to standard $\mathcal A$-equivalence of parameterized curves) split into exactly $2k+1$ symplectic singularities (orbits with respect to symplectic equivalence of parameterized curves). He distinguished different symplectic singularities by different orders of tangency of the parameterized curve to the *nearest* smooth Lagrangian submanifold. Arnold posed a problem of expressing these invariants in terms of the local algebra’s interaction with the symplectic structure and he proposed to call this interaction the local symplectic algebra.
In [@IJ1] G. Ishikawa and S. Janeczko classified symplectic singularities of curves in the $2$-dimensional symplectic space. All simple curves in this classification are quasi-homogeneous.
We recall that a subset $N$ of $\mathbb R^m$ is **quasi-homogeneous** if there exists a coordinate system $(x_1,\cdots,x_m)$ on $\mathbb R^m$ and positive numbers $w_1,\cdots,w_m$ (called weights) such that for any point $(y_1,\cdots,y_m)\in \mathbb R^m$ and any $t>0$ if $(y_1,\cdots,y_m)$ belongs to $N$ then the point $(t^{w_1}y_1,\cdots,t^{w_m}y_m)$ belongs to $N$.
The generalization of results in [@IJ1] to volume-preserving classification of singular varieties and maps in arbitrary dimensions was obtained in [@DR]. A symplectic form on a $2$-dimensional manifold is a special case of a volume form on a smooth manifold. In [@K] P. A. Kolgushkin classified the stably simple symplectic singularities of parameterized curves (in the $\mathbb C$-analytic category). Symplectic singularity is stably simple if it is simple and remains simple if the ambient symplectic space is symplectically embedded (i.e. as a symplectic submanifold) into a larger symplectic space. In [@Zh] was developed the local contact algebra. The main results were based on the notion of the algebraic restriction of a contact structure to a subset $N$ of a contact manifold.
In [@DJZ2] new symplectic invariants of singular quasi-homogeneous subsets of a symplectic space were explained by the algebraic restrictions of the symplectic form to these subsets.
The algebraic restriction is an equivalence class of the following relation on the space of differential $k$-forms:
Differential $k$-forms $\omega_1$ and $\omega_2$ have the same [**algebraic restriction**]{} to a subset $N$ if $\omega_1-\omega_2=\alpha+d\beta$, where $\alpha$ is a $k$-form vanishing on $N$ and $\beta$ is a $(k-1)$-form vanishing on $N$.
In [@DJZ2] the generalization of Darboux-Givental theorem ([@ArGi]) to germs of arbitrary subsets of the symplectic space was obtained. This result reduces the problem of symplectic classification of germs of quasi-homogeneous subsets to the problem of classification of algebraic restrictions of symplectic forms to these subsets. For non-quasi-homogeneous subsets there is one more cohomological invariant except the algebraic restriction ([@DJZ2], [@DJZ1]). The dimension of the space of algebraic restrictions of closed $2$-forms to a $1$-dimensional quasi-homogeneous isolated complete intersection singularity $C$ is equal to the multiplicity of $C$ ([@DJZ2]). In [@D] it was proved that the space of algebraic restrictions of closed $2$-forms to a $1$-dimensional (singular) analytic variety is finite-dimensional. In [@DJZ2] the method of algebraic restrictions was applied to various classification problems in a symplectic space. In particular the complete symplectic classification of classical $A-D-E$ singularities of planar curves and $S_5$ singularity were obtained. Most of different symplectic singularity classes were distinguished by new discrete symplectic invariants: the index of isotropy and the symplectic multiplicity.
In this paper using the method of algebraic restrictions we obtain the complete symplectic classification of the classical isolated complete intersection singularities $W_8$ and $W_9$ (Theorems \[w8-main\] and \[w9-main\]). We calculate discrete symplectic invariants for this classification (Theorems \[lagr-w8\] and \[lagr-w9\]) and we present geometric descriptions of its symplectic orbits (Theorems \[geom-cond-w8\] and \[geom-cond-w9\]).
Isolated complete intersection singularities were intensively studied by many authors (e. g. see [@L]), because of their interesting geometric, topological and algebraic properties.
In [@DT1] following ideas from [@Ar1] and [@D] new discrete symplectic invariants - the Lagrangian tangency orders were introduced and used to distinguish symplectic singularities of simple planar curves of type $A-D-E$, symplectic $T_7$ and $T_8$ singularities. In [@DT2] was obtained the complete symplectic classification of the isolated complete intersection singularities $S_{\mu}$ for $\mu>5$.
The paper is organized as follows. In Section \[discrete\] we recall discrete symplectic invariants (the symplectic multiplicity, the index of isotropy and the Lagrangian tangency orders). Symplectic classification of $W_8$ and $W_9$ singularity is studied in Sections \[sec-w8\] and \[sec-w9\] respectively. In Section \[proofs\] we recall the method of algebraic restrictions and use it to classify symplectic singularities.
Discrete symplectic invariants. {#discrete}
===============================
We can use discrete symplectic invariants to characterize symplectic singularity classes.
The first invariant is a symplectic multiplicity ([@DJZ2]) introduced in [@IJ1] as a symplectic defect of a curve.
Let $N$ be a germ of a subvariety of $(\mathbb R^{2n},\omega)$.
\[def-mu\] The [**symplectic multiplicity**]{} $\mu_{symp}(N)$ of $N$ is the codimension of the symplectic orbit of $N$ in the orbit of $N$ with respect to the action of the group of local diffeomorphisms.
The second invariant is the index of isotropy [@DJZ2].
The [**index of isotropy**]{} $ind(N)$ of $N$ is the maximal order of vanishing of the $2$-forms $\omega \vert _{TM}$ over all smooth submanifolds $M$ containing $N$.
This invariant has geometrical interpretation. An equivalent definition is as follows: the index of isotropy of $N$ is the maximal order of tangency between non-singular submanifolds containing $N$ and non-singular isotropic submanifolds of the same dimension. The index of isotropy is equal to $0$ if $N$ is not contained in any non-singular submanifold which is tangent to some isotropic submanifold of the same dimension. If $N$ is contained in a non-singular Lagrangian submanifold then the index of isotropy is $\infty $.
The symplectic multiplicity and the index of isotropy can be described in terms of algebraic restrictions (Propositions \[sm\] and \[ii\] in Section \[proofs\]).
There is one more discrete symplectic invariant introduced in [@D] following ideas from [@Ar2] which is defined specifically for a parameterized curve. This is the maximal tangency order of a curve $f:\mathbb R\rightarrow M$ to a smooth Lagrangian submanifold. If $H_1=...=H_n=0$ define a smooth submanifold $L$ in the symplectic space then the tangency order of a curve $f:\mathbb R\rightarrow M$ to $L$ is the minimum of the orders of vanishing at $0$ of functions $H_1\circ f,\cdots, H_n\circ f$. We denote the tangency order of $f$ to $L$ by $t(f,L)$.
The [**Lagrangian tangency order**]{} $Lt(f)$ **of a curve** $f$ is the maximum of $t(f,L)$ over all smooth Lagrangian submanifolds $L$ of the symplectic space.
The Lagrangian tangency order of the quasi-homogeneous curve in a symplectic space can also be expressed in terms of algebraic restrictions (Proposition \[lto\] in Section \[proofs\]).
In [@DT1] the above invariant was generalized for germs of curves and multi-germs of curves which may be parameterized analytically since the Lagrangian tangency order is the same for every ’good’ analytic parameterization of a curve.
Consider a multi-germ $(f_i)_{i\in\{1,\cdots,r\}}$ of analytically parameterized curves $f_i$. We have $r$-tuples $(t(f_1,L), \cdots,
t(f_r,L))$ for any smooth submanifold $L$ in the symplectic space.
For any $I\subseteq \{1,\cdots, r\}$ we define **the tangency order of the multi-germ** $(f_i)_{i\in I}$ to $L$: $$t[(f_i)_{i\in\ I},L]=\min_{i\in\ I} t(f_i,L).$$
The [**Lagrangian tangency order**]{} $Lt((f_i)_{i\in\ I})$ **of a multi-germ** $(f_i)_{i\in I}$ is the maximum of $t[(f_i)_{i\in\ I},L]$ over all smooth Lagrangian submanifolds $L$ of the symplectic space.
Symplectic $W_8$-singularities {#sec-w8}
==============================
Denote by $(W_8)$ the class of varieties in a fixed symplectic space $(\mathbb R^{2n}, \omega )$ which are diffeomorphic to $$\label{defw8} W_8=\{x\in \mathbb R ^{2n\geq
4}\,:x_1^2+ x_3^3=x_2^2+x_1x_3=x_{\geq 4}=0\}.$$
This is simple $1$-dimensional isolated complete intersection singularity $W_8$ ([@G], [@AVG][^1]). Here $N$ is a quasi-homogeneous set with weights $w(x_1)=6,\; w(x_2)=5$, $w(x_3)=4$.
We used the method of algebraic restrictions to obtain a complete classification of symplectic singularities of $(W_8)$ presented in the following theorem.
\[w8-main\] Any submanifold of the symplectic space $(\mathbb R^{2n},\sum_{i=1}^n dp_i \wedge dq_i)$ where $n\geq3$ (resp. $n=2$) which is diffeomorphic to $W_8$ is symplectically equivalent to one and only one of the normal forms $W_8^i, i = 0,1,\cdots ,8$ (resp. $i=0,1,2a,2b)$ listed below. The parameters $c, c_1, c_2$ of the normal forms are moduli.
$W_8^0: \ p_1^2 + p_2q_1 = 0, \ \ p_2^2 +q_1^3= 0, \
\ q_2 = c_1q_1 + c_2p_1, \ \ p_{\ge 3} = q_{\ge 3} = 0$;
$W_8^1: \ q_1^2 + p_1q_2 = 0, \ \ p_1^2 + q_2^3= 0, \
\ p_2 = c_1p_1 + c_2q_1q_2, \ \ p_{\ge 3} = q_{\ge 3} = 0, \; c_1\ne 0$;
$W_8^{2a}: \ p_2^2 \pm p_1q_1 = 0, \ \ p_1^2+q_1^3 = 0, \
\ q_2 = \frac{c_1}{2}q_1^2+\frac{c_2}{3}q_1^3, \ \ p_{\ge 3} = q_{\ge 3} = 0$;
$W_8^{2b}: \ q_1^2 + p_1q_2 = 0, \ \ p_1^2 + q_2^3= 0, \
\ p_2 = c_1q_1q_2 + \frac{c_2}{2}q_1^2, \ \ p_{\ge 3} = q_{\ge 3} = 0$;
$W_8^3: \ p_2^2+p_1p_3=0, \ p_1^2+p_3^3=0, \ q_1=q_2=0, q_3=-p_2p_3-\frac{c_1}{2}p_2^2-c_2p_1p_2, \ p_{>3}=q_{>3}=0$;
$W_8^4: \ p_2^2 + p_1p_3 = 0, \ p_1^2 +p_3^3 = 0, \ q_1 = q_2=0, q_3= \mp \frac{1}{2}p_2^2-c_1p_1p_2-c_2p_2p_3^2, \ p_{>3} = q_{>3} = 0$;
$W_8^5: \ p_2^2 + p_1p_3 = 0, \ p_1^2 +p_3^3 = 0, \ q_1 = q_2=0, q_3= -p_1p_2-cp_2p_3^2, \ p_{>3} = q_{>3} = 0$;
$W_8^6: \ p_2^2 + p_1p_3 = 0, \ p_1^2 +p_3^3 = 0, \ q_1 = q_2=0, q_3= -p_2p_3^2-\frac{c}{3}p_2^3, \ p_{>3} = q_{>3} = 0$;
$W_8^7: \ p_2^2 + p_1p_3 = 0, \ p_1^2 +p_3^3 = 0, \ q_1 = q_2=0, q_3= -\frac{1}{3}p_2^3, \ p_{>3} = q_{>3} = 0$;
$W_8^8: \ p_2^2 + p_1p_3 = 0, \ p_1^2 +p_3^3 = 0, \ p_{>3} = q_{>0} = 0$.
In Section \[w8-lagr\] we use symplectic invariants (in particular the Lagrangian tangency order) to distinguish symplectic singularity classes. In Section \[w8-geom\_cond\] we propose a geometric description of these singularities which confirms the classification. Some of the proofs are presented in Section \[proofs\].
Distinguishing symplectic classes of $W_8$ by Lagrangian tangency order and the index of isotropy {#w8-lagr}
-------------------------------------------------------------------------------------------------
A curve $N\in (W_8)$ can be described as a parameterized curve $C(t)$. Its parameterization is given in the second column of Table \[tabw8-lagr\]. To characterize the symplectic classes we use the following invariants:
- $L_N=Lt(N)=\max\limits _L (t(C(t),L)),$
where $L$ is a smooth Lagrangian submanifold of the symplectic space,
- $ind$ - the index of isotropy of $N$.
The invariants can be calculated knowing algebraic restrictions for symplectic classes. We used Proposition \[ii\] to calculate the index of isotropy. The Lagrangian tangency order we can calculate using Proposition \[lto\] or by applying directly the definition of the Lagrangian tangency order and finding a Lagrangian submanifold nearest to the curve $C(t)$.
\[lagr-w8\] A stratified submanifold $N\in (W_8)$ of a symplectic space $(\mathbb R^{2n}, \omega_0)$ with the canonical coordinates $(p_1, q_1, \cdots, p_n, q_n)$ is symplectically equivalent to one and only one of the curves presented in the second column of Table \[tabw8-lagr\]. The parameters $c, c_1, c_2$ are moduli. The index of isotropy and the Lagrangian tangency order of the curve $N$ are presented respectively in the third and fourth column of Table \[tabw8-lagr\].
Class Parameterization of $N$ $ind$ $L_N$
------------------------ --------------------------------------------------------------------- ---------- ----------
$(W_8)^0$ $2n\ge 4$ $(t^5,-t^4,t^6,-c_1t^4+c_2t^5,0,\cdots )$ $0$ $5$
$(W_8)^1$ $2n\ge 4$ $(t^6,t^5,c_1 t^6-c_2t^9,-t^4,0,\cdots )$ $0$ $6$
$(W_8)^{2a}$ $2n\ge 4$ $(\pm t^6,-t^4,t^5,\frac{c_1}{2}t^8-\frac{c_2}{3}t^{12},0,\cdots )$ $0$ $6$
$(W_8)^{2b}$ $2n\ge 4$ $(t^6,t^5,-c_1t^9+\frac{c_2}{2}t^{10},-t^4,,0,\cdots )$ $0$ $6$
$(W_8)^3$ $2n\ge 6$ $(t^6,0,t^5,0, -t^4,t^9-\frac{c_1}{2}t^{10}-c_2t^{11},0,\cdots )$ $1$ $9$
$(W_8)^4$ $2n\ge 6$ $(t^6,0,t^5,0, -t^4,\mp t^{10}-c_1t^{11}-c_2t^{13},0,\cdots )$ $1$ $10$
$(W_8)^5$ $2n\ge 6$ $(t^6,0,t^5,0, -t^4,-t^{11}-ct^{13},0,\cdots )$ $1$ $11$
$(W_8)^6$ $2n\ge 6$ $(t^6,0,t^5,0, -t^4,-t^{13}-\frac{c}{3}t^{15},0,\cdots )$ $2$ $13$
$(W_8)^7$ $2n\ge 6$ $(t^6,0,t^5,0, -t^4,-t^{15},0,\cdots )$ $2$ $15$
$(W_8)^8$ $2n\ge 6$ $(t^6,0,t^5,0, -t^4,0,0,\cdots )$ $\infty$ $\infty$
: The symplectic invariants for symplectic classes of $W_8$ singularity.[]{data-label="tabw8-lagr"}
The comparison of invariants presented in Table \[tabw8-lagr\] shows that the Lagrangian tangency order distinguishes more symplectic classes than the index of isotropy. Symplectic classes $(W_8)^{2a}$ and $(W_8)^{2b}$ can not be distinguished by any of the invariants but we can distinguish them by geometric conditions.
Geometric conditions for the classes $(W_8)^i$ {#w8-geom_cond}
----------------------------------------------
We can characterize the symplectic classes $(W_8)^i$ by geometric conditions without using any local coordinate system.
Let $N\in (W_8)$. Denote by $W$ the tangent space at $0$ to some (and then any) non-singular $3$-manifold containing $N$. We can define the following subspaces of this space: $\ell$ – the tangent line at $0$ to the curve $N$, $V$ – the $2$-space tangent at $0$ to the curve $N$.
The classes $(W_8)^i$ satisfy special conditions in terms of the restriction $\omega\vert_ W $, where $\omega$ is the symplectic form. For $N=W_8=$(\[defw8\]) it is easy to calculate $$\label{linesw8}
W\!=\!\operatorname{span}(\partial /\partial x_1,\partial /\partial x_2,\partial /\partial x_3), \
\ell\!=\!\operatorname{span}(\partial /\partial x_3), \
V\!=\!\operatorname{span}(\partial /\partial x_2,\partial /\partial x_3).$$
\[geom-cond-w8\]If a stratified submanifold $N\in (W_8)$ of a symplectic space $(\mathbb R^{2n}, \omega)$ belongs to the class $(W_8)^i$ then the couple $(N, \omega)$ satisfies the corresponding conditions in the last column of Table \[tabw8-geom\].
Class Normal form Geometric conditions
-------------- ----------------------------------------------------------------------- -----------------------------------------------------
$(W_8)^0$ $[W_8]^0: [\theta _1 + c_1\theta _2 + c_2\theta _3]_{W_8}$ $ \omega|_V \neq 0$
$(W_8)^1$ $[W_8]^1: [c_1\theta _2 + \theta _3 + c_2\theta _4]_{W_8}$,$c_1\ne 0$ $\omega|_V=0$ and $\ker\omega \ne \ell$
$(W_8)^{2a}$ $[W_8]^{2a}: [\pm\theta _2 + c_1\theta _4 + c_2\theta _7]_{W_8}$ $\omega|_V=0$ and $\ker\omega \ne \ell$
$(W_8)^{2b}$ $[W_8]^{2b}: [\theta _3 + c_1\theta _4 + c_2\theta _5]_{W_8}$ $\omega|_V=0$ and $\ker\omega = \ell$
$\omega\vert_ W = 0$
$(W_8)^3$ $[W_8]^3: [\theta _4 + c_1\theta_5+c_2\theta_6]_{W_8}$ $L_N=9$
$(W_8)^4$ $[W_8]^4: [\pm\theta _5 + c_1\theta_6+c_2\theta_7]_{W_8}$ $L_N=10$
$(W_8)^5$ $[W_8]^5: [\theta_6+c\theta_7]_{W_8}$ $L_N=11$
$(W_8)^6$ $[W_8]^6: [\theta_7+c\theta_8]_{W_8}$ $L_N=13$
$(W_8)^7$ $[W_8]^7: [\theta_8]_{W_8}$ $L_N=15$
$(W_8)^8$ $[W_8]^8: [0]_{W_8}$ $N$ is contained in a smooth Lagrangian submanifold
: Geometric interpretation of singularity classes of $W_8$: $W$ is the tangent space to a non-singular 3-dimensional manifold in $(\mathbb R^{2n\geq4}, \omega)$ containing $N\in(W_8)$. []{data-label="tabw8-geom"}
We have to show that the conditions in the row of $(W_8)^i$ are satisfied for any $N\in (W_8)^i$.
Each of the conditions in the last column of Table \[tabw8-geom\] is invariant with respect to the action of the group of diffeomorphisms in the space of pairs $(N,\omega)$. Because each of these conditions depends only on the algebraic restriction $[\omega ]_N$ we can take the simplest $2$-forms $\omega ^i$ representing the normal forms $[W_8]^i$ for algebraic restrictions: $\omega ^0, \ \omega ^1, \ \omega ^{2,a},\ \omega ^{2,b}, \ \omega ^3, \ \omega ^4, \ \omega ^5, \ \omega ^6, \ \omega ^7, \ \omega ^8$ and we can check that the pair $(W_8,\omega=\omega ^i)$ satisfies the condition in the last column of Table \[tabw8-geom\].
We note that in the case $N = W_8 = (\ref{defw8})$ one has the description (\[linesw8\]) of the subspaces $W, \ell$ and $V$. By simple calculation and observation of the Lagrangian tangency orders we obtain that the conditions corresponding to the classes $(W_8)^i$ are satisfied.
Symplectic $W_9$-singularities {#sec-w9}
==============================
Denote by $(W_9)$ the class of varieties in a fixed symplectic space $(\mathbb R^{2n}, \omega )$ which are diffeomorphic to $$\label{defw9} W_9=\{x\in \mathbb R ^{2n\geq
4}\,:x_1^2+ x_2x_3^2=x_2^2+x_1x_3=x_{\geq 4}=0\}.$$
This is simple $1$-dimensional isolated complete intersection singularity $W_9$ ([@G], [@AVG]). Here $N$ is quasi-homogeneous with weights $w(x_1)\!=\!5,\, w(x_2)\!=\!4,\, w(x_3)\!=\!3$.
We present a complete classification of symplectic singularities of $(W_9)$ which was obtained using the method of algebraic restrictions.
\[w9-main\] Any submanifold of the symplectic space $(\mathbb R^{2n},\sum_{i=1}^n dp_i \wedge dq_i)$ where $n\geq3$ (resp. $n=2$) which is diffeomorphic to $W_9$ is symplectically equivalent to one and only one of the normal forms $W_9^i, i = 0,1,\cdots ,9$ (resp. $i=0,1,2$) listed below. The parameters $c, c_1, c_2$ of the normal forms are moduli.
$W_9^0: \ p_1^2 + p_2q_2^2 = 0, \ \ p_2^2 +p_1q_2= 0, \
\ q_1 = c_1q_2 + c_2p_2, \ \ p_{\ge 3} = q_{\ge 3} = 0$;
$W_9^1: \ p_1^2 + p_2q_1^2 = 0, \ \ p_2^2 \pm p_1q_1= 0, \
\ q_2 = -c_1p_1 + \frac{c_2}{2}q_1^2, \ \ p_{\ge 3} = q_{\ge 3} = 0$;
$W_9^2: \ p_1^2 + q_1p_2^2 = 0, \ \ q_1^2+p_1p_2 = 0, \
\ q_2 = c_1q_1p_2-c_2p_1p_2, \ \ p_{\ge 3} = q_{\ge 3} = 0$;
$W_9^3: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, q_3=\mp p_2p_3-c_1p_1p_3-c_2p_1p_2, \ q_1 = q_2=p_{>3} = q_{>3} = 0$;
$W_9^4: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, \ q_3= -p_1p_3-c_1p_1p_2-c_2p_2p_3^2, q_1=q_2=p_{>3}=q_{>3} = 0$;
$W_9^5: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, \ q_3=\mp p_1p_2-c_1p_2p_3^2-c_2p_1p_3^2, q_1=q_2=p_{>3}=q_{>3}=0$;
$W_9^6: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, \ q_1 = q_2=0, q_3= -p_2p_3^2-cp_1p_3^2, \ p_{>3} = q_{>3} = 0$;
$W_9^7: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, \ q_1 = q_2=0, q_3=\mp p_1p_3^2-cp_2p_3^3, \ p_{>3} = q_{>3} = 0$;
$W_9^8: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, \ q_1 = q_2=0, q_3=\mp p_2p_3^3, \ p_{>3} = q_{>3} = 0$;
$W_9^9: \ p_1^2 + p_2p_3^2 = 0, \ p_2^2 +p_1p_3 = 0, \ p_{>3} = q_{>0} = 0$.
In Section \[w9-lagr\] we use the Lagrangian tangency orders to distinguish symplectic classes. In Section \[w9-geom\_cond\] we propose a geometric description of the symplectic singularities. Some of the proofs are presented in Section \[proofs\].
Distinguishing symplectic classes of $W_9$ by Lagrangian tangency orders {#w9-lagr}
------------------------------------------------------------------------
Lagrangian tangency orders were used to distinguish symplectic classes of $(W_9)$. A curve $N\in (W_9)$ may be described as a union of two parameterized branches: $C_1$ and $C_2$. The curve $C_1$ is nonsingular and the curve $C_2$ is singular. Their parameterization in the coordinate system $(p_1,q_1,p_2,q_2,\cdots,p_n,q_n)$ is presented in the second column of Tables \[tabw9-lagr\]. To characterize the symplectic classes of this singularity we use the following two invariants:
- $L_N=Lt(C_1,C_2)=\max\limits _L (\min \{t(C_1,L),t(C_2,L)\}),$
- $L_{2}=Lt(C_2)=\max\limits _L\, t(C_2,L),$
where $L$ is a smooth Lagrangian submanifold of the symplectic space.
\[lagr-w9\] A stratified submanifold $N\in (W_9)$ of a symplectic space $(\mathbb R^{2n}, \omega_0)$ with the canonical coordinates $(p_1, q_1, \cdots, p_n, q_n)$ is symplectically equivalent to one and only one of the curves presented in the second column of Table \[tabw9-lagr\]. The parameters $c, c_1, c_2$ are moduli. The Lagrangian tangency orders are presented in the third and fourth column of Table \[tabw9-lagr\].
Class Parameterization of branches $L_N$ $L_2$
----------- -------------------------------------------------------------------------------------------------------------- ---------- ----------
$(W_9)^0$ $C_1:(0,c_1t,0,t,0,0,\cdots )$, $C_2:(t^5,-c_1t^3-c_2t^4,-t^4,-t^3,0,\cdots )$ $4$ $4$
$(W_9)^1$ $C_1:(0,\pm t,0,\frac{c_2}{2}t^2,0,\cdots ),$ $C_2:(t^5,\mp t^3,-t^4,-c_1 t^5+\frac{c_2}{2}t^6,0,\cdots )$ $5$ $5$
$(W_9)^2$ $C_1:(0,0, t,0,0,\cdots ),$ $C_2:(t^5,-t^4,-t^3,-c_1t^7+c_2t^8,0,\cdots )$ $5$ $5$
$(W_9)^3$ $C_1:(0,0,0,0,t,0,\cdots ),$ $C_2\!:(t^5,0,-t^4,0, t^3,\mp t^7+c_1t^8+c_2t^9,0,\cdots )$ $7$ $7$
$(W_9)^4$ $C_1:(0,0,0,0,t,0,\cdots ),$ $C_2\!:(t^5,0,-t^4,0, t^3,t^8+c_1t^9+c_2t^{10},0,\cdots )$ $8$ $8$
$(W_9)^5$ $C_1\!:\!(0,0,0,0,t,0,\cdots),$ $C_2\!:(t^5\!,0,-t^4\!,0,t^3\!,\pm t^9+c_1t^{10}-c_2t^{11},0,\cdots )$ $9$ $\infty$
$(W_9)^6$ $C_1:(0,0,0,0,t,0,\cdots ),$ $C_2:(t^5,0,-t^4,0, t^3,t^{10}-ct^{11},0,\cdots )$ $10$ $\infty$
$(W_9)^7$ $C_1:(0,0,0,0,t,0,\cdots ),$ $C_2:(t^5,0,-t^4,0, t^3,\mp t^{11}-ct^{13},0,\cdots )$ $11$ $\infty$
$(W_9)^8$ $C_1:(0,0,0,0,t,0,\cdots ),$ $C_2:(t^5,0,-t^4,0, t^3,\mp t^{13},0,\cdots )$ $13$ $\infty$
$(W_9)^9$ $C_1:(0,0,0,0,t,0,\cdots ),$ $C_2:(t^5,0,-t^4,0, t^3,0,0,\cdots )$ $\infty$ $\infty$
: Lagrangian tangency orders for symplectic classes of $W_9$ singularity.[]{data-label="tabw9-lagr"}
The invariants can be calculated knowing the paramererization of branches $C_1$ and $C_2$. We apply directly the definition of the Lagrangian tangency order finding a Lagrangian submanifold nearest to the branches.
Geometric conditions for the classes $(W_9)^i$ {#w9-geom_cond}
----------------------------------------------
Let $N\in (W_9)$. Denote by $W$ the tangent space at $0$ to some (and then any) non-singular $3$-manifold containing $N$. We can define the following subspaces of this space:
$\ell$ – the tangent line at $0$ to both branches of $N$,
$V$ – $2$-space tangent at $0$ to the singular branch of $N$.
The classes $(W_9)^i$ satisfy special conditions in terms of the restriction $\omega\vert_ W $, where $\omega $ is the symplectic form.
\[geom-cond-w9\] A stratified submanifold $N\in (W_9)$ of a symplectic space $(\mathbb R^{2n}, \omega )$ belongs to the class $(W_9)^i$ if and only if the couple $(N, \omega )$ satisfies the corresponding conditions in the last column of Table \[tabw9-geom\].
Class Normal form Geometric conditions
------------- --------------------------------------------------------------- ----------------------------------------------------------------
$(W_9)^0$ $[W_9]^0: [\theta _1 + c_1\theta _2 + c_2\theta _3]_{W_9}$ $ \omega|_V \neq 0$ (2-space tangent to $N$ is not isotropic)
$(W_9)^1$ $[W_9]^1: [\pm\theta _2 + c_1\theta _3 + c_2\theta _4]_{W_9}$ $\omega|_V=0$ and $\ker\omega \ne \ell$
$(W_9)^{2}$ $[W_9]^2: [\theta _3 + c_1\theta_4+c_2\theta_5]_{W_9}$ $\omega|_V=0$ and $\ker\omega = \ell$
$\omega\vert_ W = 0$
$(W_9)^3$ $[W_9]^3: [\pm\theta _4 +c_1\theta_5+c_2\theta_6 ]_{W_9}$ $L_N=7$
$(W_9)^4$ $[W_9]^4: [\theta _5 + c_1\theta _6+c_2\theta _7]_{W_9}$ $L_N=8$
$(W_9)^5$ $[W_9]^5: [\pm\theta _6 + c_1\theta _7 + c_2 \theta_8]_{W_9}$ $L_N=9$
$(W_9)^6$ $[W_9]^6: [\theta _7 + c\theta _8]_{W_9}$ $L_N=10$
$(W_9)^7$ $[W_9]^7: [\pm\theta _8+c\theta_9]_{W_9}$ $L_N=11$
$(W_9)^8$ $[W_9]^8: [\pm\theta_9]_{W_9}$ $L_N=13$
$(W_9)^9$ $[W_9]^9: [0]_{W_9}$ $N$ is contained in a smooth Lagrangian submanifold
: Geometric characterization of symplectic classes of $W_9$ singularity[]{data-label="tabw9-geom"}
The conditions on the pair $(\omega, N)$ in the last column of Table \[tabw9-geom\] are disjoint. It suffices to prove that these conditions in the row of $(W_9)^{i}$, are satisfied for any $N\in (W_9)^{i}$.
We can take the simplest $2$-forms $\omega ^i$ representing the normal forms $[W_9]^i$ for algebraic restrictions and we can check that the pair $(W_9,\omega=\omega ^i)$ satisfies the condition in the last column of Table \[tabw9-geom\].
We note that in the case $N = W_9 = (\ref{defw9})$ one has
$ \ell \ = \operatorname{span}(\partial /\partial x_3), \ \
V = \operatorname{span}(\partial /\partial x_2,\partial /\partial x_3$, $W = \operatorname{span}(\partial/\partial x_1, \partial / \partial x_2, \partial / \partial x_3)$.
By simple calculation and observation of the Lagrangian tangency orders we obtain that the conditions corresponding to the classes $(W_9)^i$ are satisfied.
Proofs
======
The method of algebraic restrictions {#method}
------------------------------------
In this section we present basic facts on the method of algebraic restrictions, which is a very powerful tool for the symplectic classification. The details of the method and proofs of all results of this section can be found in [@DJZ2].
Given a germ of a non-singular manifold $M$ denote by $\Lambda ^p(M)$ the space of all germs at $0$ of differential $p$-forms on $M$. Given a subset $N\subset M$ introduce the following subspaces of $\Lambda ^p(M)$: $$\Lambda ^p_N(M) = \{\omega \in \Lambda ^p(M): \ \ \omega (x)=0 \ \text {for any} \ x\in N \};$$ $$\mathcal A^p_0(N, M) = \{\alpha + d\beta : \ \ \alpha \in \Lambda _N^p(M), \ \beta \in \Lambda _N^{p-1}(M).\}$$
\[main-def\] Let $N$ be the germ of a subset of $M$ and let $\omega \in \Lambda ^p(M)$. The [**algebraic restriction**]{} of $\omega $ to $N$ is the equivalence class of $\omega $ in $\Lambda
^p(M)$, where the equivalence is as follows: $\omega $ is equivalent to $\widetilde \omega $ if $\omega - \widetilde \omega
\in \mathcal A^p_0(N, M)$.
[**Notation**]{}. The algebraic restriction of the germ of a $p$-form $\omega $ on $M$ to the germ of a subset $N\subset M$ will be denoted by $[\omega ]_N$. Writing $[\omega ]_N=0$ (or saying that $\omega $ has zero algebraic restriction to $N$) we mean that $[\omega ]_N = [0]_N$, i.e. $\omega \in A^p_0(N, M)$.
Two algebraic restrictions $[\omega ]_N$ and $[\widetilde \omega ]_{\widetilde N}$ are called [**diffeomorphic**]{} if there exists the germ of a diffeomorphism $\Phi:
\widetilde M\to M$ such that $\Phi(\widetilde N)=N$ and $\Phi ^*([\omega ]_N) =[\widetilde \omega ]_{\widetilde N}$.
The method of algebraic restrictions applied to singular quasi-homogeneous subsets is based on the following theorem.
\[thm A\] Let $N$ be the germ of a quasi-homogeneous subset of $\mathbb
R^{2n}$. Let $\omega _0, \omega _1$ be germs of symplectic forms on $\mathbb R^{2n}$ with the same algebraic restriction to $N$. There exists a local diffeomorphism $\Phi $ such that $\Phi (x) =
x$ for any $x\in N$ and $\Phi ^*\omega _1 = \omega _0$.
Two germs of quasi-homogeneous subsets $N_1, N_2$ of a fixed symplectic space $(\mathbb R^{2n}, \omega )$ are symplectically equivalent if and only if the algebraic restrictions of the symplectic form $\omega $ to $N_1$ and $N_2$ are diffeomorphic.
Theorem \[thm A\] reduces the problem of symplectic classification of germs of singular quasi-homogeneous subsets to the problem of diffeomorphic classification of algebraic restrictions of the germ of the symplectic form to the germs of singular quasi-homogeneous subsets.
The geometric meaning of the zero algebraic restriction is explained by the following theorem.
\[thm B\] [*The germ of a quasi-homogeneous set $N$ of a symplectic space $(\mathbb R^{2n}, \omega )$ is contained in a non-singular Lagrangian submanifold if and only if the symplectic form $\omega
$ has zero algebraic restriction to $N$.*]{}
In the paper we use the following notations:
$\bullet$ $\algrestall $: the vector space consisting of the algebraic restrictions of germs of all $2$-forms on $\mathbb R^{2n}$ to the germ of a subset $N\subset \mathbb
R^{2n}$;
$\bullet$ $\algrestclosed$: the subspace of $\algrestall $ consisting of the algebraic restrictions of germs of all closed $2$-forms on $\mathbb R^{2n}$ to $N$;
$\bullet$ $\algrest $: the open set in $\algrestclosed$ consisting of the algebraic restrictions of germs of all symplectic $2$-forms on $\mathbb R^{2n}$ to $N$.
To obtain a classification of the algebraic restrictions we use the following proposition.
\[elimin1\] Let $a_1, \cdots, a_p$ be a quasi-homogeneous basis of quasi-degrees $\delta_1\le\cdots\le\delta_s<\delta_{s+1}\le\cdots\le\delta_p$ of the space of algebraic restrictions of closed $2$-forms to $N$. Let $a=\sum_{j=s}^p c_ja_j$, where $c_j\in \mathbb R$ for $j=s,\cdots, p$ and $c_s\ne 0$.
If there exists a tangent quasi-homogeneous vector field $X$ over $N$ such that $\mathcal L_Xa_s=ra_k$ for $k>s$ and $r\ne 0$ then $a$ is diffeomorphic to $\sum_{j=s}^{k-1} c_ja_j+\sum_{j=k+1}^p
b_j a_j$, for some $b_j\in \mathbb R, \ j=k+1,\cdots,p$.
Proposition \[elimin1\] is a modification of Theorem 6.13 formulated and proved in [@D]. It was formulated for algebraic restrictions to a parameterized curve but we can generalize this theorem for any subset $N$. The proofs of the cited theorem and Proposition \[elimin1\] are similar and are based on the Moser homotopy method.
For calculating discrete invariants we use the following propositions.
\[sm\] The symplectic multiplicity of the germ of a quasi-homogeneous subset $N$ in a symplectic space is equal to the codimension of the orbit of the algebraic restriction $[\omega ]_N$ with respect to the group of local diffeomorphisms preserving $N$ in the space of algebraic restrictions of closed $2$-forms to $N$.
\[ii\] The index of isotropy of the germ of a quasi-homogeneous subset $N$ in a symplectic space $(\mathbb R^{2n}, \omega )$ is equal to the maximal order of vanishing of closed $2$-forms representing the algebraic restriction $[\omega ]_N$.
\[lto\] Let $f$ be the germ of a quasi-homogeneous curve such that the algebraic restriction of a symplectic form to it can be represented by a closed $2$-form vanishing at $0$. Then the Lagrangian tangency order of the germ of a quasi-homogeneous curve $f$ is the maximum of the order of vanishing on $f$ over all $1$-forms $\alpha$ such that $[\omega]_f=[d\alpha]_f$
Proofs for $W_8$ singularity
----------------------------
### Algebraic restrictions to $W_8$ and their classification {#w8-class}
One has the following relations for $(W_8)$-singularities $$[d(x_2^2+x_1x_3)]_{W_8}=[2x_2dx_2+ x_1dx_3+x_3dx_1]_{W_8}=0,
\label{w81}$$ $$[d(x_1^2+x_3^3)]_{W_8}=[2x_1dx_1+3x_3^2dx_3]_{W_8}=0.
\label{w82}$$ Multiplying these relations by suitable $1$-forms we obtain the relations in Table \[tabw81\].
$\delta$ Relations Proof
----------------- --------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------
$14$ $[x_2dx_2 \wedge dx_3]_N=-\frac{1}{2}[x_3dx_1 \wedge dx_3]_N$ (\[w81\])$\wedge\, dx_3$
$15$ $[x_1dx_2 \wedge dx_3]_N=[x_3dx_1 \wedge dx_2]_N$ (\[w81\])$\wedge\, dx_2$
$16$ $[x_2dx_1\wedge dx_2]_N=-\frac{1}{2}[x_1dx_1\wedge dx_3]_N=0$ (\[w82\])$\wedge\, dx_3$ and (\[w81\])$\wedge\, dx_1$
$17$ $[x_3^2dx_2 \wedge dx_3]_N=\frac{2}{3}[x_1dx_1\wedge dx_2]_N$ (\[w82\])$\wedge\, dx_2$
$18$ $[x_3^2dx_1 \wedge dx_3]_N=2[x_2x_3dx_2\wedge dx_3]_N=0$ (\[w82\])$\wedge\, dx_1$ and (\[w81\])$\wedge\, x_3dx_3$
$19$ $[x_2^2dx_2\wedge dx_3]_N=-\frac{1}{2}[x_2x_3dx_1\wedge dx_3]_N$ (\[w81\])$\wedge\, x_2dx_3$
$[x_2^2dx_2\wedge dx_3]_N=-[x_1x_3dx_2\wedge dx_3]_N=$ $=-[x_3^2dx_1 \wedge dx_2]_N$ (\[w81\])$\wedge\, x_3dx_2$ and $[x_2^2+x_1x_3]_N=0$
$20$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $20$ relations for $\delta\in\{14,15,16\}$ and $[x_2^2+x_1x_3]_N=0$
$21$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $21$ relations for $\delta\in\{15,16,17\}$ and $[x_1^2+x_3^3]_N=[x_2^2+x_1x_3]_N=0$
$22$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $22$ relations for $\delta\in\{16,17,18\}$ and $[x_1^2+x_3^3]_N=[x_2^2+x_1x_3]_N=0$
$23$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $23$ relations for $\delta\in\{17,18,19\}$ and $[x_1^2+x_3^3]_N=0$
$24$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $24$ relations for $\delta\in\{18,19,20\}$ and $[x_1^2+x_3^3]_N=0$
$25$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $25$ relations for $\delta\in\{19,20,21\}$
$\delta\!>\!25$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $\delta\!>\!25$ relations for $\delta> 19$
: Relations towards calculating $[\Lambda^2(\mathbb R^{2n})]_N$ for $N=W_8$[]{data-label="tabw81"}
Using the method of algebraic restrictions and Table \[tabw81\] we obtain the following proposition:
\[w8-all\] The space $[\Lambda ^{2}(\mathbb R^{2n})]_{W_8}$ is a $9$-dimensional vector space spanned by the algebraic restrictions to $W_8$ of the $2$-forms
$\theta _1= dx_2\wedge dx_3, \;\; \theta _2=dx_1\wedge dx_3,\;\; \theta_3 = dx_1\wedge dx_2,$
$\theta _4 = x_3dx_2\wedge dx_3,\;\; \theta _5 = x_2dx_2\wedge dx_3,$ $\sigma_1 = x_1 dx_2\wedge dx_3,$ $\sigma_2 = x_2 dx_1\wedge dx_3,$
$\theta _7= x_3^2 dx_2\wedge dx_3$, $\theta _8= x_2^2 dx_2\wedge dx_3$.
Proposition \[w8-all\] and results of Section \[method\] imply the following description of the space $[Z ^2(\mathbb R^{2n})]_{W_8}$ and the manifold $[{\rm Symp} (\mathbb R^{2n})]_{W_8}$.
\[w8-baza\] The space $[Z^2(\mathbb R^{2n})]_{W_8}$ is a $8$-dimensional vector space spanned by the algebraic restrictions to $W_8$ of the quasi-homogeneous $2$-forms $\theta_i$ of degree $\delta_i$
$\theta _1= dx_2\wedge dx_3,\;\;\;\delta_1=9,$
$\theta _2=dx_1\wedge dx_3,\;\;\;\delta_2=10,$
$\theta_3 = dx_1\wedge dx_2,\;\;\;\delta_3=11,$
$\theta _4 = x_3dx_2\wedge dx_3,\;\;\;\delta_4=13,$
$\theta _5 = x_2dx_2\wedge dx_3,\;\;\;\delta_5=14,$
$\theta _6=\sigma_1+\sigma_2=x_1dx_2\wedge dx_3+x_2dx_1\wedge dx_3,\;\;\;\delta_6=15, $
$\theta _7= x_3^2 dx_2\wedge dx_3,\;\;\;\delta_7=17$,
$\theta _8= x_2^2 dx_2\wedge dx_3,\;\;\;\delta_8=19$.
If $n\ge 3$ then $[{\rm Symp} (\mathbb R^{2n})]_{W_8} = [Z^2(\mathbb R^{2n})]_{W_8}$. The manifold $[{\rm Symp} (\mathbb R^{4})]_{W_8}$ is an open part of the $8$-space $[Z^2 (\mathbb R^{4})]_{W_8}$ consisting of algebraic restrictions of the form $[c_1\theta _1 + \cdots + c_8\theta _8]_{W_8}$ such that $(c_1,c_2,c_3)\ne (0,0,0)$.
\[klasw8\] $ \ $
\(i) Any algebraic restriction in $[Z ^2(\mathbb R^{2n})]_{W_8}$ can be brought by a symmetry of $W_8$ to one of the normal forms $[W_8]^i$ given in the second column of Table \[tabw8\].
\(ii) The codimension in $[Z ^2(\mathbb R^{2n})]_{W_8}$ of the singularity class corresponding to the normal form $[W_8]^i$ is equal to $i$, the symplectic multiplicity and the index of isotropy are given in the fourth and fifth columns of Table \[tabw8\].
\(iii) The singularity classes corresponding to the normal forms are disjoint.
\(iv) The parameters $c, c_1, c_2$ of the normal forms $[W_8]^i$ are moduli.
Symplectic class Normal forms for algebraic restrictions cod $\mu ^{\rm sym}$ ind
--------------------------- ------------------------------------------------------------------------ ----- ------------------ -----------
$(W_8)^0$ $(2n\ge 4)$ $[W_8]^0: [\theta _1 + c_1\theta _2 + c_2\theta _3]_{W_8}$, $0$ $2$ $0$
$(W_8)^1$ $(2n\ge 4)$ $[W_8]^1: [c_1\theta _2 + \theta _3 + c_2\theta _4]_{W_8}, \ c_1\ne 0$ $1$ $3$ $0$
$(W_8)^{2,a}$ $(2n\ge 4)$ $[W_8]^{2,a}: [\pm\theta _2 + c_1\theta_4+c_2\theta_7]_{W_8}$, $2$ $4$ $0$
$(W_8)^{2,b}$ $(2n\ge 4)$ $[W_8]^{2,b}: [\theta _3 +c_1\theta_4+c_2\theta_5 ]_{W_8}$, $2$ $4$ $0$
$(W_8)^3$ $(2n\ge 6)$ $[W_8]^3: [\theta _4 + c_1\theta _5+c_2\theta _6]_{W_8}$ $3$ $5$ $1$
$(W_8)^4$ $(2n\ge 6)$ $[W_8]^4: [\pm\theta _5 + c_1\theta _6 + c_2 \theta_7]_{W_8}$ $4$ $6$ $1$
$(W_8)^5$ $(2n\ge 6)$ $[W_8]^5: [\theta _6 + c\theta _7]_{W_8}$ $5$ $6$ $1$
$(W_8)^6$ $(2n\ge 6)$ $[W_8]^6: [\theta _7+c\theta_8]_{W_8}$ $6$ $7$ $2$
$(W_8)^7$ $(2n\ge 6)$ $[W_8]^7: [\theta_8]_{W_8}$ $7$ $7$ $2$
$(W_8)^8$ $(2n\ge 6)$ $[W_8]^8: [0]_{W_8}$ $8$ $8$ $\infty $
: Classification of symplectic $W_8$ singularities: $cod$ – codimension of the classes; $\mu ^{sym}$– symplectic multiplicity; $ind$ – the index of isotropy.[]{data-label="tabw8"}
In the first column of Table \[tabw8\] by $(W_8)^i$ we denote a subclass of $(W_8)$ consisting of $N\in (W_8)$ such that the algebraic restriction $[\omega ]_N$ is diffeomorphic to some algebraic restriction of the normal form $[W_8]^i$, where $i$ is the codimension of the class. Classes $(W_8)^{2a}$ and $(W_8)^{2b}$ have the same codimension equal to $2$ but they can be distinguished geometrically (see Table \[tabw8-geom\]).
The proof of Theorem \[klasw8\] is presented in Section \[w8-proof\].
### Symplectic normal forms. Proof of Theorem \[w8-main\] {#w8-normal}
Let us transfer the normal forms $[W_8]^i$ to symplectic normal forms. Fix a family $\omega ^i$ of symplectic forms on $\mathbb R^{2n}$ realizing the family $[W_8]^i$ of algebraic restrictions. We can fix, for example
$\omega ^0 = \theta _1 + c_1\theta _2 + c_2\theta _3 +
dx_1\wedge dx_4 + dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^1 = c_1 \theta _2 + \theta _3 + c_2\theta _4 +
dx_3\wedge dx_4 + dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge
dx_{2n}, \ \ c_1\ne 0; $
$\omega ^{2,a} = \pm\theta_2+ c_1 \theta _4 + c_2 \theta _7 + dx_2\wedge dx_4 +
dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge dx_{2n};$
$\omega ^{2,b} = \theta_3+ c_1 \theta _4 + c_2 \theta _5 + dx_3\wedge dx_4 +
dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge dx_{2n};$
$\omega ^3 = \theta _4 + c_1\theta _5 + c_2\theta _6+ dx_1\wedge dx_4 +
dx_2\wedge dx_5 + dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots +
dx_{2n-1}\wedge dx_{2n};$
$\omega ^4 = \pm \theta _5 + c_1\theta _6 + c_2\theta _7+ dx_1\wedge dx_4 +dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge dx_{2n};$
$\omega ^5 = \theta _6 + c\theta _7+ dx_1\wedge dx_4 + dx_2\wedge dx_5 + dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^6 = \theta _7 + c\theta _8+ dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^7 = \theta _8+ dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^8 = dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n}.$
Let $\omega_0 = \sum_{i=1}^m dp_i \wedge dq_i$, where $(p_1,q_1,\cdots,p_n,q_n)$ is the coordinate system on $\mathbb R^{2n}, n\ge 3$ (resp. $n=2$). Fix, for $i=0,1,\cdots ,8$ (resp. for $i = 0,1,2)$ a family $\Phi ^i$ of local diffeomorphisms which bring the family of symplectic forms $\omega ^i$ to the symplectic form $\omega_0$: $(\Phi ^i)^*\omega ^i = \omega_0$. Consider the families $W_8^i = (\Phi ^i)^{-1}(W_8)$. Any stratified submanifold of the symplectic space $(\mathbb R^{2n},\omega_0)$ which is diffeomorphic to $W_8$ is symplectically equivalent to one and only one of the normal forms $W_8^i, i = 0,1,\cdots ,8$ (resp. $i= 0,1,2$) presented in Theorem \[w8-main\]. By Theorem \[klasw8\] we obtain that parameters $c,c_1,c_2$ of the normal forms are moduli.
### Proof of Theorem \[klasw8\] {#w8-proof}
In our proof we use vector fields tangent to $N\in W_8$. Any vector fields tangent to $N\in W_8$ can be described as $V=g_1E+g_2\mathcal{H}$ where $E$ is the Euler vector field and $\mathcal{H}$ is a Hamiltonian vector field and $g_1, g_2$ are functions. It was shown in [@DT1] (Prop. 6.13) that the action of a Hamiltonian vector field on any 1-dimensional complete intersection is trivial.
The germ of a vector field tangent to $W_8$ of non trivial action on algebraic restrictions of closed 2-forms to $W_8$ may be described as a linear combination germs of vector fields: $X_0\!=\!E, \ X_1\!=\!x_3E, \ X_2\!=x_2E, \ X_3\!=x_1E, \ X_4\!=x_3^2E, \ X_5\!=x_2x_3E$, $X_6\!=x_2^2E, \ X_7\!=x_1x_3E$, where $E$ is the Euler vector field
$E= 6 x_1 \partial /\partial x_1+5 x_2 \partial /\partial x_2+4 x_3 \partial /\partial x_3$.
\[w8-infinitesimal\]
The infinitesimal action of germs of quasi-homogeneous vector fields tangent to $N\in (W_8)$ on the basis of the vector space of algebraic restrictions of closed $2$-forms to $N$ is presented in Table \[infini-w8\].
$\mathcal L_{X_i} [\theta_j]$ $[\theta_1]$ $[\theta_2]$ $[\theta_3]$ $[\theta_4]$ $[\theta_5]$ $[\theta_6]$ $[\theta_7]$ $[\theta_8]$
------------------------------- ----------------- ------------------ -------------------------- ----------------- ----------------- ----------------- ----------------- -----------------
$X_0\!=\!E$ $9 [\theta_1]$ $10 [\theta_2]$ $11 [\theta_3]$ $13 [\theta_4]$ $14 [\theta_5]$ $15 [\theta_6]$ $17 [\theta_7]$ $19 [\theta_8]$
$X_1\!=\!x_3E$ $13[\theta_4]$ $-28 [\theta_5]$ $5[\theta_6]$ $17[\theta_7]$ $[0]$ $-57[\theta_8]$ $[0]$ $[0]$
$X_2\!=\!x_2E$ $14[\theta_5]$ $ 10[\theta_6]$ $[0]$ $[0]$ $19[\theta_8] $ $[0]$ $[0]$ $[0]$
$X_3\!=\!x_1 E$ $5 [\theta_6]$ $[0]$ $\frac{51}{2}[\theta_7]$ $-19[\theta_8]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_4\!=\!x_3^2E$ $ 17[\theta_7]$ $[0]$ $-19[\theta_8]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_5\!=\!x_2x_3 E$ $[0]$ $-38[\theta_8]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_6\!=\!x_2^2 E$ $19[\theta_8]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_7\!=\!x_1x_3 E$ $-19[\theta_8]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
: Infinitesimal actions on algebraic restrictions of closed 2-forms to $W_8$. $E=6x_1 \partial /\partial x_1+ 5x_2 \partial /\partial x_2+ 4x_3 \partial /\partial x_3$[]{data-label="infini-w8"}
Let $\mathcal{A}=[c_1 \theta_1+c_2 \theta_2+c_3 \theta_3+c_4 \theta_4+c_5 \theta_5+c_6 \theta_6 +c_7 \theta_7 +c_8 \theta_8]_{W_8}$ be the algebraic restriction of a symplectic form $\omega$.
The first statement of Theorem \[klasw8\] follows from the following lemmas.
\[w8lem0\] If $c_1\ne 0$ then the algebraic restriction $\mathcal{A}=[\sum_{k=1}^8 c_k \theta_k]_{W_8}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\theta_1+\widetilde{c}_2 \theta_2+\widetilde{c}_3 \theta_3]_{W_8}$.
Using the data of Table \[infini-w8\] we can see that for any algebraic restriction $[\theta_k]_{W_8}$, where $k\!\in\!\{4,5,\ldots,8\}$ we can find a vector field $V_k$ tangent to $W_8$ such that $\mathcal L_{V_k}[\theta_1]_{W_8}\!=\![\theta_k]_{W_8}$. We deduce from Proposition \[elimin1\] that the algebraic restriction $\mathcal{A}$ is diffeomorphic to $[c_1\theta_1+{c}_2 \theta_2+{c}_3 \theta_3]_{W_8}$.
By the condition $c_1\ne 0$ we have a diffeomorphism $\Psi \in Symm(W_8)$ of the form $$\label{proofw8lem04}
\Psi:\,(x_1,x_2,x_3)\mapsto (c_1^{-\frac{6}{9}} x_1,c_1^{-\frac{5}{9}} x_2,c_1^{-\frac{4}{9}} x_3)$$ and finally we obtain $$\Psi^*([c_1\theta_1+{c}_2 \theta_2+{c}_3 \theta_3]_{W_8})=[ \theta_1+c_2 c_1^{-\frac{10}{9}} \theta_2+c_3 c_1^{-\frac{11}{9}} \theta_3]_{W_8} =
[ \theta_1+ \widetilde{c}_2 \theta_2+\widetilde{c}_3 \theta_3]_{W_8}.$$
\[w8lem1\] If $c_1\!=\!0$and $c_2\cdot c_3\!\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\widetilde{c}_2 \theta_2+ \theta_3+\widetilde{c}_4 \theta_4]_{W_8}$.
We use the homotopy method to prove that $\mathcal{A}$ is diffeomorphic to $[\widetilde{c}_2 \theta_2+ \theta_3+\widetilde{c}_4 \theta_4]_{W_8}$.
Let $\mathcal{B}_t=[c_2 \theta_2+c_4 \theta_3+c_4 \theta_4+(1-t)c_5 \theta_5+(1-t)c_6 \theta_6+(1-t)c_7 \theta_7 +(1-t)c_8 \theta_8]_{W_8}$ for $t \in[0;1]$. Then $\mathcal{B}_0=\mathcal{A}$ and $\mathcal{B}_1=[c_2 \theta_2+c_3 \theta_3+c_4 \theta_4]_{W_8}$. We prove that there exists a family $\Phi_t \in Symm(W_8),\;t\in [0;1]$ such that $$\label{proofw8lem11} \Phi_t^*\mathcal{B}_t=\mathcal{B}_0,\;\Phi_0=id.$$ Let $V_t$ be a vector field defined by $\frac{d \Phi_t}{dt}=V_t(\Phi_t)$. Then differentiating (\[proofw8lem11\]) we obtain $$\label{proofw8lem12} \mathcal L_{V_t} \mathcal{B}_t=[c_5 \theta_5+c_6 \theta_6+c_7 \theta_7 +c_8 \theta_8]_{W_8}.$$ We are looking for $V_t$ in the form $V_t=\sum_{k=1}^5 b_k(t) X_k$ where the $b_k(t)$ for $k=1,\ldots,5$ are smooth functions $b_k:[0;1]\rightarrow \mathbb{R}$. Then by Proposition \[w8-infinitesimal\] equation (\[proofw8lem12\]) has a form $$\label{proofw8lem13}
\left[ \begin{array}{ccccc}
-28c_2 & 0 & 0 & 0 & 0 \\
5c_3 & 10c_2 & 0 & 0 & 0 \\
17c_4 & 0 & \frac{51}{2}c_3 & 0 & 0 \\
-57c_6(1-t) & 19c_5(1-t) & -19c_4 & -19c_3 &-38c_2
\end{array} \right]
\left[ \begin{array}{c} b_1 \\ b_2 \\ b_3 \\ b_4 \\ b_5 \end{array} \right] =
\left[ \begin{array}{c} c_5 \\ c_6 \\ c_7 \\ c_8 \end{array} \right]$$ If $c_2\cdot c_3\ne 0$ we can solve (\[proofw8lem13\]) and $\Phi_t$ may be obtained as a flow of the vector field $V_t$. The family $\Phi_t$ preserves $W_8$, because $V_t$ is tangent to $W_8$ and $\Phi_t^*\mathcal{B}_t=\mathcal{A}$. Using the homotopy arguments we have $\mathcal{A}$ diffeomorphic to $\mathcal{B}_1=[c_2 \theta_2+c_3 \theta_3+c_4 \theta_4]_{W_8}$. By the condition $c_3\ne 0$ we have a diffeomorphism $\Psi \in Symm(W_8)$ of the form $$\label{proofw8lem14}
\Psi:\,(x_1,x_2,x_3)\mapsto (c_3^{-\frac{6}{11}} x_1,c_3^{-\frac{5}{11}} x_2,c_3^{-\frac{4}{11}} x_3),$$ and we obtain $$\Psi^*(\mathcal{B}_1)=[c_2 c_3^{-\frac{10}{11}} \theta_2+ \theta_3+c_4 c_3^{-\frac{13}{11}} \theta_3]_{W_8} =
[ \widetilde{c}_2 \theta_2+ \theta_3+\widetilde{c}_4 \theta_4]_{W_8}.$$
\[w8lem2a\] If $c_1\!=\!c_3\!=\!0$ and $c_2\!\neq\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\pm\theta_2+\widetilde{c}_4 \theta_4+\widetilde{c}_7 \theta_7]_{W_8}$.
We can see from Table \[infini-w8\] that for any algebraic restriction $[\theta_k]_{W_8}$, where $k\in \{5,6,8\}$ there exists a vector field $V_k$ tangent to $W_8$ such that $\mathcal L_{V_k}[\theta_2]_{W_8}=[\theta_k]_{W_8}$. Using Proposition \[elimin1\] we obtain that $\mathcal{A}$ is diffeomorphic to $ [c_2\theta_2+{c}_4 \theta_4+\widehat{c}_7 \theta_7]_{W_8}$ for some $\widehat{c}_7\in\mathbb{R}$.
By the condition $c_2\ne 0$ we can use a diffeomorphism $\Psi \in Symm(W_8)$ of the form $$\label{proofw8lem2a4}
\Psi:\,(x_1,x_2,x_3)\mapsto (|c_2|^{-\frac{6}{10}} x_1,|c_2|^{-\frac{5}{10}} x_2,|c_2|^{-\frac{4}{10}} x_3)$$ and we obtain $$\Psi^*([c_2\theta_2+c_4 \theta_4+\widehat{c}_7\theta_7]_{W_8})\!=\![\frac{c_2}{|c_2|}\theta_2+c_4 |c_2|^{-\frac{13}{10}} \theta_4+\widehat{c}_7 |c_2|^{-\frac{17}{10}} \theta_7]_{W_8}\!=\! [\pm\theta_2+\widetilde{c}_4 \theta_4+\widetilde{c}_7\theta_7]_{W_8}.$$
The algebraic restrictions $[ \theta_2+ \widetilde{c}_4 \theta_4+\widetilde{c}_7 \theta_7]_{W_8}$ and $[-\theta_2+ \widetilde{b}_4 \theta_4+\widetilde{b}_7 \theta_7]_{W_8}$ are not diffeomorphic. Any diffeomorphism $\Phi=(\Phi_1,\ldots,\Phi_{2n})$ of $(\mathbb{R}^{2n},0)$ preserving $W_8$ has to preserve a curve $C(t)=(t^6,t^5,-t^4,0,\ldots,0)$ which means that
$\Phi_1(t^6,t^5,-t^4,0,\ldots,0)=(\psi(t))^6$,
$\Phi_2(t^6,t^5,-t^4,0,\ldots,0)=(\psi(t))^5$,
$\Phi_3(t^6,t^5,-t^4,0,\ldots,0)=-(\psi(t))^4$,
$\Phi_k(t^6,t^5,-t^4,0,\ldots,0)=0$ for $k>3$,
where $\psi(t)=a_1t+a_2t^2+a_3t^3+\ldots$ is a diffeomorphism of $(\mathbb{R},0)$.
Hence $\Phi$ has a linear part
$$\label{lindyfw8}
\begin{array}{cccccccccccccccc}
\Phi_1: & A^6x_1& &+ & & &&A_{14}x_4&+& \cdots &+&A_{1,2n}x_{2n}\\
\Phi_2: & A_{2,1}x_1 &+& A^5x_2& &+& &A_{24}x_4&+& \cdots &+&A_{2,2n}x_{2n} \\
\Phi_3: & A_{3,1}x_1&+ &A_{3,2}x_2 &+ & A^4x_3&+&A_{34}x_4&+& \cdots &+&A_{3,2n}x_{2n}\\
\Phi_4: & & & & & & &A_{44}x_4&+& \cdots &+&A_{4,2n}x_{2n}\\
\vdots & & & & & & & \vdots & \vdots & \vdots & \vdots\\
\Phi_{2n}: & & & & & & &A_{2n,4}x_4&+& \cdots &+&A_{2n,2n}x_{2n},\\
\end{array}$$
where $A, A_{i,j}\in \mathbb R$.
If we assume that $\Phi^*([ \theta_2+ \widetilde{c}_4 \theta_4+\widetilde{c}_7 \theta_7]_{W_8})= [-\theta_2+ \widetilde{b}_4 \theta_4+\widetilde{b}_7 \theta_7]_{W_8}$, then
$A^{10}dx_1\wedge dx_3|_0=-dx_1\wedge dx_3|_0$, which is a contradiction.
\[w8lem2b\] If $c_1\!=\!c_2\!=\!0$ and $c_3\!\neq\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\theta_3+\widetilde{c}_4 \theta_4+\widetilde{c}_5 \theta_5]_{W_8}$.
\[w8lem3\] If $c_1=c_2=c_3=0$ and $c_4\ne 0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\theta_4+\widetilde{c}_5 \theta_5+\widetilde{c}_6 \theta_6]_{W_8}$.
\[w8lem4\] If $c_1=0,\ldots,c_4=0$ and $c_5\ne 0$, then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\pm\theta_5+\widetilde{c}_6 \theta_6+\widetilde{c}_7 \theta_7]_{W_8}$.
\[w8lem5\] If $c_1=0,\ldots,c_5=0$ and $c_6\ne 0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\theta_6+\widetilde{c}_7 \theta_7]_{W_8}$.
\[w8lem6\] If $c_1=0,\ldots,c_6=0$ and $c_7\ne 0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\theta_7+\widetilde{c}_8 \theta_8]_{W_8}$.
\[w8lem7\] If $c_1=0,\ldots,c_7=0$ and $c_8\ne 0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_8$ to an algebraic restriction $[\theta_8]_{W_8}$.
The proofs of Lemmas \[w8lem2b\] – \[w8lem7\] are similar to the proofs of Lemmas \[w8lem0\] – \[w8lem2a\] and are based on Table \[infini-w8\].
Statement $(ii)$ of Theorem \[klasw8\] follows from the conditions in the proof of part $(i)$ (codimension) and from Theorem \[thm B\] and Proposition \[sm\] (symplectic multiplicity) and Proposition \[ii\] (index of isotropy).
To prove statement $(iii)$ of Theorem \[klasw8\] we have to show that singularity classes corresponding to normal forms are disjoint. The singularity classes that can be distinguished by geometric conditions obviously are disjoint. From Theorem \[geom-cond-w8\] we see that only classes $(W_8)^1$ and $(W_8)^{2,a}$ can not be distinguished by the geometric conditions. To prove that these classes are disjoint we compare the tangent spaces to the orbits of the respective algebraic restrictions. From Table \[infini-w8\] we see that the tangent space to the orbit of $[c_1\theta _2 + \theta _3 + c_2\theta _4]_{W_8}$ at $[c_1\theta _2 + \theta _3 + c_2\theta _4]_{W_8}$ is spanned by the linearly independent algebraic restrictions $[10c_2\theta _2 + 11\theta _3 + 13c_2\theta _4]_{W_8}$, $[\theta _5]_{W_8}$, $[\theta _6]_{W_8}$, $[\theta _7]_{W_8}$, $[\theta _8]_{W_8}$ and the tangent space to the orbit of $[\pm\theta _2 + b_1\theta _4 + b_2\theta _7]_{W_8}$ at $[\pm\theta _2 + b_1\theta _4 + b_2\theta _7]_{W_8}$ is spanned by the linearly independent algebraic restrictions $[\pm 10\theta _2 + 13b_1\theta _4 + 17b_2\theta _7]_{W_8}$, $[\pm 28\theta _5+17b_1\theta_7]_{W_8}$, $[\theta _6]_{W_8}$ and $[\theta _8]_{W_8}$.
To prove statement $(iv)$ of Theorem \[klasw8\] we have to show that the parameters $c, c_1, c_2$ are moduli in the normal forms. The proofs are very similar in all cases. We consider as an example the normal form with two parameters $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_8}$. From Table \[infini-w8\] we see that the tangent space to the orbit of $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_8}$ at $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_8}$ is spanned by the linearly independent algebraic restrictions $[9\theta_1+10c_1\theta_2+11c_2\theta_3]_{W_8}$, $[\theta_4]_{W_8},[\theta_5]_{W_8}, [\theta_6]_{W_8}, [\theta_7]_{W_8}, [\theta_8]_{W_8}.$ Hence the algebraic restrictions $[\theta_2]_{W_8}$ and $[\theta_3]_{W_8}$ do not belong to it. Therefore the parameters $c_1$ and $c_2$ are independent moduli in the normal form $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_8}$.
Proofs for $W_9$ singularity
----------------------------
### Algebraic restrictions to $W_9$ and their classification {#w9-class}
$ \ $
One has the following relations for $(W_9)$-singularities $$[d(x_1^2+x_2x_3^2)]_{W_9}=[2x_1dx_1+ 2x_2x_3dx_3+x_3^2 dx_2]_{W_9}=0,
\label{w91}$$ $$[d(x_2^2+x_1x_3)]_{W_9}=[2x_2dx_2+x_3dx_1+x_1dx_3]_{W_9}=0.
\label{w92}$$ Multiplying these relations by suitable $1$-forms we obtain the relations in Table \[tabw91\].
$\delta$ Relations Proof
---------- ------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------
$11$ $[x_3dx_1 \wedge dx_3]_N=-2[x_2dx_2 \wedge dx_3]_N$ (\[w92\])$\wedge\, dx_3$
$12$ $[x_1dx_2 \wedge dx_3]_N=[x_3dx_1 \wedge dx_2]_N$ (\[w92\])$\wedge\, dx_2$
$13$ $[x_3^2dx_2\wedge dx_3]_N=-2[x_1dx_1 \wedge dx_3]_N=4[x_2dx_1 \wedge dx_1]_N$ (\[w91\])$\wedge dx_3$ and (\[w92\])$\wedge dx_1$
$14$ $[x_3^2dx_1\!\wedge dx_3]_N\!=-2[x_1dx_1\!\wedge dx_2]_N\!=\!-2[x_2x_3dx_2\!\wedge dx_3]_N$ (\[w91\])$\wedge dx_2$, (\[w92\])$\wedge x_3dx_3$
$15$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $15$ relations for $\delta\in\{11,12\}$ and (\[w91\])$\wedge dx_1$ and $[x_2^2+x_1x_3]_N=0$
$16$ $[x_3^3dx_2\!\wedge dx_3]_N\!=\!-2[x_1x_3dx_1\!\wedge dx_3]_N\!=4[x_2x_3dx_1\!\wedge dx_2]_N$ relations for $\delta=13$
$[x_1x_3dx_1\!\wedge dx_3]_N\!=\!-2[x_1x_2dx_2\!\wedge dx_3]_N\!=\!-[x_2^2dx_1\!\wedge dx_3]_N$ relations for $\delta\in\{11,12\}$ and $[x_2^2+x_1x_3]_N=0$
$17$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $17$ relations for $\delta\in\{12,13,14\}$ and $[x_1^2+x_2x_3^2]_N=0$ i $[x_2^2+x_1x_3]_N=0$
$18$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $18$ relations for $\delta\in\{13,14,15\}$ and $[x_1^2+x_2x_3^2]_N=0$
$19$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $19$ relations for $\delta\in\{14,15,16\}$ and $[x_1^2+x_2x_3^2]_N=0$
$20$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $20$ relations for $\delta\in\{15,16,17\}$
$21$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $21$ relations for $\delta\in\{16,17,18\}$
$>\!21$ $[\alpha]_N=0$ for all 2-forms $\alpha$ of quasi-degree $\delta>21$ relations for $\delta> 16$
: Relations towards calculating $[\Lambda^2(\mathbb R^{2n})]_N$ for $N=W_9$[]{data-label="tabw91"}
Using the method of algebraic restrictions and Table \[tabw91\] we obtain the following proposition:
\[w9-all\] The space $[\Lambda ^{2}(\mathbb R^{2n})]_{W_9}$ is a $10$-dimensional vector space spanned by the algebraic restrictions to $W_9$ of the $2$-forms $\theta _1= dx_2\wedge dx_3, \;\; \theta _2=dx_1\wedge dx_3,\;\; \theta_3 = dx_1\wedge dx_2,$
$\theta _4 = x_3dx_2\wedge dx_3,\;\; \theta _5 = x_3dx_1\wedge dx_3,$ $\sigma_1 = x_1 dx_2\wedge dx_3,$ $\sigma_2 = x_2 dx_1\wedge dx_3,$
$\theta _7= x_3^2 dx_2\wedge dx_3$, $\theta _8= x_3^2 dx_1\wedge dx_3$, $\theta _9= x_3^3 dx_2\wedge dx_3$.
Proposition \[w9-all\] and results of Section \[method\] imply the following description of the space $[Z ^2(\mathbb R^{2n})]_{W_9}$ and the manifold $[{\rm Symp} (\mathbb R^{2n})]_{W_9}$.
\[w9-baza\] The space $[Z^2(\mathbb R^{2n})]_{W_9}$ is a $9$-dimensional vector space spanned by the algebraic restrictions to $W_9$ of the quasi-homogeneous $2$-forms $\theta_i$ of degree $\delta_i$
$\theta _1= dx_2\wedge dx_3,\;\;\;\delta_1=7,$
$\theta _2=dx_1\wedge dx_3,\;\;\;\delta_2=8,$
$\theta_3 = dx_1\wedge dx_2,\;\;\;\delta_3=9,$
$\theta _4 = x_3dx_2\wedge dx_3,\;\;\;\delta_4=10,$
$\theta _5 = x_3dx_1\wedge dx_3,\;\;\;\delta_5=11,$
$\theta _6=\sigma_1+\sigma_2=x_1dx_2\wedge dx_3+x_2dx_1\wedge dx_3,\;\;\;\delta_6=12, $
$\theta _7= x_3^2 dx_2\wedge dx_3,\;\;\;\delta_7=13$,
$\theta _8= x_3^2 dx_1\wedge dx_3,\;\;\;\delta_8=14$,
$\theta _9= x_3^3 dx_2\wedge dx_3,\;\;\;\delta_8=16$,
If $n\ge 3$ then $[{\rm Symp} (\mathbb R^{2n})]_{W_9} = [Z^2(\mathbb R^{2n})]_{W_9}$. The manifold $[{\rm Symp} (\mathbb R^{4})]_{W_9}$ is an open part of the $9$-space $[Z^2 (\mathbb R^{4})]_{W_9}$ consisting of algebraic restrictions of the form $[c_1\theta _1 + \cdots + c_9\theta _9]_{W_9}$ such that $(c_1,c_2,c_3)\ne (0,0,0)$.
\[klasw9\] $ \ $
\(i) Any algebraic restriction in $[Z ^2(\mathbb R^{2n})]_{W_9}$ can be brought by a symmetry of $W_9$ to one of the normal forms $[W_9]^i$ given in the second column of Table \[tabw9\].
\(ii) The codimension in $[Z ^2(\mathbb R^{2n})]_{W_9}$ of the singularity class corresponding to the normal form $[W_9]^i$ is equal to $i$, the symplectic multiplicity and the index of isotropy are given in the fourth and fifth columns of Table \[tabw9\].
\(iii) The singularity classes corresponding to the normal forms are disjoint.
\(iv) The parameters $c, c_1, c_2$ of the normal forms $[W_9]^i$ are moduli.
Symplectic class Normal forms for algebraic restrictions cod $\mu ^{\rm sym}$ ind
----------------------- --------------------------------------------------------------- ----- ------------------ -----------
$(W_9)^0$ $(2n\ge 4)$ $[W_9]^0: [\theta _1 + c_1\theta _2 + c_2\theta _3]_{W_9}$, $0$ $2$ $0$
$(W_9)^1$ $(2n\ge 4)$ $[W_9]^1: [\pm\theta _2 + c_1\theta _3 + c_2\theta _4]_{W_9}$ $1$ $3$ $0$
$(W_9)^2$ $(2n\ge 4)$ $[W_9]^2: [\theta _3 + c_1\theta_4+c_2\theta_5]_{W_9}$, $2$ $4$ $0$
$(W_9)^3$ $(2n\ge 6)$ $[W_9]^3: [\pm\theta _4 +c_1\theta_5+c_2\theta_6 ]_{W_9}$, $3$ $5$ $1$
$(W_9)^4$ $(2n\ge 6)$ $[W_9]^4: [\theta _5 + c_1\theta _6+c_2\theta _7]_{W_9}$ $4$ $6$ $1$
$(W_9)^5$ $(2n\ge 6)$ $[W_9]^5: [\pm\theta _6 + c_1\theta _7 + c_2 \theta_8]_{W_9}$ $5$ $7$ $1$
$(W_9)^6$ $(2n\ge 6)$ $[W_9]^6: [\theta _7 + c\theta _8]_{W_9}$ $6$ $7$ $2$
$(W_9)^7$ $(2n\ge 6)$ $[W_9]^7: [\pm\theta _8+c\theta_9]_{W_9}$ $7$ $8$ $2$
$(W_9)^8$ $(2n\ge 6)$ $[W_9]^8: [\pm\theta_9]_{W_9}$ $8$ $8$ $3$
$(W_9)^9$ $(2n\ge 6)$ $[W_9]^9: [0]_{W_9}$ $9$ $9$ $\infty $
: Classification of symplectic $W_9$ singularities: $cod$ – codimension of the classes; $\mu ^{sym}$– symplectic multiplicity; $ind$ – the index of isotropy.[]{data-label="tabw9"}
In the first column of Table \[tabw9\] by $(W_9)^i$ we denote a subclass of $(W_9)$ consisting of $N\in (W_9)$ such that the algebraic restriction $[\omega ]_N$ is diffeomorphic to some algebraic restriction of the normal form $[W_9]^i$.
The proof of Theorem \[klasw9\] is presented in Section \[w9-proof\].
### Symplectic normal forms. Proof of Theorem \[w9-main\] {#w9-normal}
$ \ $
Let us transfer the normal forms $[W_9]^i$ to symplectic normal forms. Fix a family $\omega ^i$ of symplectic forms on $\mathbb R^{2n}$ realizing the family $[W_9]^i$ of algebraic restrictions. We can fix, for example
$\omega ^0 = \theta _1 + c_1\theta _2 + c_2\theta _3 +
dx_1\wedge dx_4 + dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^1 = \pm \theta _2 + c_1\theta _3 + c_2\theta _4 +
dx_2\wedge dx_4 + dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge
dx_{2n}; $
$\omega ^2 = \theta_3+ c_1 \theta _4 + c_2 \theta _5 + dx_3\wedge dx_4 +
dx_5\wedge dx_6 + \cdots + dx_{2n-1}\wedge dx_{2n};$
$\omega ^3 = \pm\theta _4 + c_1\theta _5 + c_2\theta _6+ dx_1\wedge dx_4 +
dx_2\wedge dx_5 + dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots +
dx_{2n-1}\wedge dx_{2n};$
$\omega ^4 = \theta _5 + c_1\theta _6 + c_2\theta _7+ dx_1\wedge dx_4 +dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge dx_{2n};$
$\omega ^5 = \pm\theta _6 + c_1\theta _7+ c_2\theta _8+ dx_1\wedge dx_4 + dx_2\wedge dx_5 + dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^6 = \theta _7 + c\theta _8+ dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^7 = \pm\theta _8+ c\theta _9+ dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^8 = \pm\theta _9+ dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n};$
$\omega ^9 = dx_1\wedge dx_4 + dx_2\wedge dx_5 +
dx_3\wedge dx_6 + dx_7\wedge dx_8 + \cdots + dx_{2n-1}\wedge
dx_{2n}.$
Let $\omega_0 = \sum_{i=1}^m dp_i \wedge dq_i$, where $(p_1,q_1,\cdots,p_n,q_n)$ is the coordinate system on $\mathbb R^{2n}, n\ge 3$ (resp. $n=2$). Fix, for $i=0,1,\cdots ,9$ (resp. for $i = 0,1,2)$ a family $\Phi ^i$ of local diffeomorphisms which bring the family of symplectic forms $\omega ^i$ to the symplectic form $\omega_0$: $(\Phi ^i)^*\omega ^i = \omega_0$. Consider the families $W_9^i = (\Phi ^i)^{-1}(W_8)$. Any stratified submanifold of the symplectic space $(\mathbb R^{2n}, \omega_0)$ which is diffeomorphic to $W_9$ is symplectically equivalent to one and only one of the normal forms $W_9^i, i = 0,1,\cdots ,9$ (resp. $i= 0,1,2$) presented in Theorem \[w9-main\]. By Theorem \[klasw9\] we obtain that parameters $c,c_1,c_2$ of the normal forms are moduli.
### Proof of Theorem \[klasw9\] {#w9-proof}
$ \ $
In our proof we use vector fields tangent to $N\in W_9$.
The germ of a vector field tangent to $W_8$ of non trivial action on algebraic restrictions of closed 2-forms to $W_9$ may be described as a linear combination germs of the following vector fields:
$X_0\!=E$, $X_1\!=x_3E, \ X_2\!=x_2E, \ X_3\!=x_1E, \ X_4\!=x_3^2E, X_5\!=x_2x_3E$, $X_6\!=x_2^2E$, $X_7\!=x_1x_3E$, $X_8\!=x_1x_2E$, $X_9\!=x_3^3E,$
where $E$ is the Euler vector field $E= 5 x_1 \partial /\partial x_1+4 x_2 \partial /\partial x_2+3 x_3 \partial /\partial x_3$.
\[w9-infinitesimal\]
The infinitesimal action of germs of quasi-homogeneous vector fields tangent to $N\in (W_9)$ on the basis of the vector space of algebraic restrictions of closed $2$-forms to $N$ is presented in Table \[infini-w9\].
$\mathcal L_{X_i} [\theta_j]$ $[\theta_1]$ $[\theta_2]$ $[\theta_3]$ $[\theta_4]$ $[\theta_5]$ $[\theta_6]$ $[\theta_7]$ $[\theta_8]$ $[\theta_9]$
------------------------------- ---------------------------- --------------------------- --------------------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------
$X_0\!=\!E$ $7 [\theta_1]$ $8 [\theta_2]$ $9 [\theta_3]$ $10 [\theta_4]$ $11 [\theta_5]$ $12 [\theta_6]$ $13 [\theta_7]$ $14 [\theta_8]$ $16[\theta_9]$
$X_1\!=\!x_3E$ $10[\theta_4]$ $11 [\theta_5]$ $4[\theta_6]$ $13[\theta_7]$ $14[\theta_8]$ $[0]$ $16[\theta_9]$ $[0]$ $[0]$
$X_2\!=\!x_2E$ $-\frac{11}{2} [\theta_5]$ $ 8[\theta_6]$ $\frac{13}{4} [\theta_7]$ $-7 [\theta_8]$ $ [0]$ $12[\theta_9]$ $[0]$ $[0]$ $[0]$
$X_3\!=\!x_1 E$ $4 [\theta_6]$ $-\frac{13}{2}[\theta_7]$ $-7[\theta_8]$ $[0]$ $-8[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_4\!=\!x_3^2E$ $ 13[\theta_7]$ $14[\theta_8]$ $[0]$ $16[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_5\!=\!x_2x_3 E$ $-7[\theta_8]$ $[0]$ $4[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_6\!=\!x_2^2 E$ $[0]$ $8[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_7\!=\!x_1x_3 E$ $[0]$ $-8[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_8\!=\!x_1x_2 E$ $4[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
$X_9\!=\!x_3^3 E$ $16[\theta_9]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$ $[0]$
: Infinitesimal actions on algebraic restrictions of closed 2-forms to $W_9$. $E=5x_1 \partial /\partial x_1+ 4x_2 \partial /\partial x_2+ 3x_3 \partial /\partial x_3$[]{data-label="infini-w9"}
Let $\mathcal{A}=[\sum_{k=1}^9 \theta_k]_{W_9}$ be the algebraic restriction of a symplectic form $\omega$.
The first statement of Theorem \[klasw9\] follows from the following lemmas.
\[w9lem0\] If $c_1\ne 0$ then the algebraic restriction $\mathcal{A}=[\sum_{k=1}^9 c_k \theta_k]_{W_9}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\theta_1+\widetilde{c}_2 \theta_2+\widetilde{c}_3 \theta_3]_{W_9}$.
\[w9lem1\] If $c_1\!=\!0$and $c_2\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\pm \theta_2+ \widetilde{c}_3\theta_3+\widetilde{c}_4 \theta_4]_{W_9}$.
\[w9lem2\] If $c_1\!=c_2=\!0$and $c_3\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\theta_3+ \widetilde{c}_4\theta_4+\widetilde{c}_5 \theta_5]_{W_9}$.
\[w9lem3\] If $c_1\!=c_2=c_3=\!0$and $c_4\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\pm\theta_4+ \widetilde{c}_5\theta_5+\widetilde{c}_6 \theta_6]_{W_9}$.
\[w9lem4\] If $c_1\!=\ldots=c_4=\!0$and $c_5\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\theta_5+ \widetilde{c}_6\theta_6+\widetilde{c}_7 \theta_7]_{W_9}$.
\[w9lem5\] If $c_1\!=\ldots=c_5=\!0$and $c_6\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\pm\theta_6+ \widetilde{c}_7\theta_7+\widetilde{c}_8 \theta_8]_{W_9}$.
\[w9lem6\] If $c_1\!=\ldots=c_6=\!0$and $c_7\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\theta_7+\widetilde{c}_8 \theta_8]_{W_9}$.
\[w9lem7\] If $c_1\!=\ldots=c_7=\!0$and $c_8\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\pm\theta_8+\widetilde{c}_9 \theta_9]_{W_9}$.
\[w9lem8\] If $c_1\!=\ldots=c_8=\!0$and $c_9\ne\!0$ then the algebraic restriction $\mathcal{A}$ can be reduced by a symmetry of $W_9$ to an algebraic restriction $[\pm\theta_9]_{W_9}$.
The proofs of Lemmas \[w9lem0\] – \[w9lem8\] are similar to the proofs of the lemmas for the $W_8$ singularity. As an example we give the proof of Lemma \[w9lem1\].
We see from Table \[infini-w9\] that for any algebraic restriction $[\theta_k]_{W_9}$, where $k\in \{5,6,7,8,9\}$ there exists a vector field $V_k$ tangent to $W_9$ such that $\mathcal L_{V_k}[\theta_2]_{W_9}=[\theta_k]_{W_9}$. Using Proposition \[elimin1\] we obtain that $\mathcal{A}$ is diffeomorphic to $[c_2\theta_2+{c}_3 \theta_3+c_4\theta_4]_{W_9}$.
By the condition $c_2\ne 0$ we have a diffeomorphism $\Psi \in Symm(W_9)$ of the form $$\label{proofw9lem24}
\Psi:\,(x_1,x_2,x_3)\mapsto (|c_2|^{-\frac{5}{8}} x_1,|c_2|^{-\frac{4}{8}} x_2,|c_2|^{-\frac{3}{8}} x_3)$$ and we obtain $$\Psi^*([c_2\theta_2+c_3 \theta_3+c_4 \theta_4]_{W_9})\!=\![\frac{c_2}{|c_2|}\theta_2+c_3 |c_2|^{-\frac{9}{8}} \theta_3+{c}_4 |c_2|^{-\frac{10}{8}} \theta_4]_{W_9}\!=\![\pm\theta_2+ \widetilde{c}_3 \theta_3+\widetilde{c}_4 \theta_4]_{W_9}.$$
The algebraic restrictions $[ \theta_2+ \widetilde{c}_3 \theta_3+\widetilde{c}_4 \theta_4]_{W_9}$ and $[-\theta_2+ \widetilde{b}_3 \theta_3+\widetilde{b}_4 \theta_4]_{W_9}$ are not diffeomorphic. Any diffeomorphism $\Phi=(\Phi_1,\ldots,\Phi_{2n})$ of $(\mathbb{R}^{2n},0)$ preserving $W_9$ has to preserve a curve $C_2(t)=(t^5,-t^4,-t^3,0,\ldots,0)$. Hence $\Phi$ has a linear part
$$\label{lindyfw9}
\begin{array}{cccccccccccccccc}
\Phi_1: & A^5x_1& &+ & & &&A_{14}x_4&+& \cdots &+&A_{1,2n}x_{2n}\\
\Phi_2: & A_{2,1}x_1 &+& A^4x_2& &+& &A_{24}x_4&+& \cdots &+&A_{2,2n}x_{2n} \\
\Phi_3: & A_{3,1}x_1&+ &A_{3,2}x_2 &+ & A^3x_3&+&A_{34}x_4&+& \cdots &+&A_{3,2n}x_{2n}\\
\Phi_4: & & & & & & &A_{44}x_4&+& \cdots &+&A_{4,2n}x_{2n}\\
\vdots & & & & & & & \vdots & \vdots & \vdots & \vdots\\
\Phi_{2n}: & & & & & & &A_{2n,4}x_4&+& \cdots &+&A_{2n,2n}x_{2n}\\
\end{array}$$
where $A, A_{i,j}\in \mathbb R$.
If we assume that $\Phi^*([ \theta_2+ \widetilde{c}_3 \theta_3+\widetilde{c}_4 \theta_4]_{W_9})= [-\theta_2+ \widetilde{b}_3 \theta_3+\widetilde{b}_4 \theta_4]_{W_9}$, then
$A^{8}dx_1\wedge dx_3|_0=-dx_1\wedge dx_3|_0$, which is a contradiction.
Statement $(ii)$ of Theorem \[klasw9\] follows from the conditions in the proof of part $(i)$ (codimension) and from Theorem \[thm B\] and Proposition \[sm\] (symplectic multiplicity) and Proposition \[ii\] (index of isotropy).
Statement $(iii)$ of Theorem \[klasw9\] follows from Theorem \[geom-cond-w9\]. The singularity classes corresponding to normal forms are disjoint because they can be distinguished by geometric conditions.
To prove statement $(iv)$ of Theorem \[klasw9\] we have to show that the parameters $c, c_1, c_2$ are moduli in the normal forms. The proofs are very similar in all cases. We consider as an example the normal form with two parameters $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_9}$. From Table \[infini-w9\] we see that the tangent space to the orbit of $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_9}$ at $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_9}$ is spanned by the linearly independent algebraic restrictions $[7\theta_1+8c_1\theta_2+9c_2\theta_3]_{W_9}$, $[\theta_4]_{W_9},[\theta_5]_{W_9}, [\theta_6]_{W_9}, [\theta_7]_{W_9}, [\theta_8]_{W_9}, [\theta_9]_{W_9}.$ Hence the algebraic restrictions $[\theta_2]_{W_9}$ and $[\theta_3]_{W_9}$ do not belong to it. Therefore the parameters $c_1$ and $c_2$ are independent moduli in the normal form $[\theta_1+c_1\theta_2+c_2\theta_3]_{W_9}$.
[AAAA]{}
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[^1]: There is a mistake in description of $W_8$ singularity in [@AVG]. We find there
$W_8=\{x\in\mathbb R^{2n\geq
4}\,:x_1^2+ x_2^3=x_2^2+x_1x_3=x_{\geq 4}=0\}$ which is not an isolated complete intersection singularity.
|
---
abstract: 'Autonomous prediction of traffic demand will be a key function in future cellular networks. In the past, researchers have used statistical methods such as Autoregressive integrated moving average (ARIMA) to provide traffic predictions. However, ARIMA based predictions fail to give an exact and accurate forecast for dynamic input quantities such as cellular traffic. More recently, researchers have started to explore deep learning techniques, such as, recurrent neural networks (RNN) and long-short-term-memory (LSTM) to autonomously predict future cellular traffic. In this research, we have designed a LSTM based cellular traffic prediction model. We have compared the LSTM based prediction with the base line ARIMA model and vanilla feed-forward neural network (FFNN). The results show that LSTM and FFNN accurately predicted the future cellular traffic. However, it was found that LSTM train the prediction model in much shorter time as compared to FFNN. Hence, we conclude that LSTM models can be effectively even used with small amount of training data which will allow to timely predict the future cellular traffic.'
author:
-
bibliography:
- 'test\_bib.bib'
title: |
Cellular Traffic Prediction with\
Recurrent Neural Network
---
Cellular traffic prediction, recurrent neural network, LSTM, feed forward neural network.
Introduction
============
Cellular communication is the most popular and ubiquitous telecommunication technology. Recently, owing to novel use cases, such as, multimedia video download, 4K/8K streaming etc., the amount of cellular data traffic has soared exponentially. It is expected that in the near future, i.e. by 2023, the monthly mobile data demands will exceed beyond 109 Exabyte (Exa = $10^{18}$) which currently rests at a modest 20 Exabytes per month consumption [@cerwall2018ericsson]. Cellular users, nevertheless, will expect high speed and ubiquitous connectivity from the network operators. Providing unhindered, ubiquitous, and high quality of service will be a serious challenge for network operators. Network operators must update traffic planning tools so they can know in advance about the state of future traffic demands. Hence, operators will rely on data-driven self-organizing networks (SON) powered by machine learning (ML) and artificial intelligence (AI). ML and AI enabled networks can preemptively take important decisions with limited human intervention. Prediction of cellular and data traffic patterns will be a key job that SON perform.
Cellular traffic prediction will enable network operators to promptly distribute resources as per the requirement of competing users. With the informed network state, operators may also allow resource sharing between devices [@jaffry2018effective; @jaffry2018shared]. This will also enable high spectral efficiency and will prevent outages caused due to cell overload. If a network can accurately predict future traffic loads in specific cells, it may take preventive actions to avoid outages. For example, network may permit device-to-device communication to relieve the base station [@jaffry2017distributed].
Recent advances in data analytics and the availability of powerful computing machines have enabled operators to harness the power of Big data to analyze and predict network operations. Hence, advanced variations of data driven ML and AI techniques are playing an ever increasing role in all aspects of modern human lives. In particular, deep learning, a special class of ML and AI algorithms, can solve enormously complex problems by leveraging the power of very deep neural network layers [@lecun2015deep]. Deep learning algorithms can extract valuable feature information from the raw data to predict outcomes. Deep learning has made great strides recently due to advent of user friendly libraries and programming environment such as Tensorflow, Keras, PyTorch, Pandas, and Scikit etc. Deep learning algorithms such as recurrent neural network (RNN), convolutional neural network (CNN) etc. are being extensively used in application, such as, computer vision [@voulodimos2018deep], health informatics [@ravi2016deep], speech recognition [@deng2013new], and natural language processing [@young2018recent] etc. In future, it is anticipated that majority operations in sixth generation (6G) cellular network will be solely catered by AI and deep learning algorithms [@zappone2019wireless].
An AI-enabled SON network will perform long and short term analysis on the data obtained from the end users and/or network [@chen2019artificial]. This self-optimization will reduce the over all capital expenditures (CAPEX) and operational expenditure (OPEX) required for network planning and maintenance. For example, a key issue concerning increasing CAPEX and OPEX for service providers is the identification and remedy of anomalies that may arise within a cell. To learn and prevent the cell from going into the anomalous state, it is necessary for the network to predict the future traffic demands.
In the past researcher have proposed to forecast the cellular traffic using statistical models, such, as Autoregressive integrated moving average (ARIMA) and its variants [@shu2005wireless]. A known limitation of ARIMA is that it reproduce the time series patterns based on average of the past values. However, ARIMA may fail to accurately predict traffic patterns in highly dynamic environments such as cellular network. Nevertheless, ARIMA can give a descent estimate of future traffic and may serve as a baseline prediction model.
Recently, deep learning based techniques to forecast any time series traffic is getting more popular. For cellular applications, deep learning techniques learn the past history of network traffic to train models such as vanilla feed-forward neural network (FFNN), recurrent neural network (RNN), or long-short-term-memory (LSTM) etc. In [@qiu2018spatio], researchers have proposed to use RNN with multi-task learning to design a spatio-temporal prediction model for cellular networks. Researchers in [@zhao2019celltrademap] have applied neural network models on cellular traffic to analyze trade activities in urban business districts. A comparative study between LSTM and ARIMA models was conducted by researchers in [@azari2019cellular].
Inspired by the works presented earlier, in this paper we will use the real world call data record to forecast future cellular traffic using LSTM. In particular, we will compare our results with the ARIMA model and vanilla feed forward neural network (FFNN) models. We will demonstrate that LSTM models learn the traffic patterns very quickly as compared to FFNN and ARIMA models.
The rest of the paper is organized as follows. The system model is presented in Section \[system\_model\]. The cellular traffic prediction model is presented in Section \[sec\_traffic\_prediction\]. We discuss the results in Section \[sec\_results\] followed by conclusion in Section \[sec\_conclusion\].
System Model {#system_model}
============
Figure \[fig:LTEArchitecture\] shows the our system model which comprise of Long Term Evolution - Advanced (LTE-A) network. The architecture of LTE-A is broadly categorized into three layers. The core network (CN), the access network, and the end user equipment (UE) [@elnashar2014design].
The wireless communication take place between a UE and evolved NodeB (eNB) over the access network which is called UMTS terrestrial radio access network (E-UTRAN) in LTE-A nomenclature. The core network, which is formally known as evolved packet core (EPC), makes essential the network level decisions. The EPC further contain several logical entities such as serving gateway (SGW), packet data network gateway (PGW), and mobility management unit (MMU) etc. Detailed explanation of these logical entities and LTE-A architecture is out of scope of current paper. Readers can refer relevant materials, for example [@elnashar2014design]. The call data record (CDR) that we will use in this research was gathered at the EPC level layer. The execution of LSTM predictive model will also take place at this layer.
Data Record Details
-------------------
The call data record used in this research was published by Telecom Italia for Big Data Challenge competition [@telecom_ItaliaCDR]. Telecom Italia collected cellular and internet activities of its subscribers within the city of Milan in Italy. In the CDR, Milan city is divided into $100 \times 100$ square grids. Each grid has a length of 0.235 Km and an area of 0.055 Km$^2$. The data record has been collected for 62 days, starting from 1st November 2013 till 1st January 2014. Data for the single day is stored in a single file which means that there are 62 files in the dataset. Readers can refer to [@parwez2017big] for detailed explanation on the CDR.
The spatio-temporal CDR contains following fields.
- Grid ID.
- Time Stamp: Raw timestamp was recorded in milliseconds units with the interval of 10 minutes.
- Country code.
- Inbound SMS Activity: Indicates the incoming SMS activity in a particular grid observed within 10 minute interval.
- Outbound SMS Activity: Indicates the outgoing SMS activity in a particular grid observed within 10 minute interval.
- Inbound Call Activity: Indicates the incoming calling in a particular grid observed within 10 minute interval.
- Outbound Call Activity: Indicates the outgoing calling activity in a particular grid observed within 10 minute interval.
- Internet Activity: Indicates the internet usage by cellular users in a particular grid observed within 10 minute interval.\
The CDR does not specify activity in terms of particular units. However, an intuitive interpretations is that the activities are proportional to the amount of real traffic. For example, the magnitude of Inbound or outbound SMS activities are high for a greater number of SMS received or sent, respectively. The data was provided in the raw format. Hence, we will discuss the data cleansing method in the next step.
Data cleansing
--------------
The CDR, in its raw format, could not be used to extract any meaningful information. Hence we applied data cleansing and filtering over the CDR. The timestamps were changed from milliseconds to minutes. There were some missing fields which we marked as zeros (0). There were multiple entry records for each timestamp. We summed them to make a single activity record per timestamp. Figure \[fig:Activty\] shows the 24-hour Internet Activity for Grid 01.
In our prediction model, we have only used Internet traffic Activity. However, the our model can be used to predict activities for SMS and calls without any modification. We will discuss traffic prediction in the next section.
Cellular Traffic Prediction
===========================
In this section, we will first briefly describe basics of feed forward and recurrent neural network (NN) followed by the LSTM based learning model.
\[sec\_traffic\_prediction\]
Feed Forward and Recurrent NN
-----------------------------
In artificial neural networks, the nodes are connected to form a directed graph which is ideal for handling temporal sequence predictions. In vanilla feed forward network (FFNN), information flows only in forward direction. In FFNN, the input layer feeds the forward looking hidden layer for calculations and manipulations. The hidden layers forward the information to the output layer which produce regression of classification predictions.
A NN maps inputs to the outputs by learning from the examples provided during the training phase and can be used for prediction in classification and regression problems. During the training process, the predictions are compared against the expected output values (often known as ground truth data) to calculate the loss function. In the beginning of the training, the loss function is usually quite high indicating the incorrect prediction by the model. With back propagation and gradient descent method, the model adjust the weights and biases corresponding to the input value to minimize the loss function. A fully trained NN has the minimal loss (also called as error) between the predicted and the expected output value [@goodfellow2016deep]. After successful training, the model is validated and a validation error is calculated. A model is fully trained for prediction when the training and validation errors are both minimized.
In recurrent neural network (RNN), though the learning process is the same as FFNN, the architecture is slightly different. RNNs takes the output of one layer, and feed it as the input to the next layer. Hence, each layer has information from the past input values. RNN considers the current input as well as the input received in the previous time steps during training and prediction. This enables RNN to learn the knowledge from the all previous time instances to make a well informed prediction for time series data.
However, vanilla RNNs have inherent vanishing and exploding gradient problem which halts the learning process as gradient either diminishes completely or explodes to very large value. Hence Long-Short-Term-Memory (LSTM), which is a variant of RNN was proposed in [@hochreiter1997long]. LSTMs were designed to avoid the long-term dependency issue, which is the cause of the vanishing-gradient problem in Vanilla RNNs [@goodfellow2016deep].
Learning Through LSTMs
----------------------
The structure of LSTM units (often known as cells) enable a neural network to learn long term dependencies. The learning processing is strictly controlled by multiple gates that allow (or bar) the flow of incoming data from the previous cell and/or input as shown in Figure \[fig:LSTM\]. The standard LSTM unit is shown in Figure \[fig:LSTM\]. There are three main gates in any LSTM unit, the forget gate ($\Gamma_f$), the update or input gate ($\Gamma_i$), and the output gate ($\Gamma_o$). The cell state for the current unit $C^t$ is updated by the information passed through the update gate ($\Gamma_i$). The candidate value for current cell’s state (i.e. $C_u^t$) is updated based on the information from the previous hidden state (i.e. $\text{a}^{t-1}$) and input $X^t$. The update gate decides to allow or bar the flow of this candidate value to the output state. Finally the output gate $\Gamma_o$ allows the information to pass from the current cell. The forget gate lets the current cell keep or forget the state value from the previous time step. The prediction is made as $\hat{y}$ after passing through an activation function (often sigmoid or softmax).
The LSTM cells are chained to form one layer of the LSTM network as shown in Figure \[fig:LSTM\_network\]. Each cells computes operation for one time step and transfer the output to the next cell. The number of cells in a LSTM network indicates the amount of observations of data that is being considered before making any prediction. For our case, the input $X^t$ is the internet activity and the number of observations is the amount of selected time steps T.
The expression for all the gates, cell states, out of the hidden layer, and the final prediction are given as below:
$$\Large \Gamma_f^{t} = \sigma(W_f[a^{t-1}, X^t] + b_f)$$
$$\Large \Gamma_i^{t} = \sigma(W_i[a^{t-1}, X^t] + b_i)$$
$$\Large C_u^t = \varphi(W_c[a^{t-1}, X^t] + b_c)$$
$$\Large \Gamma_o^{t} = \sigma(W_o[a^{t-1}, X^t] + b_o)$$
$$\Large C^t = \Gamma_f^t \ast C^{t-1} + \Gamma \ast C_u^t$$
$$\Large a^t = \Gamma_o^t\ast\varphi(C^t)$$
The final output $Y^t$ is then calculated as :
$$\Large y^t = \sigma(W_y a^t + b_y)$$
In the equations above, symbol $\sigma$ represent the sigmoid function which is often known as the squashing function because it limits the output between 0 (gate OFF) and 1 (Gate fully ON). Formally, the sigmoid function is defined as $ \sigma(x) = \Large \frac{1}{1 + e^{-x}}$ . Symbol $\varphi$ is another squashing function and often $\tanh$ or rectified linear unit (relu) operations are used for $\varphi$. Readers can refer to relevant literature to gather further information about these functions. [@goodfellow2016deep]. The symbol $\ast$ represents the element-wise multiplication. Finally, $W_{(.)}$ and $b_{(.)}$ are the vectors of weights and biases corresponding to the respective gates, hidden layer, input, and output layer. The exact values of these weights and biases are selected by the libraries described in the next sub-section.
Training and Prediction Software
--------------------------------
We have used Matlab for data cleansing and filtering. All the algorithms are implemented in Python using Keras and Scikit libraries with Tensorflow at the backend.
Results and Discussion {#sec_results}
======================
In this section we will show the performance comparison of LSTM model with the base line ARIMA and vanilla feed forward neural network model. We have compared the performance of each technique with the ground truth test data from CDR. We have fixed the training epochs to 20 for each cycle. For LSTM model, we have used two hidden layers to make an even comparison with the FFNN and ARIMA. The first hidden layer contains 50 LSTM cells followed by a dense layer with single unit. The FFNN contains two hidden layers with first layer containing 5 activation units activated by relu operation. The second hidden layer contains one non-linear activation unit. Training and validation losses are calculated using mean absolute error.
Figure \[fig:combined\_predict\_80percent\] shows the traffic prediction by LSTM, FFNN model, and ARIMA model. We used 7142 samples for training LSTM and FFNN models. For validation and testing, we used 893 samples for each case. The LSTM and FFNN both learned the pattern in less than 5 epochs due to large number of training examples. It can be observed that LSTM and FFNN predictions match to that of the ground truth data. The ARIMA model predicts very close to the ground truth but does not exactly match the traffic pattern.
We later reduced the training samples to 3571. The training and validation errors for this case are shown in Figure \[fig:combined\_error\_40percent\] and the prediction results are presented in Figure \[fig:combined\_predic\_40percent\]. We can observe that LSTM and FFNN still predict very accurately. The ARIMA baseline model however does not exactly match the ground truth traffic. It should be noted from Figure \[fig:combined\_error\_40percent\] that when we reduced the number of training samples, the training and validation error for LSTM converges to near zero (0) only after 2 epochs. However, FFNN took at least 10 epochs to fully train the model to enable accurate predictions. Nevertheless, both models’ errors converged to zero before the 20 epochs limit.
When we further reduced the training samples to 892, we observed that after training the models for 20 epochs, the FFNN could not predict according to actual ground truth data. In fact, its performance worsened even to that of the ARIMA model. The LSTM, on the other hand, very accurately predicted the traffic activity. This is due to the fact that LSTM trained the network within 20 epochs and the training and validation error converged to zero as shown in Figure \[fig:combined\_error\_10percent\]. On the other hand, the error for the FFNN remains high even after 20 epochs. Interestingly, FFNN could estimate patterns of future traffic, however, with very low accuracy.
Conclusion {#sec_conclusion}
==========
In this paper, we presented cellular data traffic prediction using recurrent neural network, in particular, with long-short-term-memory model. We demonstrated that LSTM and vanilla feed-forward neural networks predict more accurately as compared to the statistical ARIMA model. However, the LSTM models were shown to be learning more quickly as compared to the FFNN, even with a small amount of training data sample. As our future work, we are working to design a LSTM based resource allocation method for 6G networks.
|
---
abstract: |
By assuming that the most frequently occuring color in a video or a region of a video belongs to the background of the image, I propose a new algorithm for detecting foreground objects in a video. The process of detecting the foreground objects is complicated because of the fact that there may be swaying tree’s, objects of the background being moved around or lighting changes in the video. To deal with such complexities many have come up with solutions which heavily rely on expensive floating point operations. In this paper I used a data structure called Octree which is implemented only using binary operations. Traditionally octrees were used for color quantization but here in this paper I used it as a data structure to store the most frequently occuring colors in a video as well. For each of the starting few video frames, I constructed a Octree using all the colors of that frame. Next I pruned all the trees by removing nodes below a certain height and gave the leaf nodes a color which is dependant on the topological path from that node to its parent. Hence any two leaf nodes in two different octrees with the same topological path from themselves to the root will represent the same color. Next I merged all these individual trees into a single tree retaining only those nodes whose topological path to itself from the root is most common among all the trees. The colors represented by the leaf nodes in the resultant tree will be the most frequently occuring colors in the starting video frames of the video. Hence any color of an incomming frame that is not close to any of the colors represented by the leaf node of the merged tree can be regarded as belonging to a foreground object.\
As an Octree is constructed using only binary operations, it is very fast compared to other leading algorithms.
author:
- 'AV Aditya Sastry,B.Tech, GITAM University'
title: BACKGROUND MODELING USING OCTREE COLOR QUANTIZATION
---
Background modelling, Octree color quantization, moving object detection
Introduction
============
ue to the advancement of technology and heavy reduction in costs of FPGA development, traditional video surveillance equipment has evolved from merely capturing video from a static vantage point and transmitting it to performing some preliminary analysis such as extracting objects of interest from the scene and then transmitting it.\
For most of the analysis that is done on a surveillance video, the first crucial step is to detect theforeground pixels of the video. This is done by first constructing a background model and matching each individual frame with this model. Any pixel of a frame which deviates significantly from this model is marked as a foreground pixel in that image.\
Several techniques for construction of this background model have been proposed. The most trivial of this techniques is to average the first 20 or so frames and use the averaged frame as the background model. This adequately takes care of the lighting changes but this technique fails when the background of the scene contains swaying leaves, clocks or part of the background is constantly moved around the scene etc.\
Wren et al,\[1\] tried to solve this problem by fitting a Gaussian distribution for all the colors occuring at each pixel location of a frame using the past N frames. Any pixel value from the input frame which is sufficiently close to the mean of the Gaussian distribution for that pixel location is marked as a background pixel and the Gaussian distribution is updated. This technique fails when the histogram of pixel values at any given location is bimodal or has multiple peeks.\
Stauffer et al,\[2\] attacked the short comings of the previous work by using multiple(about 3 to 5) Gaussian distributions per pixel locations. Each input frame pixel is matched with a corresponding Gaussians at that location. Any pixel which is in the range of 2.5σ of the mean of a Gaussian for that pixel location is marked as a background pixel and that Gaussian’s parameters are updated. The problem with this technique is its heavy usage of floating point operations and its huge memory requirement. Also the error rate is very high.\
The wallflower principles and practises \[3\] a pioneering work by Microsoft research team, not only came up with a background modelling technique but also surveyed all the recent background modelling techniques and listed out all the common problems that a background modelling techniques would face. It views the process of constructing the background model at pixel level, segment level and frame level. The pixel level algorithm attacks the problems with leaf swaying and background motion problems. In this work I provide a better alternative to their pixel level algorithm. The frame level and the segment level techniques listed out in the wallflower paper could be used with this paper as well.\
In this paper I tackle the problem by constructing an a octree\[5\] which is discussed in section 2. The octree is constructed by iterating over each individual pixel location of each individual frame of the training video. The colors found are quantized and form a leaf node in the octree. Each pixel of the incoming frame is quantized similarly and checked to see if a corresponding leaf node exists in the already constructed octree. This is shown in section 3. If a node does exists it is marked as a background pixel. In section 4 I show a method of merging multiple octrees a technique to be used when we don’t have a trainining video or as the wallflower paper calls it, the bootstrapping problem.\
Octree color quantization
=========================
An octree is a tree data structure in which each internal node has exactly eight children. Octrees are most often used to partition a three dimensional space by recursively subdividing it into eight octants. Octrees are also used for color quantization. The octree color quantization algorithm, invented by Gervautz and Purgathofer in 1988, encodes image color data as an octree up to nine levels deep.\
In computer graphics, color quantization or color image quantization is a process that reduces the number of distinct colors present in an image, usually with the intention that the new image should be as visually similar as possible to the original image. Various methods are available for color quantization such as the median cut algorithm, popularity algorithm. Octree\[5\] is the leading color quantization technique as it executes in $O(N)$ execution time and barely occupies memory of the rder $O(K)$. Here $N$ being the number of pixels in the image and $K$ is a number which is proportional to the number of unique colors present.\
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For a picture whose colors we wish to quantize, we consider each color and store it in a octree using the store color algorithm. Next we prune the octree. By pruning the octree we hope to group together colors which look visually same. Few methods of pruning have been covered in \[14\] and \[15\]. In this paper I followed a simple pruning algorithm that is to remove few levels from the bottom of the tree. Since each color at a level N is constructed using N most significant bits of the red, green, blue values of the color, the newly exposed leaf nodes will have N bits of the original color that they represented.One way to assign a new color to these newly exposed leaf nodes is to use their N original bits and fill out the remaining bits with 0’s. This way if any two colors had the same parents till the height of pruning will now be reprsented by the same color. This new color could be used to redraw the image. Fig 2 shows the same image redrawn using an octree pruned at different levels.
$\text{root} \gets \text{create node();}$ $\text{node temp} \gets \text{root}$ $\text{index} \gets \text{Color.red[i] + Color.green[i] + Color.blue[i];}$ $\text{temp} \gets \text{temp.child[index]}$ $\text{temp.child[index]} \gets \text{create node();}$ $\text{temp} \gets \text{temp.child[index]}$ $\textbf{End If } $ $\textbf{End For}$
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Constructing an octree containing only the background colors of an image
========================================================================
If we assume that the most frequently occuring color in a video usually belong to the background, we can construct a octree only containing the background colors. We construct an octree for each frame of a video or a group of frames and prune it using the technique presented in the previous section. Each octree obtained this way will hold the quantized representation of all the colors in each frame or a group of frames. By using the merging procedure shown below we could merge together multiple octrees to obtain a single octree which will contain the most frequently occuring colors in a quantized way. This octree could be thought of as containing all the colors of the background.\
Our assumption that the most frequently occuring colors in a video usually belong to the background is invalidated in the case when a foreground object stays static for a while and suddenly starts to move. Since that object’s color has appeared in most frames it would be treated as belonging to the background of the video. This could happen when a person sleeping infront of a camera suddenly starts to move. To remedy this, we could subdivide a video into multiple regions and obtain a octree for each of these regions.\
$\text{count} \gets 0$ $\text{count} \gets \text{count + 1;}$ $\textbf{End if}$ $\textbf{End For}$ $\text{result.child[i]} \gets \text{New Node();}$ $\textbf{End If}$ $\textbf{End For}$ $\text{tempRoot[j]} \gets \text{root[j].child[i];}$ $\textbf{End For}$ $\text{Result.child[i]} \gets \text{Construct Tree(tempRoot,}$ $\text{ threshold, numberOfTrees, Number of levels);}$ $\textbf{End If}$ $\textbf{End For}$
Foreground pixel detection and octree updation.
================================================
To identify the foreground pixels in a video, we run the procedure shown in the previous section on the first few seconds of the video or the first 100 frames or so of the video. Once we constructed an octree containing the background colors, the process of identifying foreground object locations in a incomming frame is pretty straight forward. For each pixel color of the incoming frame, we use the check color procedure to see if it is present in the octree. If it is not, it is labelled as a foreground pixel.\
In case we subdivided the video into multiple regions when checking if a pixel belongs to foreground or background, we use the octree of that region in conjugation with the procedure below to ascertain if it is foreground or background.
\[H\]
$\textbf{return} \ \text{False}$ $\text{node temp} \gets \text{root}$ $\text{index} \gets \text{Color.red[i] + Color.green[i] + Color.blue[i];}$ $\text{temp} \gets \text{temp.child[index]}$ $\textbf{return} \ \text{False}$ $\textbf{End If } $ $\textbf{End For}$ $\textbf{return} \ \text{True}$ $\textbf{End If}$
Experimental Results
====================
The following results are obtained on a second generation i5 laptop running at 2.3 GHZ with 4 GB of ram on Windows 7 SP1 and JRE 7 with no updates installed. The datasets for these experimentation is from the wallflower paper.
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The $F_0$ scores for different leading algorithms(CB\[6\], IMOG\[8\], LBPH\[9\], STBS\[11\]) on different datasets is given below in comparision with Octree. The $F_0$ scores are calculated using the formula given below.\
$F_0 = 2\frac{P_0\ R_0}{P_0 + R_0}$\
Where $P_0 = \frac{T_n}{T_n + F_n}$ and $R_0 = \frac{T_n}{T_n + F_p}$\
$T_n, T_p, F_n, F_p$ stand for True negative, True positives, False negatives and False Positives.
EE SW IND WK Avg
-------- ------- ------- ------- ------- ----------
CB 0.973 0.984 0.996 0.997 0.9875
IMOG 0.984 0.995 0.995 0.999 0.99325
LBPH 0.991 0.987 0.999 0.996 0.999325
STBS 0.992 0.993 0.99 0.998 0.99325
OCTREE 0.992 0.999 0.999 0.991 0.99525
: Comparision of $F_0$ scores on different datasets[]{data-label="my-label"}
\
[1]{}
Wren, Christopher Richard, et al. “Pfinder: Real-time tracking of the human body.” Pattern Analysis and Machine Intelligence, IEEE Transactions on 19.7 (1997): 780-785.\
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Pojala, C., and Somnath Sengupta. “Neighborhood supported model level fuzzy aggregation for moving object segmentation.” (2014): 1-1.\
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---
abstract: '=-1 The co-prime array is a sub-Nyquist acquisition scheme for the estimation of second order statistics. It cannot generate all the difference values in the co-prime range and hence, one of the sub-array is extended to enable the estimation of the second order statistics at each difference value in the co-prime range. Recently, the difference set for the prototype co-prime array was studied and low latency temporal spectrum estimation was demonstrated. In this paper, we develop the fundamentals of the difference set for the extended co-prime array, also known as the conventional co-prime array, and provide expressions for the degrees of freedom and the range of continuous difference values. The closed-form expression is developed for the number of sample pairs that contribute to estimate the second order statistics (weight function). The closed-form expression of the bias window for correlogram spectral estimation is also derived. We show that the choice of $M\approx \frac{N}{2}$ generates a bias function with a large relative amplitude between the main lobe and side-lobe peaks, where $M$ and $N$ are co-prime pairs. Simulation results demonstrate low latency spectrum estimation using the correlogram method.'
address:
author:
-
bibliography:
- 'refs.bib'
title: 'Extended (Conventional) Co-Prime Arrays and Difference Set Analysis: Low Latency Approach'
---
Ł[[L]{}]{}
Introduction {#sec1}
============
Co-prime sensing scheme provides a framework for sub-Nyquist acquisition and estimation of the second order statistics of a signal [@4.7]. It has several applications which include direction of arrival estimation [@4.45; @F2; @F3; @F5], power spectrum estimation [@4.28; @4.32], system identification [@4.40], beamforming [@F1; @F4; @16.1], cross-correlation estimation, time-delay, range, velocity, and acceleration estimation [@U_S_NCC2018].
A detailed analysis of the difference set of the prototype co-prime array is provided in [@U_S_1]. The spectral estimation using the correlogram method was demonstrated with low latency (around 10 snapshots) using the combined difference set. The closed-form expressions for the weight function and bias window were also developed.
However, the co-prime array cannot generate a difference set containing all the elements in the co-prime range $[-MN+1, MN-1]$ and hence, an extended co-prime array was proposed in [@4.62]. It has one of the sub-arrays extended by an additional co-prime period. The extended co-prime array generates all the values in the co-prime range but this range does not represent the largest continuous range. The weight function of the extended co-prime array is given in Table IV of [@4.68] for $M=4$ and $N=5$, but does not provide a general expression for arbitrary values of $M$ and $N$. This paper develops the fundamentals of the difference set for the extended (also known as conventional) co-prime array and demonstrates low latency estimation. The closed-form expressions for the weight function and the bias window for the correlogram method are also provided.
Extended Co-Prime Concept
=========================
![Extended co-prime structure[]{data-label="fig_extended_concept"}](fig_concept_extended.pdf){width="45.00000%"}
An extended co-prime array is shown in Fig. \[fig\_extended\_concept\] where $M$ and $N$ are co-prime. The Nyquist distance between the antennas is denoted by $d$. It has two sub-arrays and the antenna locations are given by: $$\begin{aligned}
\label{eq:positions}
\nonumber
P1 &=& \{Mnd,\; 0\leq n \leq N-1\} \\
P2 &=& \{Nmd,\; 0\leq m \leq 2M-1\}\end{aligned}$$ Note that the antenna at the zeroth location is common to both the sub-arrays. In this paper, we assume that the second array with inter-element spacing of $Nd$ is extended. However, the first array could have been extended instead; without affecting the concept that governs this scheme. In the temporal or spacial domain, one prototype co-prime period is $[0, MN-1]$ while the extended co-prime period is $[0, 2MN-1]$. It may be noted that the co-prime and extended co-prime range corresponding to one period for autocorrelation (or other second order statistics) is $[-(MN-1), MN-1]$ and $[-(2MN-1), 2MN-1]$ respectively. Fig. \[fig\_extended\_temporal\] describes this concept from a sampling perspective. Here $d$ represents the Nyquist sampling period and the structure implies that one of the sampler is turned off every alternate co-prime period, while the other sampler uniformly samples the incoming signal. Every extended co-prime period is referred to as a snapshot. The second order statistics (e.g. autocorrelation, power spectrum, etc.) is estimated for each snapshot. Averaging across several snapshots improves the estimate. In this paper, we demonstrate the results based on the sampling framework. Some of the contributions of this paper are:
1. We analyze the difference sets (self and cross differences) for the extended co-prime scheme.
2. Describe the number of unique difference values (degrees of freedom) and continuous range.
3. Develop the closed-form expression for the weight function for the entire difference set.
4. Develop the closed-form expression for the bias window that distorts the correlogram spectral estimate.
5. Demonstrate low latency estimation using the correlogram spectral estimation method.
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Difference Set Analysis
=======================
The difference sets defined for the prototype co-prime array [@U_S_1], is also applicable to the extended co-prime array. However, the extension of one of the sub-array needs to be taken into account. In this section, we briefly describe the difference sets before analyzing the number of unique differences or degrees of freedom ($\it{dof}$) and the range of continuous difference values.
The outputs obtained from the two sub-arrays with inter-element spacings $Md$ and $Nd$ is represented by $x(Mn)$ and $x(Nm)$ respectively. Their self difference set is given by, ${\L}^+_{SM} = \{l_s|l_s=Mn\} \;\text{and}\; {\L}^+_{SN} = \{l_s|l_s=Nm\}$. The cross difference set is given by, ${\L}^+_{C} = \{l_{c}|l_{c}=Mn-Nm\}$. Note that $0\leq m\leq 2M-1$ and $0\leq n\leq N-1$. The sets containing the mirrored locations of the elements in the above defined sets are denoted by $\L^-_{SM}$, $\L^-_{SN}$ and $\L^-_{C}$ respectively. In addition, we define two union sets as: ${\L}_{S} = {\L}^+_{S} \cup {\L}^-_{S} \;\text{and}\; {\L}_{C} = {\L}^+_{C} \cup {\L}^-_{C}$, where ${\L}^+_{S}= {\L}^+_{SM} \cup {\L}^+_{SN}$ and ${\L}^-_{S}= {\L}^-_{SM} \cup {\L}^-_{SN}$.
The self difference sets are shown in Fig. \[fig\_self\]. It is obvious that ${\L}^+_{SM}$ and ${\L}^+_{SN}$ have $N$ and $2M$ unique differences respectively. This is also true for the mirrored sets ${\L}^-_{SM}$ and ${\L}^-_{SN}$. The sets ${\L}^+_{S}$ and ${\L}^-_{S}$ have $2M+N-1$ unique differences, hence the union set $\L_{S}$ has $2(2M+N-1)-1$ unique differences. The difference value of zero is common to each set and needs to be considered only once in the union set. This justifies the negation of one in the equation for unique differences. The set $\L^+_{C}$ has $2MN$ unique differences. This is also true for the mirrored set $\L^-_{C}$. Refer Fig. \[fig\_cross\] for the discussion on the cross-difference set. As in the case of the prototype co-prime array, the self differences of the extended array are a sub-set of the cross differences. However, the number of unique differences in the union set $\L_{C}$ is not trivial. Hence, to provide a better understanding we define two new sets $\L^+_{A}$ and $\L^+_{B}$ with $\L^-_{A}$ and $\L^-_{B}$ representing their mirrored counterpart: $$\begin{aligned}
\label{eq:pos_cross_new_sets}
\nonumber {\L}^+_{A} = \{l_{c}|l_{c}=Mn-Nm,n\in[0,N-1],m\in[0,M-1]\}\\
\nonumber {\L}^+_{B} = \{l_{c}|l_{c}=Mn-Nm,n\in[0,N-1],m\in[M,2M-1]\}\end{aligned}$$ It is obvious that $\L^+_{A}$ is same as the cross-difference set of the prototype co-prime array. Hence, the properties of the cross difference set of a prototype co-prime array [@U_S_1], hold true even for the set $\L^+_{A}$.
For the extended co-prime array, $\L^+_{C}=\L^+_{A} \cup \L^+_{B}$ and $\L^-_{C}=\L^-_{A} \cup \L^-_{B}$. Since the set $\L^+_{A}$ is well understood, the focus here would be to understand the set $\L^+_{B}$. Some of the properties of the set $\L^+_{B}$ are given below:
1. $\{l_{c}<0, \forall l_{c} \in \L^+_{B} \} \;\text{and}\; \{l_{c}>0, \forall l_{c} \in \L^-_{B} \}$
2. $\L^-_{SM}-\{0\} \subseteq \L^+_{B} \;\text{and}\; \L^+_{SM}-\{0\} \subseteq \L^-_{B}$
3. $\{l_{s}|l_{s} \in \L^-_{SN} \forall m \in [M, 2M-1] \} \subseteq \L^+_{B}$\
$\{l_{s}|l_{s} \in \L^+_{SN} \forall m \in [M, 2M-1] \} \subseteq \L^-_{B}$
According to Property I-1, the sets $\L^+_{B}$ and $\L^-_{B}$ do not have any common values. But they do have overlapping values with the set $\L^-_{A}$ and $\L^+_{A}$ respectively, which is captured in Property I-2. The Property I-2 implies that $(N-1)$ values in $\L^+_{B}$ overlap with $(N-1)$ values in $\L^-_{A}$ and the same holds true for the sets $\L^-_{B}$ and $\L^+_{A}$. The self differences of the array with inter-element spacing of $Nd$ are partly present in $\L^+_{A}$ and $\L^-_{A}$ ($m \in [0, M-1]$) and partly in $\L^+_{B}$ and $\L^-_{B}$ ($m \in [M, 2M-1]$). This is also evident from Fig. \[fig\_cross\]. It implies that $2(MN-(N-1))$ unique values are present in $\L^+_{B} \cup \L^-_{B}$ which is not available in set $\L^+_{A} \cup \L^-_{A}$. Therefore, the set $\L_{C}$ has $MN+M+N-2+2(MN-(N-1))=3MN+M-N$ unique values. A summary of the unique differences in each difference set for the extended array is given in Table \[table\_unique\_diffs\_extended\]. The unique differences for the prototype co-prime array is also provided for comparison.
The prototype co-prime array had holes (missing difference values) in the co-prime range, while the extended co-prime array generates all the difference values in the co-prime range $[-MN+1, MN-1]$. However, it does not represent the maximum continuous range of the extended co-prime array. We present Proposition I which provides the expressions for the range of integers in the cross difference sets of an extended co-prime array and their continuity (i.e. the range without holes). The proposition holds for one extended co-prime period $[0, 2MN-1]$ with $0\leq n \leq N-1$ and $0\leq m \leq 2M-1$.
\[Prop1\]
1. ${\L}^+_{C}$ has $2MN$ distinct integers in the range $-N(2M-1)\leq l_c \leq M(N-1)$
2. ${\L}^-_{C}$ has $2MN$ distinct integers in the range $-M(N-1)\leq l_c \leq N(2M-1)$
3. ${\L}^+_{C}$ has consecutive integers in the range $-(MN+M-1)\leq l_c \leq N-1$
4. ${\L}^-_{C}$ has consecutive integers in the range $-(N-1)\leq l_c \leq (MN+M-1)$
5. ${\L_C}$ has consecutive integers in the range $-(MN+M-1) \leq l_c \leq (MN+M-1)$ which implies that this set has its first hole at $\mid l_c\mid$ =$(MN+M)$.
Proposition I-(1) and I-(2) can be easily proved by substituting the values of $n$ and $m$ in $(Mn-Nm)$ and $(Nm-Mn)$ respectively, to generate the maximum and minimum values in the set.\
*Proof of Proposition I-(3):* Let $l_c=Mn-Nm$ be an element in set ${\L}^+_{C}$ satisfying $-(MN+M-1)\leq l_c \leq N-1$. Since $n \in [0,N-1]$, the range of $Mn$ is $0\leq Mn \leq M(N-1)$. Using the range of $Mn$ and $l_c$, we need to prove that $m \in [0,2M-1]$, from the equation $Nm=Mn-l_c$. Therefore: $$\begin{aligned}
\label{eq:proof_propI_3}
\nonumber & -(N-1)\leq Nm\leq M(N-1)+(MN+M-1) \\
\nonumber & -1 < m < 2M \\
\nonumber & 0 \leq m \leq 2M-1
\end{aligned}$$ which satisfies the required range. Proposition I-(4) can be proved along similar lines as Proposition I-(3).\
*Proof of Proposition I-(5):* Let $l_c=\pm (Mn-Nm)$ be an element in the set ${\L_{C}}$ satisfying: $-(MN+M-1) \leq l_c \leq (MN+M-1)$, where $m \in [0,2M-1]$ and $n \in [0,N-1]$. First we need to prove that $\pm (MN+M)$ is indeed a hole and then show that the range $ -(MN+M-1) \leq l_c \leq (MN+M-1)$ is continuous. To prove that $\pm (MN+M)$ is a hole in the set ${\L_{C}}$, it is sufficient to prove that it is a hole in ${\L}^+_C$. This holds true because ${\L}^-_C$ is a flipped version of ${\L}^+_C$ and ${\L_{C}} ={\L}^+_C \cup {\L}^-_C$. Let $l_c=Mn-Nm$ be an element in set ${\L}^+_C$, and let us assume that $\pm (MN+M)$ is not a hole and exists in set ${\L}^+_C$. This implies that: $ Mn-Nm=\pm (MN+M)$ or $M(n\mp 1)=N(m\pm M)$, hence: $\frac{M}{N}=\frac{(m\pm M)}{(n\mp 1)}$. Since $n-1<N$ and $m-M<M$, the ratios $\frac{(m+M)}{(n-1)}$ and $\frac{(m-M)}{(n+1)}$ cannot be satisfied. The proof of continuity in the range $-(MN+M-1) \leq l_c \leq (MN+M-1)$ follows directly from Proposition I-(3) and I-(4).
Weight Function
===============
Proposition III in [@U_S_1] gives the expression for the weight function or the number of sample pairs that contribute to estimate the autocorrelation at each value in the difference set. This parameter can affects the convergence, accuracy, latency and bias of the estimate. In this section we present the number of sample pairs that contribute to estimate the autocorrelation for the extended co-prime array and is given as Proposition \[prop2\].
\[prop2\] Let $z(l)$ denote the number of elements contributing to the estimation at difference value $l$.
1. For $l \in {\L}^+_{SM} \cup {\L}^-_{SM}-\{0\}$: $$\label{eq:contributors_self_Mn}
\nonumber z(l)=(N-i)+1, \{1\leq i \leq N-1, l=\pm Mi\}$$
2. For $l \in {\L}^+_{SN} \cup {\L}^-_{SN}-\{0\}$: $$\label{eq:contributors_self_Nm}
\nonumber z(l)=(2M-i), \{1\leq i \leq 2M-1, l=\pm Ni\}$$
3. For $l \in \{l=0\}$: $z(l)=2M+N-1$
4. For $l \in {\L}_{C} -{\L}_{S}$: $$\label{eq:contributors_A}
\nonumber z(l)=2, \{l \in \{{\L}^+_{A} \cup {\L}^-_{A}\} -{\L}_{S}\}$$ $$\label{eq:contributors_B}
\nonumber z(l)=1, \{l \in \{{\L}^+_{B} \cup {\L}^-_{B}\} -{\L}_{S}\}$$
It is easy to conclude that the number of sample pairs that map to each difference value in the self difference set of signal $x(Mn)$ is given by $N-i$ as shown in Fig. \[fig\_self\_M\]. In addition, the cross difference set also has pairs of samples that contribute to the estimate at these self differences except at ‘0’, i.e. ${\L}^+_{SM}-\{0\}$ and ${\L}^-_{SM}-\{0\}$ in Fig. \[fig\_cross\]. Thus justifying Proposition \[prop2\]-(1). Proposition \[prop2\]-(2) can be easily inferred from Fig.\[fig\_self\_N\]. The number of contributors at difference value ‘0’, Proposition \[prop2\]-(3), is the sum of the contributors of the self difference set of the two arrays at zero minus one pair which is common to both the self difference sets. ${\L^+_{A}}$ and ${\L^-_{A}}$ represent the cross difference set of the prototype co-prime array and Proposition III-(4) in [@U_S_1] shows that $z(l)=2$ for $\{l \in \{{\L}^+_{A} \cup {\L}^-_{A}\} -{\L}_{S}\}$. On the other hand, we can establish that $z(l)=1$ for $\{l \in \{{\L}^+_{B} \cup {\L}^-_{B}\} -{\L}_{S}\}$ from Fig. \[fig\_cross\] and Property I. In Fig. \[fig:wts\_M>N\] and \[fig:wts\_M<N\], we provide some examples of the weight function of the extended co-prime array for $M>N$ and $N>M$ respectively. The closed-form expression for the weight function $z_{e}(l)$ is given by: $$\label{eq:extend_entire}
\begin{split}
z_{e}(l)=\underbrace{\sum\limits_{n=-(N-1)}^{N-1} (N-\mid n\mid +1)\delta(l-Mn)}_\text{A}\\
+\underbrace{\sum\limits_{m=-(2M-1)}^{2M-1} (2M-\mid m\mid)\delta(l-Nm)}_\text{B}\\
+\underbrace{\sum\limits_{n=1}^{N-1}\sum\limits_{m=1}^{M-1} 2 \delta(l-(Mn-Nm))-\delta(l)}_\text{C}\\
+\underbrace{\sum\limits_{n=1}^{N-1}\sum\limits_{m=M+1}^{2M-1} \delta(\mid l\mid-(Mn-Nm))-\delta(l)}_\text{D}
\end{split}$$ The expressions for $z_{e}$ is directly obtained from Proposition \[prop2\]. ‘$A$’ relates to Proposition \[prop2\]-1 and $l=0$ generates a value $N+1$. ‘$B$’ relates to Proposition \[prop2\]-2 and $l=0$ generates a value $2M$. ‘$C$’ relates to the set ${\L}^+_{A}$ in Proposition \[prop2\]-4. This term gives a value equal to $-1$ when $l=0$. ‘$D$’ represents single contributors in the set ${\L}^+_{B}$ as in Proposition \[prop2\]-4. The use of $\mid l\mid$ in $\delta(\mid l\mid-(Mn-Nm))$ is due to the fact that $(Mn-Nm)$ generates values in ${\L}^+_{B}-{\L}_{S}$ which are all negative. Note that when $l=0$, the expression gives Proposition \[prop2\]-3.
The expression for the weight function is important since its Fourier transform gives the expression for the correlogram bias window. Thus, quantifying the distortion. For Nyquist acquisition, the bias window is quantified in Chapter 2 of [@7.1], while equation (20) in [@U_S_1] quantifies it for the prototype co-prime scheme. The expression in quantifies the bias window for the correlogram estimator for the extended co-prime arrays. This is the Fourier transform of . Examples in Fig. \[fig:choiceM\_N\_1\] and \[fig:choiceM\_N\_2\] show that the derived expression in matches the simulated bias window. The simulated bias window is obtained as FFT of the weight function in MATLAB. This verifies the correctness of the derived expression.
$$\label{eq:FT_extend_full}
\begin{split}
& W_{b}(e^{j\omega})=\frac{1}{s_b}\left\{\left |\frac{sin(\frac{\omega MN}{2})}{sin(\frac{\omega M}{2})}\right|^2+\frac{sin(\frac{\omega M(2N-1)}{2})}{sin(\frac{\omega M}{2})}\right. \left.
+\left|\frac{sin(\omega MN)}{sin(\frac{\omega N}{2})}\right|^2 \right.
\left.+2(1+cos(\omega MN))\frac{sin(\frac{\omega M(N-1)}{2})sin(\frac{\omega N(M-1)}{2})}{sin(\frac{\omega M}{2})sin(\frac{\omega N}{2})}-2\right\}
\end{split}$$
Ideally, we wish to have a bias window that equals an impulse. Such a window will not cause any distortion to the spectral estimate. The relative distance between the main lobe peak and side-lobe peak of the bias window was calculated for different values of $(M, N)$, as shown in Table \[table\_extened\_comparison\_Choice\_MN\]. It is of interest to have low side-lobes and hence, large relative distance between the main-lobe and side-lobe peaks. It may be noted that $(M, N)= (7, 13)$ seems to be a good choice. This value is $M\approx \frac{N}{2}$. The next section discusses the simulation results.
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
$\textbf{(M,N)}$ **(14,13)** **(14,5)** **(7,13)** **(13,14)** **(5,14)** **(13,7)**
------------------ ------------- ------------ ------------ ------------- ------------ ------------
***R*** 0.537 0.287 **0.734** 0.580 0.641 0.387
: Relative distance ($R$) between main-lobe and side-lobe peak:\
A comparison between $M>N$ and $N>M$.[]{data-label="table_extened_comparison_Choice_MN"}
Numerical simulations {#sec4}
=====================
Simulations are performed on a temporal signal model described in [@U_S_1; @4.32]. The first step is to estimate the autocorrelation using the combined set. Next take the Fourier transform of the autocorrelation estimate to obtain the correlogram spectral estimate. Repeat this for $L$ snapshots and compute the average correlogram. It may be noted that instead of calculating the correlogram using the autocorrelation, the Fourier transform of the acquired signal with zeros inserted at the holes (missing difference values) can be calculated, i.e.Periodogram (Refer Section IV in [@U_S_1]). We demonstrate the results for the case with number of snapshots, $L=10$. Fig. \[fig:sim\_1peak\_1\] and Fig. \[fig:sim\_1peak\_2\] demonstrates single peak estimation for different values of $(M, N)$. We observe several spurious peaks and note that $M\approx \frac{N}{2}$ seems to be a good choice. Furthermore, $(M, N)=(3, 7)$ has the lowest spurious peak for the examples considered.
Next, consider Fig. \[fig:sim\_3peak\_1\] and \[fig:sim\_3peak\_2\] for the estimation of three spectral peaks. Most of the examples considered here, have spurious peaks. $(M, N)=(3, 7)$ seems to be a good choice. Based on the analysis in this paper, we have the following thoughts:
1. On the basis of the bias window expression and the examples considered, we find that $M\approx \frac{N}{2}$ seems to be a good choice to reduce the side-lobes for the extended co-prime arrays.
2. The simulation results indicate that all values of $M\approx \frac{N}{2}$ will not work. For example, we find lower valued integer, i.e. $(M, N)=(3, 7)$ to be good.
3. Since the correlogram spectral estimate is the convolution of the bias wiondow with the true spectrum (in a statistical sense), we conclude that the accuracy will also depend on the signal model. For example, $(M, N)=(3, 8)$ works for single peak estimation in Fig. \[fig:sim\_1peak\_2\] but fails for three peak estimation in Fig. \[fig:sim\_3peak\_2\].
In the next section, concluding remarks and possible questions for future research are considered.
Conclusion {#sec5}
==========
This paper describes the difference set for the extended co-prime arrays. Low latency spectral estimation is demonstrated using the correlogram method. Smaller values of co-prime parameters $(M, N)$ with $M\approx \frac{N}{2}$ is found to be a good choice for the examples considered. We develop the closed-form expressions for the weight function and correlogram bias window using the combined difference set. The bias window expression describes the distortion in the estimate. This throws up several challenges which needs further investigation. For example, can the bias distortion be reduced? What could be the ways to achieve this? In addition, other spectral estimation methods can be investigated. Several applications can be explored for low latency estimation along similar lines.
|
---
abstract: |
Рассматривается возможность сверхсветовых перемещений нетахионных (т. е. движущихся с мгновенной скоростью $v<c$) тел в рамках общей теории относительности. Обсуждается запрет, предположительно налагаемый на такие перемещения квантовой теорией поля. Демонстрируется несостоятельность этого запрета в общем случае.\
Ключевые слова: сверхсветовые перемещения, причинность, кротовина, квантовое неравенство.
author:
- 'С. В. Красников[^1]'
title: 'СВЕРХСВЕТОВЫЕ ПЕРЕМЕЩЕНИЯ В (ПОЛУ)КЛАССИЧЕСКОЙ ОТО'
---
СВЕРХСВЕТОВЫЕ ПЕРЕМЕЩЕНИЯ
=========================
Скорость тела, состоящего из обычной (нетахионной) материи не может превышать скорость света. Если под скоростью понимать мгновенную скорость, т. е. скорость в точке, то это утверждение *верно*, но *тавтологично* — если найдется тело, летящее быстрее света, его придется назвать тахионом. На практике, однако, под скоростью в данном случае понимается нечто иное (нечто вроде эффективной средней скорости, как мы увидим). Утверждение оказывается *содержательным* и *неверным*.
Считается, что расстояние между Землей и Денебом составляет $d=1500$ световых лет. Допустим, корабль сумел долететь от Земли до Денеба за время $T_1<1500\,$лет, или — ниже мы увидим, что это совершенно разные допущения — сумел долететь до Денеба и вернуться на Землю за $T_2<3000\,$лет (в обоих случаях время измеряется по земным часам). Представляется вполне законным назвать такое перемещение сверхсветовым, поскольку $d/T_1>c$. Как мы увидим чуть ниже на конкретных примерах, сверхсветовое — в этом смысле — перемещение возможно, в принципе, даже для тела, движущегося с мгновенной скоростью $v<c$. Это связано с природой величины $d$: в общем случае в ОТО едва ли можно придать разумный смысл понятию расстояния между телами, а когда это оказывается возможным, расстояние вовсе не обязано совпадать с тем, что выше было обозначено $d$. Утверждение расстояние между Землей и Денебом составляет $1500\,$световых лет следует, строго говоря, понимать следующим образом. Есть область вселенной $N$, включающий в себя часть мировой линии Земли, часть мировой линии Денеба и некоторые из соединяющих их изотропных геодезических. Область эта (приблизительно) плоская и мы можем рассматривать ее и как часть реального пространства-времени (с его, возможно, очень сложной геометрией), и как часть *фиктивного* пространства Минковского, в котором мировым линиям Земли и Денеба соответствуют параллельные прямые. Понятие расстояния между такими прямыми вводится очевидным образом. Именно это расстояние и составляет $1500\,$световых лет.
Как явствует из вышесказанного, сверхсветовые перемещения удобней обсуждать не в терминах скоростей, а сравнивая реальное пространство-время с пространством Минковского [@portal].
#### Опеределение. {#опеределение. .unnumbered}
Пусть $C$ — времениподобный цилиндр $\sum_{i=1}^3x_i^2\leqslant r_0^2$ в пространстве Минковского [IL]{}$^4$. Область $U$ глобально гиперболического пространства $M$, а иногда и само $M$, будем называть *лазом* (соответствующий английский термин — shortcut), если существует изометрия $\varkappa\colon\; (M-U)\to(\text{{I\!L}}^4-C)$ и пара точек $p,q\in (M-U)$ таких, что $$p\preccurlyeq q,\qquad\varkappa(p)\not\preccurlyeq\varkappa(q)$$
(напомню обозначения: $a\preccurlyeq b$ означает, что из точки $a$ до $b$ можно провести направленную в будущее причинную кривую; ниже нам еще понадобятся $J^+(a)\equiv\{x|\:a\preccurlyeq x\}$, $J^-(a)\equiv\{x|\:x\preccurlyeq a\}$ и $J^\pm(\gamma)\equiv
\bigcup_{a\in\gamma} J^\pm(a)$). Итак, $M$ — лаз, если его можно получить из пространства Минковского заменой цилиндра $C\subset
\text{{I\!L}}^4$ на нечто такое (а именно, на $U$), что точки, пространственноподобно разделенные в [IL]{}$^4$, становятся причинно связанными. Путешествие сквозь лаз из $p$ в $q$ как раз и есть сверхсветовое перемещение, т. к. тело попадает туда, куда свет (будь это пространство Минковского) дойти бы не успел.
**Пример .Пузырь Алькубиерре.**
\[ex:Alc\] На плоскости [ IR]{}$^2$ рассмотрим метрику $${\mathrm{d}}s^2 = \Omega^2(r)({\mathrm{d}}x^2 +{\mathrm{d}}y^2) ,\qquad r\equiv\sqrt{x^2+y^2}$$ где $\Omega$ — монотонная функция такая, что $${\Omega\,\vrule \,{}_{\displaystyle{}_{r>r_0 }}}=1,\quad
{\Omega\,\vrule \,{}_{\displaystyle{}_{r<(1-\delta)r_0}}}=\Omega_0=const,\qquad
0<\delta,\Omega_0\ll 1 .$$ Все пространство, кроме колечка $1-\delta<r/r_0<1$ является плоским и, более того, при $r>r_0 $ оно неотличимо от Евклидовой плоскости. Тем не менее, точка $(x=-r_0 $, $y=0)$ намного ближе ($\approx 2\Omega_0 r_0 $ против $2r_0 $) к диаметрально противоположной точке $(x=r_0 ,$ $y=0)$, чем в случае Евклидовой плоскости. Естественное 4-мерное обобщение (ср. [@Alc]) $${\mathrm{d}}s^2 = -{\mathrm{d}}t^2 + \Omega^2(r)({\mathrm{d}}x_1^2 + {\mathrm{d}}x_2^2+ {\mathrm{d}}x_3^2)
\qquad r\equiv\Big(\sum_{i=0}^3x_i^2\Big)^{1/2}$$ очевидно является лазом \[см. рис. \[fig:lazy\](а)\].
![(a) Световой конус внутри лаза раскрыт (в данных координатах) в $1/\Omega_0$ раз шире, чем в пространстве Минковского. Поэтому пунктирная линия времениподобна. (б) Искусственный лаз. $o$ предшествует ему. \[fig:lazy\]](lazy.ps){height="9.8"}
\(a) (б)
**Пример .Труба Красникова.**
\[ex:tube\] Пусть $M$ — это [ IR]{}$^4$ с метрикой [@FTL] $$\label{eq:tr}
{\mathrm{d}}s^2 = ({\mathrm{d}}x_1-{\mathrm{d}}t)(k{\mathrm{d}}x_1+{\mathrm{d}}t) + {\mathrm{d}}x_2^2+ {\mathrm{d}}x_3^2,$$ где $k=k(r)$ — монотонная функция такая, что $${k\,\vrule \,{}_{\displaystyle{}_{r>r_0}}}=1,\quad
{k\,\vrule \,{}_{\displaystyle{}_{r<(1-\delta)r_0}}}=k_0=const,\qquad
-1<k_0<0 .$$ Пространство, как и в предыдущем примере, искривлено только в сферическом слое толщины $\delta$. Главное — и, как мы увидим ниже, важное — отличие состоит в том, что путешествие теперь может закончиться *еще до того как началось*, см. рис. \[fig:lazy\](б). Эффект этот чисто координатный (время измеряется в разных точках) и с нарушением причинности не связан. Действительно, рассмотрим функцию $F=t+\frac{k_0-1}{2}x_1$ и любую непродолжимую причинную кривую $\lambda(\zeta)$, где $\zeta$ — натуральный параметр во вспомогательной евклидовой метрике $${\mathrm{d}}s_E^2 = \omega_0^2+\omega_1^2 + {\mathrm{d}}x_2^2+ {\mathrm{d}}x_3^2,
\qquad \omega_0\equiv {\mathrm{d}}t +\tfrac{k-1}{2}{\mathrm{d}}x_1,
\quad \omega_1\equiv \tfrac{k+1}{2}{\mathrm{d}}x_1$$ (она отличается от метрики только знаком при $\omega_0^2$). Нетрудно доказать, что $|{\mathrm{d}}F/{\mathrm{d}}\zeta|>(1+k_0)/2$, откуда ясно, что $\lambda$ должна рано или поздно пересечь (единственный раз) поверхность $F=0$. Последняя, т. о., является поверхностью Коши $M$ и, следовательно (см. [@HawEl предложение 6.6.3]), $M$ глобально гиперболично. Наличие требуемых $p$ и $q$ очевидно, и мы заключаем, что $M$ — лаз.
**Пример .Кротовина.**
\[ex:wh\] Вырежем из пространства Минковского два тонких цилиндра, см. рис. \[fig:krot\](а), склеим границы получившихся дыр и сгладим метрику в окрестности склейки так, чтобы устранить возникший там разрыв производных.
![Кротовина с меняющимся во времени расстоянием между входами.\[fig:krot\] Концы пунктирных линий склеены.](krot.ps){height="14"}
\(a) (б)
Получившееся пространство [@Tho; @viss], называют *кротовиной* (в англоязычной литературе — wormhole). Часто так же называют и его пространственноподобное сечение, рис. \[fig:krot\](б). Описанная процедура неоднозначна, т. к. точки на границах цилиндров можно отождествлять по-разному. Обычно склеивают точки с равным $\tau$, где $\tau$ есть (лоренцево) расстояние до некоторой плоскости — скажем, $t=0$ — измеренное вдоль образующей цилиндра. Такое правило нужно, если мы хотим описывать кротовину, эволюционирующую, как показано на рис. \[fig:krot\](б): расстояние между входами во внешнем пространстве меняется, а форма кротовины (и, соответственно, ее длина) — нет. Очевидно, если при $t>0$ один вход покоится, а другой движется, то у склеиваемых точек окажется разное $t$ (парадокс близнецов). Поэтому, кротовину можно (в частности) использовать и для перемещений, которые кончаются раньше, чем начались, см. рис. \[fig:krot\](а). Как и в прошлом примере, это не обязательно нарушает причинность: если $\Delta t < L$, пространство остается глобально гиперболичным и, как следствие, — лазом.
Пригодность лазов для межзвездных перелетов различна. Будем говорить, что точка $o\in M$ *предшествует* лазу, если $\varkappa$ может быть продолжена на $M- J^+(o)$, и назовем лаз *вечным*, если ему не предшествует никакая точка. Очевидно, в отсутствие тахионов лаз можно интерпретировать как созданный по решению, принятому в $p$, только если $p$ этому лазу предшествует. Вечный же лаз создать вообще нельзя, а можно только *найти*.
Все три рассмотренных выше лаза, будучи статическими, являются, конечно, вечными. В случае кротовины это обстоятельство принципиально — в глобально гиперболическом пространстве-времени кротовина не может ни появиться, ни исчезнуть (см. [@HawEl предложение 6.6.8]). В то же время, два первых лаза легко модифицировать так, чтобы они перестали быть вечными. Достаточно отменить при $t,r<r_0$ условия наложенные на $\Omega$ и $k$, а при $t<r$ потребовать, чтобы $\Omega$ и $k$ были равны 1. Таким лазам предшествует, например, начало координат.
К сожалению, точка $p$, фигурирующая в определении лаза, не может этому лазу предшествовать (иначе $q$ — в противоречии с определением — лежала бы в $M- J^+(p)$, поскольку $\varkappa(q)$ лежит в [IL]{}$^4 -
J^+[\varkappa(p)]$). Т. о., когда решение лететь к Денебу принято, строить лаз уже поздно в том смысле, что $T_1$ это не уменьшит. Что, однако, не обесценивает идею такого строительства — лазом еще не поздно воспользоваться для уменьшения $T_2$ (см. рис. \[fig:lazy\](б)), а это, похоже, для практических нужд гораздо важнее.
КВАНТОВЫЕ ОГРАНИЧЕНИЯ
=====================
До сих пор мы обсуждали возможность преодоления светового барьера с чисто геометрической точки зрения. Однако, в рамках ОТО геометрия вселенной и свойства заполняющего ее вещества связаны уравнениями Эйнштейна $G_{ik}=8\pi
T_{ik}$ (здесь и ниже по умолчанию подразумевается система единиц, в которой $G =\hbar = c = 1$), и можно поставить вопрос о том, какими свойствами должен обладать тензор энергии-импульса (ТЭИ) материи, заполняющей лаз. Оказывается, во всех рассмотренных выше примерах есть точки, где $G_{ik}t^i t^k<0$ для некоторого времениподобного вектора $\bm t$ и, значит, нарушается т. н.слабое энергетическое условие (СЭУ), требующее, чтобы $$T_{ik}t^i t^k\geqslant 0\qquad\quad \forall \,\bm t\colon \ t^i t_i\leqslant
0,$$ т. е. чтобы плотность энергии была неотрицательна в любой системе отсчета. Хотя строгое доказательство пока найдено только для кротовин [@FSW], похоже, что нарушение СЭУ присуще *любому* лазу. Грубо говоря, если бы плотность энергии всюду была неотрицательна, то естественно было бы ожидать, что на больших расстояниях (т. е. в Ньютоновском пределе) гравитационное поле убывало бы, как $1/r^2$, а не исчезало совсем, как того требует определение лаза.
В классической физике плотность энергии предположительно не может быть отрицательной (поэтому вещество, нарушающее СЭУ называют *экзотическим*) и т. о. для сверхсветовых перелетов остаются две возможности:
1). Можно исследовать ситуации, в которых нарушения СЭУ связаны просто с тем, что мы приписали нулевую кривизну области $N$. Тем самым мы *заранее* посчитали отсутствующими любые источники энергии и, в том числе, нужные для обеспечения (соблюдающего СЭУ) сверхсветового перелета. Чтобы последовательно учесть наличие потенциальных источников энергии (скажем, звезд, лежащих чуть в стороне от прямой Земля–Денеб), понятие лаза придется обобщить (это не трудно) так, чтобы $M$ сравнивалось не с [IL]{}$^4$, а с неким — тоже искривленным — $M'$. Такого рода лазы, действительно, не требуют экзотического вещества (см. Приложение в [@portal]), но вопрос о том, насколько эффективными в этом случае они могут быть, остается открытым.
2). Можно также учесть квантовые поправки к уравнениям Эйнштейна. В полуклассической гравитации [@GMM; @BirDav] последние приобретают вид $$G_{ ik }=8\pi T^{\rm C}_{ ik } + 8\pi \, T_{ ik },$$ где $T^{\rm C}_{ ik }$ — вклад классической материи, а второй член — это ренормированное среднее значение ТЭИ рассматриваемого поля. Известно, что $T_{
ik }$ может нарушать СЭУ (классический пример — эффект Казимира). Эта, вторая возможность с точки зрения квантовой теории поля гораздо интересней, на ней мы и остановимся.
Пусть $\gamma(\tau)$ — мировая линия свободно падающего наблюдателя и $\tau$ — его собственное время. Рассмотрим усредненную с $\chi$плотность энергии $\rho = T_{\hat 0\hat 0}$ (шляпки над индексами тензора означают, что его компоненты определяются в ортонормированном базисе, с нулевым ортом $\partial_\tau$)
$$\rho_\chi(\tau_0)\equiv \int_{-\infty}^{-\infty}
\rho(\tau)\chi(\tau-\tau_0)\,{\mathrm{d}}\tau,$$ где интеграл берется вдоль $\gamma$, а функция $\chi$ нормирована условием $$\int_{-\infty}^{-\infty} \chi(\tau)\,{\mathrm{d}}\tau=1.$$ В случае скалярного (с минимальной связью) и электромагнитного полей в $d$-мерном пространстве Минковского ($d=2,4$) доказано , что для функций специального вида, а именно $ h(\tau)\equiv\pi^{-1}\Delta
/(\tau^2+\Delta ^2) $, справедливо неравенство $$\label{eq:qinOR}
\rho_h> -\Delta ^{-d}.$$ Факт этот часто интерпретируется как математическое выражение некоего принципа дополнительности, связывающего нарушения СЭУ с промежутком времени $\Delta $, в течение которого его можно наблюдать: чем сильнее нарушение (т. е. чем больше $|\rho|$), тем меньше $\Delta $. Исходя из идеи, что искривленное пространство является почти плоским, если рассматривается достаточно маленькая область, Форд и Роман предположили , что подобный принцип верен универсально — т. е. для любых квантовых состояний любых свободных[^2] полей в любом пространстве-времени. Конкретно, если усреднение производится с функцией $f$: $$\label{eq:uslnaf}
f\in C^\infty,\qquad \operatorname{supp}f\in (\tau_1,\tau_2),
\qquad
\int_{\tau_1}^{\tau_2}[f'(\tau)]^2/f(\tau)\,{\mathrm{d}}\tau\lesssim 1,$$ то, предположительно [@singed], должно выполняться неравенство $$\label{eq:rrr}
\rho_f\gtrsim -\mathcal T^{-d}\qquad\forall\,\mathcal T\lesssim\mathcal T_R,$$ где $$\mathcal T\equiv|\tau_2-\tau_1|,\qquad
\mathcal T_R\equiv
\big(\max |R_{\hat \imath\hat \jmath\hat m\hat n}|\big)^{-1/2}.$$ Максимум здесь берется по всем компонентам и всем $\tau\in (\tau_1,\tau_2)$. Физический смысл квантового неравенства примерно тот же, что и , а условие на $\mathcal T$ должно гарантировать малость рассматриваемой области (иногда его приходится дополнять неким топологическим требованием, см. раздел \[subs:neodn\]). Выбор функций вызван тем, что именно для таких $f$ неравенство удалось доказать (при $d=2$ для безмассового скалярного поля в состоянии конформного вакуума [@conf2; @Few]).
Квантовое неравенство (КН) позволяет весьма эффектно продемонстрировать нефизичность лазов, рассмотренных выше [@97][^3]. Ход рассуждений приблизительно таков. Пусть $p$ такая точка, что через нее проходит времениподобный геодезический сегмент $\gamma(\tau)$ длины $\mathcal
T_R$, вдоль которого $\rho$ отрицательна и примерно постоянна. Предположим, $$\label{eq:noC}
\max |R_{\hat \imath\hat \jmath\hat m\hat n}(p)|\lesssim
\max |T_{\hat k\hat l}(p)|\approx -\rho(p)$$ (во всех трех лазах действительно можно найти $p$, удовлетворяющую предъявленным требованиям). Применяя с $\mathcal T=\mathcal T_R$ к $\gamma$, получим $$\label{eq:cep}
|\rho_f| <\mathcal T_R^{-4} = \big(\max |R_{\hat \imath\hat \jmath\hat m\hat
n}|\big)^{2}
\lesssim \rho^2(p)$$ или $$\label{eq:Pl_den}
|\rho(p)|\gtrsim \rho_f/\rho(p)\approx 1.$$ Т. о. плотность энергии в $p$ должна быть Планковской! В рассмотренных выше примерах можно представить себе, что СЭУ нарушается только в тонком — скажем, Планковской толщины — сферическом слое. Однако, если мы хотим использовать лаз для перемещения макроскопических объектов (с характерным размером $l$), то естественно ожидать, что диаметр слоя будет $\sim l$ и объем $V\sim\pi l_\text{Pl}l^2$. Поэтому при $l\sim 1\,$м суммарное количество отрицательной энергии, заключенное даже в столь тонком слое будет чудовищным: $$\label{eq:ogr}
|{\ensuremath{E_{\rm tot}^-}}|=-\rho V \sim \pi\left(\frac{2,2\times 10^{-8}\,\text{кг}}{(1,6\times
10^{-35}\text{м})^3}\right) 1\,\text{м}^2 (1,6\times 10^{-35}\text{м})\approx
3\times 10^{62}\,\text{кг}.$$ Очевидно, каков бы ни был (несколько туманный) физический смысл величины ${\ensuremath{E_{\rm tot}^-}}$, этот результат следует интерпретировать как полную невозможность создания такого рода лазов даже в очень отдаленном будущем. Сам по себе этот факт не слишком важен в случае пространств из примеров \[ex:Alc\], \[ex:tube\]. Действительно, оба пространства выбиралась настолько простыми, насколько позволяла задача, для которой они были придуманы: иллюстрация *принципиальной* совместимости сверхсветовых перемещений с запретом локального превышения скорости света. Можно предполагать поэтому, что нежелательные свойства требующихся для них источников являются следствиями именно этой простоты. Проблема, однако, в том, что рассуждения, приведшие к , представляются настолько общими, что кажется, будто они справедливы для *любого* лаза. Это означало бы, что квантовое неравенство исключает сверхсветовые перемещения.
УНИВЕРСАЛЬНОСТЬ ОГРАНИЧЕНИЙ
===========================
Общность рассуждений, обосновывающих , в действительности несколько обманчива. Существует ряд ситуаций, в которых КН, даже если оно справедливо, не требует, тем не менее, справедливости . Рассмотрим некоторые из них (подробности см. [@portal; @noQI]).
Нелокальные эффекты {#subs:neodn}
-------------------
Как известно [@BirDav], у безмассового скалярного поля на цилиндре $${\mathrm{d}}s^2 = -{\mathrm{d}}t^2 +{\mathrm{d}}x^2 \qquad
x=x+L.$$ есть квантовое состояние, в котором $\rho=-\frac{\pi}{6}L^{-2}$. Такая плотность энергии очевидно противоречит , поскольку в данном случае $\mathcal T_R=\infty$. Возможно, это связано с тем, что на масштабах $\gtrsim L$ цилиндр — хотя он и плоский — все же недостаточно похож на часть пространства Минковского. Т. о. условие на величину $\mathcal T$ в *необходимо дополнить* еще каким-нибудь требованием, которое бы учитывало такую возможность. Можно, в частности[^4], ограничить действие КН геодезическими $\gamma$ настолько короткими, что $J_\gamma\equiv J^+(\gamma)\cap J^-(\gamma)$ имеет топологию шара (как это имеет место в [IL]{}$^4$). Такое условие, однако, выводит из-под действия КН наиболее интересную разновидность кротовин. Действительно, пусть $\lambda$ — любой времениподобный геодезический отрезок в горловине статической (для простоты) кротовины. Можно показать, что при $\Delta t \to
L$ и *неизменной геометрии горловины* на $\lambda$ обязательно найдутся точки, соединимые *негомотопной* $\lambda$ причинной кривой (и значит $J_\lambda$ уж точно не шар). Т. о., КН оказывается тем менее ограничительным, чем эффективнее кротовина с точки зрения межзвездных перелетов.
Искривление пространства удаленными телами
------------------------------------------
Важную роль при выводе играло неравенство в , выражающее идею, что кривизна в $p$ возникает за счет наличия в $p$ экзотического вещества. С учетом уравнений Эйнштейна обсуждаемое неравенство приобретает вид $$\label{eq:bezVeyl}
\max |R_{\hat \imath\hat\jmath\hat m\hat n}(p)|
\lesssim \max |G_{\hat k\hat l}(p)|$$ и *на первый взгляд* должно выполняться всегда при отсутствии специальных симметрий или подгонок параметров. Это, однако, не верно. Кривизна в точке определяется (среди прочего) тензором Вейля: $$R_{\hat \imath\hat\jmath\hat m\hat n}=C_{\hat \imath\hat\jmath\hat m\hat n} -
g_{\hat \imath[\hat n}R_{\hat m]\hat\jmath}
- g_{\hat\jmath[\hat m}R_{\hat n]\hat \imath}
-\tfrac{1}{3}g_{\hat \imath[\hat m}g_{\hat n]\hat\jmath}R.$$ Т. о. основания ожидать, что справедливо, есть только, когда $$P\equiv\max |C_{\hat \imath\hat\jmath\hat m\hat n}|/ \max |R_{\hat
\imath\hat\jmath}|\lesssim 1,$$ а это, *как правило*, не выполняется. В частности, в *любой* искривленной, но пустой области (например, в окрестности любой звезды) $P=\infty$.
Неоднородное распределение материи
----------------------------------
Попробуем воспроизвести вывод , опустив условие постоянства $\rho(\tau)$. Тогда для оправдания последнего неравенства в придется потребовать $|\rho(p)|\approx
\max_\gamma|\rho|$. С другой стороны, для нужно $ |\rho_f|\gtrsim |\rho(p)|$. Значит для получения таким способом существенно, чтобы $|\rho_f|\approx \max_\gamma|\rho|$, т. е. $$|\rho_f|\gtrsim |\overline\rho|,\qquad
\overline\rho\equiv
\mathcal T^{-1}\int_{\tau_1}^{\tau_2}\rho(\tau)\,{\mathrm{d}}\tau$$ ($\overline{\rho}$, очевидно, есть *обычное* среднее). Вкупе с это значит, что необходимым условием является $$\label{eq:j,sx}
|\overline\rho|\lesssim \mathcal T^{-d} \qquad\text{при}\quad \mathcal
T\lesssim\mathcal T_R.$$
Верно ли неравенство за пределами двумерного конформно-тривиального случая неизвестно, но вот определенно нарушается даже и в этом случае [@noQI].
**Пример .**
Рассмотрим двумерное пространство де Ситтера. При подходящем выборе координат $u$, $v$ (покрывающих *все* пространство) метрика в области $W=\{\epsilon<\arctg e^{2v}<\arctg
e^{2u}<\tfrac{\pi}{2}-\epsilon\}$ примет вид $${\mathrm{d}}s^2= \alpha^2\sinh^{-2}(u-v){\mathrm{d}}u{\mathrm{d}}v.$$ ТЭИ безмассового скалярного поля в состоянии конформного вакуума, ассоциированного с так записанной метрикой [@BirDav] $$T_{vv}= T_{uu}=
-\tfrac{1}{12\pi}, \qquad
T_{uv}= \tfrac{1}{12\pi}\sinh^{-2}(u-v).$$ Пусть $\gamma$ — сегмент времениподобной геодезической, заданный условием $$v+u=0,\quad t\in[t_1,t_2],
\qquad t\equiv \tfrac{1}{2}(v-u).$$ Потребуем, обеспечив этим, в частности, $\gamma\subset W$, чтобы $$\tfrac{1}{2}\ln\tan\epsilon<t_1<t_2\ll -1.$$ Плотность энергии, измеренная наблюдателем с мировой линией $\gamma$, будет $$\rho=T_{\hat 0\hat 0}=
2(T_{uu}-T_{uv})(\alpha^2\sinh^{-2}(u-v))^{-1}
\approx
-\frac{\sinh^{2}(u-v)}{6\pi \alpha^2},$$ откуда $$-\overline\rho=\frac{\mathcal T^{-1}}{6\pi}\int^{t_2}_{t_1}
\alpha^{-1}\sinh(u-v)\,{\mathrm{d}}t \approx
\frac{\mathcal T^{-1}}{24\pi}\alpha^{-1} e^{-2t_1}.$$ В то же время $$\mathcal T=-\int^{t_2}_{t_1}\,\alpha\sinh^{-1}2t\,{\mathrm{d}}t
\approx \alpha\int^{2t_2}_{2t_1}e^{s}{\mathrm{d}}s
\approx
\alpha e^{2t_2}.$$ Таким образом, $$\overline\rho=-\tfrac{1}{24\pi}e^{2(t_2-t_1)}{\mathcal T}^{-2},
\qquad\mathcal T\ll \mathcal T_R=\alpha,$$ что очевидно нарушает (к тому же сколь угодно сильно при достаточно малых $\epsilon$ и $t_1$) неравенство .
Экономные лазы
--------------
Даже если плотность энергии экзотического вещества $\rho\sim -1$, отсюда в общем случае все еще не следует . Дело в том, что $V$, на самом деле, не обязательно должен быть $\gtrsim l^2$.
**Пример .Портал.**
Введем две положительные константы, $\rho_0$, и $\eta_\varepsilon$, связанные требованием $\rho_0\gg\eta^2_\varepsilon$, и две функции — произвольную гладкую четную функцию $\varepsilon(\eta)$ со свойством $\operatorname{supp}\varepsilon=(-\eta_\varepsilon,\eta_\varepsilon)$ и $\rho$, определенную равенством $$\rho(\eta,\psi)\equiv \rho_0 - \eta^2\cos 2\psi.$$ Рассмотрим пространство-время $$\label{eq:portmet}
\begin{split}
{\mathrm{d}}s^2=-{\mathrm{d}}t^2 + 4(\varepsilon^2 + \eta^2)({\mathrm{d}}\eta^2 +
\eta^2{\mathrm{d}}\psi^2) + \rho^2{\mathrm{d}}\phi^2,\\
\eta,\rho\geqslant 0,\qquad \phi=\phi+2\pi,\quad \psi=\psi+2\pi.
\end{split}$$ (подразумевается, что точки с $\eta=0$, различающиеся только значением $\psi$, отождествляются, так же как и точки с $\rho=0$, различающиеся только значением $\phi$). Чтобы увидеть структуру этого пространства-времени (подробнее см. [@portal]), заметим, что при $\eta>\eta_\varepsilon$ метрика имеет вид $${\mathrm{d}}s^2 =-{\mathrm{d}}t^2 +
{\mathrm{d}}z^2 + {\mathrm{d}}\rho^2 + \rho^2{\mathrm{d}}\phi^2,$$ где $z\equiv \eta^2\sin 2\psi $. Поскольку $z(\psi)=z(\pi+\pi)$, можно представить пространство , как результат следующей хирургии. Возьмем два экземпляра Евклидова пространства [IE]{}$^3$ и вырежем из каждого по диску $\mathcal D$ радиуса $\rho_0$. Склеим теперь эти разрезанные пространства: левый берег каждого разреза отождествляется с правым берегом другого. Получившееся пространство (на самом деле, это просто двулистное накрытие [IE]{}$^3 -{\ensuremath{\mathbb S}}$, где ${\ensuremath{\mathbb S}}=\operatorname{Bd}\mathcal D$) сингулярно, т. к. мы вынуждены удалить точки ветвления ${\ensuremath{\mathbb S}}$ (в них теряется структура хаусдорфова многообразия). Оказывается, однако, что если должным образом искривить его метрику в полнотории (толщины $\eta^2_\varepsilon$ в нашем случае), окружающем ${\ensuremath{\mathbb S}}$, то указанная сингулярность *устранима*. В пространство можно вклеить окружность так, что оно станет полно. Это и будет cечение $t=const$ пространства-времени .
Чтобы получить из лаз, достаточно удалить по полупространству — скажем, $z>d$ и $z<-d$ — из каждого листа, а появившиеся границы склеить. Результат показан на рис. \[fig:geom\](а).
![(а) Портал. Два серых кольца, на самом деле, представляют собой *единый* полноторий. Верхняя толстая кривая непрерывна. Нижняя — является нестягиваемой петлей. (б) Карман Ван Ден Брука. СЭУ нарушается только в горловине (более темный участок). \[fig:geom\]](geom.ps){height="14em"}
\(a) (б)
Макроскопическое тело может пройти сквозь портал с $\rho_0\sim l/2$ при том, что область ненулевой кривизны в таком портале имеет объем $V\sim\pi l$ (я выбрал $\eta_\varepsilon\sim 1$). Даже если она вся заполнена экзотическим веществом планковской плотности, то $|{\ensuremath{E_{\rm tot}^-}}|$ при $l\sim 1\,$м составит . Хотя это и очень большая величина, она все же на 35 порядков (!) меньше той, что предсказана и составляет всего лишь $\sim 2\times 10^{-3} M_\odot$.
Следует также отметить, что лаз, пригодный для транспортировки макроскопических тел не обязан *сам* быть макроскопическим, обстоятельство позволяющее, в принципе, уменьшить [$E_{\rm tot}^-$]{} еще на 35 порядков. Дело в том, что даже очень большой объем может быть ограничен сферой очень маленькой площади. Например, в пространстве, показанном на рис. \[fig:geom\](б) область заполненная экзотическим веществом — это сферический слой радиуса $r_\text{вн}$. И какою бы большой ни была область с пассажиром, $r_\text{вн}$ может оставаться микроскопическим [@Broeck]. Можно, в частности потребовать $r_\text{вн}\sim 1$. Для удержания от схлопывания такого кармана достаточно, оказывается, всего ${\ensuremath{E_{\rm tot}^-}}\sim -10^{-3}\,$г экзотической материи [@portal].
[99]{} [*S. Krasnikov*. Phys. Rev. D. 2003. V. 67. 104013]{} [*M. Alcubierre*. Class. Quantum Grav. 1994. V. 11. L73]{} [*S. Krasnikov*. Phys. Rev. D. 1998. V. 57. P. 4760]{} *С. Хокинг, Дж. Эллис*. Крупномасштабная структура пространства-времени. Пер. с англ. М.: Мир, 1977. \[S. W. Hawking and G. F. R. Ellis. The Large scale structure of spacetime. Cambridge: Cambridge University Press, 1973.\][*M. S. Morris and K. S. Thorne*. Am. J. Phys. 1988. V. 56. P. 395]{} M. Visser. Lorentzian wormholes — from Einstein to Hawking. New York: AIP Press, 1995. [*J. L. Friedman, K. Schleich, and D. M. Witt*. Phys. Rev. Lett. 1993. V. 71. P. 1486]{} *А. А. Гриб, С. Г. Мамаев, В. М. Мостепаненко*. Вакуумные квантовые эффекты в сильных полях. М.: Энергоатомиздат, 1988. \[English transl.: A. A. Grib, S. G. Mamayev, and V. M. Mostepanenko. Vacuum Quantum Effects in Strong Fields. St.Petersburg: Friedmann Laboratory Publishing, 1994.\] *Н. Биррелл, П. Девис*. Квантованные поля в искривленном пространстве-времени. Пер. с англ. М.: Мир, 1984. \[N. D. Birrell and P. C. W. Davies. Quantum Fields in Curved Space. Cambridge: Cambridge University Press, 1982.\][*L. H. Ford and T. A. Roman*. Phys. Rev. D. 1995. V. 51. P. 4277]{}; [*L. H. Ford and T. A. Roman*. Phys. Rev. D. 1997. V. 55. P. 2082]{} [*L. H. Ford and T. A. Roman*. Phys. Rev.D. 1996. V. 53. P. 5496]{} [*L. H. Ford, M. J. Pfenning, and T. A. Roman*. Phys. Rev. D. 1998. V. 57. P. 4839]{} [*D. N. Vollick*. Phys. Rev. D. 2000. V. 61. 084022]{}; [*É. É. Flanagan*. ibid. 2002. V. 66. 104007]{} [*C. J. Fewster*. Phys. Rev. D. 2004. V. 70. 127501]{} [*M. J. Pfenning and L. H. Ford*. Class. Quantum Grav. 1997. V. 14. P. 1743]{}; [*A. E. Everett and T. A. Roman*. Phys. Rev. D. 1997. V. 56. P. 2100]{} *S. Krasnikov* Электронный препринт gr/qc-0409007. [*C. Van Den Broeck*. Class. Quantum Grav. 1999. V. 16. P. 3973]{}
[^1]: Главная астрономическая обсерватория РАН
[^2]: Что для взаимодействующих полей он заведомо неверен, видно уже из того, что ему не подчиняется (даже в плоском пространстве) эффект Казимира, в котором $\rho(t)=const<0$.
[^3]: На самом деле, в этих работах рассматривались чуть более сложные метрики.
[^4]: Это обобщение условия, которое Фьюстер [@Few] использует в случае $d=2$. Других приемлемых (т. е., как минимум, координатно независимых) вариантов подобного условия пока не предлагалось.
|
---
abstract: 'We study the generation of the primordial curvature perturbation in multi-field inflation. Considering both the evolution of the perturbation during inflation and the effects generated at the end of inflation, we present a general formula for the curvature perturbation. We provide the analytic expressions of the power spectrum, spectral tilt and non-Gaussianity for the separable potentials of two inflaton scalars, and apply them to some specific models.'
author:
- 'Ki-Young Choi$^{(a)}$$^{(b)}$[^1], Soo A Kim$^{(a)}$[^2], and Bumseok Kyae$^{(c)}$[^3]'
title: |
**Primordial curvature perturbation\
during and at the end of multi-field inflation**
---
[APCTP-Pre2012-002\
PNUTP-12-A02]{}
introduction
============
The generation of the large scale structures and the anisotropy in the temperature of the cosmic microwave background (CMB) suggests that there was already small inhomogeneity in the early Universe, a few Hubble times before the observable scale enters the horizon. The time-independent curvature perturbation $\zeta$ sets the initial conditions for such inhomogeneity and the subsequent evolution of all the scalar perturbations. The resulting power spectrum and the spectral tilt of the primordial curvature perturbation are almost Gaussian, and so scale-independent with the size of ${\cal P}_\zeta = 2.43\times 10^{-9}$ [@Komatsu:2010fb].
The perturbations necessary for the inhomogeneity can arise naturally from the vacuum fluctuations of light scalar field(s) during inflation and be promoted to classical one around the time of the horizon exit. In the single field inflation with the canonical kinetic term, the curvature perturbations produced at the horizon crossing are conserved after that and can explain the primordial curvature perturbation necessary for the observation.
Inflation with multiple scalar fields [@Bassett:2005xm; @Wands:2007bd], however, can admit quite different inflationary dynamics and spectra of the primordial perturbations, which is impossible in single field inflationary models. The presence of multiple field induces non-adiabatic field perturbations and the evolution of the overall curvature perturbations during inflation, possibly leading to detectable non-Gaussianity [@Byrnes:2008wi]. The evolution of the curvature perturbation continues until the non-adiabatic perturbation is converted to the adiabatic one. The conversion must be completed before the cosmologically relevant scales enter the horizon again in the deep radiation-dominated era after inflation, and thus we need to track the evolution of the curvature perturbation until it becomes frozen.
The other dominant component of the curvature perturbation in the multi-field inflation is generated at the transition between inflation and non-inflation phase [@Lyth:2005qk], which cannot happen in the single field inflation. In the single component inflation, the inflationary trajectory is unique and the inflation ends when the inflaton field ${\varphi}$ has a value ${\varphi}_e$, which is controlled entirely by the inflation and independent of the position. With additional scalar fields, however, the trajectory is in the multi-dimensional field space and the inflation ends with different field values. Then the field values of the inflatons depend on the positions and the field space at the end of inflation is not in the uniform energy density any more. The relative differences of ending inflation make additional differences in the e-folding number and contribute to the curvature perturbations at the end of inflation.
A simple example of generating the curvature perturbation at the end of inflation is the hybrid inflation with sudden-end approximation [@Lyth:2005qk; @Lyth:2006nx]. In this case, the end of inflation occurs suddenly by the dynamics of waterfall fields, and the observable large non-Gaussianity can be generated. The realization in the string theory was considered in [@Lyth:2006nx]. A specific analytic calculation was done for two-component hybrid inflations in Ref. [@Sasaki:2008uc; @Naruko:2008sq; @Huang:2009xa] with an exponential potential, and they also found that large non-Gaussianity is possible for certain conditions at the end of inflation. The more general calculation was attempted in Ref. [@Byrnes:2008zy] for two-field hybrid inflation. They studied the generation of large non-Gaussianity both during the inflation and at the end of inflation. They found that the end condition of inflation can change the pre-existing large non-Gaussianity severely by changing the sign or the magnitude.
The generation of curvature perturbation can be understood geometrically. In [@Huang:2009vk], for simplicity, the straight trajectory was considered and the relation between the non-Gaussianity and the geometrical properties for the hyper-surface of the end of inflation were studied. The generation of the primordial statistical anisotropy at the end of inflation was also considered in [@Yokoyama:2008xw] and [@Emami:2011yi].
Both the evolution of the curvature perturbation during inflation and the contribution at the end of inflation are the general properties in multi-field inflation models. To get the correct curvature perturbation after inflation, it is necessary to consider both of them in a consistent way. In this paper, we propose the general formula for the curvature perturbation after the end of inflation, considering both effects in the case of the separable potentials by product or sum. Here we will use $\delta N$ formalism and provide analytic formulae for the power spectrum, spectral index and non-Gaussianity parameters.
This paper is organized as follows. In Section \[basic\], we present the basic formulae, and in Section \[perturbation\] we describe the background for our calculation and provide the analytic formulae for curvature perturbation. They are our main results. In Section \[application\], we apply the formulae derived in the previous section to calculate the power spectrum and non-linearity parameters in the several models and conclude in Section \[conclusion\].
Background theory {#basic}
=================
We consider an inflationary scenario implemented by multi-inflaton scalars $\phi_1,\phi_2,\cdots,\phi_n$ ($\equiv\phi_I$) with the [*canonical*]{} kinetic terms and the potential $W(\phi_1, \phi_2, \cdots,\phi_n)$. The classical background of the Universe are governed by the Friedmann equation and the equations of motion for the inflaton fields. In the slow-roll regime, the Friedmann and field equations are approximately given by [$$\begin{split}
3{M_P}^2 H^2 \simeq W\, ,~~
3H\dot\phi_I \simeq -\frac{\partial W}{\partial \phi_I}\,.
\end{split}$$]{} The slow-roll parameters are defined as $$\begin{aligned}
&&\epsilon_{\phi_I}
\equiv \frac{{M_P}^2}{2}\left[\frac{1}{W}\left( \frac{\partial W}{\partial \phi_I}\right)\right]^2
\equiv \frac{{M_P}^2}{2} {{\left(\frac{W_{\phi_I}}{W} \right) }}^2\,,\\
&&\eta_{\phi_I \phi_J}
\equiv {M_P}^2 \left[\frac{1}{W}\frac{\partial^2 W}{\partial \phi_I {{\partial}}\phi_J}\right]
\equiv {M_P}^2 \frac{W_{\phi_I \phi_J}}{W}\,. \label{slow-roll}\end{aligned}$$
The curvature perturbation $\zeta$ on super horizon scales can be evaluated using $\delta N$-formalism [@starob85; @ss1; @Sasaki:1998ug; @lms], which is the perturbation of the e-folding number $N$ defined as [$$\begin{split}
N(t_c,t_*,{\bf x}) \equiv \int_*^c H dt . \label{efoldingN}
\end{split}$$]{} It is evaluated from an initial flat hypersurface at $t=t_*$ to a final uniform density hypersurface at $t=t_c$. Here we take $t_*$ as the time of the horizon exit of the relevant scale during inflation and $t_c$ as some time later after inflation. Then the number of e-foldings $N$ can be a function of the field configuration $\phi_I(t_*,{\bf x})$ on the flat hypersurface at $t=t_*$ and the time of $t_c$. Then the perturbation of the number of e-foldings $\delta N (t_c,t_*,{\bf x})$ can be expanded in terms of the field perturbations $\delta \phi_I(t_*,{\bf x})$. The curvature perturbation is given by [$$\begin{split}\label{deltaN}
\zeta\simeq \delta N= \sum_I
N_{,I}\delta\phi_{I*}+\frac12\sum_{IJ}N_{,IJ}\delta\phi_{I*}\delta\phi_{J*}
\end{split}$$]{} up to quadratic terms, where the first and second derivatives are [$$\begin{split}
N_{,I}\equiv \frac{{{\partial}}N}{{{\partial}}\phi_*^I},\qquad N_{,IJ}\equiv \frac{{{\partial}}^2 N}{{{\partial}}\phi_*^I {{\partial}}\phi_*^J} .
\label{efoldingNij}
\end{split}$$]{} Here we assumed the slow-roll conditions and ignored the possible dependence on their first time derivatives $\dot{\phi}_I(t)$. The field perturbations $\delta \phi_{I*}$ satisfy the two-point correlation function, [$$\begin{split}
{\langle \delta \phi_{I*}({\bf k_1}) \delta \phi_{J*}({\bf k_2}) \rangle} = (2\pi)^3 \delta_{IJ}^{(3)} ( {\bf k_1} + {\bf k_2} ) \frac{2\pi^3}{k_1^3} {\cal P}_*(k_1),\qquad {\cal P}_*(k)\equiv\frac{H_*^2}{4\pi^2},
\end{split}$$]{} where $H_*$ is evaluated at Hubble exit, $k=aH$.
The power spectrum of the curvature perturbation $\zeta$, ${{\cal P}}_{\zeta}$, and the bispectrum $B_\zeta$ are defined as $$\begin{aligned}
\label{powerspectrumdefn} \langle\zeta_{\bkone}\zeta_{\bktwo}\rangle &\equiv&
\picube\,
{\!\delta^{\,3}(\mathbf{\bkone+\bktwo})}\frac{2\pi^2}{k_1^3}{{\cal P}}_{\zeta}(k_1) \, , \\
\langle\zeta_{{\mathbf k_1}}\,\zeta_{{\mathbf k_2}}\,
\zeta_{{\mathbf k_3}}\rangle &\equiv& \picube\, {\!\delta^{\,3}(\mathbf{{\mathbf
k_1}+{\mathbf k_2}+{\mathbf k_3}})} B_\zeta( k_1,k_2,k_3) \,. \end{aligned}$$ Observational limits are usually put on the non-linearity parameter ${f_{\rm NL}}$, which is defined by [@Maldacena:2002vr] [$$\begin{split}
{f_{\rm NL}}=\frac56\frac{k_1^3k_2^3k_3^3}{k_1^3+k_2^3+k_3^3}
\frac{B_{\zeta}(k_1,k_2,k_3)}{4\pi^4{{\cal P}}_{\zeta}^2}. \label{fnldefn}
\end{split}$$]{}
Using [Eq. (\[deltaN\])]{}, one obtains the power spectrum of the curvature perturbation as [$$\begin{split}
{{\cal P}}_\zeta = \sum_I N_{,I}^2 {{\cal P}}_*. \label{Pzeta}
\end{split}$$]{} From this we can define the observable quantities, the spectral index and the tensor-to-scalar ratio: &&n\_-1, \[tilt\]\
&&r==, where ${{\cal P}}_T=8{{\cal P}}_*/{M_P}^2=8H_*^2/(4\pi^2{M_P}^2)$ is the power spectrum of the tensor metric fluctuations. The non-linearity parameter ${f_{\rm NL}}$ is given by [@Lyth:2005fi] [$$\begin{split}
{f_{\rm NL}}= \frac56 \frac{\sum_{IJ}N_{,I}N_{,J}N_{,IJ}}{\left(\sum_K N_{,K}N_{,K}\right)^2}.
\label{fnl}
\end{split}$$]{}
In [Eq. (\[efoldingN\])]{}, the final time can be identified as the end of inflation, if the field values at the end of inflation are on the uniform energy density hypersurface. In multi-field inflation, however, that does not happen in general; the inflation ends at slightly different times in different places. This inhomogeneous end of inflation can generate curvature perturbation additionally [@Lyth:2005qk]. To include the effect from the end of inflation, we extend the final time $t_c$ to sometime in the deep radiation dominated era after inflation, while we denote $t_e$ as the time at the end of inflation. Thus, the e-folding number has the two contributions; one is from inflation phase $N_e(t_*,t_e)$ and the other from radiation phase $N_c(t_e,t_c)$, [$$\begin{split}
N(t_*,t_c) = N_e(t_*,t_e) + N_c(t_e,t_c).
\end{split}$$]{} Here, we take into account a sudden change from the inflation phase to the radiation era. The change in the e-folding number during the radiation dominated era between $t_e$ and $t_c$ is given by [@Byrnes:2008zy] [$$\begin{split}
N_c=\frac14 \log {{\left(\frac{W_e}{W_c} \right) }}\label{Nc},
\end{split}$$]{} where $W_e$ is the potential energy as a function of the field values at the end of inflation and $W_c$ is the energy density at a time $t_c$ during the radiation dominated era, which is uniform in space.
Thus, $\delta N$ has the two contributions from the inflation and radiation epochs, [$$\begin{split}
\delta N = \delta N_e + \delta N_c, \label{deltaNec}
\end{split}$$]{} with [$$\begin{split}
\delta N_c = \frac{1}{4W_e}\sum_I \frac{{{\partial}}W_e}{{{\partial}}\phi_I^*} \delta \phi_I^* \label{deltaNc}.
\end{split}$$]{} Here $\delta N_c$ depends only on the epoch at the transition between inflation and radiation; it does not depend on the final time. The reason is manifest since the curvature perturbation is conserved in the radiation dominated era [@lms].
Using the slow-roll parameters, [Eq. (\[slow-roll\])]{}, $\delta N_c$ becomes [$$\begin{split}
\delta N_c =\sum_{IJ} \frac{\sqrt{2\epsilon^e_{\phi_J}}}{4{M_P}} {{\left(\frac{{{\partial}}\phi_J^e}{{{\partial}}\phi_I^*} \right) }} \delta \phi_I^*,
\label{deltaNcSR}
\end{split}$$]{} where the superscript ${}^e$ denote the values evaluated at the end of inflation $t=t_e$. As we will see later, $\delta N_c$ is suppressed compared to $\delta N_e$ by the slow-roll parameters.
Curvature perturbation generated during and at the end of inflation {#perturbation}
===================================================================
In this section, we will discuss the contribution to curvature perturbation from inflationary era $\delta N_e$ and $\delta N_c$ in [Eq. (\[deltaNec\])]{} with general condition of ending inflation. We attempt to get analytic formulae for the power spectrum ($\cal{P}_\zeta$), the spectral index ($n_\zeta$), and the non-linearity parameter ($f_{NL}$) particularly in the cases that the inflaton potential is given by the product, $W({\varphi},\chi)=U({\varphi})V(\chi)$, or by sum, $W({\varphi},\chi)=U({\varphi})+V(\chi)$. Since these forms of the potentials cover the large class of inflation models, the analytic expressions for $\cal{P}_\zeta$, $n_\zeta$, and $f_{NL}$ would be very useful.
Product potential $W({\varphi},\chi)=U({\varphi})V(\chi)$
---------------------------------------------------------
In this subsection, we consider the case of a separable potential by product, [$$\begin{split}
W({\varphi},\chi)=U({\varphi})V(\chi) \label{potentialproduct}.
\end{split}$$]{} First, we derive the curvature perturbation, and then from this we obtain the analytic formulae for the power spectrum $\cal{P}_\zeta$ and the non-linear parameter ${f_{\rm NL}}$. We assume the slow-roll during inflation. From [Eq. (\[slow-roll\])]{}, the slow-roll parameters are given by$$\begin{aligned}
&&\epsilon_{\varphi}= \frac{{M_P}^2}{2}\left(\frac{U_{\varphi}}{U}\right)^2, \qquad
\epsilon_\chi
= \frac{{M_P}^2}{2}\left(\frac{V_\chi}{V}\right)^2,\\
&&~\etap
={M_P}^2 \frac{U_{{\varphi}{\varphi}}}{U} ~,~\qquad\quad
\etac
={M_P}^2 \frac{V_{\chi\chi}}{V} ~,\end{aligned}$$ and supposed to be small.
The number of e-foldings with the potential in [Eq. (\[potentialproduct\])]{} in the slow-roll regime is given by [@GarciaBellido:1995qq] [$$\begin{split}
N_e({\varphi}_*,\chi_*) = -\frac{1}{{M_P}^2} \int^e_*\frac{U}{U_{\varphi}}d {\varphi}= -\frac{1}{{M_P}^2} \int^e_*\frac{V}{V_\chi}d \chi. \label{Nproduct}
\end{split}$$]{} The integrals in [Eq. (\[Nproduct\])]{} depend on both field values at horizon crossing (${\varphi}_*$ and $\chi_*$) and at the end of inflation (${\varphi}_e$ and $\chi_e$). Note that ${\varphi}_e$ and $\chi_e$ also depend on the initial field values, ${\varphi}_*$ and $\chi_*$ along the given trajectory. The differentiation of the number of e-foldings $dN_e$ is given by [$$\begin{split}
dN _e
=\frac{1}{{M_P}^2} \left[ {{\left(\frac{U}{U_{\varphi}} \right) }}_* -{{\left(\frac{U}{U_{\varphi}} \right) }}_e \frac{{{\partial}}{\varphi}_e}{{{\partial}}{\varphi}_*}\right] d{\varphi}_*
- \frac{1}{{M_P}^2} {{\left(\frac{U}{U_{\varphi}} \right) }}_e\frac{{{\partial}}{\varphi}_e}{{{\partial}}\chi_*}d\chi_* . \label{dNproduct}
\end{split}$$]{} The derivatives of the final field values with respect to the initial field values also depend on the condition of the fields at the end of inflation. Here we suppose that the inflation ends when the fields satisfy the condition given by [$$\begin{split}
E({\varphi_e},{\chi_e})= \text{a constant} \label{endcondition}.
\end{split}$$]{} This is different from that used in the previous analyses in Ref. [@GarciaBellido:1995qq; @Choi:2007su], where the uniform energy density hypersurface is simply employed at the end of inflation.
With the condition [Eq. (\[endcondition\])]{} at the end of inflation, we find that the derivatives are given by [$$\begin{split}
\frac{{{\partial}}{\varphi_e}}{{{\partial}}{\varphi_*}}=\frac{\epsce}{\bepse}\sqrt{\frac{\epspe}{\epsps}} ~, \qquad &
\frac{{{\partial}}{\varphi_e}}{{{\partial}}{\chi_*}}=-\frac{\epsce}{\bepse}\sqrt{\frac{\epspe}{\epscs}} ~,\\
\frac{{{\partial}}{\chi_e}}{{{\partial}}{\varphi_*}}=-\frac{R\epspe}{\bepse}\sqrt{\frac{\epsce}{\epsps}} ~, \qquad &
\frac{{{\partial}}{\chi_e}}{{{\partial}}{\chi_*}}=\frac{R\epspe}{\bepse}\sqrt{\frac{\epsce}{\epscs}} ~, \label{dervsproduct}
\end{split}$$]{} where the superscript (or subscript) ${}^e$ and ${}^*$ denote the values evaluated at the end of inflation and horizon crossing, respectively. Throughout this paper it should be understood that there is additional factor of ${{\rm sign}}(U_{{\varphi}})$ or ${{\rm sign}}(V_{\chi})$ whenever square root appears. We note that there is an additional factor $R$, which contains the information about the condition of ending inflation [Eq. (\[endcondition\])]{}. In [Eq. (\[dervsproduct\])]{}, $R$ and $\beps$ are defined by [$$\begin{split}
R &\equiv \left(\frac{U/U_{\varphi}}{V/V_\chi}\right)_e\, {{\left(\frac{{{\partial}}E/{{\partial}}{\varphi_e}}{{{\partial}}E/ {{\partial}}{\chi_e}} \right) }}=\sqrt{\frac{\epsilon_\chi^e}{\epsilon_{\varphi}^e}} \frac{E_{\varphi}}{E_\chi},\qquad \\
\bepse & \equiv R\epspe + \epsce=\frac{\sqrt{\epsilon_\chi}}{E_\chi}( E_{\varphi}\sqrt{\epsilon_{\varphi}} +E_\chi\sqrt{\epsilon_\chi} ), \label{R}
\end{split}$$]{} with $E_{{\varphi}}\equiv{{\partial}}E/ {{\partial}}{\varphi}_e$ and $E_{\chi}\equiv{{\partial}}E/ {{\partial}}\chi_e$. Hence, if the condition for the end of inflation is given by the uniform energy condition, $E({\varphi}_e,\chi_e)=W({\varphi}_e,\chi_e)=$ a constant, then $E_{\varphi}$ and $E_\chi$ are given by $VU_{\varphi}|_e$ and $UV_\chi|_e$, respectively, which makes $R$ the unity. Note that $R$ can be a [*negative as well as positive*]{} value, and so $\overline{\epsilon^e}$ has possibly a vanishing limit.
Using [Eq. (\[dNproduct\])]{} and [Eq. (\[dervsproduct\])]{}, we find the first derivatives of e-folding number $N(t_e,t_*)$ with respect to the fields, [$$\begin{split}
&{M_P}\frac{{{\partial}}N_e}{{{\partial}}{\varphi_*}}=\frac{\bu}{\sqrt{2\epsps}},\\
&{M_P}\frac{{{\partial}}N_e}{{{\partial}}{\chi_*}}=\frac{\bv}{\sqrt{2\epscs}},\label{linearproduct}
\end{split}$$]{} where we defined [$$\begin{split} \label{uvbar}
\bu \equiv \frac{R\epspe}{\bepse} = \frac{E_{\varphi}\sqrt{\epsilon_{\varphi}^e}}{E_{\varphi}\sqrt{\epsilon_{\varphi}^e} +E_\chi\sqrt{\epsilon_\chi^e} } ,\qquad \bv\equiv \frac{\epsce}{\bepse}=\frac{E_\chi\sqrt{\epsilon_\chi^e}}{E_{\varphi}\sqrt{\epsilon_{\varphi}^e} +E_\chi\sqrt{\epsilon_\chi^e} } ,
\end{split}$$]{} which are evaluated at the end of inflation.
Based on the first derivatives [Eq. (\[linearproduct\])]{}, we can find the second derivatives, [$$\begin{split}
{M_P}^2\frac{{{\partial}}^2 N_e}{{{\partial}}{\varphi_*}^2}&=\left( 1-\frac{\etaps}{2\epsps} \right)\bu
- \frac{1}{\epsps}{\overline{{\cal A}_P}},\\
{M_P}^2\frac{{{\partial}}^2 N_e}{{{\partial}}{\chi_*}^2}&=\left( 1-\frac{\etacs}{2\epscs} \right)\bv
- \frac{1}{\epscs}{\overline{{\cal A}_P}},\\
{M_P}^2 \frac{{{\partial}}^2 N_e}{{{\partial}}{\varphi_*}{{\partial}}{\chi_*}}
&=
\frac{1}{\sqrt{\epsps\epscs}}{\overline{{\cal A}_P}},\label{secondproduct}
\end{split}$$]{} where [$$\begin{split}
{\overline{{\cal A}_P}}&\equiv - \frac{R\epspe\epsce}{\bepse^2}
\left[ \overline{\eta_s^e}
- \frac{2(1+R)\epspe\epsce}{\bepse}
+ {M_P}\frac{\epspe\epsce}{R\bepse}
\left\{ \frac{1}{\sqrt{2\epspe}} \frac{{{\partial}}R}{{{\partial}}{\varphi_e}}
- \frac{R}{\sqrt{2\epsce}} \frac{{{\partial}}R}{{{\partial}}{\chi_e}}
\right\}
\right] ,\\
\overline{\eta_s^e}&\equiv \frac{\epsce \etape + R\epspe \etace}{\bepse}. \label{barAp}
\end{split}$$]{}
The other contribution to $\zeta$ coming from $\delta N_c$ can be obtained from [Eq. (\[deltaNcSR\])]{} using [Eq. (\[dervsproduct\])]{}, [$$\begin{split}
\delta N_c = \frac{(1-R)}{2{M_P}}\frac{\epsilon_{\varphi}^e\epsilon_\chi^e}{\overline{\epsilon^e}} \left[ \frac{\delta {\varphi}_*}{\sqrt{2\epsilon_{\varphi}^*}} - \frac{\delta \chi_*}{\sqrt{2\epsilon_\chi^*}} \right].
\end{split}$$]{} We note that this contribution disappears with $R=1$. It means that the hypersurface of the end of inflation coincides with the uniform energy density hypersurface.
At the leading order the curvature perturbation is [$$\begin{split}
\zeta &= \delta N_e+\delta N_c\\
&= \frac{1}{ {M_P}\overline{\epsilon^e}}
\left[ \left\{ R +\frac12(1-R)\epsilon_\chi^e \right\} \frac{\epsilon_{\varphi}^e}{\sqrt{2\epsilon_{\varphi}^*}} \delta {\varphi}_*
+\left\{ 1 - \frac12(1-R)\epsilon_{\varphi}^e\right\} \frac{\epsilon_\chi^e}{\sqrt{2\epsilon_\chi^*}} \delta \chi_*
\right].
\end{split}$$]{} We find that the contribution from $\delta N_c$ is suppressed by another slow-roll parameters in general except some extreme cases of very large or small $R$. Ignoring $\delta N_c$, the curvature perturbation is determined solely by the contribution from the inflation epoch, $\zeta\simeq \delta N_e$. In this case, the power spectrum [Eq. (\[Pzeta\])]{} is given by [$$\begin{split} \label{powerPrd}
{{\cal P}}_\zeta
\simeq \frac{W_*}{24\pi^2{M_P}^4} \left( \frac{\bu^2}{\epsilon_{\varphi}^*} + \frac{\bv^2}{\epsilon_\chi^*} \right) \,,
\end{split}$$]{} and the spectral index $n_\zeta$ and the tensor-to-scalr ratio $r$ are, respectively, given by [$$\begin{split}
n_\zeta & \simeq 1 -2 \epsilon^*
- \frac{4}{ \bu^2/\epsilon_{\varphi}^* + \bv^2/\epsilon_\chi^*}
\left[ \bu^2\left(1-\frac{\eta_{{\varphi}{\varphi}}^*}{2\epsilon_{\varphi}^*}\right)
+\bv^2 \left(1-\frac{\eta_{\chi\chi}^*}{2\epsilon_\chi^*} \right) \right] \,,
\\
r
&\simeq \frac{16}{ \bu^2/\epsilon_{\varphi}^* + \bv^2/\epsilon_\chi^*},
\end{split}$$]{} where $\epsilon^* \equiv \epsilon_{\phi}^*+\epsilon_{\chi}^*$. The non-linearity parameter ${f_{\rm NL}}$ is [$$\begin{split} \label{fnlPrd}
\frac65{f_{\rm NL}}\simeq \frac{2}{(\bu^2/\epsps +\bv^2/\epscs)^2}\left[ \frac{\bu^3}{\epsps}\left(1- \frac{\etaps}{2\epsps} \right) + \frac{\bv^3}{\epscs}\left( 1-\frac{\etacs}{2\epscs} \right) - \left( \frac{\bu}{\epsps}-\frac{\bv}{\epscs} \right)^2{\overline{{\cal A}_P}}\right].
\end{split}$$]{} Note that the forms of ${{\cal P}}_\zeta$ and ${f_{\rm NL}}$ are the same as those in Ref. [@Choi:2007su], but the definitions of $\bu$, $\bv$ and ${\overline{{\cal A}_P}}$ are different from those in Ref. [@Choi:2007su], which contain the informations about the end of inflation. We will find the new features from the special ending of inflation in the next section.
Separable potential by sum
--------------------------
In this subsection we consider a separable potential by sum [@VW]; $$W({\varphi},\chi) = U({\varphi}) + V(\chi) \,.$$ We keep the general condition for the end of inflaiton Eq. (\[endcondition\]). The slow-roll parameters are given by [$$\begin{split}
&\epsilon_{\varphi}= \frac{{M_P}^2}{2}\left( \frac{U_{\varphi}}{W} \right)^2 \,,~~~
\epsilon_\chi
= \frac{{M_P}^2}{2}\left( \frac{V_\chi}{W} \right)^2 \,,
\\
& \eta_{{\varphi}{\varphi}}= {M_P}^2 \frac{U_{{\varphi}{\varphi}}}{W} , \,\,~~~~~~~
\eta_{\chi\chi}= {M_P}^2 \frac{V_{\chi\chi}}{W}\,.
\end{split}$$]{}
In the slow-roll limit the number of $e$-foldings is $$N_e
= -\frac{1}{{M_P}} \int^c_* \frac{U}{U_{\varphi}}d{\varphi}-\frac{1}{{M_P}} \int^c_* \frac{V}{V_\chi}d\chi \,,$$ and its absolute derivative is given by [$$\begin{split}
dN_e
= \frac{1}{{M_P}}&
\left[ {{\left(\frac{U}{U_{\varphi}} \right) }}_*
- \frac{\partial{\varphi}_e}{\partial {\varphi}_*} {{\left(\frac{U}{U_{\varphi}} \right) }}_e
- \frac{\partial \chi_e}{\partial {\varphi}_*} {{\left(\frac{V}{V_\chi} \right) }}_e
\right] d{\varphi}_* \\
\qquad & +
\frac{1}{{M_P}}
\left[ {{\left(\frac{V}{V_\chi} \right) }}_*
- \frac{\partial\chi_e}{\partial \chi_*} {{\left(\frac{V}{V_\chi} \right) }}_e
- \frac{\partial {\varphi}_e}{\partial \chi_*} {{\left(\frac{U}{U_{\varphi}} \right) }}_e
\right] d\chi_* \,.
\end{split}$$]{} The field derivative terms at $t_*$ are $$\begin{aligned}
\frac{\partial {\varphi}_e}{\partial {\varphi}_*}
= \frac{W_e}{W_*}\frac{\epsilon^e_\chi}{\bepse}\sqrt{\frac{\epsilon^e_{\varphi}}{\epsilon^*_{\varphi}}}\,,~~
\frac{\partial {\varphi}_e}{\partial \chi_*}
= -\frac{W_e}{W_*}\frac{\epsilon^e_\chi}{\bepse} \sqrt{\frac{\epsilon^e_{\varphi}}{\epsilon^*_\chi}}\,,~~\\
\frac{\partial \chi_e}{\partial {\varphi}_*}
= -\frac{W_e}{W_*}\frac{R\epsilon^e_{\varphi}}{\bepse}\sqrt{\frac{\epsilon^e_\chi}{\epsilon^*_{\varphi}}}\,,~~
\frac{\partial \chi_e}{\partial \chi_*}
= \frac{W_e}{W_*}\frac{R\epsilon^e_{\varphi}}{\bepse}\sqrt{\frac{\epsilon^e_\chi}{\epsilon^*_\chi}}\,,\end{aligned}$$ Similar to [Eq. (\[R\])]{}, $R$ and $\overline{\epsilon}$ are defined by [$$\begin{split} \label{Rsum}
&R= \left(\frac{V_\chi}{U_{\varphi}}\right)_e\left(\frac{E_{\varphi}}{E_\chi}\right) ,\\
&\overline{\epsilon^e} = R \epsilon^e_{\varphi}+ \epsilon^e_\chi \,,
\end{split}$$]{} for the sum potential. Hence, if the condition for the end of inflation is given by the uniform energy density condition, $E({\varphi}_e,\chi_e)=W({\varphi}_e,\chi_e)=$ a costant, then $E_{\varphi}=U_{\varphi}|_e$ and $E_\chi=V_\chi|_e$. So $R$ becomes again the unity.
Using these equaions, the first derivatives of the e-folding number with respect to the two inflaton fields at horizon exit are given by [$$\begin{split} \label{uv}
{M_P}\frac{\partial N_e}{\partial {\varphi}_*}
=& \frac{1}{\sqrt{2\epsilon^*_{\varphi}}}\left( \frac{U_* + \widetilde{Z}_e}{W_*} \right)
\equiv
\frac{\bu}{\sqrt{2\epsilon^*_{\varphi}}}\,,\\
{M_P}\frac{\partial N_e}{\partial \chi_*}
=& \frac{1}{\sqrt{2\epsilon^*_\chi}} \left( \frac{V_* - \widetilde{Z}_e}{W_*} \right)
\equiv
\frac{\bv}{\sqrt{2\epsilon^*_{\varphi}}} \,,
\end{split}$$]{} where $$\label{Ztilde}
\widetilde{Z}_e
\equiv \frac{V_e R \epsilon^e_{\varphi}- U_e \epsilon^e_\chi}{\bepse}\,.$$ The second derivatives are given by $$\begin{aligned}
{M_P}^2 \frac{\partial^2 N_e}{\partial {\varphi}_*^2}
&=& 1-\frac{\eta_{{\varphi}{\varphi}}^*}{2\epsilon_{\varphi}^*}~\bu
+ \frac{{M_P}}{W_*\sqrt{2\epsilon^*_{\varphi}}}\frac{\partial\widetilde{Z}_e}{\partial {\varphi}_*}\,,\\
{M_P}^2 \frac{\partial^2 N_e}{\partial \chi_*^2}
&=& 1-\frac{\eta_{\chi\chi}^*}{2\epsilon_\chi^*}~\bv
- \frac{{M_P}}{W_*\sqrt{2\epsilon^*_\chi}}\frac{\partial\widetilde{Z}_e}{\partial \chi_*}\,,\\
{M_P}^2 \frac{\partial^2 N_e}{\partial {\varphi}_*\partial\chi_*}
&=& \frac{{M_P}}{W_*\sqrt{2\epsilon^*_{\varphi}}}\frac{\partial\widetilde{Z}_e}{\partial \chi_*} ~~
= - \frac{{M_P}}{W_*\sqrt{2\epsilon^*_\chi}}\frac{\partial\widetilde{Z}_e}{\partial {\varphi}_*}\,,\end{aligned}$$ where ${\overline{{\cal A}_S}}$ is defend as $$\begin{aligned}
{\overline{{\cal A}_S}}&\equiv& \frac{W_e^2}{W_*^2}\frac{R\epsilon^e_{\varphi}\epsilon^e_\chi}{\bepse^2}
\left[\overline{\eta^e_{s}} - \left(R\epsilon^e_{\varphi}+ \frac{\epsilon^e_\chi}{R} \right)
+{M_P}\frac{ \epsilon^e_{\varphi}\epsilon^e_\chi}{R\bepse}
\left\{ \frac{1}{\sqrt{2\epsilon^e_{\varphi}}}\frac{\partial R}{\partial{\varphi}_e}
- \frac{R}{\sqrt{2\epsilon^e_\chi}}\frac{\partial R}{\partial \chi_e}
\right\}
\right] ,\quad \nonumber\\
\overline{\eta^e_{s}}& \equiv& \frac{R\epsilon^e_{\varphi}\eta^e_{\chi\chi} + \epsilon^e_\chi \eta^e_{{\varphi}{\varphi}}}{\bepse} ~.\end{aligned}$$ The derivatives of $\widetilde{Z}_e$ can be re-written as $$\begin{aligned}
\sqrt{\epsilon^*_{\varphi}} ~\frac{\partial \widetilde{Z}_e}{\partial {\varphi}_*}
=
- \sqrt{\epsilon^*_\chi} ~\frac{\partial \widetilde{Z}_e}{\partial \chi_*}
=
\frac{\sqrt{2}}{{M_P}^2}W_* ~{\overline{{\cal A}_S}}\,.\end{aligned}$$
The contribution from the radiation dominated phase, $\delta N_c$ can be evaluated from [Eq. (\[deltaNcSR\])]{}: [$$\begin{split}
\delta N_c =\frac{(1-R)}{2{M_P}}\frac{W_e}{W_*}\frac{\epsilon_{\varphi}^e\epsilon_\chi^e}{\overline{\epsilon^e}}
\left[ \frac{\delta {\varphi}_*}{\sqrt{2\epsilon_{\varphi}^*}}
- \frac{\delta \chi_*}{\sqrt{2\epsilon_\chi^*}} \right].
\end{split}$$]{} We note that $\delta N_c$ has one more slow-roll parameter in the numerator than $\delta N_e$ and so suppressed.
In the case of $\delta N_e \gg \delta N_c$, the curvature perturbation $\zeta\simeq \delta N_e$ and the power spectrum is $$\begin{aligned}
{\mathcal{P}}_\zeta &\simeq&
\frac{W_*}{24\pi^2 {M_P}^4}\left(\frac{{{\overline{u}}}^2}{\epsilon^*_{\varphi}}+\frac{{{\overline{v}}}^2}{\epsilon^*_\chi}\right)\,,
\label{eq:P} \end{aligned}$$ and spectral index and tensor-to-scalar ratio are $$\begin{aligned}
n_\zeta-1 &\simeq &-2\epsilon^* -
\frac{4}{\bu^2/\epsilon^*_{\varphi}+\bv^2/\epsilon^*_\chi} \left[ {{\overline{u}}}\left( 1-\frac{\eta^*_{{\varphi}{\varphi}}}{2\epsilon^*_{\varphi}}~{{\overline{u}}}\right)
+ {{\overline{v}}}\left(1-\frac{\eta^*_{\chi\chi}}{2\epsilon^*_\chi}~{{\overline{v}}}\right)\right]
\, , \label{eq:n} \\
r &\simeq& \frac{2}{\pi^2 {\mathcal{P}}_\zeta} \frac{H_*^2}{{M_P}^2}
= \frac{16}{\bu^2/\epsilon^*_{\varphi}+\bv^2/\epsilon^*_\chi}\, .\label{eq:r} \end{aligned}$$ The non-linear parameter is given by [$$\begin{split}
\frac{6}{5} {f_{\rm NL}}\simeq \frac{2}{\left({{\overline{u}}}^2/\epsilon^*_{\varphi}+{{\overline{v}}}^2/\epsilon^*_\chi \right)^2}
\left[ \frac{{{\overline{u}}}^2}{\epsilon^*_{\varphi}}\left(1-\frac{\eta^*_{{\varphi}{\varphi}}}{2\epsilon^*_{\varphi}}~{{\overline{u}}}\right)
+ \frac{{{\overline{v}}}^2}{\epsilon^*_\chi}\left(1-\frac{\eta^*_{\chi\chi}}{2\epsilon^*_\chi}~{{\overline{v}}}\right)
+ \left(\frac{{{\overline{u}}}}{\epsilon^*_{\varphi}}-\frac{{{\overline{v}}}}{\epsilon^*_\chi}\right)^2 {\overline{{\cal A}_S}}\right]
\,.
\label{fnlSum}
\end{split}$$]{} These obervables are again of the same form as in Ref. [@VW] for the sum potential. However, we note again that the definitions of parameters are different and contains the effects from the end of inflation.
Large non-Gaussianity generated during and at the end of inflation
------------------------------------------------------------------
In multiple scalar field inflation models, the large non-Gaussiniaty can be generated during inflation and the conditions for that has been studied by Byrnes, Choi and Hall [@Byrnes:2008wi] with applications to the multiple hybrid inflation in Ref. [@Byrnes:2008zy]. More general cases were studied, in the models beyond slow-roll in [@Byrnes:2009qy] and in the general $n$-field inflation models in [@Battefeld:2009ym]. Other studies on the non-Gaussianiy generated during inflation can be found in [@Alabidi:2006hg; @Kim:2010ud; @Wang:2010si; @Peterson:2010mv; @Meyers:2010rg; @Elliston:2011dr; @Byrnes:2010em].
Basically, the necessary condition for large non-Gaussianity to be generated during the inflation is that one of the slow-roll parameters at the horizon exit is very small and it increases with time [@Byrnes:2008wi]. As discussed in [@Byrnes:2008wi], for inflation with two scalars ${\varphi}$ and $\chi$, only if $\epsilon^*_{\varphi}\ll\epsilon^e_{\varphi}\ll\epsilon^*_\chi \ll 1$ (or similarly only if $\epsilon^*_\chi\ll\epsilon^e_\chi\ll\epsilon^*_{\varphi}\ll1$), the non-linearity parameter ${f_{\rm NL}}$ can be large. That is to say, large non-Gaussianity is possible by a small slow-roll parameter $\epsilon_{\varphi}$ or $\epsilon_\chi$ [*at the horizon crossing*]{}. For the product potential, for instance, this limit implies that $u \ll1$ and $v\simeq1$ so that $\eta_{{\varphi}{\varphi}^*}/\epsilon_{\varphi}^*$ term dominate and ${\cal A}_p\simeq -u \, \eta_{{\varphi}{\varphi}}^e$. Accordingly, [$$\begin{split}
{f_{\rm NL}}\simeq \frac56 \frac{u^3}{(u^2 + \epsilon_{\varphi}^*/\epsilon_\chi^*)^2} \left[ -\eta_{{\varphi}{\varphi}}^* +2 \eta_{{\varphi}{\varphi}}^e \right].
\end{split}$$]{} The sign of ${f_{\rm NL}}$ is determined by the sign of $-\eta^*_{{\varphi}{\varphi}}+2\eta^e_{{\varphi}{\varphi}}$ for the case of $\epsilon^*_{{\varphi}}\ll\epsilon^*_{\chi}$. However, this result is valid only when $R$ is the unity, i.e. the hypersurface of the end of inflation is not very different from the uniform energy density hypersurface, as in the models of Refs. [@Choi:2011me].
In a large class of multi-field inflation models, however, the hypersurface at the end of inflation can be quite different from that of the uniform energy density. In these cases there could be another possibility for large non-Gaussianity: [*the non-linearity parameter ${f_{\rm NL}}$ can be large by the ending effect of inflation.*]{}
[**(a)**]{} From the similarity of the forms of ${f_{\rm NL}}$ without the end effect given in the Ref. [@Byrnes:2008wi] and ${f_{\rm NL}}$ with end effect in [Eq. (\[fnlPrd\])]{} and in [Eq. (\[fnlSum\])]{}, we can infer the conditions for large ${f_{\rm NL}}$ in the case of $R\neq1$ including the end effect. Using the arguments in the previous paragraph, for small $\epsilon_{\varphi}^*$, the condition for large ${f_{\rm NL}}$ is $\epsilon^*_{\varphi}\ll |R| \epsilon^e_{\varphi}\ll \epsilon^*_\chi\ll 1$, therefore $\bu \ll 1$. The new effect comes in with $R$ at the end of inflation. Now there is no need for $\epsilon_{\varphi}$ to increase upto the end of inflation. Instead, the condition $\epsilon^*_{\varphi}\ll |R| \epsilon^e_{\varphi}$ should be satisfied by a relatively large $|R|$. In this case, large non-linearity parameter approximately becomes [$$\begin{split}
{f_{\rm NL}}\simeq \frac56 \frac{{{\overline{u}}}^3}{({{\overline{u}}}^2 + \epsilon_{\varphi}^*/\epsilon_\chi^*)^2} \left[ -\eta_{{\varphi}{\varphi}}^* +2 \eta_{{\varphi}{\varphi}}^e \right],
\end{split}$$]{} if ${{\partial}}_{\varphi}R$ and ${{\partial}}_\chi R$ are suppressed enough. The similar expression holds also for small $\epsilon_\chi^*$ with small $|R|$. In this case, the required condition is $\epsilon_\chi^* \ll |R|^{-1}\epsilon_\chi^e \ll \epsilon_{\varphi}^* \ll 1$.
[**(b)**]{} In the limit of $R\rightarrow\infty$ (or $0$), $\bar{u}$ (or $\bar{v}$) approach to the unity, while $\bar{v}$ (or $\bar{u}$) vanish. If ${{\partial}}_{\varphi}R$ and ${{\partial}}_\chi R$ are not divergent and [*$\epsilon^*_{\varphi}$ and $\epsilon^*_\chi$ are not hierarchical*]{}, ${f_{\rm NL}}$ becomes just of order $\epsilon^*_{{\varphi}}$ or $\epsilon^*_{\chi}$ in this limit, which is quite suppressed. However, a large ${{\partial}}_\chi R$ (or ${{\partial}}_{\varphi}R$) can result in a large ${f_{\rm NL}}$, when $\epsilon_\chi\simeq\eta_{\chi\chi}\simeq 0$, thus $R=\sqrt{\epsilon_\chi^e/\epsilon_{\varphi}^e} (E_{\varphi}/E_\chi)\simeq 0$ in [Eq. (\[fnlPrd\])]{}, [$$\begin{split} \label{endfnl}
\frac65{f_{\rm NL}}\simeq -\frac{2}{(1+ \bu^2 \epsilon_\chi^* /\bv^2 \epsilon_{\varphi}^*)^2}\frac{{\overline{{\cal A}_P}}}{\bv^2} \simeq
\frac{1}{(1+ \bu^2 \epsilon_\chi^* /\bv^2 \epsilon_{\varphi}^*)^2} \frac{2\epsilon_{\varphi}}{R}\left( \frac{1}{\sqrt{2\epsilon_{\varphi}}} \frac{{{\partial}}R}{{{\partial}}{\varphi}} - \frac{R}{\sqrt{2\epsilon_\chi}}\frac{{{\partial}}R}{{{\partial}}\chi} \right)
\,,\end{split}$$]{} for ${M_P}= 1$. In this case, [*a large ${f_{\rm NL}}$ is generated only by the end effect of inflation*]{}. We will discuss a specific example later.
[**(c)**]{} As pointed earlier, $R$ can be negative and so $\overline{\epsilon^e}$ has a vanishing limit. It opens a new possibility for large non-Gaussianity. Consider the case that $\overline{\epsilon^e}$ in Eqs. (\[R\]) and (\[Rsum\]) is extremely small, [$$\begin{split} \label{condi}
\overline{\epsilon^e}=R\epsilon^e_{\varphi}+\epsilon^e_\chi
\longrightarrow\varepsilon ~,
\end{split}$$]{} where $\varepsilon$ is assumed to be very small ($\ll\epsilon^*_{{\varphi},\chi},~\epsilon^e_{{\varphi},\chi}$). Note that the parameters appearing in the definition of $\overline{\epsilon^e}$ are all associated with the field values at the end of inflation. Under the limit of $\overline{\epsilon^e}\rightarrow\varepsilon$ or $\epsilon_\chi^e \simeq -R \epsilon_{\varphi}^e$, $\bar{u}$, $\bar{v}$ in [Eq. (\[uvbar\])]{} and $\bu$, $\bv$, $\widetilde{Z}_e$ in Eqs. (\[uv\]), (\[Ztilde\]) diverge as ${\cal O}(1/\varepsilon)$. Since ${\overline{{\cal A}_P}}$ and ${\overline{{\cal A}_S}}$ are divergent as ${\cal O}(1/\varepsilon^3)$ in this limit, the third terms in Eqs. (\[fnlPrd\]) and (\[fnlSum\]) become dominant in ${f_{\rm NL}}$, and so the large non-linearity parameter ${f_{\rm NL}}$ becomes also divergent as ${\cal O}(1/\varepsilon)$. In this case ${f_{\rm NL}}$ is approximately [$$\begin{split}
{f_{\rm NL}}\simeq -\frac56\frac{\epsilon_\chi^e}{\varepsilon}\frac{2(\epsilon_\chi^*-\epsilon_{\varphi}^*)^2}{(\epsilon^*)^2} \left[ \eta^e_{\varphi}- \eta^e_\chi +2\left(1+\frac{1}{R}\right)\epsilon_\chi^e- \frac{\sqrt{\epsilon_\chi^e}}{R^2\sqrt2 } \left(\sqrt{-R} \frac{{{\partial}}R}{{{\partial}}{\varphi}} -R \frac{{{\partial}}R}{{{\partial}}\chi} \right) \right]\,,\end{split}$$]{} setting ${M_P}= 1$.
Application
===========
In this section we apply our results obtained in the previous section to the several specific models. \[application\]
Generating curvature perturbation at the end of inflation
---------------------------------------------------------
As a simple example, let us discuss a scenario in which one field $\phi$ drives the inflation, while the other $\sigma$ is negligible during inflation, since its potential is so flat to make practically no effect on the inflation trajectory. During inflation the potential is, hence, $W(\phi,\sigma)\simeq U(\phi)$. More specific potentials are given in [@Lyth:2005qk; @Lyth:2006nx; @Alabidi:2006hg].
Inflation ends when inflaton field $\phi$ has a certain value $\phi_e$. However the $\phi_e$ depends on the field $\sigma$. Since $\phi_e(\sigma)$ would depend on the position through the perturbation $\delta\sigma(\bf x)$, curvature perturbation should have an additional contribution $\zeta_e$ coming from the end effect as well as the pre-existing perturbation by the inflaton field $\zeta_{\phi}$.
The $\sigma$ field does not move during inflation, $\sigma_e\simeq\sigma_*$, and $\epsilon_\sigma \rightarrow 0$ since its potential is so flat. As the condition of the fields at the end of inflation, we use $E(\phi_e,\sigma_e)=E(\phi_e,\sigma_*)=$ a constant from [Eq. (\[endcondition\])]{}.
For this case, [Eq. (\[dervsproduct\])]{} leads to [$$\begin{split}
\frac{{{\partial}}\phi_e}{{{\partial}}\phi_*}=\frac{{{\partial}}\sigma_e}{{{\partial}}\phi_*}=0,\quad
\frac{{{\partial}}\phi_e}{{{\partial}}\sigma_*}=-\frac{E_\sigma}{E_\phi} , \quad\text{and}\quad \frac{{{\partial}}\sigma_e}{{{\partial}}\sigma_*}=1,
\end{split}$$]{} and [$$\begin{split}
\bu =1,\qquad \bv =\frac{\epsilon_\sigma^e}{R\epsilon_\phi^e},\qquad
\text{with} \qquad
R=\sqrt{\frac{\epsilon_\sigma^e}{\epsilon_\phi^e}} \frac{E_\phi}{E_\sigma}.
\label{R-Erel}
\end{split}$$]{} Here we can see that $\bv \rightarrow 0$ and $R\rightarrow 0 $. It is easy to find the first derivatives of e-folding number, [$$\begin{split}
\frac{{{\partial}}N_e}{{{\partial}}\phi_*} = \frac{1}{{M_P}\sqrt{2\epsilon_\phi^*}},
\qquad \frac{{{\partial}}N_e}{{{\partial}}\sigma_*} = \frac{1}{{M_P}\sqrt{2\epsilon_\phi^e}} \frac{E_\sigma}{E_\phi},\label{linear_1}
\end{split}$$]{} where the small value of $\epsilon_\sigma$ was canceled and disappeared. The second derivatives are given as [$$\begin{split}
{M_P}^2\frac{{{\partial}}^2 N_e}{{{\partial}}\phi_{*}^2}&=\left( 1-\frac{\eta_{\phi\phi}^*}{2\epsilon^*_\phi} \right),\\
{M_P}^2\frac{{{\partial}}^2 N_e}{{{\partial}}\sigma_*^2}&=-\left( 1-\frac{\eta_{\phi\phi}^e}{2\epsilon_{\phi}^e} \right){{\left(\frac{E_\sigma}{E_\phi} \right) }}^2 +
\frac{{M_P}}{ \sqrt{2\epsilon_\phi^e}} \frac{{{\partial}}}{{{\partial}}\sigma_*}{{\left(\frac{E_\sigma}{E_\phi} \right) }},\\
{M_P}^2\frac{{{\partial}}^2 N_e}{{{\partial}}\sigma_*{{\partial}}\phi_*}&= {M_P}^2\frac{{{\partial}}^2 N}{{{\partial}}\phi_*{{\partial}}\sigma_*}=0.\label{second_1}
\end{split}$$]{}
The curvature perturbation $\delta N_e$ can be determined by the two contributions: one is $\zeta_{\phi}$ coming from the perturbation of inflaton field $\delta \phi$ and the other is $\zeta_\sigma$ from $\delta \sigma$, i.e. [$$\begin{split}
\zeta\simeq \delta N_e = \zeta_\phi + \zeta_\sigma,
\end{split}$$]{} where [$$\begin{split}
\zeta_\phi &= \frac{{{\partial}}N_e}{{{\partial}}\phi_*} \delta \phi_* +\frac12 \frac{{{\partial}}^2 N_e}{{{\partial}}\phi_*^2}(\delta \phi_* )^2,\\
\zeta_\sigma &= \frac{{{\partial}}N_e}{{{\partial}}\sigma_*} \delta\sigma_* +\frac12 \frac{{{\partial}}^2 N_e}{{{\partial}}\sigma_*^2} (\delta\sigma_*)^2.
\end{split}$$]{} Here the first and second derivatives are given in Eqs. (\[linear\_1\]) and (\[second\_1\]).
Then the Power spectrum is given by [$$\begin{split}
{{\cal P}}_\zeta&= \frac{1}{2{M_P}^2\epsilon_\phi^*}\left[1+ \frac{\epsilon_\phi^*}{\epsilon_\phi^e} {{\left(\frac{E_\sigma}{E_\phi} \right) }}^2 \right] {{\left(\frac{H_*}{2\pi} \right) }}^2.
\end{split}$$]{} $\zeta_\sigma$ can dominate $\zeta_\phi$, when [$$\begin{split}
\frac{E_\sigma^2}{ E_\phi^2} \gg \frac{\epsilon_\phi^e}{\epsilon_\phi^*}.
\end{split}$$]{} For $\zeta_\sigma\gg\zeta_\phi$, the non-linearity parameter ${f_{\rm NL}}$ becomes [$$\begin{split}
\frac65{f_{\rm NL}}\simeq \left. \frac{{{\partial}}^2 N_e}{{{\partial}}\sigma_*^2} \right/ \left( \frac{{{\partial}}N_e}{{{\partial}}\sigma_*} \right)^2
= -(2\epsilon_\phi^e-\eta_\phi^e) + {M_P}\sqrt{2 \epsilon_\phi^e} {{\left(\frac{E_\sigma}{E_\phi} \right) }}^{-2}\frac{{{\partial}}}{{{\partial}}\sigma_*} {{\left(\frac{E_\sigma}{E_\phi} \right) }}.
\end{split}$$]{} Thus, a large ${f_{\rm NL}}$ is possible when the second term is large. Note that it can be obtained from [Eq. (\[endfnl\])]{} using [Eq. (\[R-Erel\])]{}. Accordingly, it is classified to “[**(b)**]{}” discussed in the previous section.
Hybrid inflation with general ending
------------------------------------
In this subsection we consider the hybrid-type inflation with two scalar fields $\phi_1$ and $\phi_2$ during inflation in addition to a waterfall field $\psi$. The hybrid inflation can be realized with the potential [$$\begin{split}
W=W_0(\phi_1,\phi_2,\psi)W_{\rm inf}(\phi_1,\phi_2)~,
\end{split}$$]{} where $W_0$ is given by $$\begin{aligned}
&&W_0=\frac12 G(\phi_1,\phi_2) \psi^2 +\frac{\lambda}{4} \left( \psi^2-\frac{\sigma^2}{\lambda} \right)^2 ~.\end{aligned}$$ Here a mass term for $\psi$, $G(\phi_1,\phi_2)$, can take the form [@Naruko:2008sq], [$$\begin{split}
G(\phi_1,\phi_2) \equiv g_1^2 (\phi_1\cos\alpha +\phi_2\sin\alpha)^2 +g_2^2 (-\phi_1\sin\alpha + \phi_2\cos\alpha)^2 ~,\label{Gend}
\end{split}$$]{} where $g_1$ and $g_2$ are dimensionless constants. During inflation when $G(\phi_1,\phi_2)$ is large enough compared to $\sigma^2$, the waterfall field $\psi$ is heavy and stuck to the origin, inflating the universe with the positive vacuum energy $W_0=\sigma^4/4\lambda$. When $G$ becomes smaller than $\sigma^2$, however, $\psi$ rolls down to the true minimum ($\psi^2=\sigma^2/\lambda$), terminating inflation. Hence, $G(\phi_1,\phi_2)$ yields the condition for the end of inflation and can be identified as [Eq. (\[endcondition\])]{} introduced in the previous section. Identifying $E=G$, we find that [$$\begin{split}
E_1 &\equiv \frac{{{\partial}}E}{{{\partial}}\phi_1}= 2g_1^2 (\phi_1\cos\alpha +\phi_2\sin\alpha) \cos\alpha - 2g_2^2(-\phi_1\sin\alpha + \phi_2\cos\alpha)\sin\alpha ~, \\
E_2&\equiv \frac{{{\partial}}E}{{{\partial}}\phi_2}= 2g_1^2 (\phi_1\cos\alpha +\phi_2\sin\alpha) \sin\alpha +2g_2^2(-\phi_1\sin\alpha + \phi_2\cos\alpha)\cos\alpha ~.
\end{split}$$]{}
For a clear discussion, we consider the specific potential for $W_{\rm inf}$ used in Refs. [@Naruko:2008sq; @Alabidi:2006hg; @Byrnes:2008zy], given by [$$\begin{split}
W_{\rm inf}(\phi_1,\phi_2) = \exp\left[\frac12 m_1^2\phi_1^2+\frac12m_2^2\phi_2^2\right] .
\end{split}$$]{} For simplicity, throughout this subsection we set $M_P=1$ and drop the super- (sub-) scripts “$e$” indicating end of inflation.
Particularly, in the case of the equal mass $m_1=m_2\equiv m$, one take $\alpha=0$ in the condition of the end of inflation, because of the rotational symmetry. Thus, $W_{\rm inf}=\exp\left[ m^2(\phi_1^2+\phi_2^2)/2 \right]$ and $E=g_1^2\phi_1^2 + g_2^2\phi_2^2$. With this potential, the slow-roll parameters and $R$ defined in [Eq. (\[R\])]{} becomes [$$\begin{split} \label{SRend}
\epsilon_{1}=\frac{1}{2}m^4\phi_1^2 ~,&\qquad\epsilon_{2}=\frac{1}{2}m^4\phi_2^2 ~,\\
\eta_{1}= m^2(1+m^2\phi_1^2) ~,&\qquad\eta_{2}= m^2(1+m^2\phi_2^2) ~.
\end{split}$$]{} From $E_1= 2g_1^2 \phi_1$ and $E_2= 2g_2^2 \phi_2$ , we have [$$\begin{split}
R=\frac{m_2^2\phi_2}{m_1^2\phi_1}\, {{\left(\frac{g_1^2\phi_{_1}}{g_2^2\phi_{_2}} \right) }}= \frac{ g_1^2}{ g_2^2},
\end{split}$$]{} which is independent of $\phi_1$ and $\phi_2$. Hence, [$$\begin{split} \label{epsilonbar}
\beps =R\epsilon_1+\epsilon_2 = \frac{m^4\sigma^2}{2g_2^2},
\end{split}$$]{} where $\sigma^2=g_1^2\phi_1^2 + g_2^2\phi_2^2 $ is a constant.
For $m_1=m_2$ ($=m$) and $\alpha=0$, thus, the power spectrum is given by [$$\begin{split}
{{\cal P}}_\zeta = \frac{1}{2} \frac{E_1^2 \epsilon_1/\epsilon_1^* + E_2^2 \epsilon_2/\epsilon_2^*}{( E_1\sqrt{\epsilon_1} +E_2\sqrt{\epsilon_2} )^2} {{\left(\frac{H_*}{2\pi} \right) }}^2 = \frac{e^{-2m^2N}}{m^4\sigma^2} (g_1^4\phi_1^2 +g_2^4 \phi_2^2) {{\left(\frac{H_*}{2\pi} \right) }}^2 ,
\end{split}$$]{} and the non-linear parameter ${f_{\rm NL}}$ is [$$\begin{split}
\frac65 {f_{\rm NL}}= \frac{4m^2}{g_1^4\phi_1^2 +g_2^4 \phi_2^2} \left[ - \sigma^2(g_1^6\phi_1^2 +g_2^6\phi_2^2) +2(g_1^2-g_2^2) g_1^2g_2^2\phi_1^2\phi_2^2\right].
\end{split}$$]{}
Conclusion
==========
In this paper, we have obtained a formula for curvature perturbation in multi-field inflation models. The formula contains the contributions from the evolution of perturbation during inflation and also from the effect at the end of inflation. For clearness of discussion, we utilized the potentials separable by product or sum with two scalar fields, but it is manifest to extend our discussion to cases of more scalar fields. Using the derived expression for curvature perturbation, we could get the analytic formulae for the power spectrum, spectral index and non-linearity parameter. Especially we could find more possibilities for large non-Gaussianity ${f_{\rm NL}}$ generated due to the effects by the end of inflation in addition to the generation during inflation. This formalism can be extended to accommodate more scenarios for the generation of curvature perturbation e.g. by the curvaton or modulated reheating [@Alabidi:2010ba].
Appendix
========
In this Appendix we show that our formulae give the same results with those in two-brid inflation studied in Ref. [@Naruko:2008sq]. The relevant potential is [$$\begin{split}
W=W_0(\phi_1,\phi_2,\psi)W_{\rm inf}(\phi_1,\phi_2)~,
\end{split}$$]{} where $W_0$ and $W_{\rm inf}$ are given by $$\begin{aligned}
&&W_0=\frac12 G(\phi_1,\phi_2) \psi^2 +\frac{\lambda}{4} \left( \psi^2-\frac{\sigma^2}{\lambda} \right)^2 ~.\end{aligned}$$ [$$\begin{split}
W_{\rm inf}(\phi_1,\phi_2) = \exp\left[\frac12 m_1^2\phi_1^2+\frac12m_2^2\phi_2^2\right] .
\end{split}$$]{} For simplicity, we set $M_P=1$ and drop the super-(and sub-)scripts “$e$” indicating end of inflation. Here the mass term for $\psi$, $G(\phi_1,\phi_2)$ can take the form [@Naruko:2008sq], [$$\begin{split}
G(\phi_1,\phi_2) \equiv g_1^2 (\phi_1\cos\alpha +\phi_2\sin\alpha)^2 +g_2^2 (-\phi_1\sin\alpha + \phi_2\cos\alpha)^2 ~,\label{Gend}
\end{split}$$]{} where $g_1$ and $g_2$ are dimensionless constants. The condition for ending inflation is $E(\phi_1,\phi_2)=G(\phi_1,\phi_2)=\sigma^2$.
For comparison, we use the same notation $X,Y,W$ and $Z$ defined in Ref. [@Naruko:2008sq], [$$\begin{split} \label{phi_12}
&\phi_1=\frac{\sigma}{g_1g_2}\left(g_2{\rm cos}\alpha{\rm cos}\gamma -g_1{\rm sin}\alpha{\rm sin}\gamma\right)\equiv \left(\frac{g\sigma}{g_1g_2}\right)X ~,
\\
&\phi_2=\frac{\sigma}{g_1g_2}\left(g_2{\rm sin}\alpha{\rm cos}\gamma +g_1{\rm cos}\alpha{\rm sin}\gamma\right)\equiv \left(\frac{g\sigma}{g_1g_2}\right)Z ~,
\\
&\quad\quad\quad \frac{{{\partial}}}{{{\partial}}\gamma}X\equiv -Y ~,\quad{\rm and}\quad \frac{{{\partial}}}{{{\partial}}\gamma}Z\equiv -W ~.
\end{split}$$]{} They satisfy [$$\begin{split} \label{rel}
\frac{X}{Z} =\frac{\phi_{1}}{\phi_{2}} ~,\quad \frac{W}{Y}=- \frac{E_1}{E_2} ~,\quad
R= -\frac{m_2^2}{m_1^2}\frac{WZ}{XY}=-\frac{m_2^2\phi_2^2}{m_1^2\phi_1^2}\frac{XW}{YZ} ~,
\end{split}$$]{} and [$$\begin{split} \label{epsilonbar}
\beps =R\epsilon_1+\epsilon_2 = \frac{1}{2} m_2^2\phi_2^2 {{\left(\frac{X}{Y} \right) }} \left(m_2^2 \frac{Y}{X} - m_1^2 \frac{W}{Z} \right).
\end{split}$$]{} Other useful relations are [$$\begin{split} \label{relations}
&\qquad\qquad \frac{{{\partial}}}{{{\partial}}\phi_{1}} {{\left(\frac{E_1}{E_2} \right) }} = \frac{g_1g_2}{\sigma g}\frac{Z}{Y^2} ~, \quad
\frac{{{\partial}}}{{{\partial}}\phi_{2}} {{\left(\frac{E_1}{E_2} \right) }} = -\frac{g_1g_2}{\sigma g}\frac{X}{Y^2} ~,\\
&\qquad \frac{{{\partial}}}{{{\partial}}\phi_{1}} {{\left(\frac{E_1}{E_2} \right) }}- {{\left(\frac{E_1}{E_2} \right) }}\frac{{{\partial}}}{{{\partial}}\phi_{2}} {{\left(\frac{E_1}{E_2} \right) }} =-\frac{g_1g_2}{\sigma g} {{\left(\frac{ZX}{Y^3} \right) }} \left(\frac{W}{Z} -\frac{Y}{X}\right) ~,
\\
&\frac{{{\partial}}R}{{{\partial}}\phi_{1}}=\frac{m_2^2}{m_1^2}\frac{g_1g_2}{g\sigma}\frac{Z}{XY}\left(\frac{W}{X}+\frac{Z}{Y}\right)~,\quad{\rm and}\quad \frac{{{\partial}}R}{{{\partial}}\phi_{2}}=-\frac{m_2^2}{m_1^2}\frac{g_1g_2}{g\sigma}\frac{Z}{XY}\left(\frac{W}{Z}+\frac{X}{Y}\right)~.
\end{split}$$]{}
Using the equation of motion of the fields in the slow-roll limit, the field values at the horizon exit, $\phi_1^*$ and $\phi_2^*$ can be estimated by tracing back the field values at the end of inflation, $\phi_1$ and $\phi_2$ so as to achieve 55-65 e-folds: [$$\begin{split}
N_e\approx\int^*_e\frac{d\phi_{i}}{m_{i}^2~\phi_{i}}=\frac{1}{m_{i}^2}~{\rm log}\frac{\phi^*_{i}}{\phi_{i}} ~,
\end{split}$$]{} where $i=$ 1 or 2. The slow-roll parameters at the horizon crossing, $\epsilon^*_{1}$, $\epsilon^*_{2}$ and $\eta^*_{1}$, $\eta^*_{2}$ are given by [$$\begin{split} \label{inlVal}
&\qquad\qquad\epsilon^*_{i}=\frac12 m_{i}^4\phi^{*2}_{i}=\frac12 m^4_{i}\phi_{i}^2~e^{2N_em^2_{i}} ~,
\\
&\eta^*_{i}
=m^2_{i}\left(1+m^2_{i}\phi^{*2}_{i}\right)
=m^2_{i}\left(1+m^2_{i}\phi^{2}_{i}
~e^{2N_em^2_{i}}\right) .
\end{split}$$]{}
After some manipulations with the above relations, the $\delta N_e$ given by [Eq. (\[linearproduct\])]{} and [Eq. (\[secondproduct\])]{} is easily converted to give: [$$\begin{split}
\delta N_e=& \sum_I
N_{e,I}\delta\phi_{I*}+\frac12\sum_{IJ}N_{e,IJ}\delta\phi_{I*}\delta\phi_{J*},
\\
=& \frac{\displaystyle{-\frac{W}{Z} \frac{\delta \phi_1^*}{\phi_1^*}+\frac{Y}{X} \frac{\delta \phi_2^*}{\phi_2^*} }}{\displaystyle{m_2^2 \frac{Y}{X}- m_1^2 \frac{W}{Z} }}
+ \frac12\frac{\displaystyle{\frac{W}{Z} {{\left(\frac{\delta \phi_1^*}{\phi_1^*} \right) }}^2-\frac{Y}{X} {{\left(\frac{\delta \phi_2^*}{\phi_2^*} \right) }}^2 }}{\displaystyle{m_2^2 \frac{Y}{X}- m_1^2 \frac{W}{Z} } }
\\
& -\frac12\frac{\displaystyle{\left( 1 -\frac{Y}{X}\frac{W}{Z} \right) \left(\frac{W}{Z} -\frac{Y}{X}\right)\left(\frac{m_2^2}{\phi_1^*}\delta\phi_1^* -\frac{m_1^2}{\phi_2^*}\delta\phi_2^* \right)^2 }}{\displaystyle{\left(m_2^2 \frac{Y}{X}- m_1^2 \frac{W}{Z} \right)^3}} +\cdots.
\end{split}$$]{} which is exactly the same as the result of Ref. [@Naruko:2008sq].
More explicitly we can check out our formulae for the power spectrum and non-linear parameter. With the definitions of $\overline{u}$ and $\overline{v}$ of [Eq. (\[uvbar\])]{}, the initial values of the slow-roll parameters of [Eq. (\[inlVal\])]{}, and Eqs. (\[SRend\]), (\[epsilonbar\]), (\[phi\_12\]), (\[rel\]), we get [$$\begin{split} \label{denorm}
\frac{\overline{u}^2}{\epsilon^*_{1}}+\frac{\overline{v}^2}{\epsilon^*_{2}}=&\frac{m_2^4}{2\overline{\epsilon}^2}\left(\frac{g\sigma}{g_1g_2}\right)^2\frac{Z^2}{Y^2}\left[W^2e^{-2N_em_1^2}+Y^2e^{-2N_em_2^2}\right] \\
=&2\left(\frac{g_1g_2}{g\sigma}\right)^{2}
\frac{\left[W^2e^{-2N_em_1^2}+Y^2e^{-2N_em_2^2}\right]}
{\left(m_2^2YZ-m_1^2XW\right)^2} ~.
\end{split}$$]{} This relation recasts the expression of the power spectrum given in [Eq. (\[powerPrd\])]{} to $${\cal P}_\zeta = \frac12\left(\frac{\overline{u}^2}{\epsilon^*_{1}}+\frac{\overline{v}^2}{\epsilon^*_{2}}\right)\left(\frac{H_*}{2\pi}\right)^2
=\left(\frac{g{\rm sin}\beta{\rm cos}\beta}{\sigma}\right)^2
\frac{\left[W^2e^{-2N_em_1^2}+Y^2e^{-2N_em_2^2}\right]}
{\left(m_2^2YZ-m_1^2XW\right)^2}\left(\frac{H_*}{2\pi}\right)^2 ,$$ where ${\rm cos}\beta\equiv g_1/g$ and ${\rm sin}\beta\equiv g_2/g$. This is again exactly coincident with the expression of Ref. [@Naruko:2008sq]. Similarly, one can confirm also the spectral index $n_\zeta$ and the scalar to tensor ratio.
Using Eqs. (\[SRend\]), (\[phi\_12\]), (\[rel\]), and (\[inlVal\]), the first two terms of the numerator in [Eq. (\[fnlPrd\])]{} also can be evaluated in terms of the parameters employed in Ref. [@Naruko:2008sq] as follows; $$\begin{aligned}
\label{1st+2nd}
&&\frac{\overline{u}^3}{\epsilon^*_{1}}\left(1-\frac{\eta^*_{1}}{2\epsilon^*_{1}}\right)+\frac{\overline{v}^3}{\epsilon^*_{2}}\left(1-\frac{\eta^*_{2}}{2\epsilon^*_{2}}\right)=\frac{m_2^6}{4\overline{\epsilon}^3}\left(\frac{g\sigma}{g_1g_2}\right)^2\frac{Z^3}{Y^3}\left[\frac{W^3}{X}e^{-4N_em_1^2}-\frac{Y^3}{Z}e^{-4N_em_2^2}\right] .~~~~~\end{aligned}$$ Now let us calculate the third term of [Eq. (\[fnlPrd\])]{} piece by piece. With Eqs. (\[SRend\]), (\[phi\_12\]), (\[rel\]), (\[relations\]), and (\[inlVal\]), we get the following piece expressions for calculation of the third term of ${f_{\rm NL}}$ in [Eq. (\[fnlPrd\])]{}; $$\begin{aligned}
&&\left(\frac{\overline{u}}{\epsilon^*_{1}}
-\frac{\overline{v}}{\epsilon^*_{2}}\right)^2
=\frac{1}{\overline{\epsilon}^2m_1^4}
\frac{Z^2}{Y^2}\left(m_2^2\frac{W}{X}e^{-2N_em_1^2}+m_1^2\frac{Y}{Z}e^{-2N_2m_2^2}\right)^2 , \\
&&-\frac{\epsilon_{1}\epsilon_{2}}{\overline{\epsilon}^3}R=\frac{m_1^2m_2^6}{4\overline{\epsilon}^3}
\left(\frac{g\sigma}{g_1g_2}\right)^4\frac{XWZ^3}{Y} ~, \nonumber \\
&&\epsilon_{2}\eta_{1}+R\epsilon_{1}\eta_{2}
=\frac{m_1^2m_2^4}{2}\left(\frac{g\sigma}{g_1g_2}\right)^2\left[Z^2-\frac{XWZ}{Y}+m_1^2\left(\frac{g\sigma}{g_1g_2}\right)^2X^2Z^2\left(1-\frac{m_2^2}{m_1^2}\frac{WZ}{XY}\right)\right] , \nonumber \\
&&-2(1+R)\epsilon_{1}\epsilon_{2}=-\frac{m_1^4m_2^4}{2}\left(\frac{g\sigma}{g_1g_2}\right)^4X^2Z^2\left(1-\frac{m_2^2}{m_1^2}\frac{WZ}{XY}\right) , \nonumber \\
&&\frac{1}{R}(\partial_{\phi_1}R)\epsilon_{2}
\sqrt{\frac{\epsilon_{1}}{2}}
-(\partial_{\phi_2}R)\epsilon_{1}
\sqrt{\frac{\epsilon_{2}}{2}}=\frac{m_1^2m_2^4}{4}\left(\frac{g\sigma}{g_1g_2}\right)^2\left[-Z^2+\frac{XWZ}{Y}-\frac{XZ^3}{YW}+\frac{X^2Z^2}{Y^2}\right] . \nonumber \end{aligned}$$ Assembling the above small fragments gives the full expression for the third term of the numerator in [Eq. (\[fnlPrd\])]{}: $$\begin{aligned}
\label{3rd}
\left(\frac{\overline{u}}{\epsilon^*_{1}}
-\frac{\overline{v}}{\epsilon^*_{2}}\right)^2 {\overline{{\cal A}_P}}=
\frac{m_2^6}{4\overline{\epsilon}^3}\left(\frac{g\sigma}{g_1g_2}\right)^2\frac{Z^3}{Y^3}\left[\frac{(XZ-YW)\left(\frac{W}{Z}-\frac{Y}{X}\right)\left(m_2^2\frac{W}{X}e^{-2N_em_1^2}+m_1^2\frac{Y}{Z}e^{-2N_em_2^2}\right)^2}{\left(m_2^2\frac{Y}{X}-m_1^2\frac{W}{Z}\right)^2}\right] . \nonumber \\\end{aligned}$$ By inserting Eqs. (\[denorm\]), (\[1st+2nd\]), (\[3rd\]), and (\[epsilonbar\]), (\[phi\_12\]) into Eq. (\[fnlPrd\]), we can reproduce the result of Ref. [@Naruko:2008sq]: $$\begin{aligned}
\frac65 f_{\rm NL}&=&\frac{XZ}{\left(m_2^2\frac{Y}{X}-m_1^2\frac{W}{Z}\right)
\left(Y^2e^{2N_em_1^2}+W^2e^{2N_em_2^2}\right)^2}
\Bigg[
\left(\frac{W^3}{X}e^{4N_em_2^2}
-\frac{Y^3}{Z}e^{4N_em_1^2}\right)
\left(m_2^2\frac{Y}{X}-m_1^2\frac{W}{Z}\right)^2
\nonumber \\
&&\quad\quad\quad -\left(XZ-YW\right)\left(\frac{W}{Z}-\frac{Y}{X}\right)
\left(m_1^2\frac{Y}{Z}e^{2N_em_1^2}
+m_2^2\frac{W}{X}e^{2N_em_2^2}\right)^2\Bigg] ~.
\label{fnlquad}\end{aligned}$$ In Ref. [@Naruko:2008sq], the authors discussed the possibility of large non-Gaussianity based on the above expression [Eq. (\[fnlquad\])]{} for the cases that the two inflaton masses are degenrate ($m_1^2=m_2^2$) and hierarchical ($m_1^2\gg m_2^2$).
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[^1]: email: [email protected]
[^2]: email: [email protected]
[^3]: email: [email protected]
|
---
abstract: 'In this paper we use fixed point tilings to answer a question posed by Michael Hochman and show that every one-dimensional effectively closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construction of an aperiodic tile set and its extensions.'
author:
- 'Bruno Durand, Andrei Romashchenko, Alexander Shen'
title: |
Effective closed subshifts in 1D can be\
implemented in 2D
---
Introduction
============
Let $A$ be a finite set (*alphabet*); its elements are called *letters*. By $A$-*configuration* we mean a mapping $C\colon \mathbb{Z}^2\to A$. In geometric terms: a cell with coordinates $(i,j)$ contains letter $C(i,j)$.
A *local rule* is defined by a positive integer $M$ and a list of prohibited $(M\times M)$-*patterns* ($M\times M$ squares filled by letters). A configuration $C$ *satisfies* a local rule $R$ if none of the patterns listed in $R$ appears in $C$.
Let $A$ and $B$ be two alphabets and let $\pi\colon A\to B$ be any mapping. Then every $A$-configuration can be transformed into a $B$-configuration (its homomorphic image) by applying $\pi$ to each letter. Assume that the local rule $R$ for $A$-configurations and mapping $\pi$ are chosen in such a way that local rule $R$ prohibits patterns where letters $a$ and $a'$ with $\pi(a)\ne\pi(a')$ are vertical neighbors. This guarantees that every $A$-configuration that satisfies $R$ has an image where vertically aligned $B$-letters are the same. Then for each $B$-configuration in the image every vertical line carries one single letter of $B$. So we can say that $\pi$ maps a $2$-dimensional $A$-configuration satisfying the local rule $R$ to a $1$-dimensional $B$-configuration.
Thus, for every $A$, $B$, local rule $R$ and $\pi$ (with described properties) we get a subset $L(A,B,R,\pi)$ of $B^\mathbb{Z}$ (i.e., $L(A,B,R,\pi)$ is the set of $\pi$-images of all $A$-configurations that satisfy the local rule $R$). The following result (theorem \[main\]) characterizes the subsets that can be obtained in this way.
Consider a product topology in $B^\mathbb{Z}$. Base open sets of this topology are *intervals*. Each interval is obtained by fixing letters in finitely many places (other places may contain arbitrary letters). Each interval is therefore a finite object and we can define *effectively open* subsets of $B^\mathbb{Z}$ as unions of (computably) enumerable families of intervals. An *effectively closed* set is a complement of an effectively open one. A *subshift* is a subset of $B^\mathbb{Z}$ that is closed and invariant under left and right shifts. We are mostly interested in subshifts that are not only closed but effectively closed sets.
\[main\] The subset $L(A,B,R,\pi)$ is an effectively closed subshift. For every effectively closed subshift $S\subset \mathbb{B}^Z$ one can find $A$, $R$, and $\pi$ such that $S=L(A,B,R,\pi)$.
The first part of the statement is easy. The set $L(A,B,R,\pi)$ is evidently shift invariant; it remains to show that it is effectively closed. The set of all $A$-configurations that satisfy $R$ is a closed subset of a compact space and therefore is a compact space itself. The mapping of $A$-configurations into $B$-configurations is continuous. Therefore the set $L(A,B,R,\pi)$ is compact (as a continuous image of a compact set). This argument can be effectivized in a standard way. A $B$-string is declared *bad* if it cannot appear in the $\pi$-image of any $A$-configuration that satisfies $R$. The set of all bad strings is enumerable and $L(A,B,R,\pi)$ is the set of all bi-infinite sequences that have no bad factors.
The reverse implication is more difficult and is the main subject of this paper. It cannot be proven easily since it implies the classical result of Berger [@berger]: the existence of a local rule that makes all configurations aperiodic. Indeed, it is easy to construct an effectively closed subshift $S$ that has no periodic points; if it is represented as $L(A,B,R,\pi)$, then local rule $R$ has no periodic configurations (configurations that have two independent period vectors); indeed, those configurations have a horizontal period vector.
So it is natural to expect a proof of theorem \[main\] to be obtained by modifying one of the existing constructions of an aperiodic local rule. It is indeed the case: we use the fixed-point construction described in [@firstpart]. We do not repeat this construction (assuming that the reader is familiar with that paper or has it at hand) and explain only the modifications that are needed in our case. This is done in sections \[sec:idea\]–\[sec:degenerate\]; in the rest of this section we survey some other steps in the same direction.
M. Hochman [@hochman] proved a similar result for 3D implementations of 1D subshifts (and, in general, $(k+2)$-dimensional implementation of $k$-dimensional subshifts) and asked whether a stronger statement is true where 3D is replaced by 2D.
As we have mentioned, it is indeed true and can be achieved by the technique of fixed point self-similar tilings. The detailed exposition of this technique and its applications, including an answer to Hochman’s question, is given in our paper [@long]. Since this paper contains many other results (most boring are related to error-prone tile sets), we think that a self-contained (modulo [@firstpart]) exposition could be useful for readers that are primarily interested in this result, and provide such an exposition in the current paper.
In fact, the fixed point construction of algorithms and machines is an old and well known tool (used, e.g., for Kleene’s fixed point theorem and von Neumann’s self-reproducing automata) that goes back to self-referential paradoxes and Gödel’s incompleteness theorem. One may only wonder why it was not used in 1960s to construct an aperiodic tile set. In a context of hierarchical constructions in the plane this technique was used by P. Gács in a much more complicated situation (see [@gacs2d]); however, Gács did not bother to mention explicitly that this technique can be applied to construct aperiodic tile sets.
Fixed point tilings are not the only tool that can be used to implement subshifts. In [@dls] a more classical (Berger–Robinson style) construction of an aperiodic tile set is modified in several ways to implement one specific shift: the family of bi-infinite bit sequences $\omega$ such that all sufficiently long substrings $x$ of $\omega$ have complexity greater than $\alpha |x|$ or at least $\omega$ can be cut into two pieces (left- and right-infinite) that have this property. (Here $\alpha$ is some constant less than $1$, and $|x|$ stands for the length of $x$.) In fact, the construction used there is fairly general and can be applied to any enumerable set $F$ of forbidden substrings: one may implement a shift that consists of bi-infinite sequences that have no substrings in $F$ or at least can be cut into two parts with this property. Recently N. Aubrun and M. Sablik found a more ingenious construction that is free of this problem (splitting into two parts) and therefore provides another proof of theorem \[main\] (see [@sablik]).
The authors thank their LIF colleagues, especially E. Jeandel who pointed out that their result answers a question posed in [@hochman].
The idea of the construction {#sec:idea}
============================
We do not refer explicitly to our paper [@firstpart] but use the notions and constructions from that paper freely. In that paper we used local rules of special type (each letter was called a tile and has four colors at its sides; the local rule says that colors of the neighbor tile should match). In fact, any local rule can be reduced to this type by extending the alphabet; however, we do not need to worry about this since we construct a local rule and may restrict ourselves to tilings.
We superimpose two layers in our tiling. One of the layers contains $B$-letters; the local rule guarantees that each vertical line carries one $B$-letter. (Vertical neighbors should be identical.) For simplicity we assume that $B=\{0,1\}$, so $B$-letters are just bits, but this is not really important for the argument.
The second layer contains an aperiodic tile set constructed in a way similar to [@firstpart]. Then some rules are used to organize the interaction between the layers; computations in the second layer are fed with the data from the first layer and check that the first layer does not contain any forbidden string.
Indeed, the macro-tiles (at every level) in our construction contain some computation used to guarantee their behavior as building blocks for the next level. Could we run this computation in parallel with some other one that enumerates bad patterns and terminates the computation (creating a violation of the rules) if a bad pattern appears?
This idea immediately faces evident problems:
- The computation performed in macro-tiles (in [@firstpart]) was limited in time and space (and we need unlimited computations since we have infinitely many forbidden substrings and no limit on the computational resources used to enumerate them).
- Computations on high levels do not have access to bit sequence they need to check: the bits that go through these macro-tiles are “deep in the subconscious”, since macro-tiles operate on the level of their sons (cells of the computation that are macro-tiles of the previous level), not individual bits.
- Even if every macro-tile checks all the bits that go through it (in some mysterious way), a “degenerate case” could happen where an infinite vertical line is not crossed by any macro-tile. Imagine a tile that is a left-most son of a father macro-tile who in its turn is the left-most son of its father and so on (see Fig. \[degenerate.mps\]). They fill the right half-plane; the left half-plane is filled in a symmetric way, and the vertical dividing line between then is not crossed by any tile. Then, if each macro-tile takes care of forbidden substrings inside its zone (bits that cross this macro-tile), some substrings (that cross the dividing line) remain unchecked.
These problems are discussed in the following sections one after another; we apologize if the description of them seemed to be quite informal and vague and hope that they would become more clear when their solution is discussed.
Variable zoom factor {#sec:variable}
====================
In our previous construction the macro-tiles of all levels were of the same size: each of them contained $N\times N$ macro-tiles of the previous level for some constant zoom factor $N$. Now it is not enough any more, since we need to host arbitrarily long computations in high-level macro-tiles. So we need an increasing sequence of zoom factors $N_0,N_1,N_2,\ldots$; macro-tiles of the first level are blocks of $N_0\times N_0$ tiles; macro-tiles of the second level are blocks of $N_1\times N_1$ macro-tiles of level $1$ (and have size of $N_0N_1\times N_0N_1$ if measured in individual tiles). In general, macro-tiles of level $k$ are made of $N_{k-1}\times N_{k-1}$ macro-tiles of level $k-1$ and have side $N_0N_1\ldots N_{k-1}$ measured in individual tiles.
However, all the macro-tiles (of different levels) carry the same program in their computation zone. The difference between their behavior is caused by the data: each macro-tiles “knows” its level (consciously, as a sequence of bits on its tape). Then this level $k$ may be used to compute $N_k$ which is then used as a modulus for coordinates in the father macro-tile. (Such a coordinate is a number between $0$ and $N_k-1$, and the addition is performed modulo $N_k$.)
Of course, we need to ensure that this information is correct. Two properties are required: (1) all macro-tiles of the same level have the same idea about their level, and (2) these ideas are consistent between levels (each father is one level higher than its sons). The first is easy to achieve: the level should be a part of the side macro-color and should match in neighbor tiles. (In fact an explicit check that brothers have the same idea about their levels is not really needed, since the first property follows from the second one: since all tiles on the level zero “know” their level correctly, by induction we conclude that macro-tiles of all levels have correct information about their levels.)
To achieve the second property (consistency between level information consciously known to a father and its sons) is also easy, though we need some construction. It goes as follows: each macro-tile knows its place in the father, so it knows whether the father should keep some bits of his level information in that macro-tile. If yes, the macro-tile checks that this information is correct. Each macro-tile checks only one bit of the level information, but with brothers’ help they check all the bits.[^1]
There is one more thing we need to take care of: the level information should fit into the tiles (and the computation needed to compute $N_k$ knowing $k$ should also fit into level $k$ tile). This means that $\log k$, $\log N_k$ and the time needed to compute $N_k$ from $k$ should be much less than $N_{k-1}$ (since the computation zone is some fraction of $N_{k-1}$). So $N_k$ should not grow too slow (say, $N_k=\log k$ is too slow), should not grow too fast (say, $N_k=2^{N_{k-1}}$ is too fast) and should not be too difficult to compute. However, these restriction still leave a lot of room for us: e.g., $N_k$ can be proportional to $\sqrt{k}$, to $k$, to $2^k$, or $2^{2^k}$, or $2^{2^{2^k}}$ (any fixed height is OK). Recall that computation deals with binary encodings of $k$ and $N_k$ and normally is polynomial in their lengths.
In this way we are now able to embed computations of increasing sizes into the macro-tiles. Now we have to explain which data these computations would get and how the communication between levels is organized.
Conscious and subconscious bits
===============================
The problem of communication between levels can be explained using the following metaphor. Imagine you are a macro-tile; then you have some program, and process the data according to the program; you “blow up” (i.e., your interior cannot be correctly tiled) if some inconsistency in the data is found. This program makes you perform as one cell in the next-level brain (in the computation zone of the father macro-tile), but you do not worry about it: you just perform the program. At the same time each cell of yourself in fact is a son macro-tile, and elementary operations of this cell (the relation between signals on its sides) are in fact performed by a lower-level computation. But this computation is your “sub-conscious”, you do not have direct access to its data, though the correct functioning of the cells of your brain is guaranteed by the programs running in your sons.
Please do not took this metaphor too seriously and keep in mind that the time axis of the computations is just a vertical axis on the plane; configurations are static and do not change with time. However, it could be useful while thinking about problems of inter-level communication.
Let us decide that for each macro-tile all the bits (of the bit sequence that needs to be checked) that cross this macro-tile form its *responsibility zone*. Moreover, one of the bits of this zone may be *delegated* to the macro-tile, and in this case the macro-tile consciously knows this bit (is *responsible* for this bit). The choice of this bit depends on the vertical position of the macro-tile in its father.
More technically, recall that a macro-tile of level $k$ is a square whose side is $L_k=N_0\cdot N_1\cdot \ldots\cdot N_{k-1}$, so there are $L_k$ bits of the sequence that intersect this macro-tile. We delegate each of these bits to one of the macro-tiles it intersects. Note that every macro-tile of the next level is made of $N_k\times N_k$ macro-tiles of level $k$. We assume that $N_k$ is much bigger than $L_k$ (more about choice of $N_k$ later); this guarantees that there are enough macro-tiles of level $k$ (in the next level macro-tile) to serve all bits that intersect them. Let us decide that $i$th macro-tile of level $k$ (from bottom to top) in a $(k+1)$-level macro-tile knows $i$th bit (from the left) in its zone. Since $N_k$ is greater than $L_k$, we leave some unused space in each macro-tile of level $k+1$: many macro-tiles of level $k$ are not responsible for any bit, but this does not create any problems.
$$\includegraphics[scale=0.8]{delegation.mps}$$
This is our plan; however, we need a mechanism that ensures that the delegated bits are indeed represented correctly (are equal to the corresponding bits “on the ground”, in the sequence that forms the first level of our construction). This is done in the hierarchical way: since every bit is delegated to macro-tiles of all levels, it is enough to ensure that the ideas about bit values are consistent between father and son.
For this hierarchical check let us agree that every macro-tile not only knows its own delegated bit (or the fact that there is no delegated bit), but also knows the bit delegated to its father (if it exists) as well as father’s coordinates (in the grandfather macro-tile). This is still an acceptable amount of information (for keeping father’s coordinates we need to ensure that $\log N_{k+1}$, the size of father’s coordinate, is much smaller that $N_{k-1}$). To make this information consistent, we ensure that
- the data about the father’s coordinates and bits are the same among brothers;
- if a macro-tile has the same delegated bit as its father (this fact can be checked since a macro-tile knows its coordinates in the father and father’s coordinates in the grandfather), these two bits coincide;
- if a macro-tile is in the place where its father keeps its delegated bit, the actual father’s information is consistent with the information about what the father should have.
So the information transfer between levels is organized as follows: the macro-tile that has the same delegated bit as its father, non-deterministically guesses this fact and distributes the information about father’s coordinates and bit among the brothers. Those of the brothers who are in the correct place, check that father indeed has correct information.
On the lowest level we have direct access to the bits of the sequence, so the tile that is above the correct bit can keep its value and transmit it together with (guessed) coordinates of its father macro-tile (in the grandfather’s macro-tile) to all brothers, and some brothers are in the right place and may check these values against the bits in the computation zone of the father macro-tile.
This construction makes all bits present at all levels, but this is not enough for checking: we need to check not individual bits, but bit groups (against the list of forbidden substrings). To this end we need a special arrangement described in the next section.
Checking bit groups
===================
Here the main idea is: each macro-tile checks some substring (bit group) that is very small compared to the size of this macro-tile. However, since the size of the computation zone grows infinitely as the level increases, this does not prevent complicated checks (that may involve a long substring that appears very late in the enumeration of the forbidden patterns) from happening.
The check is performed as follows: we do some number of steps in the enumeration of forbidden patterns, and then check whether one of these patterns appears in the bit group under consideration (assigned to this macro-tile). The number of enumeration steps can be also rather small compared to the macro-tile size.
We reserve also some time and space to check that all the patterns appeared during the enumeration are not substrings of the bit group under consideration. This is not a serious time/space overhead since substring search in the given bit group can be performed rather fast, and the size of the bit group and the number of enumeration steps are chosen small enough (having in mind this overhead).
Then in the limit any violation inside some macro-tile will be discovered (and only degenerate case problem remains: substrings that are not covered entirely by any tile). The degenerate case problem is considered in the next section; it this section it remains to explain how the groups of (neighbor) bits are made available to the computation and how they are assigned to macro-tiles.
Let us consider an infinite vertical stripe of macro-tiles of level $k$ that share the same $L_k=N_0\cdot\ldots \cdot N_{k-1}$ columns. Together, these macro-tiles keep in their memory all $L_k$ bits of their common zone of responsibility. Each of them perform a check for a small bit group (of length $l_k$, which increases extremely slowly with $k$ and in particular is much less than $N_{k-1}$). We need to distribute somehow these groups among macro-tiles of this infinite stripe.
It can be done in many different ways. For example, let us agree that the starting point of the bit group checked by a macro-tile is the vertical coordinate of this macro-tile in its father (if it is not too big; recall that $N_k \gg N_0 N_1\ldots N_{k-1}$). It remains to explain how groups of (neighbor) bits are made available to the computational zones of the corresponding macro-tiles.
We do it in the same way as for delegated bits; the difference (and simplification) is that now we may use only two levels of hierarchy since all the bits are available in the previous level (and not only in the “deep unconscious”, at the ground level). We require that this group and the coordinate that determines its position are again known to all the sons of the macro-tile where the group is checked. Then the sons should ensure that (1) this information is consistent between brothers; (2) it is consistent with delegated bits where delegated bits are in the group, and (3) it is consistent with the information in the macro-tile (father of these brothers) itself. Since $l_k$ is small, this is a small amount of information so there is no problem of its distribution between macro-tiles of the preceding level.
If a forbidden pattern belongs to a zone of responsibility of macro-tiles of arbitrarily high level, then this violation is be discovered inside a macro-tile of some level, so the tiling of the plain cannot not exist. Only the degenerate case problem remains: so far we cannot catch forbidden substrings that are not covered entirely by any macro-tile. We deal with the degenerate case problem in the next section.
Dealing with the degenerate case {#sec:degenerate}
================================
The problem we need to deal with: it can happen that one vertical line is not crossed by any macro-tile of any level (see Fig. \[degenerate.mps\]). In this case some substrings are not covered entirely by any macro-tile, and we do not check them.
$$\includegraphics[scale=0.8]{degenerate.mps}$$
After the problem is realized, the solution is not difficult. We let every macro-tile check bit groups in its *extended responsibility zone* that is three times wider and covers not only the macro-tile itself but also its left and right neighbors.
Now a macro-tile of level $k$ is given a small bit group which is a substring of its extended responsibility zone (the width of the extended responsibility zone is $3L_k$; it is composed of the zones of responsibility of the macro-tile itself and two its neighbors). Respectively, a macro-tile of level $(k-1)$ keeps the information about three groups of bits instead of one: for its father, left uncle, and right uncle. This information should be consistent between brothers (since they have the same father and uncles). Moreover, it should be checked across the boundary between macro-tiles: if two macro-tiles $A$ and $B$ are neighbors but have different fathers ($B$’s father is $A$’s right uncle and $A$’s father is $B$’s left uncle), then they should compare the information they have (about bit groups checked by fathers of $A$ and $B$) and ensure it is consistent. For this we need to increase the amount of information kept in a macro-tile by a constant factor (a macro-tile keeps three bit groups instead of one, etc.), but this is still acceptable.
It is easy to see that now even in the degenerate case every substring is entirely in the extended responsibility zone of arbitrary large tiles, so all the forbidden patterns are checked everywhere.
Final adjustments
=================
We finished our argument, but we was quite vague about the exact values of parameters saying only that some quantities should be much less than others. Now we need to check again the entire construction and see that the relations between parameters that were needed at different steps could be fulfilled together.
Let us remind the parameters used at several steps of the construction: macro-tiles of level $k+1$ consist of $N_k\times N_k$ macro-tile of level $k$; thus, a $k$-level macro-tile consists of $L_k\times L_k$ tiles (of level $0$), where $L_k=N_0\cdot\ldots\cdot N_{k-1}$. Macro-tiles of level $k$ are responsible for checking bit blocks of length $l_k$ from their extended responsibility zone (of width $3L_k$). We have several constraints on the values of these parameters:
- $\log N_{k+1}\ll N_k$ and even $\log N_{k+2}\ll N_k$ since every macro-tile must be able to do simple arithmetic manipulations with its own coordinates in the father and with coordinates of the father in the grandfather;
- $N_k\gg L_k$ since we need enough sons of a macro-tile of level $k+1$ to keep all bits from its zone of responsibility (we use one macro-tile of level $k$ for each bit);
- $l_{k}$ and even $l_{k+1}$ should be much less than $N_{k-1}$ since a macro-tile of level $k$ must contain in its computational zone the bit block of length $l_k$ assigned to itself and three bit blocks of length $l_{k+1}$ assigned to its father and two uncles (the left and right neighbors of the father);
- a $k$-level macro-tile should enumerate in its computational zone several forbidden patterns and check whether any of them is a substring of the given (assigned to this macro-tile) $l_k$-bits block; the number of steps in this enumeration must be small compared to the size of the macro-tile; for example, let us agree that a macro-tile of level $k$ runs this enumeration for exactly $l_k$ steps;
- the values $N_k$ and $l_k$ should be simple functions of $k$: we want to compute $l_k$ in time polynomial in $k$, and compute $N_k$ in time polynomial in $\log N_k$ (note that typically $N_k$ is much greater than $k$, so we cannot compute or even write down its binary representation in time polynomial in $k$).
With all these constraints we are still quite free in the choice of parameters. For example, we may let $N_{k}=2^{C 2^k}$ (for some large enough constant $C$) and $l_k=k$.
Final remarks
=============
One may also use essentially the same construction to implement $k$-dimensional effectively closed subshifts using $(k+1)$-dimensional subshifts of finite type.
How far can we go further? Can we implement evert $k$-dimensional effectively closed subshifts by a tiling of the same dimension $k$? Another question (posed in [@hochman]): let us replace a finite alphabet by a Cantor space (with the standard topology); can we represent every $k$-dimensional effectively closed subshifts over a Cantor space as a continuous image of the set of tilings of dimension $k+1$ (for some finite tile set)? E. Jeandel noticed that the answers to the both questions are negative (this fact is also a corollary of results from [@dls] and [@rumush]).
[9]{}
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B. Durand, A. Romashchenko, A. Shen, Fixed point theorem and aperiodic tilings, *Bulletin of the EATCS*, no. 97 (2009), pp. 126–136 (The Logic in Computer Science Column by Yuri Gurevich). Electronic version: [arXiv:1003.2801 \[cs.LO\]]{}, [http://arxiv.org/abs/1003.2801]{}
B. Durand, A. Romashchenko, A. Shen, Fixed-point tile sets and their applications. [arXiv:0910.2415 \[cs.CC\]]{}, 2009. [http://arxiv.org/abs/0910.2415]{}
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[^1]: People sitting on the stadium during the football match and holding color sheets to create a slogan for their team can check the correctness of the slogan by looking at the scheme and knowing their row and seat coordinates; each person checks one pixel, but in cooperation they check the entire slogan.
|
---
abstract: |
In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulae in highly energetic hadronic processes. We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles. Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization. As examples we treat the $\pi$-$\gamma$ form factor ($\gamma\gamma^*\to \pi^0$), light meson form factors ($\gamma^* M \to M$), as well as deep inelastic scattering ($e^- p\to e^- X$), Drell-Yan ($p\bar p\to X
\ell^+\ell^-$), and deeply virtual Compton scattering ($\gamma^* p \to
\gamma^{(*)} p$).
author:
- 'Christian W. Bauer'
- Sean Fleming
- Dan Pirjol
- |
\
Ira Z. Rothstein
- 'Iain W. Stewart'
title: ' Hard Scattering Factorization from Effective Field Theory\'
---
Introduction
============
The principle of factorization underlies all theoretical predictions for hadronic processes. Simply put, factorization is the statement that short and long distances contributions to physical processes can be separated, up to corrections suppressed by powers of the relevant large scale in the process. The predictive power gained from this result stems from the fact that the incalculable long distance effects are universal, defined in an unambiguous way in terms of matrix elements. As a consequence, the non-perturbative long distance effects can be extracted in one process and then used in another. In general, proving factorization is a difficult task [@Ellis:ty]. The proof of factorization in Drell Yan processes, for instance, took several years to sort out [@DYrefs] (for reviews on factorization see [@Collins:1987pm; @Sterman:1995fz; @Jaffe:zw]). Indeed, there are still some processes such as $B\to \pi\pi$ where a proof of factorization only exists at one-loop [@Bpipi].
Given that we would like to retain our predictive power over the largest possible range of energies, we are compelled to understand power corrections to the factorized rates. These corrections are not necessarily universal, and as such, the relevant size of the power corrections are process dependent. In processes for which there exists an operator product expansion (OPE), there is a systematic way in which to include power corrections. However, for a majority of observables we do not have an OPE at our disposal, and the nature of the power correction is not always known. For instance, in the case of shape variables there is still some on going discussion about the form of subleading corrections [@nason; @Korchemsky:1999kt].
The purpose of this paper is to show that an effective theory framework can be used to simplify proofs of factorization and describe processes with an operator formalism. To do this we extend the soft-collinear effective theory (SCET) developed in Refs. [@bfl; @bfps; @cbis; @bpssoft; @bps], to high energy processes. It should be emphasized that there are several other useful advantages in using an effective field theory (EFT). For instance, the EFT makes any symmetries which emerge in the $Q\to \infty$ limit manifest in the Lagrangian and operators, and allow statements to be made to all orders in perturbation theory. The calculation of hard coefficients reduces to simple matching calculations, where subtracting the EFT graphs automatically removes all infrared divergences from the QCD calculation. Perhaps most importantly, it provides a framework for systematically investigating power corrections. Finally, the EFT framework allows standard renormalization group techniques to be used for the resummation of logarithms that are often necessary in calculating rates for certain high energy scattering events [@bfl; @bfps; @resum; @resum2]. The factorization formulae that we prove in this paper are not new, but serve to illustrate our approach in familiar settings. The results are valid to all orders in $\alpha_s$ and leading order in the power expansion. The simplicity of our approach lies in the fact that factorization occurs at the level of the SCET Lagrangian and operators, and is facilitated by gauge symmetry in the EFT. This provides the advantage that our proofs do not rely on making use of Ward identities and induction, or on specifying a particular gauge.[^1] Furthermore, it becomes rather simple to derive factorization formulae for a myriad of processes, since many results are universal. The examples given here serve to illustrate these simplifications. Developments on the issues of power corrections and resummations are left to future publications.[^2]
In section \[section\_effective\] we review the construction of the SCET. The formalism developed in Refs. [@bfl; @bfps; @cbis; @bpssoft] is extended to include two types of collinear particles moving in opposite directions in section \[section\_nnbar\], and factorization for $\gamma^*$ to two collinear states is discussed as an example. In section \[section\_melt\] we define the non-perturbative matrix elements such as the parton distribution functions that will be needed for the processes presented in the paper, and in section \[section\_symm\] we discuss some of the symmetries in SCET that may be used to place restrictions on matrix elements. In the remaining sections we give various examples on how factorization theorems emerge in the effective theory language. In section \[section\_exclusive\] we prove factorization theorems for two exclusive processes, namely the $\pi$-$\gamma$ form factor, and meson form factors ($\gamma^* M \to M$) for arbitrary spin and isospin structure. In section \[section\_inclusive\] two inclusive processes are treated, namely DIS ($e^- p\to e^- X$) and Drell-Yan ($p\bar p\to X \ell^+\ell^-$), and we also give results for deeply virtual Compton scattering ($\gamma^* p \to \gamma^{(*)}
p$). In these processes we include all leading power contributions in the factorization proofs (even if the operators are only matched onto at higher orders in perturbation theory such as for the gluon distribution functions). Our conclusions are given in section \[section\_conclusion\]. In appendix \[app\_fact\] we show how auxiliary fields can be used to prove the simultaneous factorization of soft fields from collinear fields for particles in back-to-back directions.
Formalism {#section_effective}
=========
Effective field theories provide a simple and elegant way of organizing physics in processes containing widely disparate energy scales. In constructing an EFT, some degrees of freedom are eliminated, and the remaining degrees of freedom must reproduce all the infrared physics of the full theory in the domain where the EFT is valid. The EFT is organized by an expansion in $\lambda$, defined as the ratio of small to large energy scales. As a useful guideline the following steps are used to identify the infrared degrees of freedom: 1) Determine the relevant scales in a problem from the size of the momenta and masses of all particles that can make up the initial and final states, 2) Construct all momenta from these scales whose components correspond to propagating degrees of freedom, and which have offshellness less than the large scale, i.e. $p^2-m^2\lesssim Q^2$. Effective theory fields are then constructed for each unique set of these momenta.
We will be interested in an EFT with particles of energy $Q$ much greater than their mass. The dynamics of these particles can be described by constructing a soft-collinear effective theory (SCET). This theory is organized as an expansion in powers of $\lambda\sim p_\perp/Q$, and offshell fluctuations with $p^2 \gg
(Q\lambda)^2$ are integrated out. In section \[section\_SCET\] we begin by describing this procedure and comparing the construction to other EFT’s. We then give a brief review of the soft-collinear effective theory developed in Refs. [@bfl; @bfps; @cbis; @bpssoft; @bps]. We do not attempt to give a comprehensive treatment, but instead emphasize the main results and refer the reader to the literature for details. In section \[section\_nnbar\] we extend the formulation of SCET to describe processes with collinear particles moving in back to back directions, and prove the factorization formula for $\gamma^*$ to two collinear states as an example. In section \[section\_melt\] we define the non-perturbative matrix elements that are needed for our examples, then in section \[section\_symm\] we discuss some of the symmetry properties of collinear fields and currents.
Soft-Collinear Effective Theory {#section_SCET}
-------------------------------
In the standard construction of an EFT one removes the short distance scales and massive fields by integrating them out one at a time. A classical example is integrating out the $W$ boson to obtain the effective electroweak Hamiltonian with 4-fermion operators. However, in some situations we are interested in integrating out large momentum fluctuations without fully removing the corresponding field. The simplest example of this is Heavy Quark Effective Theory (HQET) [@bbook], which is constructed to describe the low energy properties of mesons with a heavy quark. Here the heavy anti-quarks are integrated out and only heavy quarks with fluctuations close to their mass-shell are retained. This is accomplished by removing fluctuations of order the heavy quark mass $m_Q$ with a field redefinition[@georgi] $$\label{fd1}
\psi(x)=\sum_v e^{-im_Q v\cdot x}h_v(x) \,,$$ where $v$ is the heavy quark velocity and $h_v$ is the field in the EFT. While $\partial^\mu \psi(x)\sim m_Q\, \psi(x)$, the effective field has $\partial^\mu
h_v(x) \sim \Lambda_{\rm QCD}\, h_v(x)$, indicating that it no longer describes short-distance fluctuations about the perturbative scale $m_Q$. Instead these effects are encoded in calculable Wilson coefficients. The HQET degrees of freedom with offshellness $p^2\sim \Lambda_{\rm QCD}^2$ are the heavy quarks, soft gluons, and soft quarks.
Similarly, for collinear particles with energy $Q\gg m$, one needs to remove momentum fluctuations $\sim\!Q$ while retaining effective theory fields to describe smaller momenta. However, unlike heavy quarks the collinear particles have two low energy scales. Consider the light-cone momenta, $p^+ = n\cdot p$ and $p^- = {{\bar n}}\cdot p$ where $n^2={{\bar n}}^2=0$ and $n\cdot{{\bar n}}=2$. Here $n$ parameterizes a light-cone direction close to that of the collinear particle and ${{\bar n}}$ the opposite direction (eg. for motion in the $z$ direction $n_\mu =
(1,0,0,1)$ and ${{\bar n}}_\mu = (1,0,0,-1)$). For a particle of mass $m\lesssim
p_\perp \ll Q$, we have $p^-\sim Q$, and a small parameter $\lambda \sim
p_\perp/Q$. The scaling of the $p^+$ component is then fixed by the equations of motion $p^+p^-+p^2_\perp =m^2$, so that $(p^+,p^-,p_\perp) \sim Q(\lambda^2,
1,\lambda)$.
The appearance of two small scales, $Q\lambda^2\ll Q\lambda \ll Q$, is similar to the situation in non-relativistic QCD (NRQCD), which is an EFT for systems of two heavy quarks with an expansion in their relative velocity $\beta$. In a non-relativistic bound state the momentum of a heavy quark is ${\bf p}\sim
m_Q\beta$, but the equations of motion $E={\bf p}^2/(2 m_Q)$ make the energy $E\sim m_Q\beta^2$, giving scales $m_Q\beta^2 \ll m_Q\beta\ll m_Q$. The two low energy scales can be distinguished by following Eq. (\[fd1\]) with a further field redefinition [@LMR] $h_v(x) = \sum_{\bf p} e^{i{\bf p\cdot x}}
\psi_{\bf p}(x)$, so that derivatives on $\psi_{\bf p}$ only pick out the $m\beta^2$ scale. The on-shell degrees of freedom are then the heavy quarks, soft quarks and gluons with $p^2\sim (m_Q v)^2$, and ultrasoft quarks and gluons with $p^2\sim (m_Q v^2)^2$.
[****]{}\
For collinear particles the analogous field redefinitions are [@bfps; @cbis] $$\phi(x)=\sum_n \sum_p e^{-ip\cdot x} \phi_{n,p}(x) \,,$$ where the collinear fields $\phi_{n,p}$ are labelled by light-cone vectors $n$ and label momentum $p$. Here $p$ contains the ${{\bar n}}\cdot p\sim Q$ and $p_\perp\sim Q\lambda$ momenta so that $\partial^\mu \phi_{n,p}\sim (Q\lambda^2)
\phi_{n,p}$. The field $\phi_{p}$ can either be a quark or gluon field. Similar to $\psi_{\bf p}$, the missing $\sim Q$ fluctuations are described by Wilson coefficients and the $\sim Q\lambda$ labels simplify the power counting by distinguishing the $Q\lambda$ and $Q\lambda^2$ scales. Now $$\begin{aligned}
\label{phi+-}
\phi_{n,p} \equiv \phi^+_{n,p} + \phi^-_{n,-p} \,,\end{aligned}$$ so collinear particles and antiparticles are contained in the same effective theory field, but have momentum labels with the opposite sign. In the large energy limit the four component fermion spinors contain two large and two small components. One therefore defines collinear quark fields $\xi_{n,p}$ which only retain the large components for motion in the $n$ direction and satisfies $\nslash \xi_{n,p}=0$. For these fields $\xi_{n,p}^+/\xi_{n,p}^-$ destroy/create the particles/antiparticles with large momentum ${{\bar n}}\cdot p>0$ [@cbis]. For collinear gluons $A_{n,q}^{\mu\,\dagger}=A^\mu_{n,-q}$, and $(A^\mu_{n,q})^+$/$(A^\mu_{n,q})^-$ destroy/create gluons with ${{\bar n}}{\!\cdot\!}q>0$.
For simplicity we will ignore quark masses and only consider massless $u$ and $d$ quarks. For the processes considered here SCET then requires three types of degrees of freedom: collinear, soft, and ultrasoft (usoft) fields. These are distinguished by the scaling of the light cone components ($p^+$,$p^-$,$p^\perp$) of their momenta: $(\lambda^2,1,\lambda)$ for collinear modes in the $n$ direction ($A_{n,q}$, $\xi_{n,p}$), $(\lambda,\lambda,\lambda)$ for the soft modes ($A_{q}^s$, $q_p^s$), and $(\lambda^2,\lambda^2,\lambda^2)$ for the usoft modes ($A_{us}$, $q_{us}$). The soft modes are labelled by their order $Q\lambda$ momenta, so $A_{q}^s$ and $q_p^s$ are essentially just momentum space fields. The usoft fields have no labels and depend only on the coordinate $x$. The fields are assigned a scaling with $\lambda$ to make the action for their kinetic terms order $\lambda^0$ [@bfl; @bfps; @bpssoft]. For instance $\xi_{n,p}\sim\lambda$, $A_{n,q}^\mu \sim (\lambda^2,1,\lambda)$, $A^s_q\sim\lambda$, and $A_{us}^\mu \sim \lambda^2$. At leading order only order $\lambda^0$ vertices are necessary to correctly account for all order $\lambda^0$ Feynman diagrams.
In HQET only external currents with momenta of order $m_b$ can change the label $v$. Thus the Lagrangian has a superselection rule forbidding changes in the four-velocity of the heavy quark [@bbook; @georgi]. In NRQCD the $v$ labels are also conserved, but the smaller momentum labels ${\bf p}$ are changed by operators in the effective theory such as the Coulomb potential. A novel feature of SCET is that interactions in the leading action can change both the large and small parts of the momentum labels $p^\mu$. However, only external currents can change the direction $n$ of a collinear particle, so this label is conserved. Thus, for each distinct direction $n$ a separate set of collinear fields are needed. In the remainder of this section we will restrict ourselves to collinear particles with a single $n$. We will generalize the discussion to the case of two back-to-back directions and discuss the factorization of collinear particles with different $n$’s in section \[section\_nnbar\].
Since in SCET interactions can change the order $Q$ label momenta it turns out to be very useful to introduce a label operator, ${\cal P}^\mu$ [@cbis], for which the collinear fields satisfy ${\cal P}^\mu \xi_{n,p} = p^\mu
\xi_{n,p}$. More generally, ${\cal P}^\mu$ acts on a product of labelled fields as $$\begin{aligned}
f({\cal P}^\mu) \Big(\phi^\dagger_{q_1} \cdots \phi^\dagger_{q_m}
\phi_{p_1} \cdots \phi_{p_n}\Big)
= f( p_1^\mu\!+\!\ldots\!+\!p_n^\mu\! -\! q_1^\mu \!-\!\ldots\!-\! q_m^\mu)
\Big(\phi^\dagger_{q_1} \cdots \phi^\dagger_{q_m} \phi_{p_1} \cdots
\phi_{p_n}\Big) \,.\end{aligned}$$ so conjugate field labels come with a minus sign. The operator ${{\cal P}}_\mu$ acts to the right, while the conjugate operator ${{\cal P}}_\mu^\dagger$ acts to the left. As explained in Ref. [@cbis] the label operator allows all large phases to be moved to the front of operators with a factor $\exp({-ix{\!\cdot\!}{{\cal P}}})$. This phase and the label sums can then be suppressed if we impose that interactions conserve label momenta and that the momentum indices on fields are implicitly summed over. Basically, for labels $p$ and $p'$ and residual momenta $k$ and $k'$ $$\begin{aligned}
\label{mcons}
\int d^4x\ e^{i(p'-p+k'-k)\cdot x}
= \delta(p-p') \int d^4x\ e^{i(k'-k)\cdot x} \,,\end{aligned}$$ so that the label and residual momenta are individually conserved. (Although technically the label momenta are discrete we abuse notation and use $\delta(p\!-\!p')$ rather than $\delta_{p,p'}$ because it makes the subscripts easier to read.) For convenience we define the operator ${\bar {\cal P}}$ to pick out only the order $\lambda^0$ labels on collinear fields, and the operator ${{\cal P}}^\mu$ to pick out only the order $\lambda$ labels. For the matrix element of any collinear operator ${\cal O}$, momentum conservation constrains the sum of field labels [@cbis], giving $$\begin{aligned}
\label{MC}
\big\langle M_{n,p_1} \big| \big[ f({\bar {\cal P}})\, {\cal O} \big]
\big| M_{n,p_2}\big\rangle
&=& f({{\bar n}}{\!\cdot\!}(p_2\!-\!p_1))\ \big\langle M_{n,p_1} \big|
{\cal O} \big| M_{n,p_2}\big\rangle \,,\end{aligned}$$ for any function $f$.
For a single $n$ the Lagrangian can be broken up into three sectors: collinear, usoft, and soft. We therefore write $$\begin{aligned}
{\cal L} = {\cal L}_{c,n}[\xi_{n,p},A_{n,q}^\mu,A_{us}^\mu]
+ {\cal L}_{us}[q_{us},A^\mu_{us}]
+ {\cal L}_{s}[q_{s,p}\,, A^\mu_{s,q}] \,, \label{sclag}\end{aligned}$$ where we have made the field content of each sector explicit. We will discuss each of these terms separately.
[****]{}\
As explained in detail in Ref. [@bpssoft], gauge invariance in SCET restricts the Lagrangian and allowed form of operators. Only local gauge transformations whose action is closed on the effective theory fields need to be considered. These include collinear, soft, and usoft transformations. Each of these vary over different distance scales, with collinear gauge transformations satisfying $\partial^\mu
U_n(x) \sim Q(\lambda^2, 1, \lambda)\: U_n(x)$, soft satisfying $\partial^\mu
V_{s}(x) \sim Q\lambda\: V_{s}(x)$, and usoft transformations with $\partial^\mu
V_{us}(x) \sim Q\lambda^2\: V_{us}(x)$. All particles transform under $V_{us}(x)$ and usoft gluons act like background fields for collinear particles. Invariance under $U_n(x)$ requires a collinear Wilson line built out of the order $\lambda^0$ gluon fields [@bfps; @cbis] $$\begin{aligned}
\label{momW}
W_n(x) = \bigg[
\lower7pt\hbox{ $\stackrel{\mbox{\small$\sum$}}{\mbox{\footnotesize perms}}$ }
{\rm exp} \Big( -\!g\, \frac{1}{{\bar {\cal P}}}\, {{\bar n}}{\!\cdot\!}A_{n,q}(x) \Big) \bigg] \,.\end{aligned}$$ Here the operator ${\bar {\cal P}}$ acts only inside the square brackets, the $n$ on $W_n$ refers to the direction of the collinear quanta, and $W_n$ is local with respect to $x$ (corresponding to the residual momenta). Taking the Fourier transform of $\delta(\omega-{\bar {\cal P}}) W_n(0)$ with respect to $\omega$ gives the more familiar path-ordered Wilson line $W_n(y,-\infty)= {\rm P} \; {\rm exp}\big[ ig
\int^y_{-\infty} {\rm d}s\, {{\bar n}}{\!\cdot\!}A_n(s{{\bar n}})\big]$. Under a collinear gauge transformation $W_n$ transforms as $W_n\to U_n W_n$. An invariant under collinear gauge transformations can therefore be formed by combining a collinear fermion $\xi_{n,p}$ and the Wilson line $W_n$ in the form $$\begin{aligned}
\label{invariant}
W_n^\dagger(x)\: \xi_{n,p}(x)\,.\end{aligned}$$ This combination still transforms under an usoft gauge transformation, $W_n^\dagger\,\xi_{n,p}\to V_{us}(x)\: W_n^\dagger\,\xi_{n,p}$. We will often suppress the $x$ dependence of the combination $W_n^\dagger\: \xi_{n,p}$.
Integrating out hard fluctuations gives Wilson coefficients in the effective theory that are functions of the large ${{\bar n}}{\!\cdot\!}p_i$ collinear momenta, $C({{\bar n}}{\!\cdot\!}p_i)$. However, collinear gauge invariance restricts these coefficients to only depend on the linear combination of momenta picked out by the order $\lambda^0$ operator ${\bar {\cal P}}$ [@cbis]. In general the Wilson coefficients are then functions $C({\bar {\cal P}},{\bar {\cal P}}^\dagger)$ which must be inserted between gauge invariant products of collinear fields. In general the Wilson coefficients also depend on the large momentum scales in a process such as $Q$ and the renormalization scale $\mu$.
To construct the collinear Lagrangian one can match full QCD onto operators with collinear fields that are invariant under usoft and collinear gauge transformations. The collinear Lagrangian at order $\lambda^0$ is [@bfps; @cbis; @bpssoft] $$\begin{aligned}
\label{Lc}
{\cal L}_{c,n}
&=& \bar\xi_{n,p'}\: \bigg\{ i\, n{\!\cdot\!}D\!+\! g n{\!\cdot\!}A_{n,q}
+ \Big( \SppP\! + g \Aslash_{n,q}^\perp\Big)\, W\ \frac{1}{{\bar {\cal P}}}\ W^\dagger\,
\Big( \SppP\! + g \Aslash_{n,q'}^\perp\Big) \bigg\}
\frac{\bnslash}{2}\, \xi_{n,p} {\nonumber}\\
&& + \frac{1}{2 g^2}\, {\rm tr}\ \bigg\{
\Big[i{{\cal D}}^\mu +g A_{n,q}^\mu \,, i{{\cal D}}^\nu + g A_{n,q'}^\nu \Big]^{\,2}
\bigg\} + {\cal L}_c^{g.f.}\,,\end{aligned}$$ where ${\cal L}_c^{g.f.}$ are gauge fixing terms, $iD^\mu = i\partial^\mu +
g A^\mu_{us}$, and $$\begin{aligned}
\label{Dc}
i{{\cal D}}^\mu = \frac{n^\mu}{2}\, {\bar {\cal P}}+ {\cal P}_\perp^\mu +
\frac{{{\bar n}}^\mu}{2}\, i\, n{\!\cdot\!}D \,.\end{aligned}$$ Since usoft gluons act as background fields in the collinear gauge transformation the couplings, $g(\mu)$, for both types of gluons must be identical.
[****]{}\
The usoft and soft Lagrangians for gluons and massless quarks are the same as those in QCD. From Eq. (\[sclag\]) we see that collinear quarks and gluons interact with usoft gluons, however at order $\lambda^0$ only the $n{\!\cdot\!}A_{us}$ component appears in Eq. (\[Lc\]). In order to prove factorization formulae it is essential to disentangle the collinear and usoft modes. This can be done by introducing an usoft Wilson line $$\begin{aligned}
\label{Ydef}
Y_n(x) &=& {\rm P} \; {\rm exp}\bigg( ig \int^x_{-\infty}\!\!\! {\rm d}s\
n{\!\cdot\!}A_{us}(s n)\bigg) \,,\end{aligned}$$ where the subscript $n$ on $Y_n$ labels the direction of the Wilson line (we emphasize that this is different from the meaning of the subscript on $W_n$ in Eq. (\[momW\])). An usoft gauge transformation takes $Y_n\to V_{us} Y_n$. In Ref. [@bpssoft] it was shown that the field redefinitions $$\begin{aligned}
\label{def0}
\xi_{n,p} = Y_n\, \xi^{(0)}_{n,p} \,, \qquad\qquad
A^\mu_{n,p} = Y_n\, A^{(0)\mu}_{n,p}\, Y^\dagger_n \,, \end{aligned}$$ imply $W_n = Y_n W^{(0)}_n Y^\dagger_n$ and decouple the usoft gluons from the collinear particles in the leading order Lagrangian $$\begin{aligned}
\label{Lagrdecouple}
{\cal L}_{c,n}[\xi_{n,p},A_{n,q}^\mu,n{\!\cdot\!}A_{us}]
= {\cal L}_{c,n}[\xi_{n,p}^{(0)},A_{n,q}^{(0)\mu},0] \,.\end{aligned}$$ Thus, the new collinear fields with superscript $(0)$ no longer interact with usoft gluons or transform under an usoft gauge transformation. Since the field redefinitions do not change physical $S$ matrix elements, the new fields give an equally valid parameterization of the collinear modes. The leading SCET Lagrangian therefore factors into separate collinear and usoft sectors. This alone does not guarantee factorization in operators and currents, since after the field redefinition these operators may still contain both usoft and collinear fields. However, the field redefinition makes factorization transparent since identities such as $Y_n^\dagger Y_n=1$ may be applied directly to the operators. This will become clear in the examples in sections \[section\_exclusive\] and \[section\_inclusive\].
The coupling of soft gluons to collinear particles differs from the usoft-collinear interactions. Interactions of a soft gluon with a collinear particle results in a particle with momentum $p\sim Q(\lambda,1,\lambda)$, so soft gluons can not appear in the collinear Lagrangian. These offshell particles have $p^2\sim Q^2\lambda$ and since $Q^2\lambda \gg (Q\lambda)^2$ these offshell quarks and gluons can be integrated out. At leading order in $\lambda$ it was shown in Ref. [@bpssoft] that in operators with collinear fields this simply builds up factors of a soft Wilson line $S_n$ involving the $n\cdot A_s$ component of the soft gluon field, $$\begin{aligned}
\label{Sdef}
S_n &=& \bigg[
\lower7pt \hbox{ $\stackrel{\mbox{\small$\sum$}}{\mbox{\small perms}}$ }
\!\! \exp\bigg(-\!g\,\frac{1}{n{\!\cdot\!}{{\cal P}}}\ n{\!\cdot\!}A_{s,q} \bigg) \bigg] \,.\end{aligned}$$ The factors of $S_n$ appear outside gauge invariant products of collinear fields, and their location is restricted by soft gauge invariance.
SCET for $n$ and ${{\bar n}}$ collinear fields {#section_nnbar}
----------------------------------------------
In this section we extend SCET to include the possibility of collinear fields moving in different light-cone directions: $n_1$, $n_2$, $n_3, \dots$. These directions can be considered to be distinct provided that $n_i{\!\cdot\!}n_j\gg
\lambda^2$ for $i\ne j$. This follows from the fact that if $n_1\cdot n_2\sim
\lambda^2$ then the directions $n_1$ and $n_2$ are too close to be distinguished. For example, a momentum $p_2=Q n_2$ can be considered to be collinear in the $n_1$ direction if $n_1\cdot p_2 = Q n_1\cdot n_2 \sim
Q\lambda^2$, since this is the correct scaling for the small momentum component of an $n_1$-collinear particle.
For simplicity we will only consider the case of back-to-back jets corresponding to collinear particles moving in the $n$ and ${{\bar n}}$ directions. These are clearly distinct since $n\cdot{{\bar n}}= 2$. Collinear particles in the ${{\bar n}}$ direction have $(+,-,\perp)$ momenta $\sim Q(1,\lambda^2,\lambda)$, and the $n\cdot p\sim 1$ and $p_\perp\sim \lambda$ momenta appear as labels on the corresponding fields: $\xi_{{{\bar n}},p}$ and $A_{{{\bar n}},p}^\mu$. Emission of a collinear particle moving in the $n$ direction from a collinear particle in the ${{\bar n}}$ direction results in a particle with momentum $k\sim Q(1,1,\lambda)$ and offshellness $k^2\sim
Q^2$. These offshell modes are integrated out to construct the SCET, so collinear modes in the $n$ direction do not directly couple to collinear modes in the ${{\bar n}}$ direction. A distinct set of collinear gauge transformations is associated with each of $n$ and ${{\bar n}}$, and fields in one direction do not transform under the gauge symmetry associated with the opposite direction. Two Wilson lines $W_n(x)$ and $W_{{\bar n}}(x)$ are necessary (defined as in Eq. (\[momW\])), and they appear in a way that makes collinear operators gauge invariant. For instance the combinations $$\begin{aligned}
\label{invariant2}
W_n^\dagger \xi_{n,p}\,, \qquad W_{{\bar n}}^\dagger \xi_{{{\bar n}},p}\end{aligned}$$ are invariant under collinear gauge transformations in the $n$ and ${{\bar n}}$ directions, respectively. We also require two types of label operators, ${\bar {\cal P}}$ as before, and an operator ${{\cal P}}$ to pick out $n{\!\cdot\!}p$ labels that are order $\lambda^0$. Thus, ${\bar {\cal P}}$ and ${{\cal P}}$ act only on the $n$ and ${{\bar n}}$ collinear fields respectively. (The label operator ${{\cal P}}^\mu$ still picks out order $\lambda$ momentum components and therefore acts on both $n$ and ${{\bar n}}$ fields.) With two collinear directions, decoupling usoft gluons requires introducing both $Y_n$ and $Y_{{\bar n}}$ Wilson lines, defined as in Eq. (\[Ydef\]), but along the $n$ or ${{\bar n}}$ paths respectively. Finally, integrating out $\sim Q^2\lambda$ fluctuations at leading order induces both $S_n$ and $S_{{\bar n}}$ soft Wilson lines defined analogous to Eq. (\[Sdef\]). This is discussed in greater detail in Appendix \[app\_fact\] where we show explicitly to all orders in $g$ that integrating out the $Q^2\lambda$ fluctuations causes $$\begin{aligned}
\label{Sadd}
W_n^\dagger \xi_{n,p} \to S_{n} W_n^\dagger \xi_{n,p} \,,\qquad
\bar\xi_{n,p} W_n \to \bar\xi_{n,p} W_n S_{n}^\dagger \,, {\nonumber}\\
W_{{{\bar n}}}^\dagger \xi_{{{\bar n}},p} \to S_{{{\bar n}}} W_{{{\bar n}}}^\dagger \xi_{{{\bar n}},p} \,,\qquad
\bar\xi_{{{\bar n}},p} W_{{{\bar n}}} \to \bar\xi_{{{\bar n}},p} W_{{{\bar n}}} S_{{{\bar n}}}^\dagger \,.\end{aligned}$$ Relations for operators with collinear gluon fields are also derived in Appendix \[app\_fact\].
Note that we have not included “Glauber gluons” with momenta $p^\mu\sim
(\lambda^2,\lambda^2,\lambda)$, which are kinematically allowed in $t$-channel Coulomb exchange between $n$ and ${{\bar n}}$ collinear quarks. In determining the relevant degrees of freedom we have assumed that Glauber gluons are not necessary to describe the infrared for the processes considered in this paper. Intuitively, this can be seen from the fact these gluons are instantaneous in both time and longitudinal separation, and only could contribute when the $n$ and ${{\bar n}}$ jets overlap for a duration of order $1/(Q\lambda^2)$ in a space-time diagram. In processes with a hard interaction the overlap scale is always much shorter than this (however this need not be the case in processes such as forward scattering). For the Drell-Yan process more quantitative arguments can be found in Refs. [@DYrefs; @CSSnew].
At order $\lambda^0$ it is not possible to construct a gauge invariant kinetic Lagrangian with terms that involve both $n$ and ${{\bar n}}$ fields. Thus, the $n$ and ${{\bar n}}$ collinear modes are described by independent Lagrangians (however $n$ and ${{\bar n}}$ modes may still both appear in an external operator). The collinear sector of the SCET Lagrangian is therefore $$\begin{aligned}
{\cal L}_{c,n}[\xi_{n,p},A_{n,q}^\mu,n{\!\cdot\!}A_{us}]
+{\cal L}_{c,{{\bar n}}}[\xi_{{{\bar n}},p},A_{{{\bar n}},q}^\mu,{{\bar n}}{\!\cdot\!}A_{us}] \,.\end{aligned}$$ Making the field redefinitions $$\begin{aligned}
\label{Yfd2}
\xi_{n,p} &=& Y_n\, \xi^{(0)}_{n,p} \,, \qquad
A^\mu_{n,p} = Y_n A^{(0)\mu}_{n,p} Y^\dagger_n \,, \nonumber \\
\xi_{{{\bar n}},p} &=& Y_{{\bar n}}\, \xi^{(0)}_{{{\bar n}},p} \,, \qquad
A^\mu_{{{\bar n}},p} = Y_{{\bar n}}A^{(0)\mu}_{{{\bar n}},p} Y^\dagger_{{\bar n}}\,,\end{aligned}$$ gives $W_n = Y_n W^{(0)}_n Y^\dagger_n$, $W_{{\bar n}}= Y_{{\bar n}}W^{(0)}_{{\bar n}}Y^\dagger_{{\bar n}}$, and usoft degrees of freedom once again decouple from the collinear modes since $$\begin{aligned}
{\cal L}_{c,n}[\xi_{n,p},A_{n,q}^\mu,n{\!\cdot\!}A_{us}] \!+\!
{\cal L}_{c,{{\bar n}}}[\xi_{{{\bar n}},p},A_{{{\bar n}},q}^\mu,{{\bar n}}{\!\cdot\!}A_{us}] \!=\!
{\cal L}_{c,n}[\xi_{n,p}^{(0)},A_{n,q}^{(0)\mu},0] \!+\!
{\cal L}_{c,{{\bar n}}}[\xi_{{{\bar n}},p}^{(0)},A_{{{\bar n}},q}^{(0)\mu},0]. $$ Thus, usoft gluons are removed from the collinear Lagrangian at the expense of inducing $Y_n$ and $Y_{{\bar n}}$ factors in operators with collinear fields. In certain cases the identities $Y_n^\dagger Y_n=1$ and $Y_{{\bar n}}^\dagger Y_{{\bar n}}=1$ can be used in these operators to cancel usoft gluon interactions. Perturbatively these cancellations would occur by adding an infinite set of Feynman diagrams.
To see in more detail how this works consider the simple example of the $\gamma^*$-production of back-to-back collinear states $X_n$ and $X_{{\bar n}}$. The full theory current $\bar\psi(x) \Gamma \psi(x)$ matches onto an effective theory operator $O_{n{{\bar n}}}$. Naively one might guess that the SCET operator mediating this process is $$\begin{aligned}
O_{n{{\bar n}}} = \bar \xi_{n,p_1} \Gamma \xi_{{{\bar n}}, p_2}\,.\end{aligned}$$ However, this operator is not invariant under the collinear gauge transformations ${U}_{n}$ and ${U}_{{{\bar n}}}$, so the process is instead mediated by the invariant operator $$\begin{aligned}
O_{n{{\bar n}}} = \bar \xi_{n,p_1} W_n\, \Gamma\, W^\dagger_{{\bar n}}\, \xi_{{{\bar n}}, p_2}\,.\end{aligned}$$ A hard matching coefficient $C({\bar {\cal P}}, {\bar {\cal P}}^\dagger, {{\cal P}}, {{\cal P}}^\dagger)$ can be inserted in any location in the operator that does not break apart the gauge invariant combinations of fields in Eq. (\[invariant2\]). The operators ${\bar {\cal P}}$ and ${{\cal P}}$ in the coefficient only pick out momenta that are order $\lambda^0$ in the power counting. Thus, ${{\cal P}}$ does not act on fields in the $n$ direction and ${\bar {\cal P}}$ does not act on fields in the ${{\bar n}}$ direction, and the most general result is $$\begin{aligned}
O_{n{{\bar n}}} = \bar \xi_{n,p_1} W_n\, \Gamma\, C({\bar {\cal P}}^\dagger, {{\cal P}})
W^\dagger_{{\bar n}}\, \xi_{{{\bar n}}, p_2}\,.\end{aligned}$$ Next, we integrate out the offshell $Q^2\lambda$ fluctuations which induces additional soft Wilson lines in the operator. This is discussed in detail in Appendix \[app\_fact\] and from Eq. (\[Sadd\]) gives $$\begin{aligned}
\label{OnnWS}
O_{n{{\bar n}}} = \bar \xi_{n,p_1} W_n S_n^\dagger \, \Gamma\,
C({\bar {\cal P}}^\dagger, {{\cal P}}) S_{{\bar n}}\, W^\dagger_{{\bar n}}\, \xi_{{{\bar n}}, p_2}\,.\end{aligned}$$ Note that ${\bar {\cal P}}$ and ${{\cal P}}$ do not act on the fields in the soft Wilson lines since soft gluons carry only order $\lambda$ momenta. Finally, we can make the usoft gluon couplings explicit by switching to the $(0)$ fields using Eq. (\[Yfd2\]): $$\begin{aligned}
\label{exop}
O_{n{{\bar n}}} = \bar \xi^{(0)}_{n,p_1} W^{(0)}_n Y^\dagger_n S_n^\dagger \, \Gamma\,
C({\bar {\cal P}}^\dagger, {{\cal P}})\, S_{{\bar n}}Y_{{\bar n}}W^{(0)\dagger}_{{\bar n}}\xi^{(0)}_{{{\bar n}}, p_2}\,. \end{aligned}$$ This operator is manifestly invariant under collinear gauge transformation in the $n$ and ${{\bar n}}$ direction, as well as under soft and usoft gauge transformations.
To separate the short distance Wilson coefficient from the long-distance operator one introduces convolution variables $\omega$ and $\omega'$ to give $$\begin{aligned}
\label{Onnf}
O_{n{{\bar n}}}&=& \int\!\!{d\omega\, d\omega'} C(\omega,\omega')
O_{n{{\bar n}}}(\omega,\omega') \,,\\
O_{n{{\bar n}}}(\omega,\omega') &=& \Big[ \bar
\xi_{n,p_1}^{(0)} W_n^{(0)} \delta({\bar {\cal P}}^\dagger\!-\!\omega)
Y^\dagger_n S^\dagger_n \, \Gamma\, S_{{\bar n}}Y_{{\bar n}}\delta({{\cal P}}\!-\!\omega')
W^{\dagger(0)}_{{\bar n}}\xi_{{{\bar n}}, p_2}^{(0)} \Big] \,. {\nonumber}\end{aligned}$$ The function $C(\omega,\omega')$ contains all the short distance physics and is determined by matching the full theory onto this effective theory operator. $O_{n{{\bar n}}}(\omega,\omega')$ contains all the infrared long-distance QCD contributions at leading order in $\lambda$.
Now consider the matrix element of the production current between $\langle X_n
X_{{\bar n}}|$ and the vacuum. Taking the $\gamma^*$ to have large time-like momentum $q^\mu=(Q,0,0,0)$ (and zero residual momentum) we have $$\begin{aligned}
\label{qqff}
&& \int\!\! d^4x\, e^{-i q\cdot x}\, \big\langle X_n X_{{\bar n}}\big| \bar\psi(x)
\Gamma \psi(x) \big| 0 \big\rangle = \int\!\! d^4x\, \big\langle X_n X_{{\bar n}}\big| O_{n{{\bar n}}}(x)
\big| 0 \big\rangle {\nonumber}\\
&& \quad = \int\!\! d^4x\, e^{ik\cdot x}\, \big\langle (X_n X_{{\bar n}})(k)
\big| O_{n{{\bar n}}}(x=0) \big| 0 \big\rangle
= \big\langle (X_n X_{{\bar n}})(0)\big| O_{n{{\bar n}}}(0) \big| 0 \big\rangle \,.\end{aligned}$$ In the first step the conservation of the large label momentum $q$ was made implicit in the matrix element (c.f. Eq. (\[mcons\])). Since we are in the center-of-mass frame the $X_n$ has large momentum ${{\bar n}}\cdot p=Q$, and the $X_{{\bar n}}$ has momentum $n\cdot p'=Q$. Now using translation invariance, we see that the remaining $x$ integral forces the $|X_n X_{{\bar n}}\rangle$ state to have zero residual momentum. Using Eq. (\[Onnf\]) this matrix element is equal to $$\begin{aligned}
\int\!\!{d\omega\, d\omega'} C(\omega,\omega') \big\langle X_n\bar X_{{\bar n}}\big|
O_{n{{\bar n}}}(\omega,\omega') \big| 0 \big\rangle
=\int\!\!{d\omega\, d\omega'}\, C(\omega,\omega')\: J_n(\omega)\:
\Gamma\, S_{n{{\bar n}}} \: J_{{\bar n}}(\omega') \,,\ \ \end{aligned}$$ where the functions $C$, $J_n$, $J_{{\bar n}}$, and $S$ also depend on the renormalization point $\mu$. Here we have used the fact that both $X_n$ and $X_{{\bar n}}$ can be described entirely by collinear particles in the $n$ and ${{\bar n}}$ directions respectively. Since the Lagrangians for the collinear, soft, and usoft fields are factorized the remaining matrix element splits into distinct matrix elements for each class of modes. These matrix elements are $$\begin{aligned}
\label{JJSY}
J_n(\omega) &=& \big\langle X_n(Q/2) \big| \bar\xi_{n,p_1}^{(0)} W_n^{(0)}
\delta({\bar {\cal P}}^\dagger\!-\!\omega) \big| 0\rangle \,,{\nonumber}\\
J_{{\bar n}}(\omega') &=& \big\langle X_{{\bar n}}(Q/2) \big| \delta({{\cal P}}\!-\!\omega')
W^{\dagger(0)}_{{\bar n}}\xi_{{{\bar n}}, p_2}^{(0)} \big| 0\rangle \,,{\nonumber}\\
S_{n{{\bar n}}} &=& \big\langle 0 \big | Y_n^\dagger S_n^\dagger S_{{\bar n}}Y_{{\bar n}}\big| 0 \big\rangle \,,\end{aligned}$$ (and are matrices whose color, spin, and flavor indices are suppressed). Note that $J_n$, $J_{{\bar n}}$, and $S_{n{{\bar n}}}$ are explicitly invariant under the collinear, soft, and usoft gauge transformations [@bpssoft] of SCET, but still transform globally under a color rotation. Now using the momentum conservation identity in Eq. (\[MC\]), the large momentum of the $X_n$ and $X_{{\bar n}}$ states set ${{\cal P}}\to Q$ and ${\bar {\cal P}}^\dagger\to Q$. Label conservation also implies that the total perpendicular momentum of each of $J_n$, $J_{{\bar n}}$, and $S_{n{{\bar n}}}$ is zero. The sum over $\omega$ and $\omega'$ can then be performed to give the final factorized form $$\begin{aligned}
\label{jjrslt}
C(Q,Q)\: J_n(Q)\: \Gamma\, S_{n{{\bar n}}} \: J_{{\bar n}}(Q) \,.\end{aligned}$$
Although rather idealized, the above example illustrates the main steps needed to derive a factorization formula. Taking $X_n$ and $X_{{\bar n}}$ to be single quark states the result in Eq. (\[jjrslt\]) also agrees with the factorization formula for the onshell production form factor for $q\bar q$ [@Korchemskyff; @CollinsP].[^3] In the above example the factors of $S$ and $Y$ in the operator in Eq. (\[exop\]) do not cancel. In the examples we will consider in sections \[section\_exclusive\] and \[section\_inclusive\] there are several operators at leading order, however the factors of $S$ and $Y$ cancel in observable matrix elements. The collinear matrix elements of long-distance operators, such as those in Eq. (\[JJSY\]), are the ones that have interpretations as structure functions or wave functions. In the next section we give the operator definitions for these functions that will be needed in the remainder of the paper.
Non-perturbative Matrix Elements {#section_melt}
--------------------------------
Predictions for hadronic processes depend on universal matrix elements that are not computable in perturbation theory. For exclusive processes these include light-cone wavefunctions and form factors, while for inclusive processes they include parton distribution functions and fragmentation functions. In this section we define matrix elements in SCET that are needed for our examples. All the collinear operators considered here decouple from usoft gluons since they are local in the residual coordinate $x$ and because $Y^\dagger(x) Y(x)=1$. Thus $$\begin{aligned}
\label{decoup}
\bar\xi_{n,p'} W\Gamma W^\dagger \xi_{n,p} =
\bar\xi_{n,p'}^{(0)} W^{(0)}\Gamma W^{(0)\dagger} \xi_{n,p}^{(0)}\,,\end{aligned}$$ and expressions with and without the $(0)$ superscript are equal. For convenience we will write the collinear fields without the superscript in the remainder of this section.
Consider first the light-cone wavefunctions. For the pion isotriplet $\pi^a$, the wavefunction $\phi_{\pi}(x)$ is conventionally defined by [@brodskylepage] $$\begin{aligned}
\label{piwfn_rest}
\big\langle \pi^a(p)\big|\, \bar{\psi}(y)\gamma^\mu \gamma^5
\frac{\tau^b}{\sqrt{2}} \,Y(y,x)\, \psi(x)\, \big| 0 \big\rangle
= -i \OMIT{\sqrt{2}\,} f_\pi \delta^{ab} p^\mu\! \int_0^1\!\! dz\,
e^{i[z p\cdot y+(1-z)p \cdot x]}\, \phi_\pi(\mu,z),\end{aligned}$$ Here $f_\pi\simeq 131\,{\rm MeV}$ and the QCD field $\psi$ denotes the isospin doublet $\{\psi^{(u)}, \psi^{(d)}\}$. The coordinates satisfy $(y-x)^2=0$, which ensures that the path from $y^\mu$ to $x^\mu$ is along the light-cone, and $Y(y,x)$ is a Wilson line along this path. In SCET we require the matrix elements of highly energetic pions, which therefore have collinear constituents. Boosting the matrix element in Eq. (\[piwfn\_rest\]) and letting $y^\mu=y {{\bar n}}^\mu$, $x^\mu=x {{\bar n}}^\mu$ we have $$\begin{aligned}
\label{piwfn_collin}
\big\langle \pi^a_{n,p}\big|\, \bar{\xi}_{n,y}\,
\Gamma_\pi^b\, W(y,x)\, \xi_{n,x}\, \big| 0 \big\rangle
= -if_\pi\, {{{\bar n}}{\!\cdot\!}p}\,\delta^{ab} \,\int_0^1\!\! dz\,
e^{i {{\bar n}}\cdot p [z y+(1-z) x]}\, \phi_\pi(\mu,z),\end{aligned}$$ where $\Gamma_\pi^b={\bnslash\gamma_5\tau^b}/{\sqrt{2}}$, and $\xi_{n,z}$ is a collinear field with position space label $z{{\bar n}}^\mu$. For our purposes it is more useful to use the operator with momentum space labels [@bps] $$\begin{aligned}
\label{pimom}
\big\langle \pi_{n,p}^a \big| \bar{\xi}_{n,p_1} W \Gamma_\pi^b
\delta(\omega\!-\!{\bar {\cal P}}_+) W^\dagger \xi_{n,p_2}\big| 0
\big\rangle
&=&\int \frac{dy}{2 \pi}\, e^{-i\omega y} \big\langle \pi_{n,p}^a \big|
\bar{\xi}_{n,y} \Gamma_\pi^b W(y,-y) \xi_{n,-y} \big| 0
\big\rangle \\
&=&\!\! -{i}f_\pi\, {{{\bar n}}{\!\cdot\!}p}\,\delta^{ab}\!\!
\int_0^1\!\! dz\,
\delta[\omega\!-\!(2z\!-\!1){{\bar n}}{\!\cdot\!}p]\, \phi_\pi(\mu,z)\,, {\nonumber}\end{aligned}$$ where ${\bar {\cal P}}_\pm={\bar {\cal P}}^\dagger\pm{\bar {\cal P}}$. In Eq. (\[pimom\]) the delta function fixes $\omega$ to the sum of labels picked out by the ${\bar {\cal P}}_+$ operator. The combination picked out by ${\bar {\cal P}}_-$ is equivalent to ($-{\bar {\cal P}}$) acting on the entire operator, and using Eq. (\[MC\]) is fixed to the ${{\bar n}}{\!\cdot\!}p$ momentum of the pion state.
In some situations it is convenient to have delta functions which fix the labels of both $W^\dagger\xi_{n,p}$ and $\bar\xi_{n,p}W$. In this case a useful field is $$\begin{aligned}
\label{CH1}
{\chi_{n,\omega}}^{\,(i)} &\equiv& \big[ \delta(\omega-{\bar {\cal P}})\,
W_n^\dagger\xi_{n,p}^{(i)} \big] \,.\end{aligned}$$ Here $i$ is the flavor index and will be omitted if the flavor doublet field is implied. Note that unlike the $p$ in $\xi_{n,p}$ the label $\omega$ on ${\chi_{n,\omega}^{}}$ is not summed over. A matrix element with $\chi_{n,\omega}$ fields is related to a matrix element like the one in Eq. (\[pimom\]) through $$\begin{aligned}
\label{pimom2}
\big\langle M_{n,p} \big| {\overline\chi_{n,\omega}}\Gamma
{\chi_{n,\omega'}} \big| M_{n,p'} \big\rangle
&=& 2\, \delta(\omega_-\!-\!{{\bar n}}{\!\cdot\!}p_-)\, \big\langle M_{n,p} \big|
\bar{\xi}_{n,p_1}W \Gamma \delta(\omega_+\!-\!{\bar {\cal P}}_+)
W^\dagger \xi_{n,p_2}\big| M_{n,p'} \big\rangle \,,{\nonumber}\\\end{aligned}$$ where $\omega_\pm = \omega\pm\omega'$ and $p_-=p-p'$. Thus, with the $\chi$ notation the momentum conserving delta functions become explicit. The factor of two appears from treating the $\omega$’s as continuous variables, and in the final results cancels with a factor of $1/2$ from a Jacobian.
For inclusive processes such as DIS it is the proton parton distribution functions for quarks of flavor $i$, $f_{i/p}(z)$, antiquarks $\bar f_{i/p}(z)$, and gluons, $f_{g/p}(z)$ which are needed. The standard coordinate space definitions [@Soper] are ($y^\mu=y {{\bar n}}^\mu$) $$\begin{aligned}
f_{i/p}(z) &=& \int\!\!\frac{dy}{2\pi}\: e^{-i\, 2z{{\bar n}}\cdot p\, y}
\big\langle p \big| \,\bar\psi^{(i)}(y) Y(y,-y) \,{\bnslash}\,
\psi^{(i)}(-y)\big| p \big\rangle \Big|_{\mbox{\footnotesize spin avg.}} \,,\\
f_{g/p}(z) &=& \frac{2}{z\: {{\bar n}}{\!\cdot\!}p} \int\!\!\frac{dy}{2\pi}\:
e^{-i\, 2z{{\bar n}}\cdot p\, y}\, {{\bar n}}_\mu{{\bar n}}^\nu
\big\langle p \big| \,G_a^{\mu\lambda}(y) Y^{ab}(y,-y)
G^b_{\lambda\nu}(-y)\big| p \big\rangle\Big|_{\mbox{\footnotesize spin avg.}}
\,, {\nonumber}\end{aligned}$$ and $\bar f_{i/p}(z)=-f_{i/p}(-z)$. Here $G^a_{\mu\lambda}(y)$ is the gluon field strength, $Y(y,-y)$ and $Y^{ab}(y,-y)$ are path-ordered Wilson lines in the fundamental and adjoint representations, and $|p\rangle$ is the proton state with momentum $p$. In SCET these distribution functions can be defined by the matrix elements of collinear fields with collinear proton states $$\begin{aligned}
\label{pdfqg}
\frac12\sum_{\rm spin}
\big\langle p_{n} \big| {\overline\chi_{n,\omega}}^{\,(i)} \:{\bnslash}\:
{\chi_{n,\omega'}}^{\,(i)} \big| p_{n} \big\rangle
&=& 4{{\bar n}}{\!\cdot\!}p \int_0^1\!\!\! dz \,\delta(\omega_-)
\delta(\omega_+ \!-\! 2z \, {{\bar n}}{\!\cdot\!}p)\, f_{i/p}(z) \\
&-& 4{{\bar n}}{\!\cdot\!}p \int_0^1\!\!\! dz \,\delta(\omega_-)
\delta(\omega_+ \!+\! 2z \, {{\bar n}}{\!\cdot\!}p)\, \bar f_{i/p}(z)\,, {\nonumber}\\
\frac12\sum_{\rm spin} \big\langle p_{n} \big| {\rm Tr}\:\big[
B^\mu_{n,\omega} B_\mu^{n,\omega'} \big] \big| p_{n} \big\rangle \!
&=& -\frac{\omega_+ \,{{\bar n}}{\!\cdot\!}p}{2} \int_0^1\!\!\! dz
\,\delta(\omega_-)\delta(\omega_+ \!-\!2 z \, {{\bar n}}{\!\cdot\!}p) f_{g/p}(z) {\nonumber},\end{aligned}$$ where $\omega_\pm=\omega\pm\omega'$, and $B^\mu_{n,\omega}\equiv{{\bar n}}_\nu ({\cal
G}_{n,\omega})^{\nu\mu}$ with the collinear gauge invariant field strength $$\begin{aligned}
\label{covG}
({\cal G}_{n,\omega})^{\mu\nu} = -\frac{i}{g}\Big[ \delta(\omega-{\bar {\cal P}})
W^\dagger [i{\cal D}_n^\mu + gA_{n,q}^\mu, i{\cal D}_n^\nu+gA_{n,q'}^\nu ] W
\Big] \,.\end{aligned}$$ Both operators in Eq. (\[pdfqg\]) are order $\lambda^2$ since $\xi_{n,\omega}\sim B^{\perp}_{n,\omega} \sim \lambda$. Note that the matrix element of a single operator (${\overline\chi_{n,\omega}}^{\,(i)} \:{\bnslash}\:
{\chi_{n,\omega'}}^{\,(i)}$) contains both the quark and antiquark distributions. This is due to Eq. (\[phi+-\]), from which we see that for $\omega=\omega'>0$ ($\omega=\omega'<0$) this operator reduces to the number operator for collinear quarks (antiquarks) with momentum $\omega$.
Processes other than DIS sometimes depend on more complicated distribution functions. In deeply virtual Compton scattering (DVCS) we will need to parameterize the matrix element of an operator between proton states with different momenta. In terms of QCD fields $\psi$ the nonforward parton distribution function (NFPDF) defined by Radyushkin in Eq.(4.1) of Ref. [@Radyushkin:1997ki] are (up to a trivial translation) $$\begin{aligned}
\label{asyqpdf}
& & \big\langle p^{\prime}, \sigma^\prime \big|
\bar\psi^{(i)}(y) Y(y,-y) \,{\bnslash}\, \psi^{(i)}(-y)
\big| p,\sigma \big\rangle \\
&&\qquad = \, e(\sigma^\prime,\sigma)
\int^1_0 dz \big[ e^{i{{\bar n}}\cdot p(2z-\zeta)y}\: {\cal F}^{(i)}_\zeta(z;t)
-e^{-i{{\bar n}}\cdot p(2z-\zeta)y}\: \overline{{\cal F}}^{(i)}_\zeta(z;t) \big]{\nonumber}\\
&& \qquad\ + \, h(\sigma^\prime,\sigma)
\int^1_0 dz \big[ e^{i{{\bar n}}\cdot p(2z-\zeta)y}\: {\cal K}^{(i)}_\zeta(z;t)
- e^{-i{{\bar n}}\cdot p(2z-\zeta)y}\: \overline{{\cal K}}^{(i)}_\zeta(z;t)
\big] \,,{\nonumber}\end{aligned}$$ where $t = (p-p^\prime)^2$, and $\zeta = 1-{\bar n \!\cdot\! p}^\prime/{\bar n \!\cdot\! p}$. Here $e(\sigma,\sigma^\prime)$ and $h(\sigma,\sigma^\prime)$ are matrix elements which respectively preserve or flip the proton spin. They are defined in terms of the proton spinors $$\begin{aligned}
e(\sigma^\prime,\sigma) &=&
\bar{u}(p',\sigma')\: {\bnslash}\: u(p,\sigma)
\,,\qquad
h(\sigma^\prime,\sigma) = \frac{1}{2m_p}\,
\bar{u}(p',\sigma')\: {[\bnslash, \pslash\!-\!\pslash']}\:
u(p,\sigma) \,, \end{aligned}$$ where $m_p$ is the proton mass. The NFPDF for gluons is similarly given by [@Radyushkin:1997ki] $$\begin{aligned}
& & {{\bar n}}_\mu {{\bar n}}^\nu \big\langle p^{\prime}, \sigma^\prime \big|
\,G_a^{\mu\lambda}(y) Y^{ab}(y,-y) G^b_{\lambda\nu}(-y)
\big| p,\sigma \big\rangle \\
&& \qquad\qquad = \frac{{{\bar n}}{\!\cdot\!}p}{2}\: e(\sigma^\prime,\sigma)
\int^1_0 dz \left[ e^{i{{\bar n}}\cdot p(2z-\zeta)y} + e^{-i{{\bar n}}\cdot p(2z-\zeta)y}
\right] {\cal F}^{g}_\zeta(z;t) \nonumber \\
&& \qquad\qquad \ + \frac{{{\bar n}}{\!\cdot\!}p}{2}\: h(\sigma^\prime,\sigma)
\int^1_0 dz \left[ e^{i{{\bar n}}\cdot p(2z-\zeta)y}
+ e^{-i{{\bar n}}\cdot p(2z-\zeta)y}\right] {\cal K}^{g}_\zeta(z;t) \,. {\nonumber}\end{aligned}$$ In SCET the definition of the NFPDFs in terms of collinear fields are $$\begin{aligned}
\label{nfpdfqg}
& & \big\langle p_n^\prime,\sigma^\prime \big|
{\overline\chi_{n,\omega}}^{\,(i)} \:{\bnslash}\: {\chi_{n,\omega'}}^{\,(i)}
\big| p_n, \sigma \big\rangle = 2\delta(\omega_-+{{\bar n}}{\!\cdot\!}p\,\zeta)
\int_0^1\!\!\! dz \, \\
&& \qquad\quad
\times\Big\{
e(\sigma',\sigma) \big[
\delta(\omega_+ \!-\! (2z\!-\!\zeta) \, {{\bar n}}{\!\cdot\!}p)\,
{\cal F}^{(i)}_\zeta(z;t) - \delta(\omega_+ \!+\!
(2z\!-\!\zeta) \, {{\bar n}}{\!\cdot\!}p)\, \overline{{\cal F}}^{(i)}_\zeta(z;t)
\big]
\nonumber \\
&& \qquad\quad
+ h(\sigma^\prime,\sigma) \big[
\delta(\omega_+ \!-\! (2z\!-\!\zeta) \, {{\bar n}}{\!\cdot\!}p)\, {\cal
K}^{(i)}_\zeta(z;t)- \delta(\omega_+ \!+\! (2z\!-\!\zeta) \,
{{\bar n}}{\!\cdot\!}p)\, \overline{{\cal K}}^{(i)}_\zeta(z;t) \big] \Big\}
\,, {\nonumber}\\
&& \big\langle p_{n}',\sigma' \big| {\rm Tr}\:\big[
B^\mu_{n,\omega} B_\mu^{n,\omega'} \big] \big| p_{n},\sigma \big\rangle \! =
-\frac{{{\bar n}}{\!\cdot\!}p}{2}\: \delta(\omega_-+{{\bar n}}{\!\cdot\!}p\, \zeta)
\int_0^1\!\!\! dz \nonumber \\
&& \qquad\quad
\times \Big\{ e(\sigma^\prime,\sigma)
\,\delta(\omega_+ \!-\!(2z\!-\!\zeta)
\, {{\bar n}}{\!\cdot\!}p) {\cal F}^{g}_\zeta(z;t)
+ h(\sigma^\prime,\sigma) \delta(\omega_+ \!-\!(2z\!-\!\zeta)\,{{\bar n}}{\!\cdot\!}p)
{\cal K}^{g}_\zeta(z,t) \Big\} \,,{\nonumber}\end{aligned}$$ where the spinors in $e(\sigma',\sigma)$ and $h(\sigma',\sigma)$ are two component effective theory spinors, so that $u\to u_n$ where $\nslash u_n=0$. Note that for $p^\prime \to p$ both $\zeta \to 0$ and $ t \to 0$. In this limit the NFPDFs reduce to the standard PDFs: $$\begin{aligned}
\lim_{p \to p^\prime} {\cal F}^{(i)}_\zeta &=& f_{i/p}(z)
\,,\qquad
\lim_{p \to p^\prime} \overline{{\cal F}}^{(i)}_\zeta =
\bar{f}_{i/p}(z)
\,, \qquad
\lim_{p \to p^\prime} {\cal F}^{(g)}_\zeta = z\, f_{g/p}(z) \,.\end{aligned}$$
Symmetries for collinear fields {#section_symm}
-------------------------------
In this section we discuss spin and discrete symmetry constraints on operators involving collinear fields.
The possible spin structures of currents with $\xi_{n,p}$ fields is restricted by the fact that they have only two components, $\nslash \xi_{n,p}=0$. The four most general spin structures for currents with two collinear particles moving in the same or opposite directions are $$\begin{aligned}
\label{Jspin}
&& \bar\xi_{n,p'}\,\Gamma_1\, \, \xi_{n,p}\qquad\quad
\Gamma_1 = \big\{ \bnslash\,, \bnslash\gamma_5\,,
\bnslash\gamma^\mu_\perp \big\}\,, {\nonumber}\\
&& \bar\xi_{{{\bar n}},p'} \,\Gamma_2\, \, \xi_{n,p}\qquad\quad
\Gamma_2 = \big\{ 1 \,, \gamma_5 \,, \gamma^\mu_\perp \big\}\,.\end{aligned}$$ Other choices for $\Gamma_{1}$ and $\Gamma_2$ either vanish between the fields or are related to those in Eq. (\[Jspin\]). This result can be expressed in a compact way by the trace formulae $$\begin{aligned}
\label{tracefomulae}
&& \bar\xi_{n,p'}\,\Gamma\, \xi_{n,p}\,,\qquad\quad
\Gamma = \frac{\bnslash}{8} {\rm Tr}[\nslash \Gamma]
-\frac{\bnslash\gamma_5}{8} {\rm Tr}[\nslash\gamma_5 \Gamma]
-\frac{\bnslash\gamma_\perp^\mu}{8} {\rm Tr}[\nslash\gamma_\mu^\perp
\Gamma] \,,{\nonumber}\\
&& \bar\xi_{{{\bar n}},p'} \,\Gamma \, \xi_{n,p}\,,\qquad\quad
\Gamma = \frac{1}{8} {\rm Tr}[\nslash\bnslash \Gamma]
+\frac{\gamma_5}{8} {\rm Tr}[\gamma_5\nslash\bnslash \Gamma]
+\frac{\gamma_\perp^\mu}{8} {\rm Tr}[\gamma_\mu^\perp\nslash\bnslash
\Gamma] \,,\end{aligned}$$ which reduce a general $\Gamma$ to a linear combination of the terms in Eq. (\[Jspin\]). For instance, it implies that $2i\bar\xi_{n} \sigma^{\mu\nu}
\xi_n =n^\nu \bar\xi_{n} \bnslash\gamma_\perp^\mu\xi_n -n^\mu \bar\xi_{n}
\bnslash \gamma_\perp^\nu \xi_n$, and $\bar\xi_{{{\bar n}}} \gamma_\perp^\mu
\gamma_5\xi_n= i\epsilon_\perp^{\mu\nu} \bar\xi_{{{\bar n}}} \gamma^\perp_\nu \xi_n$ where $\epsilon_\perp^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta} {{\bar n}}_\alpha
n_\beta/2$. Furthermore, each of the two components of $\xi_n$ and also $\xi_{{{\bar n}}}$ can be chosen to be eigenstates of their helicity operators, $h={\hat p}\cdot{\vec S}$ with eigenvalues $\pm 1/2$. For these fields $h$ is equivalent to the chiral rotation, $h=\gamma_5/2$. The structures in Eq. (\[Jspin\]) split into two classes depending on whether they conserve or flip the helicity $$\begin{aligned}
\mbox{chiral even:}&& \mbox{\hspace{0.8cm}}
\bar\xi_{n,p'} \{\bnslash,\bnslash\gamma_5\} \xi_{n,p}\,,\qquad\,
\bar\xi_{{{\bar n}},p'} \gamma_\perp^\mu \xi_{n,p} \,, \\
\mbox{chiral odd:}&& \mbox{\hspace{0.8cm}}
\bar\xi_{n,p'} \bnslash \gamma_\perp^\mu\xi_{n,p}\,, \qquad\qquad
\bar\xi_{{{\bar n}},p'} \{1,\gamma_5\} \xi_{n,p}\,.{\nonumber}\end{aligned}$$ Since gluon interactions in QCD preserve helicity, integrating out hard QCD fluctuations results in effective theory operators with the same helicity structure as the original operators at leading order in $\lambda$.
The presence of labels on the effective theory fields makes their transformation properties under the discrete symmetries $C$, $P$, and $T$ slightly different than in QCD. For example under $P$ or $T$ we have $n\leftrightarrow {{\bar n}}$ so these transformations relate collinear fields for different directions. Under charge conjugation, parity, and time-reversal the collinear fields transform as $$\begin{aligned}
\label{CPT}
\begin{array}{llll}
& C^{-1} \xi_{n,p}(x) C = - \big[ \bar \xi_{n,-p}(x)\,{\cal C} \big]^T\,,
&& C^{-1} A^\mu_{n,p}(x) C = - [A_{n,p}^\mu(x)]^T \,, \\[5pt]
& P^{-1} \xi_{n,p}(x) P = \gamma_0\, \xi_{{{\bar n}},\tilde p}(x_P)\,,
&& P^{-1} A^\mu_{n,p}(x) P = g_{\mu\nu}A_{{{\bar n}},\tilde p}^\nu(x_P) \,,\\[5pt]
& T^{-1} \xi_{n,p}(x) T = {\cal T} \xi_{{{\bar n}},\tilde p}(x_T)\,,
&& T^{-1} A^\mu_{n,p}(x) T = g_{\mu\nu} A_{{{\bar n}},\tilde p}^\nu(x_T) \,,
\end{array}\end{aligned}$$ where ${\cal C}^{-1}\gamma_\mu{\cal C}=-\gamma_\mu^T$ and ${\cal
T}=\gamma^5{\cal C}$, while if $x^\mu=(x^+,x^-,x^\perp)$ and $p^\mu=(p^+,p^-,p^\perp)$ then $\tilde p^\mu\equiv (p^-,p^+,-p_\perp)$, $x_P^\mu\equiv (x^-,x^+,-\vec x^\perp)$, and $x_T^\mu\equiv (-x^-,-x^+,\vec
x^\perp)$. The transformation properties of $W_n$ can be worked out using Eq. (\[CPT\]), for instance $C^{-1} W_n C = [W_n^\dagger]^T$.
The collinear effective Lagrangian (\[Lc\]) is invariant under the transformations in Eq. (\[CPT\]) (adding the ${{\bar n}}\leftrightarrow n$ terms). These symmetries also constrain the form of non-perturbative matrix elements. As an example, for a meson which is an eigenstate of $C$ one finds $$\begin{aligned}
\label{PSunderC}
&& \big\langle M_{n}\big| \bar{\xi}_{n,p} W_n \bnslash\gamma_5
\delta(\omega-{\bar {\cal P}}_+ ) W_n^\dagger \xi_{n,p'}\big| 0 \big\rangle {\nonumber}\\
&&\qquad\qquad\quad
= (-1)^C \big\langle M_{n}\big| ({\cal C}W_n^\dagger\xi_{n,p})^T
\bnslash\gamma_5 \delta(\omega-{\bar {\cal P}}_+) (\bar\xi_{n,p'} W_n{\cal C})^T
\big| 0 \big\rangle\nonumber\\
&&\qquad\qquad\quad
= (-1)^C \big\langle M_{n}\big| \bar{\xi}_{n,p'} W_n
\bnslash\gamma_5 \delta(\omega+ {\bar {\cal P}}_+ ) W_n^\dagger \xi_{n,p} \big| 0
\big\rangle\,.\end{aligned}$$ For the isotriplet pion state $(-1)^C=+1$ so combining Eq. (\[PSunderC\]) with Eq. (\[pimom\]) gives $$\begin{aligned}
\label{piC}
&& \big\langle \pi^a_{n,p}\big| \bar{\xi}_{n,p_1} W \Gamma_\pi^b
\delta(\omega-{\bar {\cal P}}_+ ) W^\dagger \xi_{n,p_2} \big| 0
\big\rangle {\nonumber}\\
&&\qquad\qquad
= -if_\pi \delta^{ab}\, {{\bar n}}{\!\cdot\!}p \int_0^1\!\! \mbox{d}x\,
\delta[-\omega \!-\! (2x\!-\!1){{\bar n}}{\!\cdot\!}p ] \phi_\pi(x){\nonumber}\\
&&\qquad\qquad
=-if_\pi\delta^{ab}\, {{\bar n}}{\!\cdot\!}p \int_0^1\!\! \mbox{d}x\,
\delta[\omega \!-\! (2x\!-\!1){{\bar n}}{\!\cdot\!}p ] \phi_\pi(1\!-\!x) \,.\end{aligned}$$ Together with Eq. (\[pimom\]), charge conjugation therefore implies that $\phi_\pi(1-x)=\phi_\pi(x)$.
Exclusive Processes {#section_exclusive}
===================
$\pi$-$\gamma$ Form Factor {#section_pionphoton}
--------------------------
The pion-photon form factor $F_{\pi\gamma}(Q^2)$ is perhaps the simplest setting for factorization since there is only one hadron in the external state. The form factor is measurable in single-tagged two photon $e^-e^-\rightarrow e^-e^-
\pi^0$ reactions. This process involves the scattering of a highly virtual photon and a quark-anti-quark constituent pair off an on-shell photon. The photon scatters the quark pair away from the incoming photon into a pion, so that $\gamma^*\gamma\to \pi^0$. The matrix element for this transition defines the $\pi$-$\gamma$ form factor $$\begin{aligned}
\label{Fdef}
\big\langle \pi^0(p_\pi) \big| J_\mu(0) \big| \gamma(p_\gamma,\epsilon)
\big\rangle
&=& ie\, \epsilon^\nu \int\!\! d^4z\, e^{-ip_\gamma\cdot z} \big\langle
\pi^0(p_\pi) \big| {\rm T} J_\mu(0) J_\nu(z) \big| 0
\big\rangle{\nonumber}\\
&=&-ie\, F_{\pi\gamma}(Q^2)\epsilon_{\mu \nu \rho \sigma}p_\pi^\nu
\epsilon^\rho\, q^\sigma \,.\end{aligned}$$ Here $J_\mu= \bar\psi \cal{\hat Q} \gamma_\mu\psi$ is the full theory electromagnetic current with isodoublet field $\psi$ and charge matrix $\hat
{\cal Q}= {\tau_3}/2+{\bf 1}/6$, and $-q^2=Q^2\gg \Lambda_{\rm QCD}^2$ where $q=p_\pi-p_\gamma$ is the virtual photon momentum. It has been shown that the form factor can be written as a one-dimensional convolution of a hard coefficient with the light-cone pion wavefunction [@Pink]. Here we show how this factorization takes place in the SCET.
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In the Breit frame $q^\mu=Q(n^\mu-{{\bar n}}^\mu)/2$, the real photon’s momentum is $p_\gamma^\mu=E {{\bar n}}^\mu \simeq Q {{\bar n}}^\mu/2$, and the pion is made up of collinear particles with momenta ${{\bar n}}{\!\cdot\!}p_i \simeq Q$. The particles exchanged between the two currents in Eq. (\[Fdef\]) have hard momenta and can be integrated out. At leading order in $\lambda$ the time ordered product of the two currents in Eq. (\[Fdef\]) matches onto a single operator in the effective theory. For simplicity we restrict ourselves to the tensor and spin structures that are relevant when the meson is a pion [^4] $$\begin{aligned}
\label{Opig}
O &=& \frac{i}{Q}\,\epsilon_{\mu\nu}^\perp\ [\bar{\xi}_{n,p}W]\,\Gamma
C({\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} }, \mu)\, [W^\dagger \xi_{n,p^\prime}] \,,\end{aligned}$$ where $\xi_{n,p}$ is an isodoublet collinear quark field, and $2\epsilon_{\mu\nu}^\perp = \epsilon_{\mu\nu\rho\beta} {{\bar n}}^\rho n^\beta$. $
O_{1}$ is of dimension two, just like the time-ordered product in Eq. (\[Fdef\]), and a power of $1/Q$ is included to make $C(\mu,{\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} })$ dimensionless. The time-ordered product in Eq. (\[Fdef\]) is even under charge conjugation, so the operators in Eq. (\[Opig\]) must also be even. This implies $C_{\pi\gamma}(\mu,{\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} }) = C_{\pi\gamma}(\mu,-{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} },-{\bar {\cal P}})$. The location of the $W$’s in Eq. (\[Opig\]) is fixed by gauge invariance, and $\Gamma$ contains the spin and flavor structure $$\begin{aligned}
\label{Gpig}
\Gamma &=& \big( {\bnslash}\gamma_5\big) \big(3\sqrt{2}\,\hat {\cal Q}^2\big)
\,.\end{aligned}$$
Since the offshellness of the collinear particles in the pion is $p^2\sim
\Lambda_{\rm QCD}^2$ we can also integrate out offshell modes with $p^2\sim
Q\Lambda_{\rm QCD}$ which come from soft-collinear interactions. For the collinear operators $O_j$, Eq. (\[Sadd\]) implies that factors of the soft Wilson line $S_n$ are induced. However, the location is such that $S^\dagger_n
S_n=1$, so no coupling to soft gluons occur at leading order. The coupling of the collinear fields to usoft gluons can be simplified with the field redefinitions in Eq. (\[def0\]). As discussed in section \[section\_SCET\] this moves all couplings into $O_j$, and using $Y_n^\dagger Y_n=1$ gives $$\begin{aligned}
O &=& \frac{i}{Q}\,\epsilon_{\mu\nu}^\perp\ [\bar{\xi}_{n,p}^{(0)}W^{(0)}]
\,\Gamma\, C({\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} }, \mu)\, [W^{(0)\dagger} \xi_{n,p^\prime}^{(0)}]
\,.\end{aligned}$$ Thus, usoft gluons also decouple.
In the Breit frame the pion momentum satisfies $p^\mu_{\pi}= E_\pi n^\mu+{\cal
O}(\lambda)$, and comparing Eq. (\[Fdef\]) with the SCET matrix element $i\langle \pi^0_{n,p_\pi} | O_{\pi\gamma} |0\rangle$, gives $$\begin{aligned}
\label{pig2}
\frac{Q^2}{2} F_{\pi\gamma}(Q^2)
&=& \frac{i}{Q}\, \big\langle \pi^0_{n}\big|\bar{\xi}^{(0)}_{n,p}
W^{(0)}\,\Gamma\, C({\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} },\mu)\, W^{(0)\dagger}
\xi^{(0)}_{n,p^\prime} \big| 0 \big\rangle \,.\end{aligned}$$ Defining ${\bar {\cal P}}_{\pm}={\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} }\pm \bar{\cal P}$, the operator ${\bar {\cal P}}_-$ is related to ${\bar {\cal P}}$ acting from the outside on the fields. Using Eq. (\[MC\]) it can therefore be set equal to the momentum label of the state, ${\bar {\cal P}}_- ={{\bar n}}\cdot
p_\pi = Q$. Suppressing this dependence we write $C({\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} },\mu)=C_1({\bar {\cal P}}_+,\mu)$ leaving $$\begin{aligned}
\label{pig3}
F_{\pi\gamma}(Q^2) &=& \frac{2i}{Q^3} \big\langle \pi^0_{n} \big| \,
\bar{\xi}_{n,p}^{(0)} W^{(0)} \Gamma \, C_1({\bar {\cal P}}_+,\mu) W^{(0)\dagger}
\xi_{n,p\prime}^{(0)} \, \big| 0 \big\rangle {\nonumber}\\[3pt]
&=& \frac{2i}{Q^3} \int \!\!{d\omega}\,C_1(\omega,\mu)
\big\langle \pi^0_{n} \big| \bar{\xi}_{n,p}^{(0)} W^{(0)}
\Gamma \delta(\omega-{\bar {\cal P}}_+) W^{(0)\dagger} \xi^{(0)}_{n,p\prime}
\big| 0 \big\rangle \,.\end{aligned}$$ Using Eq. (\[pimom\]) the remaining matrix identity in Eq. (\[pig3\]) can be written in terms of the light-cone pion wavefunction $$\begin{aligned}
\label{resultpionformfactor}
F_{\pi\gamma}(Q^2) &=& \frac{2 f_\pi}{Q^2} \int\!\! d\omega\!
\int_0^1\!\! dx\ \delta(\omega-(2x-1)2E_\pi)\:
C_1(\omega,\mu)\: \phi_{\pi}(x,\mu) {\nonumber}\\
&=& \frac{2f_\pi}{Q^2} \int_0^1\!\! dx\: C_1((2x-1)Q,\mu)
\: \phi_{\pi}(x,\mu) \,.\end{aligned}$$ This is the final result and is valid to leading order in $\lambda$ and all orders in $\alpha_s$. From Eq. (\[piC\]) charge conjugation implies that $\phi_\pi(x) = \phi_\pi(1-x)$ and $C_1(\omega) = C_1(-\omega)$. Eq. (\[resultpionformfactor\]) agrees with the Brodsky-Lepage [@brodskylepage] result that the form factor can be written as the convolution of a short distance function with the light-cone pion wavefunction. The SCET formalism gives a concise derivation of this result and defines the short distance function in terms of the Wilson coefficient of an effective theory operator.
As an illustrative example consider the tree level matching onto $C$ illustrated in Fig. \[fig\_gamgampi\]. Since the location of the $W$’s in $O$ are fixed by gauge invariance, $C(\mu,{\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} })$ can be determined by matching with $W=1$. Expanding the full theory graphs to leading order gives $$\begin{aligned}
\label{1ab}
i\: (\mbox{Fig.}1) =
\frac{ie}{2}\,\epsilon_{\mu\nu\rho\beta}\epsilon^\nu {{\bar n}}^\rho n^\beta
\Big(\frac{\bnslash}{2}\gamma_5\Big) \big({\cal{\hat Q}}^2\big)
\left( \frac{1}{{{\bar n}}\cdot p}-\frac{1}{{{\bar n}}\cdot p^\prime} \right),\end{aligned}$$ where we have dropped isosinglet terms, contributions with opposite parity to the pion, as well as those proportional to $\nslash\gamma_5$ since $\nslash
\xi_{n,p}=0$. Comparing Eq. (\[1ab\]) to Eq. (\[Opig\]) gives $$\begin{aligned}
\label{Cpig_tree}
C(\mu,{\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} })
=\frac{Q}{6\sqrt{2}} \Big( \frac{1}
{{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} }}-\frac{1}{{\bar {\cal P}}}\Big) + {\cal O}\big(\alpha_s(Q)\big) \,,\end{aligned}$$ so that $$\begin{aligned}
\label{Cpig2}
C_1(\mu,\omega=(2x-1)Q)
=\frac{1}{6\sqrt{2}} \Big( \frac{1}{x}+\frac{1}{1-x}\Big)
+ {\cal O}\big(\alpha_s(Q)\big) \,.\end{aligned}$$ This result is again in agreement with Ref. [@brodskylepage], and the order $\alpha_s(Q)$ corrections to this Wilson coefficient can be read off from the results in Ref. [@delAguila:1981nk; @Braaten:yp]. An identical analysis applies for operators with different spin structures such as the ones contributing to $\gamma^*\gamma\to\rho^0$.
The large $Q^2$ meson form factor
---------------------------------
Another example of an exclusive process which can be treated in the effective theory is the classic case of the electromagnetic pion form factor at large space-like momentum transfer. For generality we consider in this section the electromagnetic form factors for arbitrary mesons (pseudoscalar $P$ or vector $V$), defined as $$\begin{aligned}
\label{piff}
& &\big\langle P^\prime(p^\prime)\big| J_\mu \big| P(p) \big\rangle
= F_P(Q^2) (p_\mu+p^\prime_\mu)\,,\\
& &\big\langle V^\prime(p^\prime,\varepsilon^\prime) \big| J_\mu
\big| P(p) \big\rangle
= G(Q^2) i\epsilon_{\mu\nu\alpha\beta}\, p^\nu p^{\prime\alpha}
\varepsilon^{\prime\beta} {\nonumber}\,, \\
& &\big\langle V^\prime(p^\prime,\varepsilon^\prime) \big|J_\mu
\big| V(p,\varepsilon)\big\rangle
= F_1(Q^2) (\varepsilon^{\prime*}\cdot\varepsilon)(p+p^\prime)_\mu +
F_2(Q^2) [(\varepsilon^{\prime*}\cdot p)\varepsilon_\mu +
(\varepsilon \cdot p^\prime)\varepsilon^{\prime*}_\mu]\,,{\nonumber}\end{aligned}$$ where $q^2 = -Q^2, q=p-p^\prime$. For simplicity we suppress the dependence of the form factors on the isospin of the two mesons. We will restrict ourselves in the following to the case of hadrons made up only of $u,d$ quarks. The electromagnetic current is defined as usual by $ J_\mu = \bar q\:\hat {\cal
Q}\,\gamma_\mu\: q$, with charge matrix $\hat {\cal Q} = \mbox{diag}(2/3,-1/3)$, which can be written in terms of the up and down quark charges as $\hat {\cal
Q}=(Q_u-Q_d)\tau_3/2+(Q_u+Q_d){\bf 1}/2$.
We will be interested in the asymptotic form of the form factors in the region with $Q^2 \gg m_P^2$, where it can be expanded in a power series in $1/Q^2$ [@brodskylepage]. It is convenient to work in the Breit frame, where the momentum transfer has the light-cone components $q=(q^+, q^-, q_\perp) = (Q, -Q,
0)$. In this frame the meson momenta are $p = (Q, {m_P^2}/{Q}, 0)$, $p'=
({m_P^2}/{Q}, Q, 0)$, so the partons in the incoming/outgoing meson are collinear along the ${{\bar n}}_\mu$/$n_\mu$ direction.
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The electromagnetic current in Eq. (\[piff\]) is matched in the effective theory onto the most general combination of operators constructed from collinear fields which are compatible with collinear gauge invariance. Operators such as the dimension three current $$\begin{aligned}
\label{Feyn}
[\bar\xi_{{{\bar n}}}W_{{{\bar n}}}] \Gamma\: C(\mu,{{\cal P}}^\dagger,{\bar {\cal P}})\:
[W^\dagger_n \xi_n] \,,\end{aligned}$$ can contribute, but only overlap with the asymmetric meson states with one energetic collinear quark and one usoft or soft quark. Often this overlap is referred to as the tail of the wavefunction contribution or the Feynman mechanism of generating the form factor [@brodskyfeynman; @isgurfeynman]. There are other operators with significant overlap with more symmetric meson states (where all the constituents are allowed to be energetic). The leading such operators have the form [^5] $$\begin{aligned}
\label{BLmech}
\frac{1}{Q^3}\,
\big[\bar \xi_{n,p_1} W_n \Gamma W^\dagger_{{{\bar n}}} \xi_{{{\bar n}},p_2} \big]
C(\mu,{\bar {\cal P}},{\bar {\cal P}^{\raisebox{0.8mm}{\scriptsize$\dagger$}} },{{\cal P}},{{\cal P}}^\dagger)
\big[ \bar \xi_{{{\bar n}},p_3} W_{{{\bar n}}} \Gamma' W^\dagger_n \xi_{n,p_4}
\big]\,,\end{aligned}$$ with $C$ a dimensionless Wilson coefficient. As usual, collinear gauge invariance is enforced by the location of the $W$’s in Eqs. (\[Feyn\]) and (\[BLmech\]). There is some argument about the relative size of Eqs. (\[Feyn\]) and (\[BLmech\]) in the literature [@brodskyfeynman; @isgurfeynman]. Often it is argued that the tail of the wavefunction is suppressed by an extra $\Lambda_{\rm
QCD}^2/Q^2$ [@brodskyfeynman], in which case the operator in Eq. (\[BLmech\]) dominates by two powers of $Q$. An analysis of the tail of wavefunction contributions has not yet been performed in the effective theory framework. Therefore, we choose to ignore the operator in Eq. (\[Feyn\]), and below only analyze the operator in Eq. (\[BLmech\]). We emphasize that we do not claim to have shown that this is justified by the effective theory power counting.
There are two different structures possible for the operator in Eq. (\[BLmech\]), and we write the general matching for the electromagnetic current as $$\begin{aligned}
\label{Jmatch}
J^\nu \to \frac{1}{Q^3}\,\int\!\!{d\omega_j} \:\Big[
C_0(\mu,\omega_j)\: {\cal J}^\nu_0(\omega_j,\mu) +
C_8(\mu,\omega_j)\: {\cal J}^\nu_8(\mu,\omega_j) \Big] \,,\end{aligned}$$ where $j=1,2,3,4$. The SCET currents are dimension-6 operators $$\begin{aligned}
\label{J12}
{\cal J}_0^\nu &=& {\overline\chi_{n,\omega_1}} \Gamma {\chi_{{{\bar n}},\omega_2}}\:
{\overline\chi_{{{\bar n}},\omega_3}} \Gamma' {\chi_{n,\omega_4}}
-(\Gamma\leftrightarrow\Gamma',
\omega_{1,2}\leftrightarrow-\omega_{4,3}) \,,\\
{\cal J}_8^\nu &=& {\overline\chi_{n,\omega_1}} \Gamma\, T^a {\chi_{{{\bar n}},\omega_2}}\:
{\overline\chi_{{{\bar n}},\omega_3}} \Gamma' T^a {\chi_{n,\omega_4}}
- (\Gamma\leftrightarrow\Gamma',
\omega_{1,2}\leftrightarrow-\omega_{4,3}) {\nonumber}\,,\end{aligned}$$ where the $\chi$ fields are defined in Eq. (\[CH1\]). In terms of the charge matrix $\hat {\cal Q}$, the spin and flavor structure is $$\begin{aligned}
\Gamma\otimes \Gamma' = (n^\nu + {{\bar n}}^\nu) \Big(
\gamma_\alpha^\perp \hat {\cal Q} \otimes \gamma^\alpha_\perp {\bf 1}
\Big) \,.\end{aligned}$$ The Wilson coefficients $C_{0,8}$ can be computed in a power series in $\alpha_s(Q)$. They are functions of $\mu$, $Q$, and the $\omega_j$ which are the sum of momentum labels for gauge invariant products of collinear fields in the SCET currents.
The currents operators in Eq. (\[J12\]) are the most general allowed operators which are gauge invariant, transform the same way as $J^\mu$ under charge conjugation and satisfy current and helicity conservation. To see how these properties constrain the form of the allowed operators, we begin by noting that Eq. (\[Jspin\]) implies that $\Gamma,\Gamma'=\{1,\gamma_5,\gamma_\perp^\mu\}$ are the most general allowed spin structures. For massless quarks the electromagnetic and QCD couplings preserve helicity, whereas $\bar \xi_n
\{1,\gamma_5\} \xi_{{{\bar n}}}$ cause the helicity to flip. Thus, only the structure $\bar \xi_n \gamma_\perp^\mu \xi_{{{\bar n}}}$ is allowed. Current conservation $q^\nu {\cal J}_\nu=0$, together with $q^\nu = Q({{\bar n}}^\nu-n^\nu)/2$ implies ${\cal J}_\nu\propto (n_\nu+{{\bar n}}_\nu)$. Under charge conjugation $J^\mu\to -J^\mu$ so the same must be true for the SCET currents. In the current operators, charge conjugation switches $\omega_1\leftrightarrow -\omega_4$, $\omega_2 \leftrightarrow -\omega_3$, and $\Gamma \leftrightarrow \Gamma'$, as can be seen from Eq. (\[CPT\]). Thus, the second term in ${\cal J}_{0,8}^\nu$ is required to make these operators odd under charge conjugation. The operators ${\cal J}_{0,8}^\nu$ and the full electromagnetic current are invariant under a combined $PT$ transformation. This requires that the Wilson coefficients are real.
The operators ${\cal J}_{0,8}$ are responsible for the $P_{{\bar n}}\to P_n$ transition, while the reverse transition $P_n\to P_{{\bar n}}$ is described by similar operators with ${{\bar n}}\leftrightarrow n$. Parity invariance requires the Wilson coefficients of these operators to be identical to $C_{0,8}(\omega_i)$. Demanding hermiticity of the electromagnetic current in the effective theory then gives the relation $C_{0,8}(\omega_1, \omega_2, \omega_3, \omega_4)=
C_{0,8}^*(\omega_2, \omega_1, \omega_4, \omega_3)$. Since the coefficients are real they must therefore satisfy $C_{0,8}(\omega_1, \omega_2, \omega_3,
\omega_4)= C_{0,8}(\omega_2, \omega_1, \omega_4, \omega_3)$.
To compute the matrix elements in the effective theory, it is convenient to Fierz transform the four-quark operators in Eq. (\[J12\]). This gives $$\begin{aligned}
\label{J12f}
C_0 {\cal J}_0 + C_8 {\cal J}_8 = (n^\nu + {{\bar n}}^\nu)\:
\sum_{j=1}^4 C_j {\cal J}_j \,,\end{aligned}$$ where $$\begin{aligned}
\label{JemC}
{\cal J}_j &=& \big[{\overline\chi_{n,\omega_1}} \Gamma_j
{\chi_{n,\omega_4}}\big] \big[ {\overline\chi_{{{\bar n}},\omega_3}} \Gamma_j^\prime
{\chi_{{{\bar n}},\omega_2}} \big] \,.\end{aligned}$$ The spin, flavor, and color structures are $$\begin{aligned}
\label{G12}
\Gamma_1 \otimes \Gamma_1^\prime &=& -\frac14 (Q_u-Q_d) i\epsilon^{3bc}
(\tau^b\otimes\tau^c)
\big[ \bnslash\otimes \nslash + \bnslash\gamma_5\otimes
\nslash\gamma_5 \big]
\,,{\nonumber}\\
\Gamma_2 \otimes \Gamma_2^\prime &=& \frac14 \Big[
(Q_u+Q_d) ({\bf 1}\otimes {\bf 1} + \tau^a \otimes \tau^a) +
(Q_u-Q_d) ({\bf 1}\otimes \tau^3 + \tau^3 \otimes {\bf 1}) \Big]{\nonumber}\\
&& \times \big[ \bnslash\otimes \nslash
+ \bnslash\gamma_5\otimes \nslash\gamma^5 \big] \,,\end{aligned}$$ while $\Gamma_{3,4}=T^a\Gamma_{1,2}$ and $\Gamma_{3,4}'=T^a\Gamma_{1,2}'$. The new Wilson coefficients are $$\begin{aligned}
\label{C12}
C_1(\mu,\omega_j) &=& \bigg[ \frac18 \Big(1-\frac{1}{N_c^2}\Big)
C_8(\mu,\omega_j) + \frac{1}{4N_c} C_0(\mu,\omega_j) \bigg]
+ (\omega_{1,2}\leftrightarrow -\omega_{4,3}) \,,{\nonumber}\\
C_2(\mu,\omega_j) &=& \bigg[ \frac18 \Big(1-\frac{1}{N_c^2}\Big)
C_8(\mu,\omega_j) + \frac{1}{4N_c} C_0(\mu,\omega_j) \bigg]
- (\omega_{1,2}\leftrightarrow -\omega_{4,3}) \,,\end{aligned}$$ with similar relations for $C_{3,4}$ which are also in terms of $C_{0,8}$.
A few general predictions follow from the form of the operators in Eq. (\[JemC\]).[^6] For mesons with spin, only helicity conserving form factors appear, and furthermore no off-diagonal (e.g., $P\to V$) matrix elements are present at leading order in $1/Q^2$. These results agree with Ref. [@Pink]. We also see that the form factors between arbitrary meson states are determined at leading power by only two hard coefficients, $C_0$ and $C_8$.
Now consider what factorization tells us about the matrix element of the operators in Eq. (\[JemC\]). For the decoupling of usoft and soft gluons we will follow section \[section\_SCET\]. Integrating out offshell modes with $p^2\sim Q\Lambda_{\rm QCD}$ induces soft Wilson lines $S_n$ and $S_{{\bar n}}$, while the field redefinitions in Eq. (\[def0\]) make all couplings to usoft gluons explicit in the operators. Together these give $$\begin{aligned}
{\cal J}_j^\nu &=& \big[ {\overline\chi_{n,\omega_1}}^{(0)} Y_n^\dagger S_n^\dagger
\Gamma_j S_n Y_n {\chi_{n,\omega_4}}^{(0)}\big]\:
\big[ {\overline\chi_{{{\bar n}},\omega_3}}^{(0)} Y_{{\bar n}}^\dagger S_{{\bar n}}^\dagger \Gamma_j^\prime
S_{{\bar n}}Y_{{\bar n}}{\chi_{{{\bar n}},\omega_2}}^{(0)}\big] \,.\end{aligned}$$ Consider first the color singlet currents $j=1,2$. Here the $Y$’s and $S$’s all cancel using unitarity of the Wilson lines. Since the $A_{n,q}^\mu$ and $A_{{{\bar n}},q}^\mu$ gluons only interact with fields in the $n$ and ${{\bar n}}$ directions respectively, collinear gluons are not exchanged between the $n$ and ${{\bar n}}$ quark bilinears. Thus, the matrix element between states with particles moving in the $n$ and ${{\bar n}}$ directions factors $$\begin{aligned}
\label{Jfactor}
\langle n | {\cal J}_{1,2} | {{\bar n}}\rangle =
\langle n | {\overline\chi_{n,\omega_1}}^{(0)} \Gamma_{1,2} {\chi_{n,\omega_4}}^{(0)}
| 0\rangle
\langle 0 | {\overline\chi_{{{\bar n}},\omega_3}}^{(0)} \Gamma_{1,2}^\prime
{\chi_{{{\bar n}},\omega_2}}^{(0)}| {{\bar n}}\rangle \,.\end{aligned}$$ Next consider the currents ${\cal J}_{3,4}$, which have color structure $T^a
\otimes T^a$ in $\Gamma_j\otimes \Gamma_j^\prime$. In this case the usoft and soft gluons do not cancel, but can all be moved into one quark bilinear using the color identity $Y_n^\dagger S_n^\dagger T^a S_n Y_n \otimes Y_{{\bar n}}^\dagger
S_{{\bar n}}^\dagger T^a S_{{\bar n}}Y_{{\bar n}}= T^a \otimes Y_{{\bar n}}^\dagger S_{{\bar n}}^\dagger S_n Y_n
T^a Y^\dagger_n S^\dagger_n S_{{\bar n}}Y_{{\bar n}}$. After this rearrangement it is clear that the (u)soft gluons and $A_{n,q}^\mu$ and $A_{{{\bar n}},q}^\mu$ gluons only interact with the fields in one of the quark bilinears. Thus, the matrix element $\langle n |{\cal J}_{3,4}|{{\bar n}}\rangle$ factors, similar to Eq. (\[Jfactor\]). For color singlet states, however, the matrix element of an octet operator vanishes identically since $$\begin{aligned}
\langle n | {\overline\chi_{n,\omega_1}}^{(0)} T^a {\chi_{n,\omega_4}}^{(0)}
| 0\rangle =0 \,.\end{aligned}$$ Thus, the effective theory currents ${\cal J}_{3,4}$ do not contribute to the form factors at any order in perturbation theory.
Eq. (\[Jfactor\]) shows that for arbitrary meson states factorization occurs. It remains to show that the matrix elements in Eq. (\[Jfactor\]) are given by a two-dimensional convolution with the light-cone meson wavefunctions. To do this we consider the simple example of the $0^{-+}\to 0^{-+}$ form factor for the charged pion. It should be obvious that the same steps go through for other meson states.
The symmetry of the pion wavefunction $\phi_\pi(x)$ under charge conjugation ($x\to 1-x$) implies that only the ${\cal J}_1$ current contributes. Thus, $$\begin{aligned}
F_{\pi^\pm}(Q^2) = \frac{2}{Q^4} \int\!\!{d\omega_j}\,
C_1(\mu,\omega_j) \big\langle \pi_n^\pm(p') \big| {\overline\chi_{n,\omega_1}}^{(0)}
\Gamma_1 {\chi_{n,\omega_4}}^{(0)} \big| 0 \big\rangle
\big\langle 0 \big| {\overline\chi_{{{\bar n}},\omega_3}}^{(0)} \Gamma_1'
{\chi_{{{\bar n}},\omega_2}}^{(0)} \big| \pi_{{\bar n}}^\pm(p) \big\rangle \,.\end{aligned}$$ The required matrix elements can be obtained from Eqs. (\[pimom\]) and (\[pimom2\]) with $|\pi^\pm\rangle = \mp(\pi^1 \pm i\pi^2)/\sqrt{2}$. The momentum conserving delta functions fix $\omega_1 -\omega_4 = {{\bar n}}\cdot p'=Q$ and $\omega_2-\omega_3={{\bar n}}\cdot p=Q$, while the $\omega=\omega_1+\omega_4$ and $\omega'=\omega_3+\omega_2$ integrations can be done with the delta functions. This leaves $$\begin{aligned}
F_{\pi^\pm} &=& \pm(Q_u\!-\!Q_d)\, \frac{f_\pi^2}{Q^2}\,
\int_0^1\!\!\mbox{d}x\!
\int_0^1\!\!\mbox{d}y \: T_1(x,y,\mu) \phi_\pi(x,\mu) \phi_\pi(y,\mu)\,,\end{aligned}$$ where $T_1(x,y)$ is defined in terms of $C_1(\omega_1,\omega_2,\omega_3,\omega_4)$ as $$\begin{aligned}
T_1(x,y) = C_1\big(xQ,yQ,(y-1)Q,(x-1)Q)
\,.\end{aligned}$$ The coefficients $C_{j}(\mu,\omega_j)$, and therefore also $T_j(x,y)$, can be obtained at the scale $\mu=Q$ by a matching calculation, as illustrated in Fig. \[fig\_pipi\]. For this purpose, it is sufficient to compute the matrix element of the currents with free collinear quarks. To lowest order in $\alpha_s(Q)$, only $C_8(\omega_j,\mu=Q)$ is nonvanishing $$\begin{aligned}
\label{piff20}
C_0(\omega_j,\mu=Q) =0\,,\qquad\quad
C_8(\omega_j,\mu=Q) = 4\pi \alpha_s(Q)\: \frac{Q^2}{\omega_3\,
\omega_4 }\,.\end{aligned}$$ This implies $$\begin{aligned}
T_1(x,y,\mu=Q) = \frac{4\pi \alpha_s(Q)}{9} \Big[
\frac{1}{xy}+\frac{1}{(1-x)(1-y)} \Big] \,,\end{aligned}$$ Using the asymptotic light-cone pion wavefunction $\phi_\pi(x)=6x(1-x)$ we find agreement with Ref. [@brodskylepage], $$\begin{aligned}
\label{piff2}
F_{\pi^\pm}(Q^2) = \pm \frac{8\pi f_\pi^2 \alpha_s(Q)}{9Q^2}
\left[ \int_0^1\!\!\mbox{d}x\, \frac{\phi_\pi(x)}{x} \right]^2 \to \pm
\frac{8\pi f_\pi^2 \alpha_s(Q)}{Q^2}\,.\end{aligned}$$ The order $\alpha_s^2(Q)$ corrections to Eq. (\[piff20\]) can be found in Refs. [@Field:wx; @Dittes:aw; @Braaten:yy].
Inclusive Processes {#section_inclusive}
===================
Deep inelastic scattering
-------------------------
DIS is a process which is both simple and rich in physics. As such it provides an ideal introduction to inclusive factorization in QCD, which we study from an effective field theory point of view in this section. The aim is to prove that to all orders in $\alpha_s$ and leading order in $\lambda$ the DIS forward scattering amplitude can be written as an integral over hard coefficients times the parton distribution functions. This is done by matching onto local operators in SCET. The properties of SCET are used to show that matrix elements of the leading local operator can be written as a convolution of a hard coefficient with the parton distribution functions for the proton.
The first step is to understand the kinematics of the process. The hard scale $Q^2 =-q^2$ is set by the invariant mass of the photon, and $x = Q^2/(2 p{\!\cdot\!}q)$ is the Bjorken variable. In the Breit frame the momentum of the virtual photon is $q^\mu=Q({{\bar n}}^\mu-n^\mu)/2$, and the incoming proton momentum is $p^\mu= n^\mu {{\bar n}}{\!\cdot\!}p/2 + {{\bar n}}^\mu m_p^2/(2{{\bar n}}{\!\cdot\!}p) \simeq n^\mu Q/(2x)
+{{\bar n}}^\mu x m_p^2/(2Q)$ up to terms $\sim m_p^2/Q^2$, where $m_p$ is the proton mass. By momentum conservation the final state momentum is $P^\mu_X =
q^\mu+p^\mu$, which gives an invariant mass $P_X^2 = (Q^2/x)
(1-x)+m_p^2$. Values $1-x\simeq \Lambda_{\rm QCD}/Q$ correspond to the endpoint region where the particles in $X$ are collimated into a jet, while values $1-x\simeq \Lambda_{\rm QCD}^2/Q^2$ correspond to the resonance region. We will consider the standard OPE region where $1-x \gg \Lambda_{\rm QCD}/Q$ so that the final state has virtuality of order $Q^2$ and can be integrated out. In contrast, although the incoming proton has a large momentum component in the $n^\mu$ direction it has a small invariant mass $p^2 = m_p^2 \sim \Lambda_{\rm
QCD}^2$, and therefore is described by collinear fields in the effective theory.
Consider the spin-averaged cross section for DIS which can be written as $$\begin{aligned}
d\sigma = \frac{d^3 \mathbf{k}'}{2|\mathbf{k}'|(2\pi)^3}\:
\frac{\pi e^4}{s Q^4}\: L^{\mu\nu}(k,k')\: W_{\mu\nu}(p,q) \,,\end{aligned}$$ where $k$ and $k'$ are the incoming and outgoing lepton momenta with $q=k'-k$, $L^{\mu\nu}$ is the lepton tensor, and $s=(p+k)^2$. The hadronic tensor $W^{\mu\nu}$ can be related to the imaginary part of the DIS forward scattering amplitude: $$\begin{aligned}
W_{\mu\nu}(p,q) &=& \frac{1}{\pi}\, {\rm Im}\, T_{\mu\nu}(p,q) \,,\\
T_{\mu\nu}(p,q) &=& \frac12 \sum_{\rm spin} \big\langle p \big|
\hat T_{\mu\nu}(q) \big| p \big\rangle\,,\qquad\quad
\hat T_{\mu\nu}(q)=i\int d^4z\: e^{iq\cdot z}\,{\rm T}[J_\mu(z)J_\nu(0)]\,,{\nonumber}\end{aligned}$$ where for an electromagnetic current $J_\mu$ we can write $$\begin{aligned}
\label{tprod}
T_{\mu\nu}(p,q) = \Big(-g_{\mu\nu} + \frac{q_\mu q_\nu}{q^2}\Big)
T_1(x,Q^2)+\left(p_\mu+\frac{q_\mu}{2x}\right)
\left(p_\nu+\frac{q_\nu}{2x}\right)T_2(x,Q^2) \,. \end{aligned}$$
As explained above, the intermediate hadronic state has invariant mass $P_X^2
\sim Q^2$. Therefore, one can perform an OPE and match $\hat T^{\mu\nu}(q)$ onto operators in SCET. All fields in the resulting operators are evaluated at the same residual space time point, however, the presence of Wilson lines and label momenta make the operators nonlocal along a particular light cone direction. These nonlocal operators sum the infinite set of purely local operators of a given twist, however this is built into the formalism automatically.
![Tree level matching onto the operator $O_j^{(i)}$ in DIS. \[disopefig\]](figs/DISnew3){width="5.5in"}
To match we write down the most general leading operator in SCET which contains collinear fields moving in the $n^\mu$ direction, and enforce the condition from current conservation $q^\mu \hat T_{\mu\nu} = 0$. This leads to $$\begin{aligned}
\label{Tmatch}
\hat T^{\mu\nu} &\to& \frac{g_\perp^{\mu\nu}}{Q} \Big( \sum_i O_{1}^{(i)}
+ \frac{O_{1}^g}{Q} \Big) + \frac{(n^\mu\!+\!{{\bar n}}^\mu)(n^\nu\!+\!{{\bar n}}^\nu)}{Q}
\Big( \sum_i O_{2}^{(i)} + \frac{O_{2}^g}{Q} \Big) \,,\end{aligned}$$ where $$\begin{aligned}
\label{disop}
O_j^{(i)} &=& \bar{\xi}^{(i)}_{n,p'} W\, \frac{\bnslash}{2}\,
C^{(i)}_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)\, W^\dagger \xi^{(i)}_{n,p} \,, {\nonumber}\\
O_j^g &=& {{\bar n}}_\mu {{\bar n}}_\nu {\rm tr} \big[ W^\dagger (G_n)^{\mu\lambda} W
\: C^g_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)\: W^\dagger (G_n)_{\lambda}^{\:\nu}\, W \big],\end{aligned}$$ where $i$ labels the flavor of the fermions and $ig\, G_n^{\mu\lambda}= [i{\cal
D}_n^\mu + gA_{n,q}^\mu, i{\cal D}_n^\lambda+gA_{n,q'}^\lambda ]$. The Wilson coefficients are dimensionless functions of ${\bar {\cal P}}_+$, ${\bar {\cal P}}_-$, $Q$, and $\mu$. As in previous sections we can separate the hard coefficients from the long distance operators by introducing trivial convolutions. This gives $$\begin{aligned}
\label{disop1}
O_j^{(i)} &=& \int\!\!{d\omega_1\,d\omega_2} \,
C^{(i)}_j(\omega_+,\omega_-)\: \big[ {\overline\chi_{n,\omega_1}}^{(i)}
\frac{\bnslash}{2} {\chi_{n,\omega_2}}^{(i)} \big] \,,{\nonumber}\\
O_j^g &=& -\int\!\!{d\omega_1\,d\omega_2}\, C^{g}_j(\omega_+,\omega_-)\:
{\rm tr}\big[ B_{n,\omega_1}^{\mu} B^{n,\omega_2}_\mu
\big] \,,\end{aligned}$$ where $\omega_\pm = \omega_1 \pm \omega_2$, and $B^\mu_{n,\omega}\equiv{{\bar n}}_\nu
({\cal G}_{n,\omega})^{\nu\mu}$ with $({\cal G}_{n,\omega} )^{\mu\lambda}$ defined in Eq. (\[covG\]). Next we factor the coupling of usoft gluons from the collinear fields using the field redefinitions in Eq. (\[def0\]). The operator $O_j^{(i)}$ has the structure in Eq. (\[decoup\]) so the $Y$’s cancel trivially, while for $O_j^g$ we find $$\begin{aligned}
B_n^\mu = Y_n B_n^{\mu(0)} Y_n^\dagger\,,\end{aligned}$$ and the factors of $Y$ cancel in the trace. It is easy to see that soft gluons also decouple using Eq. (\[Sadd\]) or by noting that there is no non-trivial soft gauge invariant way of adding soft Wilson lines $S_n$ to $O_j^{(i)}$ or $O_j^g$. Under charge conjugation the full theory electromagnetic current $J_\mu\to -J_\mu$ and therefore the operator $\hat T_{\mu\nu}\to \hat
T_{\mu\nu}$. This implies relations for the effective theory Wilson coefficients since the operators $O_j^{(i)}$ must also respect this symmetry. Thus charge conjugation gives $$\begin{aligned}
\label{DISchg}
\int\!\!{d\omega_1\, d\omega_2}\, C_j^{(i)}(\omega_+,\omega_-)\
{\overline\chi_{n,\omega_1}}^{(i)} {\bnslash} {\chi_{n,\omega_2}}^{(i)}
&\stackrel{\mbox{\tiny C}}{=}&
-\int\!\!{d\omega_1\, d\omega_2}\, C_j^{(i)}(\omega_+,\omega_-)\
{\overline\chi_{n,-\omega_2}}^{(i)} {\bnslash} {\chi_{n,-\omega_1}}^{(i)}{\nonumber}\\
&=& \int\!\!{d\omega_1\,d\omega_2} \big[ - C_j^{(i)}(-\omega_+,\omega_-)\big] \
{\overline\chi_{n,\omega_1}}^{(i)} {\bnslash} {\chi_{n,\omega_2}}^{(i)} \,.\end{aligned}$$ In the second line we changed variable $\omega_1\to -\omega_2$ and $\omega_2\to
-\omega_1$ which takes $\omega_+\to -\omega_+$ and $\omega_-\to \omega_-$. Thus, to all orders in perturbation theory $C_j^{(i)}(-\omega_+,\omega_-) =
-C_j^{(i)}(\omega_+,\omega_-)$. This relates the Wilson coefficients for quarks and anti-quarks. Note that the above results are all independent of the collinear hadron on which DIS is performed.
Next we take the matrix element between proton states. Using the definitions of the nonperturbative matrix elements given in Section \[section\_melt\], and picking out the coefficients of the tensor structures we find that the delta functions in Eq. (\[pdfqg\]) set $\omega_+=\pm 2Q\xi/x$ and $\omega_-=0$. Since charge conjugation relates negative and positive values of $\omega_+$ only coefficients, $C_j(\omega_+,0)$, with positive $\omega_+$ are needed in the formulae for DIS. Therefore we define $$\begin{aligned}
H_j(z) \equiv C_j(2Qz,0,Q,\mu) \,,\end{aligned}$$ where here we have made the dependence on $Q$ and $\mu$ explicit. Combining this with Eqs. (\[tprod\]), (\[Tmatch\]), (\[disop1\]), and (\[pdfqg\]), gives the final result $$\begin{aligned}
T_1(x,Q^2) &=& - \frac{1}{x} \int_0^1 \!\!d\xi \,
\bigg\{ H^{(i)}_1\Big(\frac{\xi}{x}\Big)\,
\big[ f_{i/p}(\xi) + \bar f_{{i}/p}(\xi) \big]
+\frac{\xi}{2x} H^{g}_1\Big(\frac{\xi}{x}\Big)\,
f_{g/p}(\xi) \bigg\}\,, {\nonumber}\\
T_2(x,Q^2) &=& \frac{4x}{Q^2} \int_0^1 \!\!d\xi
\bigg\{ \Big[4\,H^{(i)}_2\Big(\frac{\xi }{x}\Big)
\!-\! H^{(i)}_1\Big(\frac{\xi}{x}\Big) \Big]
\big[ f_{i/p}(\xi)+ \bar f_{{i}/p}(\xi) \big] {\nonumber}\\
&& \qquad\qquad\ \ +\frac{\xi}{2x} \Big[4\,H^{g}_2\Big(\frac{\xi }{x}\Big)
\!-\! H^{g}_1\Big(\frac{\xi}{x}\Big) \Big] f_{g/p}(\xi) \bigg\} \,,\end{aligned}$$ where a sum over $i$ is implicit. The hadronic tensor components $W_{1,2}(x,Q^2)= {\rm Im}\: T_{1,2}(x,Q^2)/\pi$ and therefore are determined by the imaginary part of the Wilson coefficients. The Wilson coefficients are dimensionless and therefore can only have $\alpha_s(Q)\ln(\mu/Q)$ dependence on $Q$. This reproduces the Bjorken scaling of the structure functions.
Finally, consider the tree level matching onto the Wilson coefficients shown in Fig. \[disopefig\]. From these graphs only the quark coefficient functions $C^{(i)}_{j}$ can be non-zero and we find $$\begin{aligned}
&& {\rm Im}\, H^{(i)}_1(z) = {-\,Q_i^2}\,\pi\: \delta(z-1)
\,, \qquad\quad {\rm Im}\, H^{(i)}_2 = 0 \,,\end{aligned}$$ where $Q_i$ is the charge of parton $i$. The vanishing of ${\rm Im}\,H^{(i)}_2$ at tree level reproduces the Callan-Gross relation $W_1/W_2= Q^2/(4x^2)$.
Drell-Yan, $p \bar p \to \ell^+\ell^- X $
-----------------------------------------
Next we will extend the DIS analysis to the Drell-Yan (DY) process: $p \bar p
\to \ell^+\ell^- X $. Specifically we consider the $Q^2$ distribution, where $Q^2$ is the invariant mass of the lepton pair. Drell-Yan is more complicated than DIS because one has two hadrons in the initial state. In the center-of-mass frame the incoming proton and anti-proton move in opposite lightlike directions, and to prove factorization we use the fact that collinear modes in different lightlike directions can only couple to each other in external operators in SCET. We take the incoming proton to move in the $n^\mu$ direction and the incoming antiproton to move in the $\bar{n}^\mu$ direction. The hard scales in DY are $Q^2$ and the invariant mass of the colliding $p \bar p $ pair $s=(p+\bar{p})^2$. The lepton pair has an invariant mass $Q^2$, and the invariant mass of the final hadronic state is $$\begin{aligned}
p_X^2 = Q^2\left(1+\frac{1}{\tau} - \frac{1}{x_1} - \frac{1}{x_2} \right)\,,\end{aligned}$$ where $$\begin{aligned}
\tau = \frac{Q^2}{s}\,, \qquad x_1 = \frac{Q^2}{2p \cdot q}\,, \qquad
x_2 = \frac{Q^2}{2\bar p \cdot q}\,.\end{aligned}$$ We are interested in the kinematic region where $p_X^2 \sim Q^2$, which implies that both $x_1$ and $x_2$ are far away from one. As $\tau$ approaches one the invariant mass becomes too small for the treatment given here to apply. However, the effective theory can be used to deal with this region as well. It is also possible to study the $q_\perp$ distribution, but this again requires a generalization of the discussion given below.
The spin averaged cross section for Drell-Yan is$$\begin{aligned}
\label{sDY}
d\sigma = \frac{32\pi^2 \alpha^2}{Q^4\,s}\:
L_{\mu \nu}\,W^{\mu \nu}\frac{d^3k_1}{(2\pi)^3(2k_1^0)}
\frac{d^3k_2}{(2\pi)^3 (2k_2^0)} \label{DY} \,,\end{aligned}$$ where $$\begin{aligned}
W^{\mu\nu} &=& \frac{1}{4}
\sum_{{\rm spins}}\sum_X (2\pi)^4 \delta^{(4)}(p+\bar{p}-q-p_X)
\langle p \bar{p} | J^\mu(0) | X \rangle \langle X| J^\nu(0)|p \bar{p} \rangle
\nonumber\\
&=& \frac{1}{4} \sum_{{\rm spins}} \int d^4 x\, e^{-iq\cdot x}
\langle p \bar{p} | J^\mu(x) \, J^\nu(0)|p \bar{p} \rangle \,.\end{aligned}$$ The sum over spins refers to the initial hadron spins (the sum over final hadron spins is included in the sum over $X$). Integrating Eq. (\[sDY\]) over the emission angles of the final leptons one obtains $$\begin{aligned}
\label{dyqsqrdist}
\frac{d \sigma}{d Q^2} = \frac{2\alpha^2}{3\, Q^2\,s}\ \frac{1}{4}
\sum_{\rm spins}\: \langle p\bar p |\, \hat W\, | p\bar p \rangle \,, \end{aligned}$$ where we have neglected the lepton masses and defined the operator $$\begin{aligned}
\label{What}
\hat W(\tau,Q^2) = - \int\!\! \frac{d^4q}{(2\pi)^3}\: \theta(q^0)
\delta(q^2\!-\!Q^2) \int\!\! d^4x\: e^{-iq\cdot x}\, J^\mu(x)\, J_\mu(0) \,.\end{aligned}$$ As we discussed above in the region of phase space under consideration $p_X^2\sim Q^2$, so these hard fluctuations can be integrated out. Operationally this means we match $\hat W$ onto local operators in the effective theory. We would like to show that the minimal set of order $\lambda^4$ operators that contribute to Drell-Yan are $$\begin{aligned}
\label{dymatching}
\hat W \to & &
\frac{1}{Q^2} \int\!\!{d\omega_i}\: C_{qq}(\omega_i,Q)
\big[{\overline\chi_{n,\omega_1}}^{\,(i)} \:{\bnslash}\: {\chi_{n,\omega_2}}^{\,(i)}\big]
\big[{\overline\chi_{{{\bar n}},\omega_3}}^{\,(i)} \:{\nslash}\: {\chi_{{{\bar n}},\omega_4}}^{\,(i)}\big]
\, {\nonumber}\\[3pt]
&-& \frac{1}{Q^3} \int\!\!{d\omega_i}\: C_{qg}(\omega_i,Q)\,
\big[{\overline\chi_{n,\omega_1}}^{\,(i)} \:{\bnslash}\: {\chi_{n,\omega_2}}^{\,(i)}\big]
{\rm Tr}\big[ {\cal B}_{{{\bar n}},\omega_3}^{\beta}
{\cal B}^{{{\bar n}},\omega_4}_{\beta} \big] \, {\nonumber}\\[3pt]
&-& \frac{1}{Q^3} \int\!\!{d\omega_i}\: C_{gq}(\omega_i,Q)
{\rm Tr}\big[ {\cal B}_{n,\omega_1}^\nu {\cal B}^{n,\omega_2}_{\nu} \big]
\big[ {\overline\chi_{{{\bar n}},\omega_3}}^{\,(i)} \:{\nslash}\: {\chi_{{{\bar n}},\omega_4}}^{\,(i)}\big]
\,{\nonumber}\\[3pt]
&+& \frac{1}{Q^4} \int\!\!{d\omega_i}\: C_{gg}(\omega_i,Q)\,
{\rm Tr}\big[ {B}_{n,\omega_1}^\nu {B}^{n,\omega_2}_{\nu} \big]
{\rm Tr}\big[ {\cal B}_{{{\bar n}},\omega_3}^{\beta}
{\cal B}^{{{\bar n}},\omega_4}_{\beta} \big] \,,\end{aligned}$$ where the powers of $Q$ are included to make the coefficients dimensionless. The operators displayed in Eq. (\[dymatching\]) are just products of the operators that occured in DIS, so for these terms the decoupling of soft and usoft gluons occurs in a straightforward manner. To show that the operators in Eq. (\[dymatching\]) are the most general set needed we must show that all other operators that are order $\lambda^4$ either reduce to these or vanish between the matrix elements in Eq. (\[dyqsqrdist\]). For instance, $\lambda^4$ operators also exist where a $B_{n,\omega}^\nu$ field is contracted with a $B_{{{\bar n}},\omega'}^\mu$ field, or the color structures of the operators in Eq. (\[dymatching\]) could be arranged in a different way.
We now give a general argument for why we can always rewrite an arbitrary operator in the form of Eq. (\[dymatching\]) or show that it does not contribute to DY. All operators relevant for DY contain four order $\lambda$ collinear fields chosen from $\xi_{n,p}$, $\xi_{{{\bar n}},p}$, $B_{n,p}^\mu$, or $B_{{{\bar n}},p}^\mu$. Furthemore, two must move in direction $n$ and two in the direction ${{\bar n}}$ (other possibilities end up vanishing by baryon number conservation or because they involve a set of fields between physical states that can not possibly form a color singlet operator). For operators with 4 quark fields, Fierz transformations can always be made to arrange the fields such that those in the same direction sit in the same bilinear. Using as an example the operator with two collinear quarks in the $n$ directions and two gluons in the ${{\bar n}}$ direction and leaving out the soft Wilson lines for the moment, the most general matrix element is $$\begin{aligned}
\label{generalDY}
\langle p_n \bar{p}_{{\bar n}}|\, {\overline\chi_{n,\omega_1}}^{a,\alpha}
{\chi_{n,\omega_2}}^{b,\beta} \, {\cal B}_{{{\bar n}},\omega_3}^{A,\mu}
{\cal B}_{{{\bar n}},\omega_4}^{B,\nu} \, |p_n \bar{p}_{{\bar n}}\rangle \;
\Delta_{\mu\nu;\alpha\beta}^{ab;AB} \,,\end{aligned}$$ where $a,b$ are quark colors, $A,B$ are gluon colors, and $\alpha,\beta$ are spinor indices for the quarks. $\Delta_{\mu\nu;\alpha\beta}^{ab;AB}$ is some tensor that connects the indices in an arbitrary way. In the contraction of $a,b$ and $A,B$ there are two possible ways to make an overall color singlet, one where both the quarks and gluons are in a color singlet, and another where both the quarks and gluons are in a color octet. We will discuss both of these possibilities in turn.
![Tree level matching onto the operators in Drell-Yan. \[dyfig\]](figs/DYnew3){width="3.5in"}
In the color singlet case, including the soft and ultrasoft Wilson lines is trivial, since using Eqs. (\[Sadd\]), (\[Yfd2\]), and (\[eom2\]) we see that they cancel due to unitarity/orthogonality of the various Wilson lines in the fundamental/adjoint representations. Thus, there are no soft, usoft, or collinear interactions that connect the $n$ and the ${{\bar n}}$ fields. As in previous sections this leads to a factorization of the matrix element in Eq. (\[generalDY\]), namely $$\begin{aligned}
\langle p_n |\, {\overline\chi_{n,\omega_1}}^{(0)a,\alpha} {\chi_{n,\omega_2}}^{(0)b,\beta}
\, |p_n \rangle \, \langle \bar{p}_{{\bar n}}| {\cal B}_{{{\bar n}},\omega_3}^{(0)A,\mu}
{\cal B}_{{{\bar n}},\omega_4}^{(0)B,\nu} \, | \bar{p}_{{\bar n}}\rangle
\; \Delta_{\mu\nu;\alpha\beta}^{ab;AB} \,.\end{aligned}$$ Since the proton spins are summed over, we can write (with the help of Eq. (\[tracefomulae\])) $$\begin{aligned}
\label{quarkmatrix}
\langle p_n |\, {\overline\chi_{n,\omega_1}}^{(0)a,\alpha} {\chi_{n,\omega_2}}^{(0)b,\beta}
\, |p_n \rangle \propto \; \delta^{ab} (\nslash)_{\alpha\beta}\:
\langle {p}_n |\, {\overline\chi_{n,\omega_1}}^{(0)c,\gamma}\bnslash
{\chi_{n,\omega_2}}^{(0)c,\gamma} \, | {p}_n \rangle \,,\end{aligned}$$ so that spin and color are summed over in the matrix element. Similarly the antiproton matrix element can be simplified to $$\begin{aligned}
\label{gluonmatrix}
\langle \bar{p}_{{\bar n}}| \, {\cal B}_{{{\bar n}},\omega_3}^{(0)A,\mu}
{\cal B}_{{{\bar n}},\omega_4}^{(0)B,\nu} \, | \bar{p}_{{\bar n}}\rangle\propto \; \delta^{AB}
g^{\mu\nu}_\perp
\langle \bar{p}_{{\bar n}}| \, {\rm Tr}\:\big[ {\cal B}_{{{\bar n}},\omega_3}^{(0)\alpha}
{\cal B}^{(0){{\bar n}},\omega_4}_{\alpha} \big] \, | \bar{p}_{{\bar n}}\rangle\,.\end{aligned}$$ Here we used the fact that the matrix element is symmetric in $\mu$ and $\nu$, and that only the perpendicular index $\mu$ of the field $B^\mu$ is order $\lambda$. Using Eqs. (\[quarkmatrix\]) and (\[gluonmatrix\]) the original matrix element in Eq. (\[generalDY\]) can be written as $$\begin{aligned}
\label{endDYeg}
&& \langle p_n \bar{p}_{{\bar n}}|\, {\overline\chi_{n,\omega_1}}^{a,\alpha}
{\chi_{n,\omega_2}}^{b,\beta}
\, {\cal B}_{{{\bar n}},\omega_3}^{A,\mu} {\cal B}_{{{\bar n}},\omega_4}^{B,\nu}
\, |p_n \bar{p}_{{\bar n}}\rangle \; \Delta_{\mu\nu;\alpha\beta}^{ab;AB}
\\[4pt]
&& \qquad\qquad
\propto \; {\rm Tr}[\Delta^{\:\mu}_\mu\, \nslash]\,
\langle \bar{p}_{{\bar n}}| \, {\rm Tr}\:\big[ {\cal B}_{{{\bar n}},\omega_3}^{(0)\nu}
{\cal B}^{(0){{\bar n}},\omega_4}_{\nu} \big] \, | \bar{p}_{{\bar n}}\rangle\,
\langle p_n |\, {\overline\chi_{n,\omega_1}}^{(0)} \bnslash
{\chi_{n,\omega_2}}^{(0)} \, |p_n \rangle \,,{\nonumber}\end{aligned}$$ where the trace of $\Delta$ is over spin and color, and just gives an overall constant. The final result in Eq. (\[endDYeg\]) is identical to the matrix element of the second operator in Eq. (\[dymatching\]).
If each of the $n$ and ${{\bar n}}$ field bilinears involve color octet structures, then the soft and usoft Wilson lines don’t cancel, since they don’t commute with the SU(3) generators. However, one can use the color identity $$\begin{aligned}
\label{cident}
Y_n^\dagger S_n^\dagger {\cal T}^x S_{{\bar n}}Y_{{\bar n}}\otimes Y_{{\bar n}}^\dagger
S_{{\bar n}}^\dagger {\cal T}^x S_n Y_n = {\cal T}^x \otimes S_n Y_n Y_{{\bar n}}^\dagger
S_{{\bar n}}^\dagger {\cal T}^x S_n Y_n Y_{{\bar n}}^\dagger S_{{\bar n}}^\dagger\end{aligned}$$ where each ${\cal T}^x$, $S$, and $Y$ factor is in the appropriate representation of the color group. Eq. (\[cident\]) moves all the soft and ultrasoft interactions between either the $n$ or the ${{\bar n}}$ collinear fields. Thus, again the fields in one bilinear can not be contracted with fields in the other bilinear and the matrix element factors. However this time the factored matrix element vanishes. For the example discussed above, $$\begin{aligned}
\label{quarkmatrixoctet}
\langle p_n |\, {\overline\chi_{n,\omega}}^{(0)\alpha}\:
T^C{\chi_{n,\omega'}}^{(0)\beta} \, |p_n \rangle= 0\,,\end{aligned}$$ since an color octet operator vanishes between color singlet states. The same holds true for the matrix element of an octet gluon operator.
An identical proof of decoupling goes through for the case of 4 quarks, where we again either have two color singlet or two color octet $n$ and ${{\bar n}}$ bilinears. With 4 gluon fields we can either have the two $n$ and two ${{\bar n}}$ fields coupled as singlets, or coupled in the same higher representation (an [**8**]{}, {[**10**]{}, $\overline{\mathbf{ 10}}$}, or [**27**]{}). In the latter case the matrix element between color singlet states still vanishes so the proof for the 4 gluon operators also goes through in an identical way.
Thus we have shown that the matrix element of an operator with an arbitrary contraction of indices either vanishes or can be written in terms of a product of a matrix element which is related to a proton pdf and a matrix element which is related to an antiproton pdf as in the example in Eq. (\[endDYeg\]). This is the result we want. To see how the final formulae are derived note that we can write the matrix element of Eq. (\[dymatching\]) in the form of a convolution $$\begin{aligned}
\label{dyconv}
\frac{1}{4}\sum_{{\rm spins}} \langle p_n \bar{p}_{{\bar n}}|\, \hat W \,
|p_n \bar{p}_{{\bar n}}\rangle
&=& \sum_{a,b} \int\!\!{d\omega_i}\:C_{a,b}(\omega_i)
\langle p_n |O^a_n(\omega_+, \omega_-)| p_n \rangle
\langle \bar{p}_{{\bar n}}|O^b_{{\bar n}}(\omega'_+, \omega'_-)| \bar{p}_{{\bar n}}\rangle \,,{\nonumber}\\\end{aligned}$$ where $\omega_\pm = \omega_1\pm \omega_2$ and $\omega_\pm'=\omega_3\pm\omega_4$. The operators here are the same as in DIS, with $a=(i)$ for the quark operator, and $a=g$ for the gluon operator $$\begin{aligned}
\label{dyop1}
O^{(i)}_n(\omega_+, \omega_-) &=& \frac{1}{Q}
\big[ {\overline\chi_{n,\omega_1}}^{(i)} \frac{\bnslash}{2}
{\chi_{n,\omega_2}}^{(i)} \big] \,,{\nonumber}\\
O^g_n(\omega_+, \omega_-) &=& -\frac{1}{Q^2}
{\rm tr}\big[ B_{n,\omega_1}^\mu B^{n,\omega_2}_\mu \big] \,.\end{aligned}$$ Apart from the dependence on the labels, the Wilson coefficients in Eq. (\[dyconv\]) can also depend on the renormalization point $\mu$ and the kinematic variable $Q$. Using Eq. (\[pdfqg\]) we see that the matrix elements in Eq. (\[dyop1\]) set $\omega_-=\omega_-'=0$ and $\omega_+=2\sqrt{s} z_1$, $\omega_+'=2\sqrt{s} z_2$ where $z_1$ and $z_2$ are the convolution variables. Since all kinematic variables aside from $Q^2$ are integrated over in Eq. (\[What\]) the only other variable that can appear in the Wilson coefficient is the center of mass energy which produces the $\ell^+\ell^-$ pair, namely $4\hat{s} = \omega_+ \omega'_+$. Thus, the Wilson coefficients only depend on $\omega_+\omega_+'=4s z_1 z_2$. Defining new coefficients $$\begin{aligned}
H^{ab}(z_1 z_2) &=& C^{ab}(\omega_+\omega_+'=4s z_1 z_2,Q,\mu)\end{aligned}$$ we can replace the matrix elements in Eq. (\[dyop1\]) with parton distribution functions using Eq. (\[pdfqg\]) to obtain: $$\begin{aligned}
\frac{1}{4}\sum_{{\rm spins}} \langle p_n\bar p_{{\bar n}}|\,
\hat W\, | p_n\bar p_{{\bar n}}\rangle
&=& \frac{1}{\tau} \int_0^1 \!dz_1\: dz_2 \,
\bigg\{ \!-\!H^{(i)(j)}(-\!z_1 z_2) \Big[ f_{i/p}(z_1)\bar{f}_{j/\bar{p}}(z_2)+
\bar{f}_{i/p}(z_1) f_{j/\bar{p}}(z_2) \Big]{\nonumber}\\[3pt]
&&\hspace{-1cm}+ H^{(i)(j)}(z_1 z_2) \Big[ f_{i/p}(z_1) f_{j/\bar{p}}(z_2)+
\bar{f}_{i/p}(z_1) \bar{f}_{j/\bar{p}}(z_2) \Big]\nonumber\\[3pt]
&&\hspace{-1cm} + \frac{z_2}{2\sqrt{\tau}}\: H^{(i)g}(z_1 z_2) f_{i/p}(z_1)
f_{g/\bar{p}}(z_2)- \frac{z_2}{2\sqrt{\tau}}\: H^{(i)g}(-z_1 z_2)
\bar{f}_{i/p}(z_1) f_{g/\bar{p}}(z_2) {\nonumber}\\[3pt]
&&\hspace{-1cm} + \frac{z_1}{2\sqrt{\tau}}\: H^{g(j)}(z_1 z_2) f_{g/p}(z_1)
f_{j/\bar{p}}(z_2) -\frac{z_1}{2\sqrt{\tau}}\: H^{g(j)}(-z_1 z_2)
f_{g/p}(z_1)\bar{f}_{j/\bar{p}}(z_2) {\nonumber}\\[3pt]
&&\hspace{-1cm} + \frac{z_1 z_2}{4\tau}\: H^{gg}(z_1 z_2) f_{g/p}(z_1)
f_{g/\bar{p}}(z_2) \bigg\} \,.\end{aligned}$$ This is the final convolution formulae for Drell-Yan and is valid to all orders in $\alpha_s$ and leading order in the power expansion. At tree level the matching calculation shown in Fig. \[dyfig\] yields zero for all the Wilson coefficients except $$\begin{aligned}
H^{(i)(i)}(-z_1 z_2) =-\frac{2\pi\tau}{3}\: {Q_i^2} \,\delta(\tau - z_1 z_2)
\,,\end{aligned}$$ where $Q_i$ is the charge of parton $i$. The coefficients $H^{(i)g}(\pm z_1
z_2)$ and $H^{g(j)}(\pm z_1 z_2)$ start at order $\alpha_s(Q)$, while $H^{(i)(i)}(z_1 z_2)$ and $H^{gg}(z_1 z_2)$ start at order $\alpha_s^2(Q)$.
Deeply Virtual Compton Scattering, $\gamma^* p\to \gamma^{(*)} p'$
------------------------------------------------------------------
Next we examine deeply virtual Compton scattering (DVCS). To be more precise we examine the exclusive reaction $\gamma^* p \to \gamma^{(*)} p$, where the incoming photon is highly virtual, the final photon is either off-shell or real, and the incoming and outgoing protons have different momenta. The reason we have included this process in the inclusive section is that DVCS has the remarkable property that the nonperturbative physics is described by a so called non-forward parton distribution function (NFPDF). The NFPDF is a more general distribution function that reduces to the standard parton distribution functions (familiar from DIS) for some values of the momentum fraction, and behaves like a lightcone wavefunction (familiar from the pion examples) for other values. Deeply virtual Compton scattering was first studied in perturbative QCD in Refs [@Ji:1996ek; @Ji:1996nm; @Radyushkin:1996nd], and proofs of factorization to all orders in perturbation theory were later presented in Refs [@Collins:1998be; @Ji:1998xh]. In addition properties of NFPDFs were studied in Ref. [@Radyushkin:1997ki]. Here we present a proof of factorization for DVCS based on SCET.
As with the previous proofs it is important to understand the kinematics of the process. We take the incoming proton and photon momenta to be $p$ and $q$ respectively, with $x = Q^2/(2 p\cdot q)$ and $q^2 = - Q^2 \gg \Lambda_{{\rm
QCD}}^2$. The outgoing proton and photon momentum are $p'$ and $q'$ respectively, with $0 \geq q^{\prime 2} \geq -Q^2$. It is convenient to define a parameter $\zeta\equiv 1-{{\bar n}}{\!\cdot\!}p^\prime/{{\bar n}}{\!\cdot\!}p$, which measures the change to the proton’s large momentum. Working in the Breit frame and neglecting contributions that are $\ll \Lambda_{{\rm QCD}}^2/Q$ we have: $$\begin{aligned}
\label{DVCSkin}
\begin{tabular}{llll}
& & \mbox{\normalsize Label Momenta}
& \mbox{\normalsize Residual Momenta}\\ \hline \\[-12pt]
& $q^\mu=$\hspace{0.8cm} & ${\frac{\mbox{\normalsize $Q$}}{\mbox{\normalsize $2$}} } ({{\bar n}}^\mu\!-\! n^\mu)$
& $+0$ \\[5pt]
& $p^\mu=$ & ${\frac{\mbox{\normalsize $Q$}}{\mbox{\normalsize $2 x$}} } \, n^\mu$
& $+{\frac{\mbox{\normalsize $x$}}{\mbox{\normalsize $2Q$}} }\, m_p^2(n^\mu\!+\!{{\bar n}}^\mu)$ \\[5pt]
& $p^{\prime\mu}=$ & ${\frac{\mbox{\normalsize $Q$}}{\mbox{\normalsize $2 x$}} } (1\!-\!\zeta) \, n^\mu
\!+\! p^{\prime\mu}_\perp$ & $+{\frac{\mbox{\normalsize $x$}}{\mbox{\normalsize $2Q$}} } \big[ m_p^2(1\!-\! \zeta)n^\mu
\!+\! \{m_p^2(1\!+\!\zeta)\!-\! t\}{{\bar n}}^\mu\big]$\hspace{0.3cm} \\[5pt]
& $q^{\prime\mu}=$ & ${\frac{\mbox{\normalsize $Q$}}{\mbox{\normalsize $2$}} } \big({\frac{\mbox{\normalsize $\zeta$}}{\mbox{\normalsize $x$}} }-\! 1 \big)
\, n^\mu + {\frac{\mbox{\normalsize $Q$}}{\mbox{\normalsize $2$}} }\, {{\bar n}}^\mu \!-\! p_\perp^{\prime\mu}$\hspace{0.7cm}
& $+{\frac{\mbox{\normalsize $x$}}{\mbox{\normalsize $2Q$}} } \big[ \zeta m_p^2 n^\mu
+ (t\!-\!\zeta m_p^2){{\bar n}}^\mu\big]$ \\[5pt] \hline
\end{tabular} \hspace{0.5cm} \vspace{0.2cm}\end{aligned}$$ Here the label momenta are order $Q$ or $Q\lambda$, while the residual momenta are order $Q\lambda^2$ and depend on $p^2=m_p^2$ and $t=(p^\prime-p)^2=(p_\perp^{\prime 2}\!-\!\zeta^2 m_p^2)/(1\!-\!\zeta)$, which are both $\sim \Lambda_{\rm QCD}^2$. The invariant mass of the intermediate hadronic state is $(p+q)^2 \approx Q^2(1-x)/x$ just like DIS, so for $1-x \gg
\Lambda_{{\rm QCD}}/Q$ the intermediate state can be integrated out.
We will proceed in a manner analogous to the analysis for DIS. The amplitude (up to an overall momentum conserving $\delta$-function) is given by a time ordered product of currents: $$\begin{aligned}
T_{\mu\nu}(p,q,q') &=& \langle p',\sigma' | \hat{T}_{\mu\nu}(q,q')
| p,\sigma \rangle \nonumber \\
\hat{T}_{\mu\nu}(q,q') &=& i \int d^4z \, e^{i(q+q')\cdot z/2}\,
T\big[J_\mu(-z/2) J_\nu(z/2)\big] \,.\end{aligned}$$ This time ordered product is contracted with a lepton tensor to obtain the amplitude. Now current conservation requires $q^\mu T_{\mu\nu}= q^{\prime\nu}
T_{\mu\nu}=0$, however the DVCS $T_{\mu\nu}$ is not symmetric under $\mu
\leftrightarrow \nu$. For electromagnetic currents $J_\mu$ we have $$\begin{aligned}
\label{TDVCS}
T_{\mu\nu} &=&
-\Big( g_{\mu\nu} -\frac{q^\prime_\mu q_\nu}{q{\!\cdot\!}q'} \Big)\: T_1
+ \Big( p_\mu - \frac{q^\prime_\mu\,p{\!\cdot\!}q}{q{\!\cdot\!}q'} \Big)
\Big( p_\nu - \frac{q_\nu\,p{\!\cdot\!}q'}{q{\!\cdot\!}q'} \Big)\: T_2 {\nonumber}\\
&& + \ell_\mu \ell^{\prime}_\nu\: T_3
+ \ell_\mu \Big( p_\nu - \frac{q_\nu\,p{\!\cdot\!}q'}{q{\!\cdot\!}q'} \Big)\: T_4
+ \Big( p_\mu - \frac{q^\prime_\mu\,p{\!\cdot\!}q}{q{\!\cdot\!}q'} \Big)\ell_\nu
\: T_5
+ \ldots\,,\end{aligned}$$ where the functions $T_i=T_i(x,\zeta,Q^2,t)$, and the vectors $\ell_\mu\equiv
q^\prime_\mu-q_\mu + p_\mu (q^2\!-\!q^{\prime 2}\!+\!t) /(2p\cdot q)$ and $\ell_\mu^\prime\equiv q_\mu-q^\prime_\mu + p_\mu (q^{\prime 2}\!-\!q^{2}\!+\!t)
/(2p\cdot q^\prime)$ are defined so that $q\cdot\ell = q^\prime\cdot\ell^\prime
= 0$. In Eq. (\[TDVCS\]) and below the ellipses denote spin dependent terms. For simplicity we will show how factorization is achieved for the spin independent contributions shown in Eq. (\[TDVCS\]) with the understanding that it is no more difficult to also include the other terms.
It is convenient to define a parameter $0\leq\alpha\leq 1$, by $q^{\prime
2}\equiv -\alpha Q^2$. The DIS hadronic time-ordered product is obtained in the limit $p' \to p$, where $\alpha \to 1$ and $\zeta \to 0$. From Eq. (\[DVCSkin\]) we see that $$\begin{aligned}
\zeta=x(1-\alpha)+{\cal O}\Big(\frac{t}{Q^2},\frac{m_p^2}{Q^2}\Big)\,,\end{aligned}$$ so these parameters are not independent. Since the intermediate hadronic state has invariant mass ${\cal O}(Q^2)$ we can match $\hat{T}_{\mu\nu}$ onto operators in SCET. Requiring $q^\mu \hat{T}_{\mu\nu}=0$ and $q^{\prime\nu}
\hat{T}_{\mu\nu}=0$ for the order $Q$ label momenta leads to $$\begin{aligned}
\hat{T}_{\mu\nu} \to \frac{g^{\mu\nu}_\perp}{Q}
\Big(O^{(i)}_1+\frac{O^{g}_1}{Q} \Big)+
\frac{1}{Q} \left( n^\mu + {{\bar n}}^\mu \right)
\left( \alpha n^\nu + {{\bar n}}^\nu \right)
\Big(O^{(i)}_2+\frac{O^{g}_2}{Q} \Big) +\ldots \,,\end{aligned}$$ where the ellipses are spin dependent terms and the displayed operators are $$\begin{aligned}
\label{dvcsop}
O_j^{(i)} &=& \bar{\xi}^{(i)}_{n,p'} W\, \frac{\bnslash}{2}\,
C^{(i)}_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)\, W^\dagger \xi^{(i)}_{n,p} \,, {\nonumber}\\
O_j^g &=& {{\bar n}}_\mu {{\bar n}}_\nu\: {\rm tr} \big[ W^\dagger (G_n)^{\mu\lambda} W
\: C^g_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)\: W^\dagger (G_n)_{\lambda}^{\:\nu}\, W \big] \,.\end{aligned}$$ We have suppressed the dependence of the Wilson coefficients $C({\bar {\cal P}}_+,{\bar {\cal P}}_-)$ on $Q$, $\alpha$, and $\mu$. The form of the operators in Eq. (\[dvcsop\]) looks the same as the DIS operators given in Eq. (\[disop\]), however the operators here are more general because the Wilson coefficients depend on $\alpha$. In the limit $\alpha \to 1$ the DVCS operators reduce to the DIS operators. However, since the field structure of the DVCS operators is identical to DIS several results follow immediately. For instance, the steps which factorize soft and usoft gluons and leave fields with superscript $(0)$ are the same and are not repeated here, $$\begin{aligned}
\label{dvcsop2}
O_j^{(i)} &=& {\overline\chi_{n,\omega}}^{\,(0)(i)} \frac{\bnslash}{2}\,
C^{(i)}_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)\, {\chi_{n,\omega'}}^{\,(0)(i)} \,, {\nonumber}\\
O_j^g &=& -{\rm tr} \Big[ B_{n,\omega}^{(0)\mu}
\: C^g_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)\: (B_{n,\omega'}^{(0)})_\mu \Big] \,.\end{aligned}$$ The restrictions on the DVCS Wilson coefficients from charge conjugation are the same as in Eq. (\[DISchg\]), $C_j({\bar {\cal P}}_+,{\bar {\cal P}}_-)=-C_j(-{\bar {\cal P}}_+,{\bar {\cal P}}_-)$, however because $p\ne p'$ this is not simply a relation between quark and anti-quark Wilson coefficients. The way in which DVCS is unique is that the matrix elements involve nucleon states with different momenta. This is what leads to results in terms of non-forward parton distribution functions.
The definition of the NFPDFs are given in Eq. (\[nfpdfqg\]), and can be used along with the relations above to obtain expressions for the $T_i$ in terms of the NFPDFs. Before we give this result we note that the Wilson coefficients depend on the operators ${\bar {\cal P}}_\pm$ which become the variables $\omega_\pm$ after introducing trivial convolutions and the $\chi_{n,\omega_i}$ fields in Eq. (\[CH1\]). The delta functions in Eq. (\[nfpdfqg\]) then set $\omega_-=
-Q \zeta / x$ and $\omega_+= \pm Q (2 \xi - \zeta)/x$, where $\xi$ is the convolution variable. Note that $\zeta/x = 1-\alpha$, and just like DIS it is the combination $\xi/x$ which appears. Since charge conjugation relates the Wilson coefficients for $\omega_+>0$ and $\omega_+<0$ it is convenient to define $$\begin{aligned}
\label{Hdvcs}
{\cal H}_j(\xi/x) \equiv C_j\big(Q(2\xi/x\!- \!1\!+\!\alpha),Q(\alpha\!-\!1),Q,\alpha,\mu\big) \,,\end{aligned}$$ where in the last three arguements we have made the dependence on $Q$, $\alpha$, and $\mu$ explicit. Combining Eqs. (\[nfpdfqg\]), (\[dvcsop2\]), and (\[Hdvcs\]) then gives $$\begin{aligned}
\label{tmatch}
T_1 &=& -\frac{e(\sigma^\prime,\sigma)}{2Q}
\int_0^1 \!\!d\xi \,
\bigg\{ {\cal H}^{(i)}_1 \! \Big(\frac{\xi}{x}\Big)\:
\Big[ {\cal F}^{i}_\zeta(\xi;t) + \overline{{\cal F}}^{i}_\zeta(\xi;t) \Big]
+ \frac{1}{2x} {\cal H}^{g}_1 \Big(\frac{\xi}{x}\Big)\:
{\cal F}^{g}_\zeta(\xi;t) \bigg\} +\ldots \,, {\nonumber}\\[3pt]
T_2 &=& \frac{x^2(1+\alpha)}{Q^3}\: e(\sigma^\prime,\sigma)
\int_0^1 \!\!d\xi \bigg\{ \Big[ 2(1+\alpha) {\cal H}^{(i)}_2 \!
\Big(\frac{\xi}{x}\Big) \!-\! {\cal H}^{(i)}_1 \! \Big(\frac{\xi}{x}\Big)
\Big] \Big[{\cal F}^{i}_\zeta(\xi;t)+\overline{{\cal F}}^{i}_\zeta(\xi;t)\Big]
{\nonumber}\\
&& + \frac{1}{2x} \Big[ 2(1+\alpha)\,
{\cal H}^{g}_2 \Big(\frac{\xi}{x}\Big)
\!-\! {\cal H}^{g}_1 \Big(\frac{\xi}{x}\Big) \Big]
{\cal F}^{g}_\zeta(\xi;t) \bigg\} +\ldots \,, {\nonumber}\\[3pt]
T_3 &=& 0\,, \qquad T_4 =0\,, \qquad T_5 =0 \,,\end{aligned}$$ which are the final convolution results valid to all orders in $\alpha_s$ and leading order in the power expansion. The structure functions $T_{3,4,5}$ vanish since the vectors $\ell^\mu=\ell^{\prime\mu}=0$ at leading order in the power expansion. The terms with ellipses are for the spin flip terms and involve the NFPDF ${\cal K}$ defined in Eq. (\[nfpdfqg\]). The results for these terms have a similar form to those in Eq. (\[tmatch\]).
Finally we match at tree level. The tree level diagram in QCD is the same as in Fig. \[disopefig\] except the outgoing photon and proton have momenta $q^\prime$ and $p^\prime$ respectively. Only the quark Wilson coefficients are nonzero at tree level. We find $$\begin{aligned}
C_1^{(i)}(\omega_+,\omega_-,Q,\alpha) &=& e^2 Q^2_i
\bigg( \frac{2 Q}{2 Q +\omega_+ - \omega_-} -
\frac{2 Q}{2Q-\omega_+ - \omega_-} \bigg)
\\
C_2^{(i)}(\omega_+,\omega_-,Q,\alpha) &=& 0 \,,
{\nonumber}\end{aligned}$$ which gives $$\begin{aligned}
{\cal H}_1^{(i)}\left(\frac{\xi}{x}\right) &=& -e^2 Q^2_i
\bigg( \frac{1}{1-\xi/x} - \frac{1}{1+(\xi-\zeta)/x}
\bigg) \\
{\cal H}_2^{(i)}\left(\frac{\xi}{x}\right) &=& 0 \,.{\nonumber}\end{aligned}$$ Since ${\cal H}_2^{(i)}=0$ at tree level, DVCS also obeys a Callan-Gross relation.
Conclusion {#section_conclusion}
==========
What we hope we have demonstrated here is the power of effective field techniques in the context of factorization for hard scattering processes. The explicit separation of modes and the implementation of gauge invariance for these modes greatly simplifies seemingly complex problems. What is normally accomplished by diagramatic Ward identities and induction techniques now falls out as a consequence of the gauge symmetry of operators in a low energy soft-collinear effective theory.
As we have emphasized the factorization formulae derived in this paper are not new. The purpose here was simply to extend the formalism introduced in [@bfl; @bfps; @cbis; @bpssoft; @bps] to cases with back-to-back collinear particles, and apply these ideas to more general processes than previously considered. Furthermore, the factorization proofs presented are perhaps simpler than those previously given (certainly they are more concise). We believe that within the confines of the SCET more difficult, and unresolved problems can be addressed, such as power corrections in cases without an OPE, and proofs of factorization for more complex processes.
This work was supported in part by the Department of Energy under the grant DOE-FG03-97ER40546.
Factorization of soft and collinear $n$ & ${{\bar n}}$ modes {#app_fact}
============================================================
This Appendix discusses the simultaneous factorization of the soft $(\lambda,\lambda,\lambda)$ modes, $n$-collinear $(\lambda^2,1,\lambda)$ modes, and ${{\bar n}}$-collinear $(1,\lambda^2,\lambda)$ modes. These three classes of modes can not interact with each other in a local manner and therefore do not couple through the SCET Lagrangian. However, they can couple in a gauge invariant way through external operators and currents. These interactions in currents are built up by integrating out offshell fluctuations with $p^2\gg (Q\lambda)^2$. For the special case of factorization of soft from $n$-collinear modes this was shown in detail in the Appendix of Ref. [@bpssoft]. There it was shown that integrating out certain modes with offshellness $p^2\sim Q^2\lambda$ causes the Wilson lines $W_n$ and $S_n$ to appear in operators in a gauge invariant way. Here we will extend this approach to the factorization of modes for cases involving two classes of collinear particles. For simplicity we restrict ourselves to the case where the original operators involve only collinear quark or gluon fields. This type of factorization was used for the pion form factor example discussed in Sec. III.B and the Drell-Yan process presented in Sec. IV.B.
Momenta $(+,-,\perp)$ Fields Wilson lines
------------------ ------------------------ --------------------------------------------- ------------------------------------------------ ------------------------------------ -- --
onshell collinear-$n$ $p_1^\mu\sim (\lambda^2,1,\lambda)$ $\xi_{n}$, $A_{n}^\mu$ $W_n$
collinear-${{\bar n}}$ $p_2^\mu\sim (1,\lambda^2,\lambda)$ $\xi_{{{\bar n}}}$, $A_{{{\bar n}}}^\mu$ $W_{{\bar n}}$
soft $q^\mu\sim (\lambda,\lambda,\lambda)$ $q_{s}$, $A_{s}^\mu$ $S_n$, $S_{{\bar n}}$
usoft $k^\mu\sim (\lambda^2,\lambda^2,\lambda^2)$ $q_{us}$, $A_{us}^\mu$ $Y_n$, $Y_{{\bar n}}$
\[3pt\] offshell $p=p_1+p_2$ $p^\mu \sim (1,1,\lambda)$ $\psi_Q$, $A_Q^\mu$ $X_n$, $X_{{\bar n}}$
$p=p_1+q$ $p^\mu\sim (\lambda,1,\lambda)$ $\psi^L_{n}$, $A^{X\mu}_{n}$ $W_n^X$, $S_n^X$
$p=p_2+q$ $p^\mu\sim (1,\lambda,\lambda)$ $\psi^L_{{{\bar n}}}$, $A_{{{\bar n}}}^{X\mu}$ $W_{{\bar n}}^X$, $S_{{\bar n}}^X$
: Summary of the onshell modes discussed in section \[section\_SCET\], and the auxiliary fields introduced to represent the offshell fluctuations that are integrated out in this appendix. \[table\_off\]
The basic idea is to first match onto a Lagrangian with couplings between onshell and offshell modes that give all order $\lambda^0$ diagrams. The offshell modes (with $p^2\gg (Q\lambda)^2$) can then be integrated out, so that all operators are expressed entirely in terms of the onshell degrees of freedom. In table \[table\_off\] a summary is given of the three types of offshell momenta that are induced by adding soft, $n$-collinear, and ${{\bar n}}$-collinear momenta. For each type auxiliary quark and gluon fields are defined, and for convenience momentum labels are suppressed in this section. For example, the $\psi_Q$ quarks are created by the interaction of a $n$-collinear quark with an ${{\bar n}}$-collinear gluon, whereas the $\psi_n^L$ quarks are created when a collinear quark $\xi_n$ is knocked offshell by a soft gluon. For the field $\psi_Q$ we write $\psi_Q = \psi^Q_n + \psi_{{\bar n}}^Q$, where $\psi^Q_n =
\frac{1}{4}{\nslash\bnslash}\, \psi_Q$ and $\psi^Q_{{\bar n}}=
\frac{1}{4}\bnslash\nslash\, \psi_Q$. Then we have $\nslash \psi_n^Q = \bnslash
\psi_{{\bar n}}^Q =0$ and $\nslash \psi_n^L =\bnslash \psi_{{\bar n}}^L =0$. Various Wilson lines are also required and are listed in the table. It is convenient to define a generic Wilson line $L[a,A]$ along direction $a$ with field $A$ by the solution of $$\begin{aligned}
\big( a{\!\cdot\!}{{\cal P}}+ g\, a{\!\cdot\!}A\big) L[ a, A] = 0 \,.\end{aligned}$$ With this notation the on-shell Wilson lines are $W_n=L[{{\bar n}},A_n]$, $W_{{\bar n}}=L[n,A_{{\bar n}}]$, $S_n=L[n,A_s]$, and $S_{{\bar n}}=L[{{\bar n}},A_s]$. (Recall that the subscripts on $W$ and $S$ mean different things.) The Wilson lines involving offshell fields that we will require are $$\begin{aligned}
X_n &=& L[{{\bar n}},A_Q\!+\!A_n^X\!+\!A_n] \,, \qquad
X_{{\bar n}}=L[n,A_Q\!+\!A_{{\bar n}}^X\!+\!A_{{\bar n}}] \,,\\
W_n^X &=& L[{{\bar n}},A_n^X\!+\!A_n] \,, \qquad\qquad\!
W_{{\bar n}}^X=L[n,A_{{\bar n}}^X\!+\!A_{{\bar n}}] \,,{\nonumber}\\
S_n^X &=& L[n,A_n^X\!+\!A_s]\,, \qquad\qquad\
S_{{\bar n}}^X = L[{{\bar n}},A_{{\bar n}}^X\!+\!A_s] \,.{\nonumber}\end{aligned}$$ Below we discuss the results which allow us to integrate out offshell fluctuations. The structure of the auxiliary Lagrangians and construction of their solutions are very similar to the case presented in Ref. [@bpssoft], to which we refer for a more detailed presentation.
From table \[table\_off\] we see that adding $n$ and ${{\bar n}}$-collinear momenta gives $p^2\sim Q^2$, whereas adding soft and collinear momenta gives $p^2\sim
Q^2\lambda$. Loops that are dominated by offshell momenta only modify Wilson coefficients and not the infrared physics. Therefore, to determine the structure of SCET fields in an operator it sufficient to integrate out the offshell fields at tree level. For convenience we can integrate out the fluctuations starting with those with the largest offshellness. Recall that we only wish to consider offshell propagators connected to external operators. A subtlety for quarks is that distinct auxiliary fields are needed for the incoming and outgoing offshell propagators. However, the solution for the outgoing field turns out to be the conjugate of the incoming field, so to avoid a proliferation of notation we simply denote the outgoing terms in the Lagrangian by $+\mbox{h.c.}$, and present a solution for the incoming fields. Finally, note that for the gluon field $A_Q$ the fields $A_n$, $A_{{\bar n}}$, $A^X_n$, $A_{{\bar n}}^X$, and $A_s$ appear as background fields while for the fields $A_n^X$ and $A_{{\bar n}}^X$ it is $A_n$, $A_{{\bar n}}$, and $A_s$ that appear as background fields.
The terms in the auxiliary Lagrangian involving the $p^2\sim Q^2$ fields are $$\begin{aligned}
\label{Laux}
{\cal L}_{\rm aux}^Q &=& \bar \psi^Q_n gn{\!\cdot\!}(A_Q\!+\!A_{{{\bar n}}}^X\!+\!A_{{{\bar n}}})
\frac{\bnslash}{2}( \psi_n^L+\xi_n)
+ \bar \psi^Q_n \big[ n{\!\cdot\!}{\cal P} + gn{\!\cdot\!}(A_Q\!+\!A^X_{{{\bar n}}}\!+\!A_{{{\bar n}}}) \big] \frac{\bnslash}{2} \psi^Q_n {\nonumber}\\
&& + (n\leftrightarrow {{\bar n}}) + \mbox{h.c.} {\nonumber}\\
&& + \frac{1}{2g^2}\mbox{tr}\left\{
[iD_Q^\mu + gA_Q^\mu\,, iD_Q^\nu + gA_Q^\nu]^2\right\} +
\frac{1}{\alpha_Q}\mbox{tr} \{[iD_Q^\mu, A_{Q\mu}]^2\}\,.\end{aligned}$$ where $iD_Q^\mu = \frac12 n^\mu [{\bar {\cal P}}+ g{{\bar n}}{\!\cdot\!}(A_n^X\!+\! A_n)] + \frac12
{{\bar n}}^\mu [{{{\cal P}}} + gn\cdot (A^X_{{\bar n}}\!+\! A_{{\bar n}})]$. The solution of the equations of motion for these modes are $$\begin{aligned}
\label{eom1}
&& \psi_n^Q = (X_{{\bar n}}\!-\!1) (\psi_n^L+\xi_n) \,,\qquad
\psi_{{\bar n}}^Q=(X_n\!-\!1) (\psi_{{\bar n}}^L+\xi_{{\bar n}}) \,,{\nonumber}\\[3pt]
&& X_{{\bar n}}^\dagger X_n = W_n^X W_{{\bar n}}^{X\dagger}\,.\end{aligned}$$ (In addition to the last equation a constraint on the components $n{\!\cdot\!}A_Q$ and ${{\bar n}}{\!\cdot\!}A_Q$ also comes from the gauge fixing term, but will not be needed.) The terms in the auxiliary Lagrangian involving the $p^2\sim
Q^2\lambda$ fields are [@bpssoft] $$\begin{aligned}
\label{Laux2}
{\cal L}_{\rm aux}^X &=&
\bar \psi^L_n gn{\!\cdot\!}(A_{n}^X\!+\!A_{s}) \frac{\bnslash}{2}\xi_n
+ \bar \psi^L_n \big[ n{\!\cdot\!}{\cal P} + gn{\!\cdot\!}(A^X_{n}\!+\!A_{s}) \big]
\frac{\bnslash}{2} \psi^L_n +(n\leftrightarrow {{\bar n}})+ \mbox{h.c.} \\
&+& \frac{1}{2g^2}\mbox{tr}\Big\{ [iD_{nX}^\mu \!+\! gA_{n}^{X\mu}\,,
iD_{nX}^\nu \!+\! gA_n^{X\nu}]^2\Big\} + \frac{1}{\alpha_{n}}\mbox{tr}
\Big\{[iD_{nX}^\mu, A_{n\mu}^{X}]^2\Big\} + (n\leftrightarrow {{\bar n}}) \,,{\nonumber}\end{aligned}$$ where $iD_{nX}^\mu =\frac12 n^\mu [{\bar {\cal P}}+g{{\bar n}}{\!\cdot\!}A_n]+\frac12 {{\bar n}}^\mu
[n{\!\cdot\!}{{{\cal P}}}+gn{\!\cdot\!}A_s]$. The solutions for these modes are $$\begin{aligned}
\label{eom2}
\psi_n^L= (S_n^X\!-\!1) \xi_n \,,\qquad
S_n^{X\dagger} W_n^X = W_n S_n^\dagger \,,{\nonumber}\\
\psi_{{\bar n}}^L = (S_{{\bar n}}^X\!-\! 1)\xi_{{\bar n}}\,, \qquad
S_{{\bar n}}^{X\dagger} W_{{\bar n}}^X = W_{{\bar n}}S_{{\bar n}}^\dagger \,.\end{aligned}$$ Together Eqs. (\[eom1\]) and (\[eom2\]) can be used at leading order to eliminate the fields representing offshell fluctuations with $p^2\gg(Q\lambda)^2$. For collinear quarks this leads to the rules in Eq. (\[Sadd\]). Note that we did not need to use the gauge fixing term to resolve the ambiguity in the implicit solution for the ${{\bar n}}{\!\cdot\!}A$ and $n{\!\cdot\!}A$ auxillary fields.
As an illustration of these results we discuss the soft-collinear factorization for the production of a $q_n \bar q_{{\bar n}}$ pair with a large invariant mass $Q^2$. This process is mediated in the full theory by the electromagnetic current $J=
\bar \psi\Gamma \psi$ ($\Gamma$ a color singlet). This current will match onto a current in SCET that is built entirely out of onshell fields. Using the results in this appendix this current can be systematically derived. To start the quark field in $J$ matches onto $\xi_n$ plus all possible fields which the auxiliary Lagrangian can create starting from $\xi_n$, so $$\begin{aligned}
\label{J1}
J\to (\bar \xi_n + \bar \psi^L_n+\bar\psi^Q_n)\, \Gamma\, (\xi_{{\bar n}}+
\psi^L_{{\bar n}}+\psi^Q_{{\bar n}}) \,.\end{aligned}$$ Integrating out the $p^2\sim Q^2$ fluctuations with Eq. (\[eom1\]) and inserting a hard Wilson coefficient $C$ which depends on label operators turns Eq. (\[J1\]) into $$\begin{aligned}
(\bar\xi_n+\bar\psi^L_n) X_{{\bar n}}^\dagger\,C\Gamma\, X_n (\xi_{{\bar n}}+\psi^L_{{\bar n}})=
(\bar\xi_n+\bar\psi^L_n) W_n^X\,C\Gamma\, W_{{\bar n}}^{X\dagger}(\xi_{{\bar n}}+\psi^L_{{\bar n}})
\,.\end{aligned}$$ To construct the first operator we used the equations of motion for $\psi_n^Q$ and $\psi_{{\bar n}}^Q$, and in the second operator we used the equation of motion identity for the gluons in $X_n$ and $X_{{\bar n}}$. In a similar fashion we can now integrate out the $p^2\sim Q^2\lambda$ fluctuations with Eq. (\[eom2\]) to give $$\begin{aligned}
\bar\xi_n S_n^{X\dagger} W_n^X\,C\Gamma\, W_{{\bar n}}^{X\dagger} S_{{\bar n}}^X \xi_{{\bar n}}= \bar\xi_n W_n S_n^\dagger \,C\Gamma\, S_{{\bar n}}W_{{\bar n}}^\dagger \xi_{{\bar n}}\,. \end{aligned}$$ The operator on the right is the final result used in Eq. (\[OnnWS\]), and is soft, collinear, and usoft gauge invariant. It should be obvious from this example how the equations of motion in Eqs. (\[eom1\]) and (\[eom2\]) can be used to determine the factorized form of a general leading order operator.
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[^1]: In fact our factorization proofs rely heavily on the gauge symmetry structure of SCET. When a gauge fixing term is required for explicit calculations we use general covariant gauges.
[^2]: For recent work on subleading corrections in SCET for heavy-to-light transitions see Ref. [@Chay:2002vy].
[^3]: In this case depending on the choice of infrared regulator(s), it may not be possible to distinguish the $Y_n$ and $S_n$ Wilson lines in $S_{n{{\bar n}}}(\mu)$. For instance if one chooses $\Lambda_{\rm
IR}\sim Q\lambda$ then the usoft gluons give scaleless loop integrals and can be dropped, so that $Y_n^\dagger S_n^\dagger S_{{\bar n}}Y_{{\bar n}}\to S_n^\dagger S_{{\bar n}}$. If instead one chooses $\Lambda_{\rm IR}\sim Q\lambda^2$ then the soft gluons give scaleless loop integrals (they simply act to pull-up the ultraviolet divergences in the usoft integrals to the hard scale [@amis3; @hms1]), so the soft Wilson lines can be suppressed. This is why one only finds $S_n^\dagger S_{{\bar n}}$ for this operator in the literature. For typical regulator choices the other gluons are simply not required to reproduce the infrared structure of the full theory result.
[^4]: Note that a pure glue operator would not have the same isospin as the pion state.
[^5]: There are also gluon operators that can contribute when one or more of the mesons is a neutral isosinglet, however for simplicity these are not discussed here.
[^6]: These predictions depend on the dominance of the operators in Eq. (\[BLmech\]) over those in Eq. (\[Feyn\]).
|
---
abstract: 'In this paper, we investigate the performance of a dual-hop block fading cognitive radio network with underlay spectrum sharing over independent but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ fading channels. The primary network consists of a source and a destination. Depending on whether the secondary network which consists of two source nodes have a single relay for cooperation or multiple relays thereby employs opportunistic relay selection for cooperation and whether the two source nodes suffer from the primary users’ (PU) interference, two cases are considered in this paper, which are referred to as Scenario (a) and Scenario (b), respectively. For the considered underlay spectrum sharing, the transmit power constraint of the proposed system is adjusted by interference limit on the primary network and the interference imposed by primary user (PU). The developed new analysis obtains new analytical results for the outage capacity (OC) and average symbol error probability (ASEP). In particular, for Scenario (a), tight lower bounds on the OC and ASEP of the secondary network are derived in closed-form. In addition, a closed from expression for the end-to-end OC of Scenario (a) is achieved. With regards to Scenario (b), a tight lower bound on the OC of the secondary network is derived in closed-form. All analytical results are corroborated using Monte Carlo simulation method.'
author:
- |
Saeed Vahidian$^{1}$, Maryam Najafi$^{2}$, Marzieh Najafi$^{3}$, Fawaz S. Al-Qahtani$^{4}$\
$^{2}$Electrical & Computer Engineering, Shiraz University of Technology, Shiraz, Iran\
$^{3}$Electrical & Computer Engineering, K. N. Toosi University of Technology, Tehran, Iran\
$^{4}$Electrical & Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar
bibliography:
- 'refrence.bib'
title: 'Power Allocation and Cooperative Diversity in Two-Way Non-Regenerative Cognitive Radio Networks '
---
Introduction
============
In the last decade, a significant growth in wireless cellular network has been observed [@DBLP:conf/infocom/KiskaniS16; @VahidMarkTCOM2016]. However, the available technologies and the limited radio spectrum are unable to address the stringent demand for high data rate wireless communication [@DBLP:journals/wicomm/KiskaniWSOG15; @saeedhaj1; @Vahid]. It is desired to have a solution that can mitigate the problem of inadequate cellular spectrum [@sani11]. To cope with this, cognitive radio, has been proposed to alleviate the problem of resource congestion via utilization of frequency bands through dynamic spectrum management [@DBLP:conf/icseng/KiskaniK11; @DBLP:conf/mswim/KiskaniKV10]. There are three main cognitive radio protocols: underlay, overlay and interweave. The focus of this paper is on the underlay systems. The underlay paradigm allows secondary user (SU) utilize the spectrum as long as its interference to the primary network remains below a tolerable level [@Goldsmith:2009].
On the other hand, cooperative communication has appeared as a powerful spatial diversity technology for wireless communications to overcome the damaging effects of multipath fading [@VahidRabieiBeaulieuCommLetter]. Relay technology is one of these methods that are used to enhance the performance of wireless networks. Among various relaying schemes, best-relay selection scheme has drawn increasing research interests while being spectrally more efficient than repetition-based schemes [@Bletesas:JSAC:2006]. Best relay selection [@Vahidian:WCL:2015] is an ideal protocol to reach the best performance. There are two sorts of classical relay communication protocols, i.e., non-regenerative and regenerative protocols [@Vahidian:2015:WPC].
Two-way relay networks have aroused great interest due to their potential of significantly enhancing the network throughput as well as the spectrally efficient benefits. Thereby two end users exchange information simultaneously with each other in just two time phases via a half-duplex or multiple relays [@Vahidian1:TVT:2016].
Literature on Cognitive Relaying Network
----------------------------------------
Opportunistic relaying has been extensively studied for one-way relaying systems. For instance, in [@Duong:2012:TVT; @Shao:CommunL:2104; @VahidMarkWirelessLetter], the main purpose was to explore the effect of interference in unidirectional cognitive relay networks in terms of outage performance. In [@Duong:2012:TVT], the outage capacity (OC) of a dual-hop cognitive relay network has been explored where the power constraint of the secondary network has been derived with two different strategies. However, in this work the power constraint of the SUs depends on the knowledge of the instantaneous channel state information (CSI) of the link between the SUs and the primary users (PUs). The OC for incremental and DF relaying in underlay spectrum sharing systems over Nakagami-$m$ fading channels has been discussed in [@Shao:CommunL:2104]. Some studies have extended the opportunistic relaying to two-way relaying systems [@li:glob:2012; @tao:TSP:2013; @Vahidian:TVT:DF:2016; @Kong:Conf:2012; @pan:wcnc:2013; @Saeed.Vahidian:IET:2014; @saeed2]. In [@li:glob:2012] and [@tao:TSP:2013], analog network coding (ANC) and physical-layer network coding (PNC) were adopted in a two-way spectrum sharing relaying network, respectively. In [@Vahidian:TVT:DF:2016], a two-way cognitive radio system with a regenerative scheme has been studied in terms of average error rate, average sum-rate. The authors in [@pan:wcnc:2013] investigated an underlay cognitive two-way relaying network in multiple access broadcasting (MABC) and time division broadcasting (TDBC)[^1], respectively, where in [@Kong:Conf:2012] the interference caused by the PU was neglected.
Considerable efforts have been spent on addressing the performance of the cognitive radio networks. However, most of the prior works studied the performance of the cognitive radio networks for one-way relaying systems in Nakagami-$m$ environment or assumed Rayleigh fading channels. This may not be useful in a wide range of fading scenarios that are typical in realistic wireless relay applications. The performance of two-way relaying in spectrum sharing systems over Nakagami-$m$ fading channels is still much less understood. Moreover, in most of the previous works the power limitation of the secondary network depends on the knowledge of instantaneous CSI of the links between nodes [@sani123; @sani1234] while in some practical communication setups with high terminal speed the channel varies rapidly and experiences fast fading. Owing to these facts, the aim of this paper is to investigate the performance of dual-hop cognitive radio networks with non-regenerative relay/relays and subject to i.n.i.d. Nakagami-$m$ fading channels for two Scenarios. To our best knowledge, no similar studies have been reported despite the importance of such an issue. A detailed description of the considered Scenarios follows.
Considered Scenarios and Technical Contributions
------------------------------------------------
Scenario (a): Scenario (a) is composed of a pair of cognitive sources communicating with each other with the aid of a relay, while sharing the spectrum with a one-way primary network. In cognitive radio network, the source nodes and the relay are affected by all existed interferers and additive white Gaussian noise (AWGN). Scenario (b): Scenario (b) is the extension of Scenario (a) to a cognitive two-way multi-non-regenerative-relaying $({{{R}}_{{k}}}~~k = 1,...,K)$ dual-hop configuration with best relay selection strategy, where the source nodes only suffer from AWGN, while the relays are affected by all existed interferers as well as AWGN. In this Scenario the interference from the PU to the secondary sources is assumed to be neglected as in [@Lee:TEC:2011]. This can be justified if the primary transmitter is located far away from the secondary sources, or the interference is modeled as the noise term [@Lee:TEC:2011].
- In an attempt to assess the performance, the cumulative distribution function (cdf) of the SINR at primary destination is obtained from which a closed-form expressions for the OC is derived over Nakagami-$m$ fading channels, which build the relationship between the outage performance and the related system parameters. From a realistic point of view, the choice of Nakagami-$m$ fading includes some other fading channels that are more or less severe than Rayleigh fading via the $m$ fading parameter. For example, it includes the Rayleigh fading $(m=1)$ as a special case. The Nakagami-$m$ fading also approximates the Hoyt fading, for $m<1$, and the Rician fading, for $m>1$. On the basis of the outage performance of the primary network the power allocation of the SUs are presented.
- For both Scenarios, the exact equivalent SINR at the secondary sources are provided and upper bounded. Then, the cdfs of the upper bounded SINRs are explored. In addition, the selection of the best relay, $k$, is discussed. According to these results, tight lower bounds on the OC of Scenarios (a) and (b) are achieved. Furthermore, a closed-form expression for the end-to-end OC of the network in Scenario (a) is obtained.
- The tight lower bound closed-form expression for the ASEP of Scenario (a) is derived over Nakagami-$m$ fading channels.
The rest of this paper is organized as: Section II, introduces channel statistics and system models. The secondary network and its performance is examined in Section III. Numerical results are provided in Section IV and conclusions are given in Section V.
Channel Statistics and System Model {#sec:System_Model}
===================================
Let us consider a spectrum-sharing cooperative non-regenerative relaying model. In this model, $PT$, $PX$, ${S}_i~i=1, 2$ and $R$/$R_k$ represent a primary transmitter, a primary receiver, secondary sources and secondary relay/relays, respectively. Let us consider that there is no direct link between two sources due to poor channel condition [@hosseini2016real]. Further, we assume that the channels are reciprocal. A time-division multiple-access scheme is employed for the secondary nodes which is performed into two consecutive time intervals, namely the multiple-access (MA) phase and the broadcast (BC) phase. In the first phase, $PT$ sends its signal ${x_1}$, ${\rm{\mathbb{E}\{ | }}{{{x}}_1}{{{|}}^2}{{\} }} = 1$, drawn with equal probability from MPSK constellation of size $M$ to $PX$ with power ${P_{{PT}}}$. Meanwhile $S_i$ ${i=1, 2}$ sends its information $\hat x_i$ ${\rm{\mathbb{E}\{ | }}{{{\hat x}}_i}{{{|}}^2}{{\} }} = 1,~i=1, 2$ meant for the other with transmit power of ${P_{{{S_i}}}}$ under a transmit power constraint and the relay/relays receive(s) the signals. As was previously mentioned, we assume that the instantaneous channel gain of each link ($\left| h \right|$) follows the Nakagami-$m$ distribution, where integer $m$ stands for the fading severity parameter. As such, the distribution of the ${\left| h \right|^2}$ is Gamma random variable (RV), where $\frac{{\bar \gamma }}{m}$ is scale parameter, while ${{\bar \gamma }}$ is the mean value of Gamma RVs. Therefore, we have ${\left| {{h_{PT - R}}} \right|^2}\mathop \sim \limits^d {\rm{G}}\left( {{m_y},\frac{{{{\bar \gamma }_y}}}{{{m_y}}}} \right)$, ${\left| {{h_{R - {S_1}}}} \right|^2}\mathop \sim \limits^d {\rm{G}}\left( {{m_w},\frac{{{{\bar \gamma }_w}}}{{{m_w}}}} \right)$, ${\left| {{h_{R - {S_2}}}} \right|^2}\mathop \sim \limits^d {\rm{G}}\left( {{m_x},\frac{{{{\bar \gamma }_x}}}{{{m_x}}}} \right)$, ${\left| {{h_{PT - {S_1}}}} \right|^2}\mathop \sim \limits^d {\rm{G}}\left( {{m_z},\frac{{{{\bar \gamma }_z}}}{{{m_z}}}} \right)$, and ${\left| {{h_{PT - PX}}} \right|^2}\mathop \sim \limits^d {\rm{G}}\left( {{m_e},\frac{{{{\bar \gamma }_e}}}{{{m_e}}}} \right)$, where the symbol $\mathop \sim \limits^d $ denotes “distributed as”. Then, the signal received by ${R}$/${R_k}$ in Scenarios (a) and (b) and also by $PX$ in the MA phase can be written as $$\begin{aligned}
\label{gamma_SR}
{\scalebox{0.97}{${y_R} \hspace{-1mm}=\hspace{-1mm} \sqrt {{P_{{PT}}}} {h_{{PT} - R}}{x_1} \hspace{-1mm}+\hspace{-1mm} \sqrt {{P_{{S_1}}}} {h_{{S_1} - R}}{{\hat x}_1} \hspace{-1mm}+\hspace{-1mm} \sqrt {{P_{{S_2}}}} {h_{{S_2} - R}}{{\hat x}_2} \hspace{-1mm}+\hspace{-1mm} {n_2}$}},\end{aligned}$$ $$\begin{aligned}
\label{Y_SR}
{\scalebox{0.97}{${y_{{R_k}}} \hspace{-1.2mm}=\hspace{-1.2mm} \sqrt {{P_{{PT}}}} {h_{{PT} \hspace{-0.5mm}- \hspace{-0.5mm}{R_k}}}{x_1} \hspace{-1.2mm}+\hspace{-1.2mm} \sqrt {{P_{{S_1}}}} {h_{{S_1} \hspace{-0.6mm}- \hspace{-0.6mm} {R_k}}}{{\hat x}_1} \hspace{-1.2mm}+\hspace{-1.2mm} \sqrt {{P_{{S_2}}}} {h_{{S_2} \hspace{-0.6mm}- \hspace{-0.6mm}{R_k}}}{{\hat x}_2} \hspace{-1.2mm}+ \hspace{-1.2mm}{n_{{2_k}}}$}},\end{aligned}$$ $$\begin{aligned}
{\scalebox{0.97}{${y_{{PX}}} \hspace{-1.2mm}=\hspace{-1.3mm} \sqrt {{P_{PT}}} {h_{{PT} \hspace{-0.9mm}- \hspace{-0.9mm} {PX}}}{x_1}\hspace{-1.3mm} + \hspace{-1.2mm}\sqrt {{P_{{S_1}}}} {h_{{S_1} \hspace{-0.9mm}- \hspace{-0.9mm} {PX}}}{{\hat x}_1} \hspace{-1.2mm}+ \hspace{-1.2mm}\sqrt {{P_{{S_2}}}} {h_{{S_2} \hspace{-0.9mm}- \hspace{-0.9mm} {PX}}}{{\hat x}_2} \hspace{-1.2mm}+ \hspace{-1.2mm}{n_1}$}},
\label{gamma_PD1}\end{aligned}$$where ${h_{{{PT - PX}}}}$, ${h_{{{S_1 - PX}}}}$, ${h_{{{S_2 - PX}}}}$, ${h_{PT - R/{R_k}}}$, ${h_{S_1- R/{R_k}}}$ and ${h_{S_2 - R/{R_k}}}$ are the channel coefficients of ${{PT}} \hspace{-1.3mm} \to\hspace{-1.3mm} {{PX}},{\mkern 1mu} {\mkern 1mu} \,{{S_1}} \hspace{-1.3mm}\to\hspace{-1.3mm} {{PX}},{\mkern 1mu} {\mkern 1mu} \,{{S_2}} \hspace{-1.3mm}\to \hspace{-1.3mm} {{PX}},{\mkern 1mu} \,PT \to R/{R_k}$, ${S_1} \to R/{R_k}$ and ${S_2} \to R/{R_k}$ links, respectively, while ${n_1}$ and ${n_2}/{n_{{2_k}}}$ are AWGN at the primary receiver terminal and the relay/relays, respectively. Noise signals at all nodes are zero mean with power spectral density of ${N_0}$. In the second phase, PT, sends another information message $x_2$ ${\rm{\mathbb{E}\{ | }}{{{x}}_2}{{{|}}^2}{{\} }} = 1$ to $PX$. Meanwhile in the second phase, $\rm{R}$/${{\rm{R}}_k}$, simply scales the received signal before broadcasting to the destination with transmit power ${P_R}/{P_{{R_k}}}$. Accordingly, the received signal at $PX$ and $S_1$ in Scenario (a) and (b) are given respectively by $$\begin{aligned}
&{\scalebox{0.97}{${y_{{PX}}} = \sqrt {{P_{{\rm{PT}}}}} {h_{{\rm{PT - PX}}}}{x_2}+ {{{G}}_k}\sqrt {{P_{{\rm{}}{{\rm{R}}_k}}}} {h_{{\rm{}}{{\rm{R}}_k}{\rm{ - PD}}}}{y_{{{\rm{R}}_k}}} + {n_{\rm{1}}}$}},
\label{Y_PX2}\end{aligned}$$ $$\begin{aligned}
&{\scalebox{0.97}{${y_{{S_1}}} = \sqrt {{P_{PT}}} {h_{PT - {S_1}}}{x_2} + G\sqrt {{P_R}} \sqrt {{P_{PT}}} {h_{PT - R}}{h_{R - {S_1}}}{x_1}$}}
\notag \\
&\;+ {\scalebox{0.97}{$G\sqrt {{P_R}} \sqrt {{P_{{S_2}}}} {h_{{S_2} - R}}{h_{R - {S_1}}}{{\hat x}_2} + G\sqrt {{P_R}} {h_{R - {S_1}}}{n_2} + {n_3}$}},\label{Y_ST1}\end{aligned}$$ $$\begin{aligned}
&{\scalebox{0.97}{${y_{{S_{1,\,k}}}} = {G_k}\sqrt {{P_{{R_k}}}} \sqrt {{P_{PT}}} {h_{{R_k} - {S_1}}}{h_{PT - {R_k}}}{x_1} +
{G_k}\sqrt {{P_{{R_k}}}}$}} \notag
\\
&{\scalebox{0.97}{$\times\sqrt {{P_{{S_2}}}} {h_{{R_k} - {S_1}}}{h_{{S_2} - {R_k}}}{{\hat x}_2} \hspace{-1mm}+\hspace{-1mm} {G_k}\sqrt {{P_{{R_k}}}} {h_{{R_k} - {S_1}}}{n_{{2_k}}} \hspace{-1.2mm}+\hspace{-1.2mm} {n_{3_k}}$}},
\label{Y_ST1_Selection}\end{aligned}$$where ${h_{{{PT - S_1}}}}$ is the channel coefficient of ${\mkern 1mu} {{PT}} \hspace{-1.3mm}\to\hspace{-1.3mm} {{S_1}}$ link, while ${n_3}$/${n_{3_k}}$, is AWGN at $S_1$/$S_{1,k}$ and $G$ and $G_k$, are the relay amplification gains defined as $$\begin{aligned}
&{\scalebox{0.93}{${G^{ - 1}}\hspace{-0.5mm} \buildrel \Delta \over =\hspace{-0.5mm} \sqrt {{P_{PT}}{{\left| {{h_{PT - R}}} \right|}^2} \hspace{-0.5mm}+ \hspace{-0.5mm} {P_{{S_1}}}{{\left| {{h_{{S_1} - R}}} \right|}^2} \hspace{-0.5mm}+\hspace{-0.5mm} {P_{{S_2}}}{{\left| {{h_{R - {S_2}}}} \right|}^2} \hspace{-0.5mm}+\hspace{-0.5mm} {N_0}}$}}, \label{G}\end{aligned}$$ $$\begin{aligned}
{\scalebox{0.93}{$G_k^{ - 1} \hspace{-1mm}\buildrel \Delta \over = \hspace{-1mm}{\rm{ }}\sqrt {{P_{PT}}{{\left| {{h_{PT - {R_k}}}} \right|}^2} \hspace{-1mm}+\hspace{-1mm} {P_{{S_1}}}{{\left| {{h_{{S_1} - {R_k}}}} \right|}^2} \hspace{-1mm}+\hspace{-1mm} {P_{{S_2}}}{{\left| {{h_{{R_k} - {S_2}}}} \right|}^2} \hspace{-1mm}+\hspace{-1mm} {N_0}}$}}. \label{G_k}\end{aligned}$$
In the following sections, we will evaluate the network performance in key operation metrics.
Performance Analysis
====================
In this section the statistics of the equivalent SINR will be explored, tight lower bounds on the most important performance metrics of the secondary network will be quantified and some useful observations in this realm will be provided.
Primary OC and Power Constraint of the Secondary Nodes
------------------------------------------------------
In this subsection, we present some of statistics for the OC of the primary network. The OC is defined as the probability that the instantaneous SINRs falls below a predetermined rate $R$ bits/sec/Hz. Hence the primary OC can be explored by the following theorem:
\[Thm-outage probability-DMM\] *The OC of the primary network is obtained in closed-form expression as shown in ,*
$$\begin{aligned}
&{\scalebox{0.97}{${P_{{\rm{out,}}\,{\rm{pri}}}} = 1 -
\frac{{\left( {{m_e} - 1} \right)!{{\left( {{m_f}} \right)}^{{m_f}}}{{\left( {{m_g}} \right)}^{{m_g}}}}}{{\Gamma \left( {{m_e}} \right)\Gamma \left( {{m_g}} \right){{\left( {{{\bar \gamma }g}} \right)}^{{m_g}}}\Gamma \left( {{m_f}} \right){{\left( {\bar \gamma f^{}} \right)}^{{m_f}}}}} \exp \left( { - \left( {\frac{{{m_e}}}{{{{\bar \gamma }e}{{\bar \gamma }{\rm{P}}}}}{\gamma {{\rm{th}}}}} \right)} \right) \sum\limits_{\ell = 0}^{{m_e} - 1} {\sum\limits_{{\tau 1=0}}^\ell {\sum\limits_{{\tau 2=0}}^{\ell - {\tau 1}} {\frac{{{{\left( {\frac{{{m_e}{\gamma {{\rm{th}}}}}}{{{{\bar \gamma }e}{{\bar \gamma }{\rm{P}}}}}} \right)}^\ell }{{\left( {{{\bar \gamma }{{{\rm{S}}_1}}}} \right)}^{{\tau 1}}}{{\left( {{{\bar \gamma }{{{\rm{S}}_2}}}} \right)}^{{\tau _2}}}}}{{\ell !}}} } }$}}
\nonumber\\&
{\scalebox{0.97}{$\times \left( \begin{array}{l}
\ell - {\tau _1}\\
\,\,{\tau _2}
\end{array} \right)\,\left( \begin{array}{l}
\ell \\
{\tau _1}
\end{array} \right) \Gamma \left( {{m_f} + {\tau _1}} \right)
\Gamma \left( {{m_g} + {\tau 2}} \right){\left( {\frac{{{m_f}}}{{{{\bar \gamma }f}}} + \frac{{{m_e}{{\bar \gamma }{{{\rm{S}}_1}}}}}{{{{\bar \gamma }e}{{\bar \gamma }{\rm{P}}}}}{\gamma {{\rm{th}}}}} \right)^{ - \left( {{m_f} + {\tau 1}} \right)}}{\left( {\frac{{{m_g}}}{{{{\bar \gamma }g}}} + \frac{{{m_e}{{\bar \gamma }{{{\rm{S}}_2}}}}}{{{{\bar \gamma }e}{{\bar \gamma }{\rm{P}}}}}{\gamma {{\rm{th}}}}} \right)^{ - \left( {{m_g} + {\tau _2}} \right)}}$}},
\label{OP of PN}\end{aligned}$$
------------------------------------------------------------------------
where ${\gamma _{{\rm{th}}}}=2^{R_{\rm{P}}}-1$ and ${R_{\rm{P}}}$ is the primary transmission rate.
*See Appendix \[apx1\]*\
In both Scenarios, by assuming ${\bar \gamma _{{S_1}}} = {\bar \gamma _{{S_2}}}$ and without loss of generality, the power limitation of $S_i,~i=1, 2$ are derived under the constraint which guarantees that the interference on $PX$ does not exceed a threshold. Therefore, the power constraint of $S_i~i=1, 2$ is obtained by solving ${P_{{\rm{out,}}\,{\rm{pri}}}} \le P_{{\rm{out}},{\rm{pri}}}^{{\rm{Thr}}}$ w.r.t. ${P _{{S_1}}} $.
Similar to *Theorem \[Thm-outage probability-DMM\]* and according to the transmit power of $k$th secondary relay regulated alone by the primary outage constraint can be readily achieved by solving w.r.t. ${P _{{R}}} $. $$\begin{aligned}
&{\scalebox{0.97}{${P_{{\rm{out,}}\,{\rm{{pri}_2}}}} =
1 - \Gamma(m_e)\exp \left( { - \frac{{{m_e}\Theta }}{{{{\bar \gamma }_e}{{\bar \gamma }_{\rm{P}}}}}} \right){\sum\limits_{\ell = 0}^{{m_e} - 1} {\left( { - \frac{{{m_e}\Theta }}{{{{\bar \gamma }_e}{{\bar \gamma }_{\rm{P}}}}}} \right)} ^\ell }\frac{{{{\left( {{m_l}} \right)}^{{m_l}}}}}{{\Gamma \left( {{m_l}} \right)\bar \gamma _l^{{m_l}}}} $}}\nonumber\\&
{\scalebox{0.97}{$\frac{{\sum\limits_{i = 0}^\ell {l \choose i} }}{{\ell !}}\left( {{m_l} - 1 + \ell - i} \right)!{\left( {\frac{{{m_e}{{\bar \gamma }_{\rm{R}}}\Theta }}{{{{\bar \gamma }_e}{{\bar \gamma }_{\rm{P}}}}} + \frac{{{m_l}}}{{{{\bar \gamma }_l}}}} \right)^{ - \left( {{m_l} + \ell - i} \right)}} \le P_{{\rm{out}},{\rm{pri}}}^{{\rm{Thr}}}$}}.
\label{OP of PN2}\end{aligned}$$
cdf of the Equivalent SINRs
---------------------------
By substituting into and after some algebraic manipulations of the SINR at $S_1$ in Scenario (a) can be obtained as in Eq. (11) of [@Vahidian1:TVT:2016]. To simplify the ensuing derivations, ${\gamma _{{{S_1}}}}$ and ${\gamma _{{S_1},\,k}}$ should be addressed in more mathematically tractable forms. As such, we use the tight upper bounds for ${\gamma _{{{S_1}}}}$ and ${\gamma _{{S_1},\,k}}$ as presented in Eqs. (9) and (24) of [@Vahidian1:TVT:2016]. In the next step we proceed to explore the OC of both Scenarios.
### cdf of the SINR of Scenario (a)
The closed-form of the cdf of $\gamma _{{{S_1}}}^{{\rm{up}}}{\mkern 1mu}$ for Scenario (a) can be presented in the following key result.
\[op-secanrio-a\] *The cdf of the SINR of Scenario (a) is obtained in closed-form expression as shown in*
$$\begin{aligned}
&{\scalebox{0.97}{${F_{\gamma _{{S_1}}^{{\rm{up}}}}}\left( \Theta \right)\approx 1 - \left( {{m_x} - 1} \right)!\left( {{m_w} - 1} \right)!\frac{{{{\left( {{m_Y}} \right)}^{{m_Y}}}{{\left( {{m_z}} \right)}^{{m_z}}}{{\left( {{m_z}} \right)}^{{m_z}}}}}{{\Gamma \left( {{m_x}} \right)\Gamma \left( {{m_y}} \right)\Gamma \left( {{m_z}} \right)\Gamma \left( {{m_w}} \right)\Gamma \left( {{m_z}} \right){{\left( {\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }y}} \right)}^{{m_Y}}}{{\left( {{{\bar \gamma }{\rm{P}}}{{\bar \gamma }_z}} \right)}^{2{m_z}}}}} $}}\nonumber\\&
{\scalebox{0.97}{$\times \exp \left( { - \left( {\frac{{\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right){m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_w}\Theta }}{{{{\bar \gamma }W}{{\bar \gamma }{\rm{R}}}}}} \right)} \right) \times \sum\limits_{n = 0}^{{m_x} - 1} {\sum\limits_{{i_1} = 0}^n {\sum\limits_{{i_2} = 0}^{n - {i_1}} {\sum\limits_{k = 0}^{{m_w} - 1} {\sum\limits_{{k_1} = 0}^k {\left( \begin{array}{l}
k\\
{k_1}
\end{array} \right)} } \left( \begin{array}{l}
n\\
{i_1}
\end{array} \right)} } \left( \begin{array}{l}
n - {i_1}\\
\,\,\,{i_2}
\end{array} \right)} \times$}} \nonumber\\&
{\scalebox{0.95}{$\frac{{{{\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right)}^{n - {i_1} - {i_2}}}{{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}}} \right)}^n}{{\left( {\frac{{{m_w}\Theta }}{{{{\bar \gamma }W}{{\bar \gamma }{\rm{R}}}}}} \right)}^k}}}{{n!k!}}\left( {{m_y} + {i_1} - 1} \right)!\left( {{m_z} + {i_2} - 1} \right)!\left( {{m_z} + {k_1} - 1} \right)!{\left( {\frac{{{m_w}\Theta }}{{{{\bar \gamma }W}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_z}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}}}} \right)^{ - \left( {{m_z} + {k_1}} \right)}} $}} \nonumber\\&
{\scalebox{0.97}{$\times{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_z}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}}}} \right)^{ - \left( {{m_z} + {i_2}} \right)}}\,{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_y}}}{{\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }y}}}} \right)^{ - \left( {{m_y} + {i_1}} \right)}}$}},
\label{CDF}
\end{aligned}$$
------------------------------------------------------------------------
where $\Theta$ is the outage threshold.
*See Appendix \[apx2\].*
### End-to-End cdf of the SINR of Scenario (a)
The end-to-end cdf of the SINR of Scenario (a) is derived as
$$\begin{aligned}
{F_{\gamma _{{\rm{e2e}}}^{{\rm{up}}}}}\left( \Theta \right) \approx 1 - \left[\Upsilon _1+\Upsilon _2 \right] \left[\Upsilon _3+\Upsilon _4 \right]
\end{aligned}$$ where $\Upsilon _1$ and $\Upsilon _2$ are given respectively in and . The terms $\Upsilon _3$ and $\Upsilon _4$ can be obtained respectively by replacing appropriately the parameters in $\Upsilon _1$ and $\Upsilon _2$, i.e., ${m_v} \leftrightarrow {m_z}$, ${m_x} \leftrightarrow {m_w}$, ${\bar \gamma _x} \leftrightarrow {\bar \gamma _w}$ and ${\bar \gamma _v} \leftrightarrow {\bar \gamma _z}$.
$$\begin{aligned}
\label{Upsilon1}
&{\scalebox{0.97}{$\Upsilon 1 =\frac{{{{\left( {{m_y}} \right)}^{{m_y}}}{{\left( {{m_z}} \right)}^{{m_z}}}}}{{\Gamma \left( {{m_y}} \right)\Gamma \left( {{m_z}} \right){{\left( {{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}} \right)}^{{m_z}}}{{\left( {\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }y}} \right)}^{{m_y}}}}}\rm{exp}\left( { - \left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right)} \right)\sum\limits_{\rho = 0}^{{m_x} - 1} {\sum\limits_{{\varepsilon 1} = 0}^\rho {\sum\limits_{{\varepsilon 2} = 0}^{\rho - {\varepsilon 1}} {\left( \begin{array}{l}
\rho \\
{\varepsilon _1}
\end{array} \right)} } \left( \begin{array}{l}
\rho - {\varepsilon _1}\\
\,\,\,{\varepsilon _2}
\end{array} \right)} $}} \nonumber\\&
{\scalebox{0.97}{$\times {{\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right)}^{\rho - {\varepsilon 1} - {\varepsilon 2}}} \frac{{{{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}}} \right)}^\rho }}}{{\rho !}}\Gamma \left( {{m_z} + {\varepsilon 2}} \right){\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_z}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}}}} \right)^{{m_z} + {\varepsilon 2}}}\left( {{m_y} + {\varepsilon 1} - 1} \right)!{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_y}}}{{\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }_y}}}} \right)^{{m_y} + \varepsilon }} - $}}\nonumber\\&
{\scalebox{0.95}{$\frac{{{{\left( {{m_y}} \right)}^{{m_y}}}{{\left( {{m_z}} \right)}^{{m_z}}}}}{{\Gamma \left( {{m_y}} \right)\Gamma \left( {{m_z}} \right){{\left( {{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}} \right)}^{{m_z}}}{{\left( {\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }y}} \right)}^{{m_y}}}}}\exp \left( { - \left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}}\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right)} \right) + \left( {\frac{{{m_v}{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}{{\bar \gamma }{\rm{S}}}}}} \right)} \right) $}} \nonumber\\&
{\scalebox{0.97}{$\times\sum\limits_{\varpi = 0}^{{m_v} - 1} {\sum\limits_{\rho = 0}^{{m_x} - 1} {\sum\limits_{{\varepsilon 1} = 0}^\rho {\sum\limits_{{\varepsilon 2} = 0}^{\rho - {\varepsilon _1}} {\sum\limits_{{t_1} = 0}^\varpi {\sum\limits_{{t_2} = 0}^{\varpi - {t_1}} {\left( \begin{array}{l}
\varpi \\
{t_1}
\end{array} \right)} } \left( \begin{array}{l}
\varpi - {t_1}\\
\,\,\,{t_2}
\end{array} \right)\left( \begin{array}{l}
\rho \\
{\varepsilon _1}
\end{array} \right)} } \left( \begin{array}{l}
\rho - {\varepsilon _1}\\
\,\,\,{\varepsilon _2}
\end{array} \right){{\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right)}^{\rho - {\varepsilon 1} - {\varepsilon 2}}}{{\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}}} \right)}^{\varpi - {t_1} - {t_2}}} \times } }$}}\nonumber\\&
{\scalebox{0.97}{$\frac{{{{\left( {\frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}}} \right)}^\varpi }{{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}}} \right)}^\rho }}}{{\varpi !\rho !}}\Gamma \left( {{t_2} + {\varepsilon 2} + {m_z}} \right){\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}} + \frac{{{m_z}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}}}} \right)^{-\left( {t_2} + {\varepsilon 2} + {m_z}\right)}}\Gamma \left( {{t_1} + {\varepsilon _1} + {m_y}} \right) $}}
{\scalebox{0.97}{$\times{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}} + \frac{{{m_y}}}{{\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }y}}}} \right)^{-\left( {t_1} + {\varepsilon _1} + {m_y}\right)}}$}},
\end{aligned}$$
------------------------------------------------------------------------
$$\begin{aligned}
\label{Upsilon2}
&{\scalebox{0.99}{$\Upsilon _2=
\frac{{{{\left( {{m_v}} \right)}^{{m_v}}}}}{{\Gamma \left( {{m_v}} \right){{\left( {{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}} \right)}^{{m_v}}}}}\frac{{{{\left( {{m_x}} \right)}^{{m_x}}}}}{{\bar \gamma x^{{m_x}}}}\exp \left( { - \frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}}} \right)\exp \left( { - \frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}}\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}}} \right)} \right)\frac{{{{\left( {{m_y}} \right)}^{{m_y}}}}}{{\Gamma \left( {{m_y}} \right){{\left( {\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }_y}} \right)}^{{m_y}}}}}$}} \nonumber\\&
{\scalebox{0.99}{$\times\frac{{{{\left( {{m_z}} \right)}^{{m_z}}}}}{{\Gamma \left( {{m_z}} \right){{\left( {{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}} \right)}^{{m_z}}}}}\sum\limits_{i = 0}^{{m_x} - 1} {\sum\limits_{j = 0}^i {\sum\limits_{k = 0}^{{m_v} + j - 1} {\sum\limits_{{k_1} = 0}^k {\sum\limits_{{k_2} = 0}^{{k_1}} {\left( \begin{array}{l}
{k_1}\\
{k_2}
\end{array} \right)} \left( \begin{array}{l}
k\\
{k_1}
\end{array} \right)} \,{{\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}}} \right)}^{k - {k_1}}}\left( \begin{array}{l}
i\\
j
\end{array} \right)} } } $}} \nonumber\\&
{\scalebox{0.99}{$\times\frac{{\Gamma \left( {{m_v} + j} \right){{\left( {\frac{\Theta }{{{{\bar \gamma }{\rm{R}}}}}} \right)}^i}\Gamma \left( {{k_2} + {m_z}} \right)\Gamma \left( {{k_1} + {m_y} - {k_2}} \right)}}{{\left( i \right)!\left( k \right)!{{\left( {\frac{{{m_x}}}{{{{\bar \gamma }x}}}} \right)}^{{m_x} - i}}{{\left( {\frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}}} \right)}^{{m_v} + j - k}}}} {\left( {\frac{{{m_z}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }z}}} + \frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}}} \right)^{ - \left( {{k_2} + {m_z}} \right)}}
{\left( {\frac{{{m_y}}}{{\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }y}}} + \frac{{{m_x}\Theta }}{{{{\bar \gamma }x}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_v}}}{{{{\bar \gamma }{\rm{P}}}{{\bar \gamma }v}}}} \right)^{ - \left( {{k_1} + {m_y} - {k_2}} \right)}}.$}}
\end{aligned}$$
------------------------------------------------------------------------
### cdf of the SINR of Scenario (b)
In general, by applying the best relay selection strategy, the relay which has the highest value of end-to-end SINR is viewed as the “best” one. Hence, the end-to-end received SINR at terminals is given by $$\begin{aligned}
\gamma _{{\rm{e2e}}}^{{\rm{up}}} \hspace{-1mm}=\hspace{-1mm} \mathop {\max }\limits_{k = 1,2,...,K} \left\{ {\gamma _{}^{{\rm{up}}}} \right\} = \hspace{-1mm}\mathop {\max }\limits_{k = 1,2,...,K} \left\{ {\min \left( {\gamma _{{{S_{1,k}}}}^{{\rm{up}}},\gamma _{{{S_{2,k}}}}^{{\rm{up}}}} \right)} \right\}.
\end{aligned}$$ Since the exact cdf of the end-to-end SINR is intractable, in the following, we obtain the cdf of the end-to-end upper bounded SINR.
\[op-secanrio-b\] *The cdf of the end-to-end SINR of Scenario (b) is derived in .*
$$\begin{aligned}
&{\scalebox{0.95}{${F_{\gamma _{{\rm{e2e}}}^{{\rm{up}}}}}\left( \Theta \right) =
\sum\limits_{k = 0}^K {\frac{{{{\left( { - 1} \right)}^k}}}{{k!}}} \,\,\,\,\sum\limits_{{\varpi 1},...,{\varpi k}}^K {\,\,\prod\limits_{\tau = 1}^k {\left[ {\frac{{{{\left( {{m_{{y_{{\varpi \tau }}}}}} \right)}^{{m_{{y_{{\varpi \tau }}}}}}}}}{{\Gamma \left( {{m_{{y_{{\varpi \tau }}}}}} \right){{\left( {\frac{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}}}{{{{\bar \gamma }{\rm{S}}}}}{{\bar \gamma }{{y_{{\varpi \tau }}}}}} \right)}^{{m_{{y_{{\varpi _\tau }}}}}}}}}} \right. \times } } $}}\nonumber\\&
{\scalebox{0.97}{$\exp \left( { - \left( {\left( {\frac{{{m_{{x_{{\varpi \tau }}}}}}}{{{{\bar \gamma }{{x_{{\varpi \tau }}}}}{{\bar \gamma }{\rm{R}}}}}} \right)\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right) + \left( {\frac{{{m_{{w_{{\varpi \tau }}}}}}}{{{{\bar \gamma }{{w_{{\varpi \tau }}}}}{{\bar \gamma }{\rm{R}}}}}} \right)\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}} + 1} \right)} \right)\Theta } \right) \times \sum\limits_{n = 0}^{{m_{{x_k}}} - 1} {\sum\limits_{{n_1} = 0}^{{m_{{x_k}}} - 1} {\sum\limits_{{i_1} = 0}^{n + {n_1}} {\sum\limits_{{i_2} = 0}^{n + {n_1} - {i_1}} {\frac{{\Gamma \left( {{i_1} + {m_{{y_{{\varpi _\tau }}}}}} \right)}}{{{n_1}!n!}}} } } } $}}\nonumber\\&
{\scalebox{0.97}{$ \times {{\left( {\frac{{{m_{{w_{{\varpi \tau }}}}}\Theta }}{{{{\bar \gamma }{{w_{{\varpi \tau }}}}}{{\bar \gamma }{\rm{R}}}}}} \right)}^{{n_1}}} {\left( {\frac{{{m_{{x_{{\varpi \tau }}}}}\Theta }}{{{{\bar \gamma }{{x_{{\varpi \tau }}}}}{{\bar \gamma }{\rm{R}}}}}} \right)^n}\binom{n+n_1}{i_1}\binom{n + {n_1} - {i_1}}{i_2} {\left( {\frac{{{{\bar \gamma }{\rm{R}}}}}{{{{\bar \gamma }{\rm{S}}}}}} \right)^{{i_2}}}\left. {{{\left( {\frac{{{m_{{x_{{\varpi \tau }}}}}\Theta }}{{{{\bar \gamma }{{x_{{\varpi \tau }}}}}{{\bar \gamma }{\rm{R}}}}} + \frac{{{m_{{w_{{\varpi \tau }}}}}\Theta }}{{{{\bar \gamma }{{w_{{\varpi \tau }}}}}{{\bar \gamma }{\rm{R}}}}} + \frac{{{{\bar \gamma }{\rm{S}}}{m_{{y_{{\varpi \tau }}}}}}}{{{{\bar \gamma }{\rm{R}}}{{\bar \gamma }{\rm{P}}}{{\bar \gamma }{{y_{{\varpi \tau }}}}}}}} \right)}^{ - \left( {{i_1} + {m_{{y_{{\varpi _\tau }}}}}} \right)}}} \right]$}}.
\label{CDF_Selection}
\end{aligned}$$
------------------------------------------------------------------------
*See Appendix \[apx3\].*
By using and , we can now obtain the following lower bounds on the exact OC of the Scenarios (a) and (b) respectively as $$\begin{aligned}
{P_{{\rm{ou}}{{\rm{t}}_{{\rm{e2e}}}}}} = {F_{\gamma _{{\rm{e2e}}}^{{\rm{up}}}}}\left( {{\gamma _{{\rm{th}}}}} \right),\label{OP-Scenario a}\\
{P_{{\rm{ou}}{{\rm{t}}_{{S_1}}}}} = {F_{\gamma _{{{\rm{S}}_{\rm{1}}}}^{{\rm{up}}}}}\left( {{\gamma _{{\rm{th}}}}} \right).\label{OP-Scenario b}
\end{aligned}$$
Error Probability
------------------
In this subsection we concentrate on the ASEP of the secondary network for Scenario (a) as another important performance evaluation metric. For several modulation schemes, in practical systems ASEP can be lower bounded as [@Duong:Let:2013 Eq. (9)]. The following theorem summarizes the ASEP of Scenario (a).
*We derive a closed-form expression for the ASEP of Scenario (a) as in shown on the top of the next page,*
$$\begin{aligned}
&{\scalebox{0.98}{$\bar P_{{S_1}}^e = \frac{a}{2} - \frac{{a\sqrt b }}{{2\sqrt \pi }}\left( {{m_x} - 1} \right)!\left( {{m_w} - 1} \right)!\frac{{{{\left( {{m_Y}} \right)}^{{m_Y}}}{{\left( {{m_z}} \right)}^{{m_z}}}{{\left( {{m_z}} \right)}^{{m_z}}}}}{{\Gamma \left( {{m_x}} \right)\Gamma \left( {{m_y}} \right)\Gamma \left( {{m_z}} \right)\Gamma \left( {{m_w}} \right)\Gamma \left( {{m_z}} \right){{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{{\bar \gamma }_y}} \right)}^{{m_Y}}}{{\left( {{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_z}} \right)}^{2{m_z}}}}} \times $}} \nonumber\\&
{\scalebox{0.98}{$\sum\limits_{n = 0}^{{m_x} - 1} {\sum\limits_{{i_1} = 0}^n {\sum\limits_{{i_2} = 0}^{n - {i_1}} {\sum\limits_{k = 0}^{{m_w} - 1} {\sum\limits_{{k_1} = 0}^k {\left( \begin{array}{l}
k\\
{k_1}
\end{array} \right)} } \left( \begin{array}{l}
n\\
{i_1}
\end{array} \right)} } \left( \begin{array}{l}
n - {i_1}\\
\,\,\,{i_2}
\end{array} \right)\frac{{{{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)}^{n - {i_1} - {i_2}}}{{\left( {\frac{{{m_x}}}{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}}}} \right)}^n}{{\left( {\frac{{{m_w}}}{{{{\bar \gamma }_W}{{\bar \gamma }_{\rm{R}}}}}} \right)}^k}}}{{n!k!}}} \times $}} \nonumber\\&
{\scalebox{0.98}{$\left( {{m_y} + {i_1} - 1} \right)!\left( {{m_z} + {i_2} - 1} \right)!\left( {{m_z} + {k_1} - 1} \right)!\, \times {\left( {\frac{{{m_w}}}{{{{\bar \gamma }_W}{{\bar \gamma }_{\rm{R}}}}}} \right)^{ - \left( {{m_z} + {k_1}} \right)}}{\left( {\frac{{{m_x}}}{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}}}} \right)^{ - \left( {{m_z} + {i_2}} \right)}}{\left( {\frac{{{m_x}}}{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}}}} \right)^{ - \left( {{m_y} + {i_1}} \right)}}$}} \times \nonumber\\&
{\scalebox{0.98}{$ \times \left[ {\sum\limits_{{j_1} = 1}^{{m_z} + {k_1}} {{A_{{j_1}}}} \Gamma \left( {n + k + \frac{1}{2}} \right){\alpha _1}^{n + k + \frac{1}{2} - {j_1}}\Psi \left( {n + k + \frac{1}{2},n + k - \frac{1}{2} - {j_1},\left( {\frac{{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right){m_x}}}{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}}} + \frac{{{m_w}}}{{{{\bar \gamma }_W}{{\bar \gamma }_{\rm{R}}}}} + b} \right){\alpha _1}} \right)} \right.$}} \nonumber\\&
{\scalebox{0.98}{$ + \sum\limits_{{j_2} = 1}^{{m_z} + {i_2}} {{A_{{j_2}}}} \Gamma \left( {n + k + \frac{1}{2}} \right){\alpha _1}^{n + k + \frac{1}{2} - {j_2}}\Psi \left( {n + k + \frac{1}{2},n + k - \frac{1}{2} - {j_2},\left( {\frac{{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right){m_x}}}{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}}} + \frac{{{m_w}}}{{{{\bar \gamma }_W}{{\bar \gamma }_{\rm{R}}}}} + b} \right){\alpha _2}} \right)$}} \nonumber\\&
{\scalebox{0.98}{$\left. { + \sum\limits_{{j_3} = 1}^{{m_y} + {i_1}} {{A_{{j_3}}}} \Gamma \left( {n + k + \frac{1}{2}} \right){\alpha _1}^{n + k + \frac{1}{2} - {j_3}}\Psi \left( {n + k + \frac{1}{2},n + k - \frac{1}{2} - {j_3},\left( {\frac{{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right){m_x}}}{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}}} + \frac{{{m_w}}}{{{{\bar \gamma }_W}{{\bar \gamma }_{\rm{R}}}}} + b} \right){\alpha _3}} \right)} \right]$}},
\label{SEP_Final}
\end{aligned}$$
------------------------------------------------------------------------
where ($a$,$b$) are constants specified by the type of modulation; $\Gamma \left( . \right)$ is the gamma function [@Gradshteyn:Table:2007 Eq. (8.310.1)], $\Psi \left( {.,.,.} \right)$ is the Tricomi confluent hypergeometric function [@Gradshteyn:Table:2007 Eq. (9.210.2)]; ${A_{{m_z+k_1-j_1+1}}} \hspace{-1mm}=\hspace{-1mm} \frac{{\psi _1^{\left( {{m_z} + {k_2} - {j_1}} \right)}\left( { - {\alpha _1}} \right)}}{{\left( {{j_1} - 1} \right)!}}$, ${A_{{j_2}}} \hspace{-1mm}=\hspace{-1mm} \frac{{\psi _2^{\left( {{m_z} + {i_2} - {j_2}} \right)}\left( { - {\alpha _2}} \right)}}{{\left( {{m_z} + {i_2} - {j_2}} \right)!}}$, ${A_{{j_3}}} \hspace{-1mm}=\hspace{-1mm} \frac{{\psi _3^{\left( {{m_y} + {i_1} - {j_3}} \right)}\left( { - {\alpha _3}} \right)}}{{\left( {{m_y} + {i_1} - {j_3}} \right)!}}$; ${\alpha _1} \hspace{-1mm}=\hspace{-1mm} \frac{{{{\bar \gamma }_W}{{\bar \gamma }_{\rm{R}}}{m_z}}}{{{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_z}{m_w}}}$, ${\alpha _2} \hspace{-1mm}=\hspace{-1mm} \frac{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{R}}}{m_z}}}{{{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_z}{m_x}}}$, $\,{\alpha _3} \hspace{-1mm}=\hspace{-1mm} \frac{{{{\bar \gamma }_x}{{\bar \gamma }_{\rm{S}}}{m_y}}}{{{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_y}{m_x}}}$, ${\psi _1}\left( \Theta \right) = \frac{1}{{{{\left( {\Theta + {\alpha _2}} \right)}^{{m_z} + {i_2}}}{{\left( {\Theta + {\alpha _3}} \right)}^{{m_y} + {i_1}}}}}$, ${\psi _2}\left( \Theta \right) \hspace{-1mm}=\hspace{-1mm} \frac{1}{{{{\left( {\Theta + {\alpha _1}} \right)}^{{m_z} + {k_1}}}{{\left( {\Theta + {\alpha _3}} \right)}^{{m_y} + {i_1}}}}}$ and ${\psi _3}\left( \Theta \right) \hspace{-1mm}=\hspace{-1mm} \frac{1}{{{{\left( {\Theta + {\alpha _1}} \right)}^{{m_z} + {k_1}}}{{\left( {\Theta + {\alpha _2}} \right)}^{{m_z} + {i_2}}}}}$. *In order to facilitate error derivation, should be addressed in a more mathematically tractable form. For this purpose, by substituting into [@Duong:Let:2013 Eq. (9)] and utilizing partial fraction expansions [@Gradshteyn:Table:2007 Eq. (2.102)], and based on [@Gradshteyn:Table:2007 Eq. (9.210.2)], [@Gradshteyn:Table:2007 Eq. (3.361.2)], a closed-form expression for the SEP over i.n.i.d. Nakagami-$m$ fading channels can be derived as in .*
Simulation Results and Discussions
==================================
In this section, we perform Monte carlo simulation to verify the accuracy of our analysis where 4PSK is employed for the data symbols and assuming ${\gamma _{{\rm{th}}}} = 3\,{\rm{dB}}$ to determine the OC. Monte-Carlo simulations are averaged over $10^5$ independent one. In Fig. 1, we present the effect of primary transmit SNR on the secondary OC of Scenario (a) for fading severity parameter equal to $m = 2$. In the same figure, the simulation results of the OC is depicted along with the results related to the lower bound, given in for different values of ${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$. It can be noticed that the decrease in threshold (${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$) enhance the outage performance of the PUs, in turn, provides an intensive limitation for transmit powers of SUs which further leads in an increase of the secondary OC. However, if SNR of the primary transmitter is increased beyond a certain level, the maximum allowed power is reached for SUs, which does not allow further increase in $P_{S_i}$ and $P_R$. Thus, with an additional increase in the primary power, an error floor occurs for the outage performance of the secondary network.
In Fig. 2, we analyze the SEP of Scenario (a). It is worth noting that an increase in ${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$ relaxes the limitation of the power of the SUs. However, if we increase ${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$ beyond a certain level, the power of the SU nodes will reach it maximum admissible power. Hence, in such a case regardless of the value of the ${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$ the secondary outage performance remains constant. We can see that our analytical results match well with the simulation results, which validates our theoretical analysis. In addition, the outage performance improves with the increase of the fading severity parameters (compared with [@Vahidian1:TVT:2016] where $m=1$).
In Fig. 3, we present the outage performance of Scenario (b) versus the SNR. In this figure the simulation results of the error probability is illustrated along with the results pertaining to the lower bound, given in for various values of ${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$. As can be seen the outage performance improves when the number of relays increases specially, in the median and high SNR regions. As observed, the error and outage performance of cognitive radio network in Scenario (a) and (b), deteriorate with the decrease of ${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}$. More importantly, when strict limits on the transmit power of the secondary nodes is imposed (${P_{{\rm{ou}}{{\rm{t}}_{{\rm{Pri}}}}}^{\rm{Thr}}}=0$), in such a case, no transmission allowed for the secondary network, therefore, the OC of the secondary network equals one. Furthermore, in Scenario (b) the selection diversity enables the secondary system to perform better by changing both the choice of the relay and its transmit power depending on the channel fades.
Conclusion
==========
In this paper, we have developed new analytical results that can be used to investigate the OC and ASEP of non-regenerative cognitive two-way relaying networks with underlay spectrum sharing in Nakagami-$m$ fading environment. Two scenarios are treated including Scenario (a) in which two transmit nodes communicate via a single relay, or communicate through a selected relay as considered in Scenario (b). For both scenarios, tight closed-form expressions for OC were derived which provided an efficient means to investigate the impact of the PU’s interference and fading parameters on the outage performance of the system. Moreover, a lower bound expression of ASEP for Scenario (a), were derived. Simulations results were provided to testify the correctness of the derive analytical results and to study the effect of many parameters such as the number of relays, constellation size, and channel statistics on the considered performance measures.
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Proof of Theorem \[Thm-outage probability-DMM\] {#apx1}
===============================================
We assume the following change of variables ${\left| {{h_{R - {S_2}}}} \right|^2} = X$, ${\left| {{h_{R - {S_1}}}} \right|^2} = W$, $\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{\left| {{h_{PT - R}}} \right|^2} = Y$, ${\bar \gamma _{\rm{P}}}{\left| {{h_{PT - {S_1}}}} \right|^2} = Z$, ${\bar \gamma _{\rm{P}}}{\left| {{h_{PT - {S_2}}}} \right|^2} = V,{\left| {{h_{PT - PX}}} \right|^2} = E$, ${\left| {{h_{{S_1} - PX}}} \right|^2} = F$, ${\left| {{h_{{S_2} - PX}}} \right|^2} = G$ and ${\left| {{h_{R - PX}}} \right|^2} = L$. The OC of the primary network can be obtained as $$\begin{aligned}
&{P_{out,\Pr i}} \hspace{-1mm}=\hspace{-1mm} \left\{ {\log \left( {1 \hspace{-1mm}+\hspace{-1mm} \frac{{{{\bar \gamma }_P}{{\left| {{h_{PT \hspace{-0.5mm}-\hspace{-0.5mm} PX}}} \right|}^2}}}{{{{\bar \gamma }_{{S_1}}}{{\left| {{h_{{S_1} \hspace{-0.5mm}-\hspace{-0.5mm} PX}}} \right|}^2} \hspace{-1mm}+\hspace{-0.9mm} {{\bar \gamma }_{{S_2}}}{{\left| {{h_{{S_2} \hspace{-0.5mm}- \hspace{-0.5mm} PX}}} \right|}^2} \hspace{-1.1mm}+\hspace{-1.2mm} 1}}} \right)\hspace{-1mm} \le\hspace{-1.2mm} {R_P}} \right\}\nonumber\end{aligned}$$ $$\begin{aligned}
& = {{\rm{E}}_{F,G}}\left[ {\Pr \left\{ {E \le \frac{\Theta }{{{{\bar \gamma }_{\rm{P}}}}}\left( {{{\bar \gamma }_{{{\rm{S}}_1}}}F + {{\bar \gamma }_{{{\rm{S}}_2}}}G + 1} \right)\left| {F,G} \right.} \right\}} \right] = \nonumber\\&
{{\rm{E}}_{F,G}}\left[ {{F_E}\left( {\frac{\Theta }{{{{\bar \gamma }_{\rm{P}}}}}\left( {{{\bar \gamma }_{{{\rm{S}}_1}}}F + {{\bar \gamma }_{{{\rm{S}}_2}}}G + 1} \right)\left| {F,G} \right.} \right)} \right].\end{aligned}$$
The cdf of a Gamma RV, $X$, is defined as
$$\begin{aligned}
F_X\left( x \right)=\frac{{\Gamma \left( {{m_x},\left( {\frac{{{m_x}}}{{{{\bar \gamma }_x}}}} \right)} \right)}}{{\Gamma \left( {{m_x}} \right)}}\label{cd}.\end{aligned}$$ Therefore, we have $$\begin{aligned}
{\scalebox{0.97}{${P_{out,\Pr i}} \hspace{-1mm}=\int {\int_{Z,Y>0} {} } 1 - $}}&{\scalebox{0.97}{$\frac{{\Gamma \left( {{m_e},\left( {\frac{{{m_e}}}{{{{\bar \gamma }_e}}}} \right)\frac{\Theta }{{{{\bar \gamma }_{\rm{P}}}}}\left( {{{\bar \gamma }_{{{\rm{S}}_1}}}F + {{\bar \gamma }_{{{\rm{S}}_2}}}G + 1} \right)} \right)}}{{\Gamma \left( {{m_e}} \right)}}$}}\nonumber\\& \times {f_F}\left( f \right){f_G}\left( g \right) dFdG,\label{pdf-3}\end{aligned}$$ with the pdf of the RVs $F$ and $G$ at hand as in $$\begin{aligned}
&{\scalebox{0.97}{${f_F}\left( f \right) = \frac{{{{\left( {{m_f}} \right)}^{{m_f}}}{F^{{m_f} - 1}}}}{{\Gamma \left( {{m_f}} \right)\bar \gamma _f^{{m_f}}}}\exp \left( {\frac{{ - F{m_f}}}{{{{\bar \gamma }_f}}}} \right)$}},\notag\\& {\scalebox{0.97}{${f_G}\left( g \right) = \frac{{{{\left( {{m_g}} \right)}^{{m_g}}}{G^{{m_g} - 1}}}}{{\Gamma \left( {{m_g}} \right){{\left( {{{\bar \gamma }_g}} \right)}^{{m_g}}}}}\exp \left( {{\raise0.7ex\hbox{${ - G{m_g}}$} \!\mathord{\left/
{\vphantom {{ - G{m_g}} {{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_g}}}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{${{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_g}}$}}} \right)$}},\label{pdf-2}\end{aligned}$$
and by substituting in and making use of the identities [@Gradshteyn:Table:2007] as in the following we arrive at .
$$\begin{aligned}
{\left( {{\varepsilon _1} + {\varepsilon _2} + {\varepsilon _3}} \right)^n} = \sum\limits_{{i_1} = 0}^n {\sum\limits_{{i_2} = 0}^{n - {i_1}} {\left( \begin{array}{l}
n\\
{i_1}
\end{array} \right)} } \left( \begin{array}{l}
n - {i_1}\\
\,\,\,{i_2}
\end{array} \right){\varepsilon _1^{{i_2}}}{\varepsilon _2^{{i_1}}}{\varepsilon _3^{n - {i_1} - {i_2}}},
\label{I1}
\end{aligned}$$
$$\begin{aligned}
\int_0^\infty {{x^s}\exp \left( { - hx} \right)dx = s!{{\left( {\frac{1}{h}} \right)}^{s + 1}}} = s!{\left( h \right)^{ - \left( {s + 1} \right)}},\label{I2}
\end{aligned}$$ $$\begin{aligned}
\Gamma \left( {1 + n,x} \right) = n!{e^{ - x}}\sum\limits_{m = 0}^n {\frac{{{x^m}}}{{m!}}}. \label{I3}\end{aligned}$$
Proof of Theorem \[op-secanrio-a\] {#apx2}
==================================
$$\begin{aligned}
&{\scalebox{0.97}{${F_{\gamma _{{{S1}}}^{{\rm{up}}}}}\left( \Theta \right) = \Pr \left\{ {{{\bar \gamma }_{\rm{R}}}\min \left( {\frac{X}{{Z + Y + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1}},\frac{{\rm{W}}}{{Z + 1}}} \right) < \Theta } \right\} \approx 1 -$}}\nonumber\\&
{\scalebox{0.94}{$\underbrace {{{\rm{E}}_{Z,Y}}\hspace{-2mm}\left[\hspace{-1mm} {\Pr \left\{ {\frac{X}{{Z \hspace{-1mm}+\hspace{-1mm} Y \hspace{-1mm}+\hspace{-1mm} \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} +\hspace{-1mm} 1}} >\hspace{-1mm} \frac{\Theta }{{{{\bar \gamma }_{\rm{R}}}}}\left| Z \right.,Y} \right\}} \right]}_{{\chi _1}} \hspace{-1mm}\times \hspace{-1mm}\underbrace {{{\rm{E}}_Z}\left[ {\Pr \left\{ {\frac{{\,{\rm{W}}}}{{Z \hspace{-1mm}+\hspace{-1mm} 1}} > \hspace{-1mm}\frac{\Theta }{{{{\bar \gamma }_{\rm{R}}}}}\left| Z \right.} \right\}} \right]}_{{\chi _2}}$}}.\label{we}\end{aligned}$$
First, we focus on evaluating ${{\chi _1}}$ which with the concepts of probability and with the help of as well as the pdf of the RVs $Z$ and $Y$ can be given as $$\begin{aligned}
{\chi _1}\hspace{-1mm} =&\hspace{-1mm} \int \hspace{-1mm}{\int_{Z,Y>0} {} } \hspace{-4mm} \frac{{\Gamma \left( {{m_x},\left( {\frac{{{m_x}}}{{{{\bar \gamma }_x}}}} \right)\left( {Z \hspace{-1mm}+ \hspace{-1mm}Y \hspace{-1mm}+\hspace{-1mm} \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} \hspace{-1mm}+ \hspace{-1mm}1} \right)\frac{\Theta }{{{{\bar \gamma }_{\rm{R}}}}}} \right)}}{{\Gamma \left( {{m_x}} \right)}}\frac{{{{\left( {{m_Y}} \right)}^{{m_Y}}}{Y^{{m_Y} - 1}}}}{{\Gamma \left( {{m_Y}} \right){{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{{\bar \gamma }_y}} \right)}^{{m_Y}}}}}\nonumber\\&
\exp \left( {\frac{{ - Y{m_Y}}}{{\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{{\bar \gamma }_Y}}}} \right)
\frac{{{{\left( {{m_z}} \right)}^{{m_z}}}{Z^{{m_z} - 1}}}}{{\Gamma \left( {{m_z}} \right){{\left( {{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_z}} \right)}^{{m_z}}}}}\exp \left( {\frac{{ - Z{m_z}}}{{{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_z}}}} \right)dYdZ.\end{aligned}$$
Next, utilizing the identities in , and and also after some calculations we achieve ${\chi _1}$. In addition, ${\chi _2}$ can be computed taking the same steps as that of ${\chi _1}$. Finally, by substituting the former and the latter in we arrive at ${F_{\gamma _{{{S1}}}^{{\rm{up}}}}}\left( \Theta \right)$.
Proof of Theorem \[op-secanrio-b\] {#apx3}
==================================
$$\begin{aligned}
&{\scalebox{0.97}{${F_{\gamma _{{\rm{e2e}}}^{{\rm{up}}}}}\left( \Theta \right) =
\mathop {\max }\limits_{k = 1,2,...,K} \left\{ {\min \left( {\gamma _{{\rm{S_1}}}^{{\rm{up}}},\gamma _{{\rm{S_2}}}^{{\rm{up}}}} \right)} \right\} = $}}\nonumber\\&
{\scalebox{0.97}{$\Pr \hspace{-1mm}\left[\hspace{-1mm} {\mathop {\max }\limits_{k = 1,2,...,K} \hspace{-1mm}\left\{ {\min \hspace{-1mm}\left\{ \hspace{-1mm}{\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}}\min\hspace{-1mm} \left( \hspace{-1mm}{\frac{{{{\bar \gamma }_{\rm{S}}}{{\left| {{h_{{S_2} \hspace{-1mm}-\hspace{-1mm} {R_k}}}} \right|}^{\rm{2}}}}}{{\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{h_{PT - {R_k}}} \hspace{-1mm}+\hspace{-1mm} \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} \hspace{-1mm}+\hspace{-1mm} 1}},\,{{\bar \gamma }_{{{\rm{S}}_1}}}{{\left| {{h_{{\rm{ST1 - S}}{{\rm{R}}_k}}}} \right|}^{\rm{2}}}} \right),}\hspace{-1mm} \right.} \hspace{-1mm}\right.}\hspace{-1mm} \right.}\nonumber\\&
{\scalebox{0.97}{$\left. \hspace{-1mm}{\left. {\left. {\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}}\min \left( {\frac{{{{\bar \gamma }_{\rm{S}}}{{\left| {{h_{{S_1} - {R_k}}}} \right|}^{\rm{2}}}}}{{\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{h_{PT - {R_k}}} + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1}},\,{{\bar \gamma }_{\rm{S}}}{{\left| {{h_{{{\rm{S}}_{\rm{2}}}{\rm{ - }}{{\rm{R}}_k}}}} \right|}^{\rm{2}}}} \right) \hspace{-1mm}<\hspace{-1mm} \Theta } \right\}} \right\$}}} \right].$}}\end{aligned}$$
Let ${\cal{V}}=\min \left( {\gamma _{{\rm{S_1}}}^{{\rm{up}}},\gamma _{{\rm{S_2}}}^{{\rm{up}}}} \right)$. Due to the independency of the RVs, we first proceed to obtain the cdf of the RV ${\cal{V}}$ as in the following. $$\begin{aligned}
&{\scalebox{0.97}{${F_{\cal{V}}}\left( \Theta \right)=1 - {E_{{Y_k}}}\Bigg[ \Pr \left( {{X_k} \ge \frac{{\left( {Y + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)\Theta }}{{{{\bar \gamma }_{\rm{R}}}}}\,\,,\,\,{X_k} \ge \frac{\Theta }{{{{\bar \gamma }_{{{\rm{R}}_k}}}}}} \right)$}}\nonumber\\&{\scalebox{0.97}{$ \times
\Pr \left( {{W_k} \ge \frac{{\left( {Y_k + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)\Theta }}{{{{\bar \gamma }_{\rm{R}}}}}\,\,,\,\,{W_k} \ge \frac{\Theta }{{{{\bar \gamma }_{\rm{R}}}}}} \right) \Bigg]$}}\label{cdv}.\end{aligned}$$ Using concepts of probability and having in mind, we can rewrite as
$$\begin{aligned}
&{\scalebox{0.97}{${F_{\cal{V}}}\left( \Theta \right) \hspace{-1mm}= \hspace{-1mm}1 \hspace{-1mm}-\hspace{-1mm} {E_{{Y_k}}}\left[ {{\Gamma ^{ - 1}}\left( {{m_{{x_k}}}} \right)\Gamma \left( {{m_{{x_k}}},\left( {\frac{{{m_{{x_k}}}}}{{{{\bar \gamma }_{{x_k}}}}}} \right)\frac{{\left( {{Y_k} + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)\Theta }}{{{{\bar \gamma }_{\rm{R}}}}}} \right)} \right.$}}\nonumber\\& {\scalebox{0.97}{$\times
\left. {{\Gamma ^{ - 1}}\left( {{m_{{w_k}}}} \right)\Gamma \left( {{m_{{w_k}}},\left( {\frac{{{m_{{w_k}}}}}{{{{\bar \gamma }_{{w_k}}}}}} \right)\frac{{\left( {{Y_k} + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)\Theta }}{{{{\bar \gamma }_{\rm{R}}}}}} \right)} \right]$}}.\end{aligned}$$
Making use of the identity in , we have $$\begin{aligned}
&{\scalebox{0.98}{${F_{\cal{V}}}\left( \Theta \right)=\hspace{-1mm}
1 \hspace{-1mm}-\hspace{-1mm} {E_{Y_k}}\left[ {\left( {{m_{{x_k}}}\hspace{-1mm} -\hspace{-1mm} 1} \right)!{{\left( {\Gamma \left( {{m_{{x_k}}}} \right)} \right)}^{ - 1}}\left( {{m_{{w_k}}}\hspace{-1mm} - \hspace{-1mm}1} \right)!{{\left( {\Gamma \left( {{m_{{w_k}}}} \right)} \right)}^{ - 1}}} \right.$}}\nonumber\\&
{\scalebox{0.95}{$\exp \left( { - \left( {\frac{{{m_{{x_k}}}}}{{{{\bar \gamma }_{{x_k}}}{{\bar \gamma }_{\rm{R}}}}}} \right)\left( {\frac{{{{\bar \gamma }_{{{\rm{R}}_k}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)\Theta } \right)\exp \left( { - \left( {\frac{{{m_{{w_k}}}}}{{{{\bar \gamma }_{{w_k}}}{{\bar \gamma }_{\rm{R}}}}}} \right)\left( {\frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)\Theta } \right)$}}\nonumber\\&
{\scalebox{0.98}{$\exp \left( { - \left( {\frac{{{m_{{x_k}}}\Theta }}{{{{\bar \gamma }_{{x_k}}}{{\bar \gamma }_{\rm{R}}}}} + \frac{{{m_w}\Theta }}{{{{\bar \gamma }_{{w_k}}}{{\bar \gamma }_{{{\rm{R}}_k}}}}}} \right)Y} \right)\sum\limits_{n = 0}^{{m_{{x_k}}} - 1} {\sum\limits_{{n_1} = 0}^{{m_{{x_k}}} - 1} {\frac{1}{{{n_1}!n!}}{{\left( {\frac{{{m_{{w_k}}}\Theta }}{{{{\bar \gamma }_{{w_k}}}{{\bar \gamma }_{\rm{R}}}}}} \right)}^{{n_1}}}} }$}} \nonumber\\&
{\scalebox{0.98}{$\left. {{{\left( {\frac{{{x_k}\Theta }}{{{{\bar \gamma }_{{x_k}}}{{\bar \gamma }_{\rm{R}}}}}} \right)}^n}{{\left( {{Y_k} + \frac{{{{\bar \gamma }_{\rm{R}}}}}{{{{\bar \gamma }_{\rm{S}}}}} + 1} \right)}^{n + {n_1}}}} \right]$}}.\end{aligned}$$ With pdf of $Y_k$ at hand as in the following and evoking , we reach . $$\begin{aligned}
{f_{{Y_k}}}\left( {{y_k}} \right) = \frac{{{{\left( {{m_{{y_k}}}} \right)}^{{m_{{y_k}}}}}Y_k^{{m_{{y_k}}} - 1}}}{{\Gamma \left( {{m_{{y_k}}}} \right){{\left( {\frac{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}}}{{{{\bar \gamma }_{\rm{S}}}}}{{\bar \gamma }_{{y_k}}}} \right)}^{{m_{{y_k}}}}}}}\exp \left( {\frac{{ - {Y_k}{{\bar \gamma }_{\rm{S}}}{m_{{y_k}}}}}{{{{\bar \gamma }_{\rm{R}}}{{\bar \gamma }_{\rm{P}}}{{\bar \gamma }_{{y_k}}}}}} \right).\end{aligned}$$
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by NPRP from the Qatar National Research Fund (a member of Qatar Foundation) under grant no. 8-1545-2-657.
[^1]: Two-way relaying systems can be classified into three categories: $1)$ The first model is conventional two one-way relaying to achieve a bidirectional transmission, which requires four time slots. $2)$ The alternative model is time division broadcasting (TDBC) whose number of time slots is reduced to three by using physical-layer network coding (PNC). $3)$ The final model is multiple access broadcasting (MABC) which just needs two time slots [@Shahbazpanahi:TSP:2010].
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abstract: 'In cavity quantum electrodynamics (QED) [@Mabuchi:2002a; @Walther:2006a; @Haroche:2006a], light-matter interaction is probed at its most fundamental level, where individual atoms are coupled to single photons stored in three-dimensional cavities. This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED [@Blais:2004a; @Wallraff:2004a; @Chiorescu:2004a; @Johansson:2006a; @Schuster:2007a; @Astafiev:2007a; @Deppe:2008a; @Fink:2008a; @Abdumalikov:2008; @Hofheinz:2009a], where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost. This important regime has been reached both in cavity and circuit QED, but the design flexibility and engineering potential of the latter allowed for increasing the ratio between the atom-cavity coupling rate $g$ and the cavity transition frequency $\omega_{\rm r}$ above the percent level [@Schuster:2007a; @Schoelkopf:2008a; @Bishop:2008a]. While these experiments are well described by the renowned Jaynes-Cummings model [@JaynesCummings:1963a], novel physics is expected when $g$ reaches a considerable fraction of $\omega_{\rm r}$. Promising steps towards this so-called ultrastrong coupling regime [@Ciuti:2006a; @Devoret:2007a] have recently been taken in semiconductor structures [@Guenter:2009a; @Anappara:2009a]. Here, we report on the first experimental realization of a superconducting circuit QED system in the ultrastrong coupling limit and present direct evidence for the breakdown of the Jaynes-Cummings model. We reach remarkable normalized coupling rates $g/\omega_{\rm r}$ of up to 12% by enhancing the inductive coupling of a flux qubit [@Mooij:1999a] to a transmission line resonator using the nonlinear inductance of a Josephson junction [@Bourassa:2009a]. Our circuit extends the toolbox of quantum optics on a chip towards exciting explorations of the ultrastrong interaction between light and matter.'
author:
- 'T. Niemczyk'
- 'F. Deppe'
- 'H. Huebl'
- 'E. P. Menzel'
- 'F. Hocke'
- 'M. J. Schwarz'
- 'J. J. Garcia-Ripoll'
- 'D. Zueco'
- 'T. Hümmer'
- 'E. Solano'
- 'A. Marx'
- 'R. Gross'
bibliography:
- 'myBib.bib'
title: 'Beyond the Jaynes-Cummings model: circuit QED in the ultrastrong coupling regime'
---
In the strong coupling regime, the atom-cavity coupling rate $g$ exceeds the dissipation rates $\kappa$ and $\gamma$ of both, cavity and atom, giving rise to coherent light-matter oscillations and superposition states. This regime was reached in various types of systems operating at different energy scales [@Rempe:1992a; @Mabuchi:2002a; @Walther:2006a; @Haroche:2006a; @Reithmaier:2004a; @Groebelbacher:2009a]. At microwave frequencies, strong coupling is feasible due to the enormous engineerability of superconducting circuit QED systems [@Blais:2004a; @Wallraff:2004a]. Here, small cavity mode volumes and large dipole moments of artificial atoms [@Niemczyk:2009a] enable coupling rates $g$ of about [@Bishop:2008a] 1% of the cavity mode frequency $\omega_{\rm r}$. Nevertheless, as in cavity QED, the quantum dynamics of these systems follows the Jaynes-Cummings model, which describes the coherent exchange of a single excitation between the atom and the cavity mode. Although the Hamiltonian of a realistic atom-cavity system contains so-called counterrotating terms allowing the simultaneous creation ior annihilation of an excitation in both atom and cavity mode, these terms can be safely neglected for small normalized coupling rates $g/\omega_{\rm r}$. However, when $g$ becomes a significant fraction of $\omega_{\rm r}$, the counterrotating terms are expected to manifest, giving rise to exciting effects in QED.
The ultrastrong coupling regime is difficult to reach in traditional quantum optics, but was recently realized in a solid-state semiconductor system [@Guenter:2009a; @Anappara:2009a]. There, quantitative deviations from the Jaynes-Cummings model have been observed, but a direct experimental proof of its breakdown by means of an unambiguous feature is still missing. In this report, we exploit the potential of flux-biased superconducting quantum circuits to reach the ultrastrong coupling regime [@Devoret:2007a; @Bourassa:2009a]. For this purpose, we increase $g/\omega_{\rm r}$ up to 12% utilizing the large nonlinear inductance of a Josephson junction (JJ) shared between a flux qubit and a coplanar waveguide resonator. We explicitly make use of the multimode structure of our resonator, allowing the direct observation of physics beyond the Jaynes-Cummings model. In equilibrium, our transmission spectra reveal anticrossings which can be clearly attributed to the counterrotating terms in the system Hamiltonian. These anticrossings are caused by the simultaneous creation (annihilation) of two excitations, one in the qubit and one in a resonator mode, while annihilating (creating) only one excitation in a different resonator mode.
{width="0.7\linewidth"}
{width="0.9\linewidth"}
{width="0.9\linewidth"}
Images of our quantum circuit and a schematic of the measurement setup are shown in Figure \[FIG1\]. At a current antinode for the $\lambda$-mode of a niobium superconducting resonator (Fig. \[FIG1\]a-c), a part of the center conductor is replaced with a narrow aluminum strip interrupted by a large-area JJ. This junction mediates most of the inductive coupling between a superconducting flux qubit [@Mooij:1999a] galvanically connected to the strip. The qubit consists of three nanometer-scaled JJs interrupting a superconducting loop, which is threaded by an external flux bias $\Phi_{\rm x}$. Scanning electron microscope (SEM) images of the qubit loop and the JJs are shown in Figure \[FIG1\]d-f. For suitable junction sizes, the qubit potential landscape can be reduced to a double-well potential, where the two minima correspond to states with clockwise and counter-clockwise persistent currents $|\pm I_{\rm p}\rangle$. At $\delta\Phi_{\rm x} \,=\, \Phi_{\rm x} - \Phi_{\rm 0}/2 = 0$, these two states are degenerate and separated by an energy gap $\Delta$. In the qubit eigenbasis, the qubit Hamiltonian reads $\hat{H}_{\rm q} \,=\, \hbar\omega_{\rm q}\hat{\sigma}_{\rm z}/2$. Here, $\omega_{\rm q} \,=\, \sqrt{\Delta^2 + (2I_{\rm p}\cdot\delta\Phi_{\rm x})^2}/\hbar$ is the qubit transition frequency which can be adjusted by an external flux bias. We note, that for our flux qubit the two-level approximation is well justified due to its large anharmonicity. The resonator modes are described as harmonic oscillators, $\hat{H}_{\rm n} \,=\, \hbar\omega_{\rm n}(\hat{a}_{\rm n}^{\dagger}\hat{a}_{\rm n} \,+\, 1/2)$, where $\omega_{\rm n}$ is the resonance frequency and $n$ is the resonator mode index. The operator $\hat{a}_{\rm n}^{\dagger}$ ($\hat{a}_{\rm n}$) creates (annihilates) a photon in the $n$th resonator mode. Due to the inhomogeneous transmission line geometry [@Bourassa:2009a] (see Fig. \[FIG1\]d), the higher mode frequencies of our resonator are not integer multiples of the fundamental resonance frequency $\omega_{\rm 1}$ . Throughout this work, we refer to the $n$th mode as the $n\lambda/2$-mode. Then, the Hamiltonian of our our quantum circuit can be written as $$\hat{H} = \hat{H}_{\rm q} + \sum\limits_{n} \left[\hat{H}_{\rm
n} + \hbar g_{\rm n}\left(\hat{a}_{\rm n}^{\dagger}+\hat{a}_{\rm
n} \right)\left(\cos\theta\,\hat{\sigma}_{\rm z} -
\sin\theta\,\hat{\sigma}_{\rm x}\right)\right].\label{FullH}$$ Here, $\hat{\sigma}_{\rm x,z}$ denote Pauli operators, $g_{\rm n}$ is the coupling rate of the qubit to the $n$th cavity mode, and the flux dependence is encoded in $\sin\theta = \Delta/\hbar\omega_{\rm q}$ and $\cos\theta$. The operator $\hat{\sigma}_{\rm x}$ is conveniently expressed as sum of the qubit raising ($\hat{\sigma}_{\rm +}$) and lowering ($\hat{\sigma}_{\rm -}$) operator. Thus, in contrast to the Jaynes-Cummings model, the Hamiltonian in Eq. (\[FullH\]) explicitly contains counterrotating terms of the form $\hat{a}_{\rm n}^{\dagger}\hat{\sigma}_{\rm +}$ and $\hat{a}_{\rm n}\hat{\sigma}_{\rm -}$.
Figure \[FIG1\]g shows a schematic of our measurement setup. The quantum circuit is located at the base temperature of 15 mK in a dilution refrigerator. We measure the amplified resonator transmission using a vector network analyzer (VNA). For qubit spectroscopy measurements, the system is excited with a second microwave tone $\omega_{\rm s}$ with power $P_{\rm s}$, while using the $3\lambda/2$-mode at $\omega_{\rm 3}/2\pi = 7.777$GHz for dispersive readout [@Schuster:2005a; @Abdumalikov:2008].
We first present measurements allowing the extraction of the coupling constants of the qubit to the first three resonator modes. The spectroscopy data in Fig. \[FIG2\]a shows the dressed qubit transition frequency [@Blais:2004a; @Schuster:2005a] with the expected hyperbolic flux-dependence and a minimum at $\delta\Phi_{\rm x} = 0$. Furthermore, flux-independent features corresponding to the two lowest resonator modes ($\omega_{\rm 1}$ and $\omega_{\rm 2}$) are visible. In principle, a fit to the Hamiltonian in Eq. (\[FullH\]) would yield all system parameters. However, our measurement resolution does not allow us to reliably determine the coupling constants $g_{\rm n}$ in this situation. Instead, we extract $g_{\rm n}$ from a cavity transmission spectrum with negligible photon population. For that purpose, we first measure the power-dependent ac-Zeeman shift of the qubit transition frequency at $\delta\Phi_{\rm x}\,=\,0$. The data is shown in the inset of Fig. \[FIG2\]a. The average photon number $\bar{n}_{\rm 3}$ can be estimated using the relation $P_{\rm rf} = \bar{n}_{\rm 3}\hbar\omega_{\rm 3}\kappa_{\rm 3}$ [@Astafiev:2007a; @Fink:2008a], where $\kappa_{\rm 3}/2\pi \approx 3.7$ MHz is the full width at half maximum (FWHM) of the cavity resonance and $P_{\rm rf}$ the probe power referred to the input of the resonator. Figure \[FIG2\]b shows a color coded transmission spectrum for the $3\lambda/2$-mode as a function of $\delta\Phi_{\rm x}$. The data is recorded at an input power $P_{\rm rf} \approx -140$ dBm (green data point in Fig. \[FIG2\]a, inset) corresponding to $\bar{n}_{\rm 3} = 0.18$.
We observe a spectrum with a large number of anticrossings resulting from the multimode structure of our cavity system. To extract the individual coupling constants $g_{\rm n}$, we compute the lowest nine transition frequencies of the Hamiltonian given in Eq. (\[FullH\]) incorporating the first three resonator modes. Fitting the results to the spectrum of the $3\lambda/2$-mode shows excellent agreement with the measured data as shown in Fig. \[FIG2\]c. We note that the spectrum for the $\lambda$-mode shown in Fig. \[FIG2\]d can be well described without additional fitting using the parameters extracted from the $3\lambda/2$-mode. For the qubit, we obtain $\Delta/h = 2.25$ GHz and $2I_{\rm p} = 630$ nA. Most importantly, we find coupling rates of $g_{\rm 1}/2\pi = 314$ MHz, $g_{\rm 2}/2\pi = 636$ MHz, and $g_{\rm 3}/2\pi = 568$ MHz. The values for $g_{\rm n}$ correspond to normalized coupling rates $g_{\rm n}/\omega_{\rm n}$ of remarkable 11.2%, 11.8%, and 7.3%, respectively. From these numbers, we expect significant deviations of our system from a three-mode Jaynes-Cummings model, ultimately proving that we have successfully reached the ultrastrong coupling regime.
In the following, we analyze the features in our data which constitute unambiguous evidence for the breakdown of the rotating-wave approximation inherent to the Jaynes-Cummings model. In Figure \[FIG3\], we compare the energy level spectrum of the Hamiltonian in Eq. (\[FullH\]) to that of a three-mode Jaynes-Cummings model. We note that, depending on $\delta\Phi_{\rm x}$, there are regions where our data can be well described by the Jaynes-Cummings model, and regions where there are significant deviations (see Fig. \[FIG3\]a). For our analysis we use the notation $|q,N_{\rm 1},N_{\rm 2},N_{\rm 3}\rangle = |q\rangle \otimes |N_{\rm 1}\rangle \otimes |N_{\rm 2}\rangle \otimes |N_{\rm 3}\rangle$, where $q = \{g,e\}$ denote the qubit ground or excited state, respectively, and $|N_{\rm n}\rangle = \{|0\rangle,|1\rangle,|2\rangle,\dots\}$ represents the Fock-state with photon occupation $N$ in the $n$th resonator mode. At the outermost anticrossings (Fig. \[FIG3\]b), where $\omega_{\rm 3} \approx \omega_{\rm q}$, the eigenstates $|\psi_{\pm }\rangle$ of the coupled system are in good approximation symmetric and antisymmetric superpositions of $|e,0,0,0\rangle$ and $|g,0,0,1\rangle$. This exchange of a single excitation between qubit and resonator is a characteristic of the Jaynes-Cummings model. On the contrary, the origin of the anticrossing shown in Fig. \[FIG3\]c is of different nature: the dominant contributions to the eigenstates $|\psi_{\pm}\rangle$ are approximate symmetric and antisymmetric superpositions of the degenerate states $\varphi_{\rm 1} = |e,1,0,0\rangle$ and $\varphi_{\rm 2} = |g,0,0,1\rangle$. The transition from $\varphi_{\rm 1}$ to $\varphi_{\rm 2}$ can be understood as the annihilation of two excitations, one in the $\lambda/2$-mode *and* one in the qubit, while, simultaneously, creating only one excitation in the $3\lambda/2$-mode. Such a process can only result from counterrotating terms as they are present in the Hamiltonian (\[FullH\]), but not within the Jaynes-Cummings approximation. Here, only eigenstates with an equal number of excitations can be coupled. Although counterrotating terms in principle exist in any real circuit QED system, their effects become prominent only in the ultrastrong coupling limit with large normalized couplings $g_{\rm n}/\omega_{\rm n}$ as realized in our system. Hence, the observed anticrossing shown in Fig. \[FIG3\]c is a direct experimental manifestation of physics beyond the rotating-wave approximation in the Jaynes-Cummings model. As shown in Fig. \[FIG3\]d, the latter would imply a crossing of the involved energy levels, which is not observed. A similar argument applies to the innermost anticrossings (see Fig. \[FIG3\]a), although the involved eigenstates have a more complicated character.
In conclusion, we present measurements on a flux-based superconducting circuit QED system in the ultrastrong coupling regime. The results are in excellent agreement with theoretical predictions and show clear evidence for physics beyond the Jaynes-Cummings model. Our system can act as an on-chip prototype for unveiling the physics of ultrastrong light-matter interaction. Future explorations may include squeezing, switchable ultrastrong coupling [@Peropadre:2009a], causality effects in quantum field theory [@Sabin:2009a], and the generation of bound states of qubits and photons.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank G. M. Reuther for discussions and T. Brenninger, C. Probst, and K. Uhlig for technical support. We acknowledge financial support by the Deutsche Forschungsgemeinschaft via SFB 631 and the German Excellence Initiative via NIM. E.S. acknowledges funding from UPV/EHU Grant GIU07/40, Ministerio de Ciencia e Innovación FIS2009-12773-C02-01, European Projects EuroSQIP and SOLID. D.Z. acknowledges financial support from FIS2008-01240 and FIS2009-13364-C02-0 (MICINN).
Author contributions {#author-contributions .unnumbered}
====================
T.N. fabricated the sample, conducted the experiment and analyzed the data presented in this work. F.D. provided important contributions regarding the interpretation of the results. T.N. and F.D. co-wrote the manuscript. J.J.G.-R. provided the basic idea and the techniques for the numerical analysis of the data. E.S. and J.J.G.-R. supervised the interpretation of the data. D.Z. and T.H. contributed to the understanding of the results and developed an analytical model of our system. H.H. contributed to the numerical analysis and helped with the experiment. E.P.M. contributed strongly to the experimental setup. M.J.S. and F.H. contributed to discussions and helped editing the manuscript. A.M. and R.G. supervised the experimental part of the work.
|
---
abstract: 'It is becoming increasingly clear that users should own and control their data. Utility providers are also becoming more interested in guaranteeing data privacy. As such, users and utility providers should *collaborate* in data privacy, a paradigm that has not yet been developed in the privacy research community. We introduce this concept and present explicit architectures where the user controls what characteristics of the data she/he wants to share and what she/he wants to keep private. This is achieved by collaborative learning a sanititization function — either a deterministic or a stochastic one — that retains valuable information for the utility tasks but it also eliminates necessary information for the privacy ones. As illustration examples, we implement them using a plug-and-play approach, where no algorithm is changed at the system provider end, and an adversarial approach, where minor re-training of the privacy inferring engine is allowed. In both cases the learned sanitization function keeps the data in the original domain, thereby allowing the system to use the same algorithms it was using before for both original and privatized data. We show how we can maintain utility while fully protecting private information if the user chooses to do so, even when the first is harder than the second, as in the case here illustrated of identity detection while hiding gender.'
author:
- |
Martin Bertran ^1^\
`[email protected]`\
Natalia Martinez ^1^\
`[email protected]`\
Afroditi Papadaki ^2^\
`[email protected]`\
Qiang Qiu ^1^\
`[email protected]`\
Miguel Rodrigues ^2^\
`[email protected]`\
Guillermo Sapiro ^1^\
`[email protected]`
bibliography:
- 'library\_NIPS.bib'
title: 'Learning to Collaborate for User-Controlled Privacy'
---
*Privacy is a human right.* Tim Cook, Apple CEO.
Introduction, Challenges, and Contributions {#sec:introduction}
===========================================
Advances in machine learning allow to develop products that leverage individuals data to provide valuable services to them. At the same time care must be taken to provide an appropriate protection for users in order to adhere to various privacy, legal, and ethical constraints. Critical to this is to develop a trusting-based system where the data receiver can infer from shared data information a user does not consider private but concurrently cannot infer from the data information the user desires to keep private. Each user should have the ability to define sensitive and non-sensitive information associated with her/his data, which may differ from user to user; we propose that a user and the service provider *collaborate* towards achieving *user-specific* privacy.
We address the following scenario: An entity (e.g., a bank) wants to offer a service to its users based on their data (e.g., an ATM verify the card holder is the legitimate owner). However, some users want to be able to prevent the entity from inferring certain information from their data (e.g., gender), whereas other users may wish to prevent the entity from inferring other attributes (e.g., ethnicity). We assume the entity and each individual user *collaborate* in creating such system; it is as important for the user to preserve her/his privacy as it is to the entity to guarantee it, being this a mutually beneficial system. First, the users will be more comfortable because their privacy is respected. Second, the entity offering the service increases user’s trust and provides a principled approach to guarantee that they are not using/inferring users’ sensitive information in case of legal/ethical disputes (e.g., their system can be audited to show the private information cannot be extracted).
This proposed new paradigm of collaborative privacy environment is critical since it has been shown once and again that, e.g., algorithmic or data augmentation and unpredictable correlations, can easily break privacy [@Israel2014; @Narayanan2008; @Oh2016; @Reuben2016]. The impossibility of universal privacy protection has also been studied extensively in the domain of differential privacy [@Dwork2008], where a number of authors have shown that assumptions about the data or the adversary must be made in order to be able to provide utility [@Dwork2010; @Hardt2016; @Kifer2011a; @Kifer2014].[^1]
Therefore, given each individual user specific privacy requirements, it is important to design collaborative systems where each individual user shares a sanitized version of their data with the service provider in such a way that user-defined non-sensitive tasks can be performed but user-defined sensitive ones cannot, with the set of machine learning algorithms available at the service provider. A sanitization mechanism should not alter the data format so that the service provider machine learning system, consisting of a set of algorithms to infer user attributes from their data, can work independently of the user privacy preferences.
**Contributions-** We propose a novel framework that assumes a *system*-and-*user* *collaborative* environment in order to allow a service provider to infer certain user attributes but not others depending on *user-specific* privacy preferences. The proposed framework is based on the use of user-specific privacy filters that retain important information for the entity to perform legitimate tasks but simultaneously filter out information necessary for the entity to perform sensitive non-legitimate ones. The proposed framework is agnostic to the presence or absence of such user-specific privacy filters during operation, since by design it can handle both the original and the filtered/privatized data. [The service provider uses a single processing pipeline on both sanitized and original data and still guarantee to each individual user that their privacy will be preserved, while still providing the required utility service.]{}
Our proposed framework is also flexible to allow different levels of collaboration between each user and the service provider, from re-training of all key components (utility algorithm, private data inferring algorithm, sanitization function) to re-training only some components, or to re-train only the user-specific privacy filters for ’plug-and-play’ operation. Some of these variations are exemplified here both via deterministic and stochastic sanitization mechanisms that show how we can, without affecting utility, practically achieve the (optimal privacy) prior distribution for the privacy.
The proposed framework is described in Section \[sec:model\], and learning architectures for it are given in Section \[sec:architecture\]. Examples are presented in Section \[sec:results\]. [Since data should be protected for privacy before being transmitted (even at the sensor level if needed), we present results on a prototype hardware platform (to be demonstrated at the conference) that implements the framework at the data sharing step.]{} We discuss connections with the literature in Section \[sec:literature\]. The paper is concluded in Section \[sec:conclusion\].
The Proposed Collaborative Privacy Model {#sec:model}
========================================
[L]{}[0.5]{} {width="0.4\columnwidth"}
The proposed collaborative privacy framework is based on various desiderata: **(1)** The entity must provide the user concrete guarantees. In particular, given a set of algorithms used by the entity to infer possible user attributes, the entity should be able to guarantee the user what it can and what it cannot infer given the data conveyed by the user to the entity; **(2)** The user must be able to define what she/he would like the entity to be able to infer (utility); **(3)** The user should be able to define what she/he would like the entity not to be able to infer (privacy); and **(4)** The approach should integrate seamlessly (i.e., ’plug-in’) within current systems, in the sense that the entity will not have to alter its existent state-of-the-art inference algorithms in order to allow for user-specific non-sensitive tasks and block user-specific sensitive ones. This is specially important in scenarios where an entity may use many different state-of-the-art algorithms to infer all kinds of user attributes from user data, particularly in platforms with a very large number of users with very different privacy requirements.
Following this, [from the perspective of a specific user]{}, the system has three key components, Figure \[fig:model\], **(1)** The *utility* component is the service providing function; it takes input data, either raw or sanitized data, and infers a variable of interest; **(2)** The *privacy* component also operates on the same input data (raw or sanitized, they share the same format), and attempts to infer sensitive information from it; and **(3)** The *sanitization* function is a learned transformation that maps the input data onto the same space (i.e., maps images to images with the same dimensions). Note that the sanitization function – mapping user data from the input space to the same space – is critical: it allows the user to share “filtered” data with the entity without requiring (potentially) any alteration to the state-of-the-art (utility) inference algorithms.
It is not immediately clear if it is possible to define sanitization functions that transform the data in such a way that the set of pre-trained algorithms for the legitimate tasks perform well while the set of pre-trained algorithms for the sensitive tasks under-performs when the user desires that. For example, for visual data such sanitization functions have to transform an image onto another “image” where information relevant for the utility tasks is preserved whereas information associated with the sensitive tasks is blocked. No current approaches – as reviewed in later sections – have this property. Given the utility components and privacy components (here a single utility and privacy component are considered, but the principles generalize to systems with multiple such components), our goal is to learn a sanitization function that preserves performance on the utility task and lowers performance on the privacy task.
Let $D_{KL}(p\mid\mid q)$ be the Kullback-Leibler divergence from probability distribution $q$ to probability distribution $p$; $P(u\mid x)$ the posterior probability distribution of the utility variable $u$ conditioned on the raw data $x$; $P(u\mid S(x))$ the posterior probability distribution of the utility variable $u$ conditioned on the sanitized data $S(x)$; $P(p \mid S(x))$ the posterior probability distribution of the privacy variable $p$ conditioned on the sanitized data $S(x)$; and $P(p)$ the prior distribution of the privacy variable. We propose the following loss to train the sanitization function: $$\label{eq:sanitizationLoss}
\begin{split}
\mathrm{Loss}_{S} &= (1-\alpha) D_{KL}(P(u\mid x)\mid\mid P(u\mid S(x))) + \alpha D_{KL}(P(p)\mid\mid P(p\mid S(x))).
\end{split}$$
The proposed loss function is such that it attempts to guarantee that the outputs of the utility algorithm given the true ($x$) or sanitized ($S(x)$) data are indistinguishable, and concurrently that the output of the privacy algorithm given the sanitized data is indistinguishable from the privacy variable prior. The parameter $\alpha \in (0,1)$ controls the trade-off between preserving information on the utility variable, and destroying information on the privacy variable.
We further emphasize the importance of the mapping constraint on the sanitization function. By preserving the original space of the input data, the sanitization function can work *in tandem* with both the privacy and utility algorithms without any need to modify existing architectures. It is of course not immediately clear that it is possible to simultaneously filter out a privacy variable while maintaining performance on the utility variable in the raw input space. One of the contributions of this work is to show validations of this concept.
There are several possible training approaches we can pursue, some of them exemplified in this work. We can choose to exclusively train the sanitization function using the proposed loss, leaving the existing privacy and utility algorithms untouched. This training approach is very appropriate for the simple plug-in scenario, since no modification of the pre-trained utility and privacy algorithms is needed.
We can also *adversarially* train the privacy task. In this scenario, the privacy task is attempting to defeat the sanitization function and simultaneously perform well on unsanitized data. The sanitization function is attempting to fool the privacy task (approximate the privacy prior) and simultaneously maintain performance on the utility task. This training is computationally more expensive, but opens the door to more universal privacy guarantees, being adapted to the architecture but not to its specific parameters (weights); see captions in Figure \[fig:Dkl\].
Finally, the utility function can be trained *collaboratively* with the sanitization function; the utility function is aware of the sanitized data and can collaborate with the sanitization function to better preserve utility performance.
Naturally, these approaches can be freely combined to best suit the user’s and entity’s needs. In the following section, we exemplify the plug-and-play and adversarial training approaches.
Privacy Learning Architectures {#sec:architecture}
==============================
[L]{}[0.5]{}
[0.23]{} {width="\textwidth"}
[0.25]{} {width="\textwidth"}
To illustrate the proposed framework, we chose subject recognition (harder task) and gender recognition (easier task) as our utility and privacy respectively. We propose two different sanitization architectures, a *stochastic* one and a *deterministic* one, Fig \[fig:models\]. We implement them using the plug-and-play and the adversarial approaches mentioned above. In the deterministic architecture, the data is fed directly into a fully-trainable UNET, [@ronneberger2015u], whose objective is to fulfill the sanitization task. In the stochastic approach the sanitization function first randomly overwrites the gender attribute of the input image using the pretrained FaderNet [@FaderNetworks], done by sampling the gender of each of the images from the prior gender distribution of the dataset, and then feeds the resulting image through a fully-trainable UNET.
The pretrained FaderNet was chosen as the base for the stochastic sanitization function since this network is already trained to defeat a gender discriminator in its *encoding space* (privacy variable is concealed here). This proves a suitable baseline comparison and starting point for a sanitization function that needs to fool a gender discriminator *in image space* while simultaneously preserving subject recognition performance. We show below the performance of using only the pretrained gender FaderNet and demonstrate how training a UNET on top of it improves both the utility and privatization performances.
For both architectures, the UNET portion of the sanitization function was trained using the proposed loss function in Eq. (\[eq:sanitizationLoss\]). For the adversarially trained results, the final layer of the privacy function (DEXNet) is trained using the binary cross entropy (BCE) loss function, $$\label{eq:privatizationLoss}
\begin{split}
\mathrm{Loss}_{P} &= BCE(y_p(x), p(x)) + BCE(y_p(x), p(S(x))),
\end{split}$$ where $y_p(x)$ is a one-hot encoding of the ground truth gender label. This adversarial loss equally weights the performance of the gender detector on both the raw (unsanitized) data ($x$) and the sanitized data ($S(x)$). The sanitization and privacy tasks are playing an adversarial game. Adversarial training provides stronger privacy guarantees since the privacy detector is unable to infer the privacy variable even after retraining. We alternate the training of the DEXNet and UNET every one epoch, in the supplementary methods section we show how the losses $Loss_S$ and $Loss_P$ evolve in each iteration.
Experimental Results {#sec:results}
====================
Our data consisted of 15k samples (10k train, 5k test) from the realigned FaceScrub dataset [@Ng2014]. The identification utility task is performed using the ResNet-50 implementation of VGGFace2 [@cao2017vggface2], where the final dense layer was retrained on the chosen dataset. For gender recognition (privacy) we used a pretrained WideResNet-based, [@zagoruyko2016wide], implementation of DEXNet [@rothe2018deep].
Figure \[fig:Dkl\] shows the utility/privacy trade-off achieved by the stochastic and deterministic sanitization functions over the tested $\alpha$ values, as measured by the KL divergence for both the plug-and-play and adversarial training. As $\alpha$ increases, the privacy task quickly approaches the prior distribution (KL divergence goes to zero), while the utility task correspondingly loses performance (KL divergence increases). We also observe that the baseline performance of the pretrained FaderNet comes at a considerable cost in utility performance. We also observe that the full stochastic architecture trained following Eq. (\[eq:sanitizationLoss\]) is able to simultaneously recover performance on the utility task and improve performance on the privacy task. For this particular example, the sanitization function trained with Eq. (\[eq:sanitizationLoss\]) shows a net improvement over both the privacy and utility tasks with no downside.
[cc]{}
[0.4]{} ![Trade-off achieved by the sanitization function between the KL-divergence of the utility distribution from the objective ($D_{KL}(P(u|x)||P(u|S(x)))$) and the KL-divergence of the gender distribution from the prior ($D_{KL}(P(p)||P(p|S(x)))$). Compared to the adversarial approach, we observe better performance of the tailored plug-and-play (task is inherently easier). Thereby it is important to show the still excellent performance of the “more universal” adversarial training (tested gender networks that performed the same for unfiltered data and adversarial filtered data, the privacy $D_{KL}$ deteriorated by up to 90% on the plug-and-play sanitization). For $\alpha = 0$ no sanitization occurs and there is no loss in utility performance. As $\alpha$ increases ($\alpha=0, 0.05, 0.2, 0.5, 0.8$ in the figure), the KL-divergence for the utility increases and the privacy performance quickly approaches the prior.[]{data-label="fig:Dkl"}](Figures/DKL_DET.png "fig:"){width="\textwidth"}
&
[0.4]{} ![Trade-off achieved by the sanitization function between the KL-divergence of the utility distribution from the objective ($D_{KL}(P(u|x)||P(u|S(x)))$) and the KL-divergence of the gender distribution from the prior ($D_{KL}(P(p)||P(p|S(x)))$). Compared to the adversarial approach, we observe better performance of the tailored plug-and-play (task is inherently easier). Thereby it is important to show the still excellent performance of the “more universal” adversarial training (tested gender networks that performed the same for unfiltered data and adversarial filtered data, the privacy $D_{KL}$ deteriorated by up to 90% on the plug-and-play sanitization). For $\alpha = 0$ no sanitization occurs and there is no loss in utility performance. As $\alpha$ increases ($\alpha=0, 0.05, 0.2, 0.5, 0.8$ in the figure), the KL-divergence for the utility increases and the privacy performance quickly approaches the prior.[]{data-label="fig:Dkl"}](Figures/DKL_STO.png "fig:"){width="\textwidth"}
Figures \[fig:pgenderAdv\], \[fig:pgenderPAP\] and \[fig:subjectAccuracy\] show more directly interpretable results on privacy and utility performance as a function of $\alpha$. Figures \[fig:pgenderAdv\] and \[fig:pgenderPAP\] shows how the output probabilities of the gender classification model approach the prior distribution of the dataset as $\alpha$ increases for the adversarial and plug-and-play approaches respectively. We see that images from both female and male subjects produce output gender probabilities close to the dataset prior even for relatively low $\alpha$ values. Figure \[fig:subjectAccuracy\] shows how the top-5 categorical accuracy of the subject recognition task varies across different $\alpha$ values and training modalities. These results suggest that under these conditions we can achieve almost perfect privacy while maintaining reasonable utility performance.
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\[fig:pgenderAdv\]
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\[fig:pgenderPAP\]
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\[fig:subjectAccuracy\]
[ We also show that retraining the privacy task adversarially with the sanitization function does not sacrifice performance on unfiltered data. Figure \[fig:pgenderRaw\] in the supplementary material section shows how the output probabilities of the gender classification model on unfiltered data after adversarial training either maintains or improves performance when compared with the original gender classification model. This means that if the user decides not to block the gender (private data), the same system maintains its performance (and so does the utility/identity). ]{}
We show the loss evolution of the sanitization task (Eq.\[eq:sanitizationLoss\]) and the privacy task (Eq.\[eq:privatizationLoss\]) across batch iterations for different $\alpha$ values in figures \[fig:lossesDet\] and \[fig:lossesSto\] in the supplementary material section. We observe that all losses settle quickly, and that overall the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.
**[An Onboard Implementation-]{}** The framework here proposed calls for the user to decide what to share and what not to share, and as such it renders itself for the sanitization function to be implemented as close as possible to the data acquisition, clearly before the user sharing her/his data with the service provider. To illustrate this concept, we have designed a hardware prototype, described in the supplementary information. In brief, the system consists of an end device, capturing user visual data, and an entity, processing the user data. The end device captures visual data, performs face detection, applies a privacy filter (per the user privacy requirements), and conveys the sanitized data to the service provider. The entity applies the algorithms associated with the utility and privacy tasks to the sanitized data. The respective results are displayed in a user interface. We tested the performance of the trained sanitation functions over images acquired with the proposed hardware implementation. Four new subjects provided $104$ training images total, the subject identification network was retrained to also recognize these new subjects, in addition to all the subjects already present in the FaceScrub database. The sanitization functions were not retrained on the newly acquired images. Figure \[fig:hwResults\] in the supplementary material shows representative results on the stochastic plug-and-play sanitization function over $20$ test images. The sanitization function preserved utility performance while successfully blocking inference on the privacy variable.
Connections with the Literature {#sec:literature}
===============================
Our proposed framework and perspectives are new but these also connect to various elements in the privacy, statistics, and machine learning literature. Our goal here is to describe how our concepts relate to and depart from key concepts in this vast literature.
**[Differential Privacy-]{}** Differential privacy and (local) differential privacy based data release mechanisms are often considered to be the gold standard in privacy [@Dwork2008]. Data release mechanisms based on (local) differential privacy – which are also based on the application of some form of sanitization function to the original data – ensure it will be almost impossible for a data receiver to distinguish whether or not there is a certain element in a dataset/database or a certain attribute in the data held by the data holder. However, such data release mechanisms can also result in substantial utility loss [@Dwork2010; @Hardt2016; @Kifer2011a; @Kifer2014]. Our user-specific privacy collaborative approach departs from differential privacy in various ways. In particular, by building upon the premise that it is of interest both for users and service providers to collaborate to achieve privacy (and utility), we design data release mechanisms – our privacy filter – that achieve both utility and privacy, that are user-specific, and that are transparent to the service provider existing algorithms/architectures.
**[Information Bottleneck and Privacy Funnel-]{}** The privacy funnel – a concept closely related to the information bottleneck [@IB] – has also been recently considered as a basis for the design of data release mechanisms [@funnel] (see also [@Wang2018]). In particular, this approach can lead to data release mechanisms that strike a balance between the level of utility and the level of privacy. Both our approach and the privacy funnel (and its variants) can lead to data release mechanisms that meet specific user privacy requirements. However, it is not entirely clear how to design ’plug-and-play’ privacy filters using the privacy funnel without the collaborative framework proposed here. **[Adversarial Networks and Adversarial Examples-]{}** Adversarial game-type models as the one here presented clearly remind us of the type of works represented by generative adversarial networks (GANs) [@Goodfellow2014]. GANs are designed to learn the data (feature) distribution, while in our framework the task is to preserve privacy while guaranteeing utility, maintaining only the utility-needed data structure. As such, the proposed model addresses two goals, while GANs have the single goal of distribution learning (implemented though via the game-theory approach of two competing terms, not to confuse with two competing explicit tasks). We use the same general concept in [@Goodfellow2014] of training for adversarial goals, as also exploited in some of the works described below. Similarly, the literature on adversarial examples, [@Adversarial], connects to our work, while at the same time being very different. Our goal is to preserve utility while creating uncertainty in the private data, while adversarial examples do not guarantee utility or privacy, but rather are targeted to make the system fail (note that making the system always fail in gender for example will actually reveal the gender with full certainty).
**[Removing Nuisance/Attribute Variable-]{}** A series of works have shown how to produce signals that preserve certain properties while removing others [@FaderNetworks; @Pivot; @ControllableInvariance]. While these works share certain architectural similarities with ours, and can be used as components or complementary to the framework here proposed, they are different both in goals and accomplishments. Works such as [@FaderNetworks] produce images without the nuisance, without guaranteeing neither the utility nor the privacy, e.g., producing a female version of a male face does not guarantee the utility (recognition) or even the privacy (gender). This is demonstrated in Section \[sec:results\]. The use of GANs in [@Pivot] is in order to address robustness of the classifier/regressor to nuisance variables. Same goal motivates [@ControllableInvariance], which contrary to us trains both the utility and the privacy classifiers, and both can only handle filtered data (the training is done into arbitrary spaces). The possibility to handle both filtered and unfiltered data and to not retrain all system components is critical for the utility provider adoption of a framework as the one here developed. Note also that the image example in [@ControllableInvariance] uses privacy and utility tasks that are statistically independent, whereas the pair of tasks we chose are very strongly related (there is an injective mapping from subjects to gender), thereby being a much harder problem. Finally, we should make a clear distinction between aiming at making sure the output of the predictor/classifier is independent of the nuisance variable like in these works, which is more related to the closer topic of fairness (one of the examples addressed in [@ControllableInvariance]), and the aim of privacy, which needs to make sure that the nuisance/sensitive variable is independent of the encoded features. **[Protecting Training Data-]{}** The remarkable success of machine learning has led to intense interest in the privacy of the data used for training, since the released training models contain information about it, e.g., [@Carlini2018; @Hitaj2017]. Efforts to protect such data have been made, in particular, combining tools from differential privacy with parallel and independent training using data partitions and teacher/student paradigms [@Acs2017; @PATE]. Contrary to these works, we are not concerned here with the protection of the training data but some of the attributes of the testing data, which depending on the users desires about whether these should or should not be revealed. In spite of these fundamental differences, these efforts do provide some insights for potential future directions for the challenge here addressed, in particular the use of different combinations and partitions of data for training the utility and privacy components, including potentially training them with different combinations of original and filtered data and of true and random labels. Studying the utility training with the filtered data produced by our algorithm is a subject of interest and future work ([@Carlini2018] investigated training with sanitized data in their framework). The same applies to the important subject of fairness, and investigating the possibility of training with data that preserves utility but blocks potential biasing information, as produced by our filter, is of value as well (see also [@ControllableInvariance] for a contribution in this direction).
Concluding Remarks {#sec:conclusion}
==================
We introduced a new paradigm where users and entities collaborate to achieve both utility and privacy per a user specific requirements. One salient feature of this paradigm is that it can be completely transparent – involving only the use of a simple user-specific privacy filter applied to user data – in the sense that it requires otherwise no modifications to the system infrastructure, including the service provider algorithmic capability, in order to achieve both utility and privacy.
Representative architectural and system concepts, and results, suggest that such a collaborative based user-controlled privacy approach can be achieved. While the results here presented clearly show the potential of this approach, much has yet to be done, from theoretical foundations to algorithm development to practical architectures, and extensions to scenarios involving multiple users with different utility and privacy requirements and scenarios involving other data types.
Acknowledgments {#acknowledgments .unnumbered}
===============
Work partially supported by NSF, ONR, NGA, and ARO.
**[Supplementary Material]{}**
Performance of Gender Classification Model on Unfiltered Data {#performance-of-gender-classification-model-on-unfiltered-data .unnumbered}
=============================================================
We show how the adversarially trained privacy task maintains or improves performance on unfiltered data on Figure \[fig:pgenderRaw\]. This empirically shows that there is little to no downside to this adversarial retraining of the privacy function.
[cc]{}
[0.4]{} ![*Detailed breakdown of the gender probabilities computed by the adversarially trained privacy task as a function of $\alpha$ for the test dataset over unfiltered images. Whisker plots show median and interquartile ranges, with outliers shown as circles. The first column shows the output probability of being male (M) as computed by the gender discriminator over all female (F) subjects. Second column shows the same output probabilities for all male subjects. Results are shown as a function $\alpha$, where $\alpha=0$ is the performance of the untrained gender detector over raw data.*[]{data-label="fig:pgenderRaw"}](Figures/px_g1_DET_raw.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Detailed breakdown of the gender probabilities computed by the adversarially trained privacy task as a function of $\alpha$ for the test dataset over unfiltered images. Whisker plots show median and interquartile ranges, with outliers shown as circles. The first column shows the output probability of being male (M) as computed by the gender discriminator over all female (F) subjects. Second column shows the same output probabilities for all male subjects. Results are shown as a function $\alpha$, where $\alpha=0$ is the performance of the untrained gender detector over raw data.*[]{data-label="fig:pgenderRaw"}](Figures/px_g0_DET_raw.png "fig:"){width="\textwidth"}
\
[0.4]{} ![*Detailed breakdown of the gender probabilities computed by the adversarially trained privacy task as a function of $\alpha$ for the test dataset over unfiltered images. Whisker plots show median and interquartile ranges, with outliers shown as circles. The first column shows the output probability of being male (M) as computed by the gender discriminator over all female (F) subjects. Second column shows the same output probabilities for all male subjects. Results are shown as a function $\alpha$, where $\alpha=0$ is the performance of the untrained gender detector over raw data.*[]{data-label="fig:pgenderRaw"}](Figures/px_g1_STO_raw.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Detailed breakdown of the gender probabilities computed by the adversarially trained privacy task as a function of $\alpha$ for the test dataset over unfiltered images. Whisker plots show median and interquartile ranges, with outliers shown as circles. The first column shows the output probability of being male (M) as computed by the gender discriminator over all female (F) subjects. Second column shows the same output probabilities for all male subjects. Results are shown as a function $\alpha$, where $\alpha=0$ is the performance of the untrained gender detector over raw data.*[]{data-label="fig:pgenderRaw"}](Figures/px_g0_STO_raw.png "fig:"){width="\textwidth"}
Details on Evolution of Training Losses Across Iterations {#details-on-evolution-of-training-losses-across-iterations .unnumbered}
=========================================================
Here we show the loss evolution of the sanitization task ($Loss_S$) when using the plug-and-play and adversarial training for both the deterministic (figures \[fig:lossesDet\] and \[fig:lossesDetPap\]) and stochastic architectures (figures \[fig:lossesSto\] and \[fig:lossesStoPap\]). The privacy loss ($Loss_P$) is also shown on the adversarial training approach. The privacy and sanitization tasks are trained alternatively every epoch. Overall, $Loss_P$ grows over time as it is unable to infer the sensitive variable on the sanitized data.
[cc]{}
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time.Results are shown on the deterministic model trained adversarially. We observe that the both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesDet"}](Figures/loss_det_a005.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time.Results are shown on the deterministic model trained adversarially. We observe that the both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesDet"}](Figures/loss_det_a02.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time.Results are shown on the deterministic model trained adversarially. We observe that the both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesDet"}](Figures/loss_det_a05.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time.Results are shown on the deterministic model trained adversarially. We observe that the both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesDet"}](Figures/loss_det_a08.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained adversarially. We observe that both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesSto"}](Figures/loss_STO_005_adv.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained adversarially. We observe that both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesSto"}](Figures/loss_STO_02_adv.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained adversarially. We observe that both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesSto"}](Figures/loss_STO_05_adv.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the privacy loss ($Loss_p$) and sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained adversarially. We observe that both losses settle quickly, and that overall, the privacy loss grows over time as it is unable to infer the sensitive variable on the sanitized data.*[]{data-label="fig:lossesSto"}](Figures/loss_STO_08_adv.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the deterministic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesDetPap"}](Figures/loss_det_a005_pap.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the deterministic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesDetPap"}](Figures/loss_det_a02_pap.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the deterministic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesDetPap"}](Figures/loss_det_a05_pap.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the deterministic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesDetPap"}](Figures/loss_det_a08_pap.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesStoPap"}](Figures/loss_STO_005_pap.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesStoPap"}](Figures/loss_STO_02_pap.png "fig:"){width="\textwidth"}
[cc]{}
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesStoPap"}](Figures/loss_STO_05_pap.png "fig:"){width="\textwidth"}
&
[0.4]{} ![*Evolution of the sanitization loss ($Loss_S$) as a function of training time. Results are shown on the stochastic model trained using the plug-and-play approach. The loss quickly settles to its final value.*[]{data-label="fig:lossesStoPap"}](Figures/loss_STO_08_pap.png "fig:"){width="\textwidth"}
Details on the Onboard Implementation {#details-on-the-onboard-implementation .unnumbered}
-------------------------------------
This section describes a proof-of-concept prototype of our proposed collaborative privacy based system. The main components of this system are an edge device and an entity. The edge device is responsible to obtain visual user data, pre-process the visual data, and convey it to the entity. The pre-processing phase occurs when the user chooses to protect data that she/he does not want the entity to infer. This phase includes several tasks:
1. The first task is to locate a face (or multiple faces) within the acquired visual data. In particular, we use the face detector from dlib library [@dlib09], consisting of a combination of the Histogram of Oriented Gradients (HOG) feature with a linear classifier, an image pyramid, and sliding window detection scheme. After this, the found faces are treated as individual images.
2. The second task is to filter what the user would like the entity not to be able to infer. Each detected face is filtered according to the learned sanitization function responsible for the selected feature the user wishes to protect. This process transforms the data onto another piece of data, in the same space, such that utility is preserved but privacy is protected.
3. Finally, the data is conveyed to the entity using TCP/IP protocols.
Figure \[fig:edgedevice\] summarizes these various steps.
![*Edge device and entity tasks.*[]{data-label="fig:edgedevice"}](Figures/processes2.png){width="0.8\columnwidth"}
A Raspberry Pi 3 Model B board [@Rpi3] with quad-core 1.2GHz ARM Cortex A53 (ARMv8) CPU and 1GB LPDDR2 RAM running at 900MHz was used as a system’s edge device. The operating system installed on the board is Raspbian and it is powered by a 5.1 Volts micro USB supply. In order to obtain the visual data, a Raspberry Pi camera module v2 [@Rpicamera] was attached to the Raspberry Pi board via a ribbon cable. The camera module v2 is based on the 8-Megapixels resolution Sony IMX219 sensor. The images are sent using the embedded WIFI module of the Raspberry Pi 3. See Figure \[fig:systemoverview\] for these components.
![System overview.[]{data-label="fig:systemoverview"}](Figures/SystemOverview.png){width="0.8\columnwidth"}
For the purposes of the entity component, we developed a local server. This server communicates with the Raspberry Pi 3 using a Wireless Local Area Network (WLAN) to receive the user data. There, the received information is saved and displayed on a user interface. In parallel, a number of algorithms are used in order to try to infer information from the visual data. The classification results are displayed in the same interface as well.
Figure \[fig:hwResults\] shows representative results on the stochastic plug-and-play sanitization function over $20$ test images. The sanitization function preserved utility performance while successfully blocking inference on the privacy variable.
[ccc]{}
[0.3]{} ![*Detailed breakdown of the top-5 categorical accuracy and gender probabilities computed by the utility and privacy tasks as a function of $\alpha$ on data from the hardware-acquired dataset. The sanitization function shown here is the stochastic model trained using the plug-and-play approach.*[]{data-label="fig:hwResults"}](Figures/UtilityAccuracyStoc_hw_pap.png "fig:"){width="\textwidth"}
&
[0.3]{} ![*Detailed breakdown of the top-5 categorical accuracy and gender probabilities computed by the utility and privacy tasks as a function of $\alpha$ on data from the hardware-acquired dataset. The sanitization function shown here is the stochastic model trained using the plug-and-play approach.*[]{data-label="fig:hwResults"}](Figures/px_g1_STO_hw_pap.png "fig:"){width="\textwidth"}
&
[0.3]{} ![*Detailed breakdown of the top-5 categorical accuracy and gender probabilities computed by the utility and privacy tasks as a function of $\alpha$ on data from the hardware-acquired dataset. The sanitization function shown here is the stochastic model trained using the plug-and-play approach.*[]{data-label="fig:hwResults"}](Figures/px_g0_STO_hw_pap.png "fig:"){width="\textwidth"}
[^1]: Privacy is not hacking, the first one is the challenge addressed in this work, the second is a security/encryption task. Both are closely connected, and removing private data will immediately improve security.
|
---
abstract: 'We suggest a new method for the calculation of the nonlocal part of the effective action. It is based on resummation of perturbation series for the heat kernel and its functional trace at large values of the proper time parameter. We derive a new, essentially nonperturbative, nonlocal contribution to the effective action in spacetimes with dimensions $d>2$.'
author:
- |
A.O. Barvinsky$^{1}$ and V.F.Mukhanov$^{2}$\
$^{1}$Theory Department, Lebedev Physics Institute,\
Leninsky prospect 53, Moscow 117924;\
e-mail: [email protected]\
$^{2}$Sektion Physik, LMU, Theresienstr.37,Munich,Germany;\
e-mail: [email protected]
title: '[**New Nonlocal Effective Action**]{}'
---
+3mm - 0.8cm
‘@=11 addtoreset[equation]{}[section]{} ‘@=12 ="7100
Introduction
============
Effective action is among the fundamental ideas of modern quantum field theory. Calculated analytically for a given background field, it gives information about the induced energy-momentum tensor of quantum fields and quantum corrections to the classical equations of motion. The nonlocal part of the effective action should contain, for instance, particle creation effects. For black holes it should be able to account simultaneously for vacuum polarization and asymptotic Hawking radiation. Various important applications of the effective action can also be found in fundamental string theory. The Lorentzian effective action, which we actually need, can be obtained from the Euclidian one, $\SGamma[\,\phi\,]$, via analytic continuation. In turn, $\SGamma[\,\phi\,]$ can be defined by the following path integral $$\exp\Big(-\SGamma[\,\phi(x)\,]\Big)= \int D\varphi\,
\exp \left(-S[\,\varphi\,]
+(\varphi -\phi)\frac{\delta\SGamma[\,\phi\,]}{\delta\phi}
\,\right) , \label{efacdef}$$ where $\phi(x)$ is a given mean field, and the functional integration over quantum fields $\varphi(x)$ is assumed. The general semiclassical expansion of $\SGamma[\,\phi\,]$ begins with the one-loop contribution, which is given by the gaussian path integral $$\exp(-\SGamma[\,\phi(x)\,])=
\int D\varphi \exp \left( -\frac{1}{2}\int dx\,
\sqrt{g}\, \varphi(x)\,\widehat{F}
\left(\nabla,\phi(x)\right)\varphi(x)\,\right) \label{oneloop}$$ The operator $\widehat{F}(\nabla,\phi(x))$ here determines the propagation of small field disturbances $\varphi(x)$ on the background of $\phi(x)$ and in bosonic case can generically be written down as $$\widehat{F}=-\Box +V(x), \label{opF}$$ where $\Box =$ $\nabla ^{2}\equiv g^{\mu \nu }\nabla _{\mu }\nabla
_{\nu }$ is the Laplacian in the Euclidean field theory, which becomes the d’Alembertian when analytically continued to the Lorentzian sector, and $V(x)$ is the potential term. Note that, for some fields, the one-loop contribution is exact, for instance, for the scalar field without self-coupling. For a properly defined measure the gaussian integral (\[oneloop\]) can be formally calculated as $$\SGamma=\frac{1}{2}\ln \left( \prod_\lambda \lambda\right)
=\frac{1}{2}\sum_\lambda \ln
\lambda =\frac12\,{\rm Tr}\,\ln \widehat{F}, \label{act1}$$ where $\lambda$ are the eigenvalues of the operator $\widehat{F}$ corresponding to appropriately normalized eigenfunctions $\phi_\lambda(x)$, $\int dx\,\sqrt{g}\,\phi _\lambda(x)
\phi_{\lambda'}(x)=\delta _{\lambda\lambda'}$. Here the functional trace ${\rm Tr}$ does not depend on a particular basis in the functional space of disturbances $\varphi$ and, therefore, in an appropriate representation it reduces to the integral over spatial coordinates $x$ of the diagonal element of the operator kernel.
The effective action (\[act1\]) is, of course, ultraviolet divergent and should be regularized, with the subsequent interpretation of explicitly isolated divergences in terms of infinite renormalizations of the coupling constants of the theory. These divergences are well understood and it is unlikely that anything new can be added here. Therefore, we concentrate on more interesting finite, and generally nonlocal, contributions to one-loop effective action. These contributions depend on infrared properties of the theory and contain nontrivial information about real physical effects. Analytical calculational schemes for $\SGamma$ are usually based on the following integral representation of the functional trace of $\widehat{F}$ $${\rm Tr}\,\ln \widehat{F}=
-\int\limits_{0}^{\infty }\frac{ds}{s}\,
{\rm Tr}\,e^{-s\widehat{F}}, \label{intrepr}$$ where all local divergences can be easily isolated with the aid of dimensional regularization. The kernel $$K(s|\,x,y)\equiv
\exp(-s\widehat{F})\,\delta ^{(d)}(x,y), \label{kern}$$ where $d$ is spacetime dimensionality, obviously satisfies the heat kernel equation $$\frac{\partial }{\partial s}K(s|\,x,y)=
-\widehat{F}K(s|\,x,y)\equiv (\Box
-V(x))K(s|\,x,y), \label{heateq}$$ with the initial condition $$K(0|\,x,y)=\delta ^{(d)}(x,y) \label{incond}$$at $s=0.$ The auxiliary parameter $s$ is usually called the proper time. Thus, calculating the effective action can be reduced to the solution of the Cauchy problem for $K(s|\,x,y)$. In fact, what we actually need is the coincidence limit of this function, since in the representation we used, the functional trace of the operator $e^{-s\widehat{F}}$ corresponds to the integration of the diagonal elements of $K$ over the spacetime coordinates $x$, so that $$\SGamma=-\frac{1}{2}\int dx\,
\left( \int_{0}^{\infty }\frac{ds}{s}
K(s|\,x,x)\right). \label{act2}$$
It is clear that the success of the calculation of mainly depends on our ability to find an analytical solution of the heat kernel equation and carry out the integration over the proper time in (\[act2\]). The integral is obviously divergent as $s
\rightarrow 0$. As we have already mentioned above, this divergence can be easily isolated and interpreted in terms of the local ultraviolet properties of the theory. On the other hand, the behavior of the integral at infinity, $s\rightarrow\infty$, determines infrared properties of the theory and carries the physical information, e.g., particle creation. If the field has a big positive mass the proper time integral is convergent at $s\rightarrow \infty$. However, in the case of massless fields, the situation is much less trivial. The infrared convergence here depends on the approximation scheme used to calculate $K(s|\,x,y)$. Then, it is often even unclear to what extent the obtained effective action reflects the physical properties of the theory rather than the features of the approximation scheme used.
Below, we discuss the known calculational techniques, namely, the local Schwinger-DeWitt expansion [@DeWitt; @PhysRep], nonlocal covariant perturbation theory [@CPTI; @CPTII; @CPTIII] and the modified gradient expansion [@MWZ] and point out why all of them fail when applied to interesting physical problems. Instead of them, we suggest a new method based on resummation of perturbation series and calculate new, essentially nonperturbative terms in the effective action. This method becomes indispensable in low-dimensional models $(d\leq 2)$ where all previously known techniques are inapplicable. In this paper, we demonstrate how our method works in flat space of dimension $d>2,$ while the generalization to the curved space and low-dimensional case will be considered in [@infrII].
One of the main results of this paper is an exact (nonperturbative in $V$) late-time asymptotics for the heat kernel which in a spacetime of dimension $d>2$ for the coincidence limit takes the following form $$K(s|\,x,x)=\frac{1}{(4\pi s)^{d/2}}
\left( 1+\frac{1}{\Box -V}V(x)\right)^{2},
\,\,\,s\rightarrow \infty . \label{ass1a}$$ This asymptotics can also be found when the arguments of $K(s|\,x,y)$ are different - see Sect.4 for details. Another important result is the nonperturbative in potential expression $${\rm Tr}\,K(s)=\frac{1}{(4\pi s)^{d/2}}\int dx\,
\left\{ 1-s\Big(V(x)+V\frac{1}{\Box -V}V(x)\Big)\right\} \,\,\,s\rightarrow \infty
\label{latetr}$$ for the functional trace of $K(s)$[^1].
To avoid excessive use of integration signs we employ here and throughout the paper the following shorthand notation $$\frac{1}{\Box -V}\,J(x)\equiv \int dy\,G(x,y)\,J(y), \label{notation}$$where $G(x,y)$ is the Green’s function of the operator $-\widehat{F}=\Box-V$ with zero boundary conditions at spacetime infinity, that is $$(\Box-V)\,G(x,y)=\delta ^{(d)}(x,y),\,\,\,G(x,y)\rightarrow
0,\,\,\,|x|\rightarrow \infty , \label{GR}$$and $J(x)$ can be any function of various field quantities like powers of potential, its derivatives, etc. We always presume that the spacetime has the positive definite (Euclidean) signature, so that the Laplacian $\Box $ is negative definite assuming zero boundary conditions at infinity. Moreover we consider only non-negative potentials $V\left( x\right) \geq 0,$ so that the whole operator $\widehat{F}=-\Box +V$ is positive definite. Therefore, the Green’s function (\[GR\]) is uniquely defined and guarantees that the nonlocal expression (\[notation\]) makes sense for $d>2$.
As we shall see, the asymptotics (\[ass1a\]) and (\[latetr\]) are the corner stone of the technique we develop for the calculation of nonlocal contributions to the effective action. In particular, they lead to essentially nonperturbative terms which can be explicitly calculated for two broad classes of potentials with compact support, namely, for those ones which are, respectively, very small or very big in units of the inverse size of their support. For small potentials we get the terms which [*replace*]{} the conventional Coleman-Weinberg contribution to the effective action. In four dimensions, for instance, these terms read $$\Delta \SGamma\!=\!\frac{1}{64\pi ^{2}}\!\int d^4x\,V^{2}
\ln \Big(\int d^4y\,V^{2}\Big)\!
-\!\frac{1}{64\pi ^{2}}\!\int d^4x\,V^{2}
\ln \Big(\int d^4y\,
V\frac{\mu ^{2}}{V-\Box}V\Big), \label{4deff}$$ where the mass parameter $\mu ^{2}$ reflects the usual ultraviolet renormalization ambiguity. On the contrary, in the case of big potentials the Coleman-Weinberg action is [*supplemented*]{} by the nonlocal term of the form $$\Delta \SGamma=\frac{1}{64\pi ^{2}}
\int\limits_{|x|\leq R}\!d^4x\,
\Big<V+V\frac{1}{\Box -V}V\Big>^{2}.$$ Here $R$ is the size of the compact support of $V(x)$, that is, $V(x)=0$ at $|x|>R$, and $\big<...\big>$ denotes the spacetime averaging of the corresponding quantity over this compact domain. The obtained expressions are both nonlocal and non-analytic in the potential $V(x)$.
The paper is organized as follows. In Sect.2 we consider the known approximation schemes and discuss their applicability in the infrared region. Sect.3 is devoted to the nonlocal and nonlinear resummation of the Schwinger-DeWitt perturbation series, corresponding to the so-called connected graph expansion of the heat kernel. In Sect.4, as an extention of this resummation, we derive the asymptotics (\[ass1a\]) and discuss its relation to the functional trace (\[latetr\]) of the heat kernel. The nonperturbative, nonlocal contributions to the effective action are obtained in Sect.5 with the aid of the new technique based on a piecewise smooth approximation for the heat kernel. In two appendices we give details of the resummation technique and derive from the covariant perturbation theory the asymptotics of the heat kernel trace (\[latetr\]) up to the first subleading order in $1/s$ inclusive.
Approximation schemes and infrared properties of the effective action
=====================================================================
In flat space, which we consider in this paper, the solution of the heat kernel equation can be easily found if the potential vanishes. For an arbitrary spatially dependent potential the analytical expressions are, of course, available only in certain approximations. In the general case, it is convenient to factorize the ”zero potential” part of the solution explicitly and use the following ansatz for $K(s|\,x,y)$ $$K(s|\,x,y)=\frac{1}{\left( 4\pi s\right)^{d/2}}
\exp \left[ -\frac{\left|x-y\right|^{2}}{4s}\right]
\Omega (s|\,x,y), \label{ansatz}$$ where the factor singular in $s$ guarantees that the initial condition (\[incond\]) is satisfied, provided that $\Omega$ is analytic in $s$ at $s=0$ and $\Omega (0|\,x,y)=1$. If $V=0$, then $\Omega \equiv 1$ and hence all nontrivial information about the potential is encoded in the deviation of $\Omega$ from unity.
The most well-known approximation used for the calculation of $K(s|\,x,y)$ is the so-called local Schwinger-DeWitt expansion, where $\Omega$ is written down as a series in growing powers of the proper time $s$. This expansion is a very powerful tool for revealing local ultraviolet properties of the theory. However, when applied in the infrared region, it gives a finite result only for massive fields. If the potential $V(x)$ has a large positive constant part, that is, $$V(x)=m^{2}+v(x), \label{pot1}$$ where $m^{2}$ is the squared mass of the field, then the function $\Omega(s|\,x,y)$ contains an overall exponential factor $e^{-sm^{2}}$ and the independent of mass part is expanded in powers of $s$ $$\Omega (s|\,x,y)=e^{-sm^{2}}
\sum\limits_{n=0}^{\infty }a_{n}(x,y)\,s^{n}. \label{dewittexp}$$ Here $a_{n}(x,y)$ are the two-point Schwinger-DeWitt coefficients, whoes coincidence limits $\left( x\rightarrow y\right)$ are explicitly calculable in general field theories, including gravity. Substituting (\[dewittexp\]) in (\[ansatz\]) and then the obtained expression at $x=y$ in (\[act2\]) one gets: $$\SGamma=-\frac{1}{2\left( 4\pi \right) ^{d/2}}\int
dx\sum\limits_{n=0}^{\infty }\,\left( \int_{0}^{\infty}
s^{n-d/2-1}e^{-sm^{2}}ds\right) a_{n}(x,x). \label{dewittact1}$$
It is important that the exponent $e^{-sm^{2}}$ is not expanded here in powers of $s$. Therefore, in the proper time integral it provides a cutoff at the upper limit, so that the powers of $s$ in this expansion get effectively replaced by powers of $1/m^{2}$. The first $(d/2+1)$ integrals in (\[dewittact1\]) diverge at $s\rightarrow 0$ and should be regularized. To do that, we apply the dimensional regularization method. Namely, by replacing the dimensionality $d$ by $2\omega$, we calculate the integrals in the domain of their convergence and then analytically continue the result to $\omega \rightarrow d/2$. In spaces with even number of dimensions, which we mainly consider in what follows, this gives rise to the contribution $\SGamma_{{\rm div,\,log}}$ containing the pole at $\omega =d/2$ and the term logarithmic in $m^{2}$ $$\SGamma_{{\rm div,log }}\!=\!\frac{1}{2(4\pi )^{d\over 2}}\int
dx\sum\limits_{n=0}^{d/2}\!
\frac{(-m^{2})^{{d\over 2}-n}}{\left({d\over2}
-n\right) !}\left[\frac{1}{\omega -\frac{d}{2}}\!
-\!\Gamma ^{\prime }\Big(
\frac{d}{2}-n+1\Big)\!
+\!\ln \frac{m^{2}}{4\pi \mu ^{2}}\right] a_{n}(x,x), \label{dix}$$ where $\omega \rightarrow d/2.$ The pole corresponds to an infinite ultraviolet renormalization of the terms proportional to $a_{0},...,a_{d/2}$ in the original Lagrangian. Other terms in the expansion (\[dewittact1\]) are finite and give the infrared contribution to the total action $$\SGamma=\SGamma_{{\rm div,\ln }}-\frac{1}{2}
\left( \frac{m^{2}}{4\pi}\right)^{d/2}
\int dx\,\sum\limits_{n=d/2+1}^{\infty} \,
\frac{\Gamma(n-d/2)}{(m^{2})^{n}}\,a_{n}(x,x). \label{infdewitt}$$ The Schwinger-DeWitt coefficients $a_{n}(x,x)$ are the homogeneous polynomials of dimensionality $2n$ in the units of inverse length, which are built of $v(x)$ and its multiple derivatives . Therefore, on dimensional grounds, they can be symbolically written down as $$a_{n}(x,x)\sim v^{k}(x)(\nabla ^{i}v^{j})(x),\,\,$$ where $i$ denotes an overall number of derivatives acting in all possible ways on $j$ factors of $v(x)$, and $k$ powers of $v(x)$ stay undifferentiated. The positive integers $(k,j,i)$ are related to $n$ as $2(k+j)+i=2n$. It is clear that the infinite series in ([infdewitt]{}) represents the expansion in growing powers of the following dimensionless quantities $$\frac{v\left( x\right) }{m^{2}}\ll 1,\,\,\,\,
\frac{\nabla ^{i}v(x)}{m^{2+i}}\ll 1, \label{exppar}$$ which obviously should be much smaller than unity. Only in this case the first few terms in the asymptotic series (\[infdewitt\]) are reliable.
Thus, the Schwinger-DeWitt expansion is applicable only in theories with [*small and slowly varying fields*]{} as compared to a big mass parameter. This expansion contains only local terms. This is not surprising because all nonlocal effects, e.g., particle creation, are very small for heavy particles in a weak external field and cannot be handled by this method. The Schwinger-DeWitt technique can be easily extended to curved spacetime and to theories with covariant derivatives built with respect to an arbitrary fibre-bundle connection. In this case, the perturbation potential $v(x)$ will also depend on the spacetime curvature tensor and fibre-bundle curvatures (commutator of covariant derivatives). The smallness of fields and their derivatives includes the requirement of the smallness of these curvatures and their derivatives as well. Despite its universality, the Schwinger-DeWitt expansion becomes inefficient when the ratios in (\[exppar\]) become of the order of unity and completely fails for massless fields. In the last case all integrals over the proper time integral are infrared divergent. This divergence has, of course, no physical meaning and is an artifact of the approximation technique used.
There are two known ways to proceed with massless fields. One possibility is the resummation of all terms which contain the undifferentiated potential $V(x)$ in the local Schwinger-DeWitt series (\[dewittexp\]). They are summed up to form an exponent similar to $e^{-sm^{2}}$ $$\Omega (s|\,x,x)=e^{-sV(x)}\sum\limits_{n=0}^{\infty }
\tilde{a}_{n}(x,x)\,s^{n}. \label{moddewitt}$$ This method was suggested in [@MWZ], where a regular technique for the calculation of the modified Schwinger-DeWitt coefficients $\tilde{a}_{n}(x,y)$ was also presented. The proper time integral in (\[act2\]) has now an infrared cutoff at $s\sim 1/V(x)$ and in this case the effective action is similar to (\[dix\]) - (\[infdewitt\]), where $m^{2}$ is replaced by $V(x)$ and $a_{n}(x,x)$ by $\tilde{a}_{n}(x,x)$. It is convenient to write this action as a sum of three terms $$\SGamma=\SGamma_{{\rm div}}+\SGamma_{{\rm CW}}
+\SGamma_{{\rm fin}}, \label{totmodact}$$ where the divergent part is equal to $$\SGamma_{{\rm div}}=\frac{1}{2(4\pi )^{\frac{d}{2}}}\int
dx\sum\limits_{n=0}^{d/2}\frac{(-V)^{{d\over2}-n}}
{\left({d\over2}-n\right)}\left[
\frac{1}{\omega-{d\over 2}}-\Gamma^\prime
\left({d\over 2}-n+1\right)-\ln 4\pi \right]
\tilde{a}_{n}(x,x). \label{moddiv}$$ The pole part of this action coincides with that of (\[dix\]) if we take $m^{2}\to 0$ limit of (\[dix\]). Actually, in this case, only the term proportional to $a_{d/2}$ survives in $\SGamma_{{\rm
div,\,log}}$ and by virtue of the relation between twiddled and untwiddled coefficients, namely, $$a_{d/2}(x,x)=\sum\limits_{n=0}^{d/2}
\frac{(-V)^{d/2-n}}{(d/2-n)!}\,
\tilde{a}_{n}(x,x),\,\,\, \label{rel}$$ the pole parts of (\[dix\]) and (\[moddiv\]) are the same. The terms proportional to $\Gamma ^{\prime }(d/2-n+1)$ perform finite renormalization of the local terms $V^{d/2-n}\,\tilde{a}_{n}.$ The logarithmic terms of (\[dix\]) are replaced in the modified action (\[totmodact\]) by $$\SGamma_{{\rm CW}}=\frac{1}{2(4\pi )^{d/2}}
\int dx\sum\limits_{n=0}^{d/2} \frac{(-V)^{d/2-n}}{(d/2-n)!}\,
\ln \frac{V}{\mu ^{2}}\,\,\tilde{a}_{n}=
\frac{1}{2(4\pi )^{d/2}}\int dx\,
\ln \frac{V}{\mu ^{2}}\,a_{d/2}. \label{CW}$$ This is nothing but the space-time integral of the Coleman-Weinberg effective potential. For instance, in four dimensions the leading term is the original Coleman-Weinberg effective potential, $V^{2}\ln (V/\mu^{2})/64\pi ^{2}$, while the rest represents corrections due to the derivatives of $V(x)$. Similarly to (\[infdewitt\]), the finite part $\SGamma_{{\rm
fin}}$ is an infinite series $$\SGamma_{{\rm fin}}=
-\frac{1}{2}\int dx\,
\left( \frac{V(x)}{4\pi }\right)^{d/2}
\sum\limits_{n=d/2+1}^{\infty }\Gamma (n-d/2)\,
\frac{\tilde{a}_{n}(x,x)}{V^{n}(x)}. \label{modfin}$$ The modified Schwinger-DeWitt coefficients do not contain the undifferentiated potential and the typical structure of the terms entering $\tilde{a}_{n}(x,x)$ is $\nabla ^{m}V^{j}(x)$, where $m+2j=2n$. Every $V$ here should be differentiated at least once and therefore $m\geq j$. Thus the coefficients $\tilde{a}_{n}$ can be symbolically written down as $$\tilde{a}_{n}(x,x)\sim \sum\limits_{j=1}^{[2n/3]}
\nabla ^{2n-2j}V^{j}, \label{struc}$$ where the upper value of $j$ is the integer part of $2n/3$.
This perturbation theory is efficient as long as the potential is slowly varying or bounded from below by a large positive constant, so that $$\frac{\nabla ^{2}V(x)}{V^{2}(x)}\ll 1,\,\,\,
\frac{\left( \nabla V(x)\right) ^{2}}
{V^{3}(x)}\ll 1,...\, . \label{parexp}$$ The case of bounded potentials reproduces the original Schwinger-DeWitt expansion for nonvanishing mass. Therefore, let us consider the potentials which vanish at spacetime infinity $(|x|\rightarrow \infty )$. Namely, we assume the case of power-like falloff $$V(x)\sim \frac{1}{|x|^{p}},\,\,\,\,
\nabla ^{m}V(x)\sim \frac{1}{|x|^{p+m}},
\,\,\,\,|x|\rightarrow \infty \label{potfall}$$ for some positive $p$. For such a potential terms of the perturbation series (\[modfin\]) behave as $$\frac{\tilde{a}_{n}}{V^{n}}\sim
\sum\limits_{j=1}^{[2n/3]}|x|^{(p-2)(n-j)} \label{exppar1}$$ and thus decrease with $n$ only if $p<2$. For $p\geq 2,$ the modified gradient expansion completely breaks down. It makes sense only for slowly decreasing potentials of the form (\[potfall\]) with $p<2$. In this case the potential $V(x)$ is not integrable over the whole spacetime $\left( \int dx\,V(x)=\infty \right)$ and moreover even the operation $(1/\Box )V(x)$ is not well defined[^2]. Therefore the above restriction is too strong to account for many interesting physical problems. In addition, similarly to (\[infdewitt\]), the asymptotic expansion (\[modfin\]) is completely local. It does not allow us to capture nonlocal effects, which are exponentially small for potentials satisfying (\[parexp\]).
The way to overcome this difficulty and to take into account nonlocal effects was suggested in the form of covariant perturbation theory (CPT) [@CPTI; @CPTII; @CPTIII]. In this theory the full potential $V(x)$ is treated as a perturbation and the solution of the heat equation is found as a series in its powers. From the viewpoint of the Schwinger-DeWitt expansion it corresponds to an infinite resummation of all terms with a given power of the potential and arbitrary number of derivatives. The result reads as $${\rm Tr}\,K(s)\equiv \int dx\,K(s|\,x,x)=
\sum\limits_{n=0}^{\infty }{\rm Tr}
\,K_{n}(s), \label{CPT1}$$ where $${\rm Tr}\,K_{n}(s)=\int
dx_{1}dx_{2}...dx_{n}\,
F_{n}(s|\,x_{1},x_{2},...x_{n}) \,V(x_{1})V(x_{2})...V(x_{n}), \label{Kn}$$ and the nonlocal form factors $F_{n}(s|\,x_{1},x_{2},...x_{n})$ were explicitly obtained in [@CPTI; @CPTII; @CPTIII]. It was shown that at $s\rightarrow \infty $ the terms in this expansion behave as $${\rm Tr}\,K_{n}(s)=
O\left( \frac{1}{s^{d/2-1}}\right) ,\,\,\,
n\geq 1, \label{CPTas}$$ and, therefore in spacetime dimension $d\geq 3$ the integrals in (\[act2\]) are infrared convergent $$\int\limits^{\infty }\frac{ds}{s}\,
O\left( \frac{1}{s^{d/2-1}}\right)
<\infty . \label{infCPT}$$ In one and two dimensions this expansion for $\SGamma$ does not exist except for the special case of the massless theory in curved two-dimensional spacetime, when it reproduces the so-called Polyakov action[@FrolVilk; @Polyakov; @CPTII][^3]. CPT should always be applicable whenever $d\geq 3$ and the potential $V$ is sufficiently small[^4], so that its effective action explicitly features analyticity in the potential at $V=0$. Therefore, its serious disadvantage is that this theory does not allow one to overstep the limits of perturbation scheme and, in particular, discover non-analytic structures in the action if they exist.
All this implies the necessity of a new approximation technique that would allow us to overcome disadvantages of the above methods. In the rest of this paper we develop such a technique, involving further resummation of the perturbation series. We develop an infrared improved perturbation theory for the heat kernel and reveal new nonlocal and non-analytical structures in the effective action.
Resummation of proper time series
=================================
We use the exponential ansatz for the function $\Omega (s|\,x,y)$ defined by Eq.(\[ansatz\]): $$\Omega (s|\,x,y)=\exp \Big[-W(s|\,x,y)\Big].$$ Our goal is to develop an approximation technique for $W$ similar to CPT, which is an alternative to the expansion in $s$. By virtue of (\[heateq\]) and (\[incond\]) the function $W(s|\,x,y)$ satisfies the equation $$\frac{\partial W}{\partial s}+\frac{(x-y)^{\mu }}
{s}\nabla _{\mu }W-\Box
W=V-(\nabla W)^{2}, \label{eqw}$$ with the initial condition $$W(s=0|\,x,y)=0. \label{iniW}$$ This equation is nonlinear in $W$ and we solve it by iteration, considering $(\nabla W)^{2}$ as a perturbation. For this purpose it is convenient to rewrite (\[eqw\]) - (\[iniW\]) as an integral equation. In Appendix A, it is shown that this integral equation takes the following form $$W(s|\,x,y)=s\int_{0}^{1}d\alpha \,e^{s\alpha (1-\alpha )
\overset{-}{\Box }}\left( V(\bar{x})-
(\nabla W(s\alpha |\,\bar{x},y))^{2}\right), \label{intform}$$ where the operator $\overset{-}{\Box }$ acts on the argument $\bar{x}\equiv \bar{x}(\alpha |x,y)=\alpha x+(1-\alpha )y.$ The equation (\[intform\]) can be easily solved if $(\nabla
W)^{2}\ll V$. As we will see later, this condition is satisfied for a broad class of potentials $V$. The lowest order approximation for $W$ is obtained by omitting $(\nabla W)^{2}$ in eq. (\[intform\]) $$W_{0}(s|\,x,y)=s\int_{0}^{1}d\alpha \,
e^{s\alpha (1-\alpha )\overset{-}{\Box}}\left.
V(\bar{x})\,\right| _{\bar{x}=\alpha x+(1-\alpha )y}. \label{lowapp}$$ This is a linear but essentially nonlocal functional of the potential. Further terms of the perturbation theory, $W_{n}=O\left[ (\nabla W_{0})^{n}\right] $, can be graphically represented by connected tree graphs with two derivatives in the vertices, internal lines associated with the nonlocal operator $$f(-s\Box )=\int_{0}^{1}d\alpha \,
e^{s\alpha (1-\alpha )\Box }, \label{oper}$$ and external lines given by (\[lowapp\])[^5]. Note that this connected graph structure arises in the exponential and when expanded gives rise to the disconnected graphs. In context of the heat kernel expansion this property was observed in [@connected]. Resummation of the perturbation series in $V$ explicitly features exponentiation of the quantities containing only connected graphs. Here we showed how this exponentiated set of connected graphs directly arises from the solution of the simple nonlinear equation (\[eqw\]). The ”propagator” (\[oper\]) was worked out within the covariant perturbation theory in [@CPTI; @CPTII; @CPTIII] and was also obtained in [@MWZ] by direct summation of gradient series.
At this stage the efficiency of the connected graph expansion is not yet obvious. Crudely, it runs in powers of the dimensionless quantity $\left( sf(-s\Box )\nabla \right) ^{2}V(x)$ which, at least naively, should be small for slowly varying or/and small potentials. Apart from this, infrared properties of the effective action strongly depend on the lowest order approximation for $W$ (\[lowapp\]). The effective action involves only diagonal elements of the two-point function $W(s|\,x,y),$ which look much simpler than (\[lowapp\]) $$W_{0}(s|\,x,x)=s\int_{0}^{1}d\alpha \,
e^{s\alpha (1-\alpha )\Box }V(x). \label{coinarg}$$ Note that, at small $s,$ the function $W_{0}$ can be expanded as $W_{0}=sV+O\left( s^{2}\right) .$ The only term with undifferentiated potential entering $W_{0}$ is linear in $s,$ while all other terms contain derivatives of the potential $V.$ The same is also true for the exact $W,$ which differs from $W_{0}$ by higher powers of the differentiated potential. This completely agrees with the modified gradient expansion discussed in Sect.2. However, the expression (\[coinarg\]) directly involves the nonlocal operator and its late time behavior is very different from that naively expected in the modified gradient expansion and in CPT.
To show that, let us first find the coordinate representation of the operator (\[oper\]), that is, the kernel $f(-s\Box)\delta^{(d)}(x,y)$. Using a well-known result for the exponentiated $\Box $-operator $$e^{s\alpha (1-\alpha )\Box }\delta^{(d)}(x,y)=\frac{1}{(4\pi s\alpha
(1-\alpha ))^{d/2}}\exp \left( -\frac{|x-y|^{2}}{4s\alpha (1-\alpha )} \right), \label{expbox}$$ one can write $$\begin{aligned}
&&\int_{0}^{1}d\alpha \,
e^{s\alpha(1-\alpha)\Box}
\delta ^{(d)}(x,y) \nonumber \\
&&\qquad\qquad\quad
=\frac{e^{-|x-y|^{2}/2s}}{(4\pi s)^{d/2}}
\int_{0}^{\infty}d\beta \,
\frac{(1+\beta )^{d-2}}{\beta ^{d/2}}
\exp \left[ -\frac{|x-y|^{2}}{4s}
\left( \frac{1}{\beta }+\beta \right) \right] , \label{kern1}
\end{aligned}$$ where we have changed the integration variable, $\alpha=\beta/(1+\beta )$, $0\leq \beta <\infty $. For $d\geq 3,$ the integral can be easily calculated and the result is expressed as a sum of McDonald functions of the argument $|x-y|^{2}/2s$ $$f(-s\Box )\delta ^{(d)}(x,y)=
\frac{2e^{-|x-y|^{2}/2s}}{(4\pi s)^{d/2}} \sum\limits_{k=1}^{d-1}C_{k-1}^{d-2}
K_{k-\frac{d}{2}}\left(
|x-y|^{2}/2s\right) , \label{kern2}$$ where $C_{k-1}^{d-2}$ are the binomial coefficients. For very large $s$, the argument of $K_{k-\frac{d}{2}}(z)$ is small. Using the asymptotics: $K_{\nu }(z)\simeq \Gamma (|\nu |)(2/z)^{|\nu
|}/2,\,\,\,z\rightarrow 0$, we find that at large $s$ the form factor is dominated by the following term $$f(-s\Box )\delta ^{(d)}(x,y)
=\frac{1}{2s}\frac{\Gamma (d/2-1)}{\pi^{d/2}|x-y|^{d-2}}
+O\left( \frac{1}{s^{2}}\right) . \label{kern3}$$ Taking into account the fact that $$\frac{1}{\Box }\delta ^{(d)}(x,y)=
-\frac{\Gamma (d/2-1)}{4\pi^{d/2}|x-y|^{d-2}}, \label{invbox}$$ we finally obtain: $$W_{0}(s|\,x,x)=sf(-s\Box )V=
-\frac{2}{\Box }V(x)+O\left( \frac{1}{s}\right) . \label{asW0}$$ This behavior agrees with the formal asymptotics found by the Laplace method in [@CPTII]. Therefore, at $s\rightarrow\infty$, the function $W_{0}$ approaches a constant. The nonlocal functional (\[asW0\]) is well-defined for $d\geq 3$ only if the potential $V$ vanishes fast enough at $|x|\rightarrow
\infty$.
For splitted arguments the asymptotics of $W_{0}(s|\,x,y)$ is more intricate. In this case the form factor (\[oper\]) no longer arises as a whole because the integration parameter $\alpha $ appears in eq.(\[lowapp\]) also in the argument $\bar{x}=\alpha
x+(1-\alpha )y$ of the potential $V(\bar{x})$. Applying the Laplace method, one can show that the integral (\[lowapp\]) is dominated by the contribution of the end points, $\alpha =0$ and $\alpha =1$. These contributions are different because $\bar{x}(\alpha =0)=y$ and $\bar{x}(\alpha =1)=x$, whence $$W_{0}(s|\,x,y)=-\frac{1}{\Box }V(x)
-\frac{1}{\Box }V(y)
+O\left( \frac{1}{s}\right) . \label{assplitarg}$$ Substituting this expression in (\[intform\]) and solving the integral equation by iterations one can find the late time asymptotics $W_\infty(x,y)$ for the exact $W(s|\,x,y)$ $$W(s|\,x,y)=W_\infty(x,y)
+O\left( \frac{1}{s}\right) \label{exas1}$$ as a nonlocal gradient series $$W_\infty(x,y)=
-\frac{1}{\Box }V(x)+
\frac{1}{\Box }\left( \frac{1}{\Box }
\nabla V(x)\right)^{2}+...
+(x\leftrightarrow y). \label{exas}$$ It is remarkable, however, that this series can be ”summed up” and the nonlocal expression for $W_{\infty }(x,y)$ can be found exactly in terms of the Green’s function of the original operator $\widehat{F}=-\Box +V$.
Late time asymptotics of the heat kernel
========================================
By substituting the late time ansatz (\[exas1\]) in equation (\[eqw\]), it is easy to see that the first two terms in its left hand side vanish at $s\rightarrow \infty $, while the rest reduce to the equation for $W_{\infty }(x,y)$ $$(\Box -V)\,e^{-W_{\infty }(x,y)}=0. \label{eqWinf}$$ Despite the positivity of the operator $-\Box +V$ with Dirichlet boundary conditions at $|x|\to\infty$, this equation admits non-trivial solutions. In fact, $e^{-W_{\infty }(x,y)}$ does not have to go to zero at infinity[^6]. In view of the iterative solution (\[exas\]) it should tend to some unknown function of $y$ $$e^{-W_{\infty }(x,y)}\rightarrow C(y),\,\,\,
|x|\rightarrow \infty . \label{bound1}$$ The equation (\[eqWinf\]) with this boundary condition is then solved by $$e^{-W_{\infty }(x,y)}=C(y)\,\Phi (x), \label{ans}$$ if the new function $\Phi (x)$ satisfies the equation $$(\Box -V)\,\Phi (x)=0, \label{eqF}$$ and $\Phi (x)\rightarrow 1$ at $|x|\rightarrow\infty$. The solution of this problem for $\Phi(x)$ is uniquely determined in terms of the Green’s function (\[GR\]) $$\Phi (x)=1+\frac{1}{\Box-V}\,V(x). \label{solF}$$ This function of $x$ is a nonlocal and essentially nonlinear functional of the potential, and it will play a very important role in what follows.
The heat kernel is symmetric in its arguments $x,y$ and, therefore, the unknown function $C(y)$ should coincide with $\Phi(y)$. Thus, finally we obtain the following exact late time asymptotics $$e^{-W_{\infty }(x,y)}=\Phi (x)\,\Phi (y). \label{finalas}$$ Expanding $W_{\infty }(x,y)$ in powers of the potential $V$ one gets $$\begin{aligned}
W_{\infty }(x,y) &=&-\ln \Phi (x)-\ln \Phi (y) \nonumber\\
&=&-\frac{1}{\Box }V(x)
-\frac{1}{\Box }V\frac{1}{\Box }V(x)
+\frac{1}{2}\left( \frac{1}{\Box }V(x)\right)^{2}+...
+(x\leftrightarrow y). \label{exexp}
\end{aligned}$$ The first term here is in agreement with the perturbative asymptotics (\[exas\]). However, beyond that, the iterative solution (\[exas\]) seems to be in contradiction with (\[exexp\]). The series (\[exas\]) runs in powers of the differentiated potential, while the expansion (\[exexp\]) contains only undifferentiated potential. However, integration by parts in the second term of (\[exas\]) exactly reproduces the second and third terms of (\[exexp\]). Thus, both expansions are equivalent, but the first one reveals more explicitly the gradient of the potential as a small parameter, while in (\[exexp\]) the smallness is a result of non-trivial cancellations between different terms.
Finally we write down the exact late time asymptotics for the heat kernel, advocated in Introduction, $$K(s|\,x,y)=\frac{1}{(4\pi s)^{d/2}}\,
\Phi (x)\Phi (y),\,\,\,s\rightarrow\infty . \label{hkas}$$ Its heuristic interpretation is rather transparent. The heat kernel can be decomposed in the series $$K(s|\,x,y)=\sum\limits_{\lambda }
e^{-\lambda s}\Phi _{\lambda }(x)\,
\Phi_{\lambda }(y), \label{eigenexp}$$ where $\lambda$ and $\Phi _\lambda$ are, respectively, the eigenvalues and eigenfunctions of the operator $\widehat{F}=-\Box
+V(x)$. Since $\widehat{F}$ is a positive-semidefinite operator, only the lowest eigenmode with $\lambda =0$ survives in this expression in the limit $s\rightarrow \infty$. The appropriate eigenfunction satisfies the equation $\widehat{F}\Phi _{\lambda
=0}=0$, which coincides with (\[eqF\]) and therefore $\Phi
_{\lambda =0}(x)=\Phi (x)$. The spectrum of the operator is continuous and the eigenmodes are not square integrable $\left(\Phi _{0}(x)\rightarrow 1\text{ at } |x|\rightarrow \infty
\right)$. This is why the integral over the spectrum, denoted above by $\sum_{\lambda }$, yields within the steepest decent approximation the power-like asymptotics, $1/s^{d/2}$, rather than the exponential one. Of course, these arguments are not very rigorous. The zero mode $\Phi (x)$ with unit boundary condition at infinity, does not even belong to the continuous spectrum of modes normalized to the delta function. Nevertheless, as we have seen, this particular mode gives the leading contribution to the late time asymptotics of the heat kernel.
If we want to calculate the contribution to the effective action due to the late time behavior of the heat kernel we need its functional trace – spacetime integral of the coincidence limit $K(s|\,x,x)$. Unfortunately, the expression (\[hkas\]) cannot be used directly to calculate ${\rm Tr}\,K(s)$ for a given $s$. Point is that this asymptotic expression taken at a fixed large $s$ is applicable only for $|x|^{2}<s$ and fails at $|x|^{2}\gg
s$[^7]. At the same time, when calculating the trace we have to integrate over the whole spacetime up to $|x|\rightarrow \infty $ and therefore need the heat kernel behavior for $|x|^2\gg s$. The attempt to disregard this subtlety and integrate the coincidence limit of (\[hkas\]) over $x$ results in a poorly defined quantity – the spacetime integral strongly divergent at infinity. Nevertheless, one can use the expression (\[hkas\]) to find ${\rm Tr}\,K(s)$ with the aid of the following somewhat subtler procedure.
First let us write the variational relation $$\delta \,{\rm Tr}K(s)=-s{\rm Tr}\,
\big(\delta V\,K(s)\big)
=-s\int dx\,\delta V(x)\,K(s|x,x), \label{var}$$ where, of course, $K(s)=\exp [s(\Box -V)]$. Then it follows that $$\frac{\delta \,{\rm Tr}K(s)}{\delta V(x)}
=-sK(s|x,x). \label{vareq}$$ Substituting $K(s|x,x)$ from (\[hkas\]) in the right hand side of this relation we obtain the following functional differential equation $$\frac{\delta \,{\rm Tr}K(s)}{\delta V(x)}=
-\frac{s}{(4\pi s)^{d/2}}\,\Phi^{2}(x). \label{feq}$$ This equation satisfies the intergrability condition, because the variational derivative $$\frac{\delta\Phi^2(x)}{\delta V(y)}=2\Phi(x)\Phi(y)
\frac1{\Box-V}\delta(x,y)$$ is symmetric in $x$ and $y$. Therefore (\[feq\]) can be integrated to determine ${\rm Tr}K(s)$. The solution subject to the obvious boundary conditions at $V=0$ reads $${\rm Tr}\,K(s)=\frac{1}{(4\pi s)^{d/2}}
\int dx\,\Big( 1-sV\Phi \Big),
\,\,\, s\to\infty. \label{sol}$$ One can easily check that this expression satisfies the equation (\[feq\]).
It is remarkable that in the covariant perturbation theory the leading and next subleading (in $s$) terms of the heat kernel trace can be explicitly calculated for $s\rightarrow\infty $ in every order of expansion in powers of $V$. The corresponding infinite series can be explicitly summed up to yield essentially nonlocal and nonlinear in $V(x)$ expression for ${\rm Tr}\,K(s)$. This is done in Appendix B to the first subleading order inclusive. The following very simple and concise result reads as $$\begin{aligned}
{\rm Tr}\,K(s)=\frac1{(4\pi s)^{d/2}}
\int dx\,\left\{1-sV\Phi
-2\nabla_\mu\Phi\frac1{\Box-V}\nabla^\mu\Phi
+O\left(\frac1s\right)\right\} \label{LATETR}
\end{aligned}$$ in terms of the function $\Phi(x)$ and its derivatives. As we see, it exactly reproduces the leading order term of (\[sol\]), $O(s/s^{d/2})$, and also gives a nontrivial $O(1/s^{d/2})$-correction. Below we use this asymptotics for obtaining new types of the nonlocal effective action.
Effective action
================
The functional trace of the heat kernel is everything we need for the calculation of the effective action. Unfortunately, only its asymptotics are known. Namely, at small $s$ one can use the modified gradient expansion (\[ansatz\]), (\[moddewitt\]) and at large $s$ – the nonlocal and nonlinear expression (\[LATETR\]). The goal of this section is to unify both of these approximations to get the expression for the effective action which would incorporate both the ultraviolet and infrared properties of the theory. The calculation will be explicitly done in four dimensional case. The generalization to other dimensions $d>2$ is straightforward.
The key idea is to replace ${\rm Tr}\,K(s)$ in (\[act2\]) by some approximate function ${\rm Tr}\,\bar{K}(s)$ such that the integral over $s$ $$\bar{\SGamma}=-\frac{1}{2}\,\int
\frac{ds}{s}{\rm Tr}\,\bar{K}(s) \label{act1a}$$ becomes explicitly calculable. The difference ${\rm
Tr}\,K(s)-\bar{K}(s)$ can then be treated as a perturbation. Certainly, the efficiency of this procedure very much depends on the successful choice of $\bar{K}(s)$. Here we exploit the simplest possibility – namely, let us take two simple functions ${\rm Tr}\,\bar{K}_{<}(s)$ and ${\rm Tr}\,\bar{K}_{>}(s)$, which coincide with the leading asymptotics of ${\rm Tr}\,K(s)$ at $s\rightarrow 0$ and $s\rightarrow \infty$ and use them to approximate ${\rm Tr}\,K(s)$ respectively at $0\leq s\leq
s_{\ast}$ and $s_{\ast }\leq s<\infty$. In turn, the value of $s_{\ast}$ will be determined from the requirement that these two functions match at $s_{\ast}$. This will guarantee the stationarity of $\bar{\SGamma}$ with respect to the choice of $s_{\ast}$, $\partial \bar{\SGamma}/\partial s_{\ast}=0$. We will discuss the justification of this procedure a little later, while now let us proceed with the calculation of $\bar{\SGamma}$.
At small $s$ we use the lowest order term of the modified gradient expansion $${\rm Tr}\,K_{<}(s)=\frac{1}{(4\pi s)^{2}}
\int dx\,\exp \left( -Vs\right),\,\,\,s<s_{\ast } \label{ass1}$$ and disregard all terms containing derivatives of the potential $V$. The rest of the asymptotic series (\[moddewitt\]), containing $\tilde{a}_{n}$, will be treated by perturbations. Correspondingly, at large $s>s_{\ast }$ we use the late time asymptotics (\[sol\]) $${\rm Tr}\,K_{>}(s)=\frac{1}{(4\pi s)^{2}}
\int dx\,(1-sV\Phi),\,\,\,s>s_{\ast }. \label{lass}$$
The requirement of stationarity of $\bar{\SGamma}$ with respect to $s_{\ast}$ leads to the equation $$\int dx\,\exp \left( -Vs_{\ast }\right)
=\int dx\,(1-s_{\ast }V\Phi ), \label{eqfors}$$ which determines the value of $s_{\ast}$ as some nontrivial functional of the potential, $s_{\ast }=s_{\ast }[\,V(x)\,]$. Unfortunately this functional is not calculable explicitly in general, but nevertheless, as we will see below, it can be obtained for two rather broad classes of potentials. The action (\[act1a\]) can be written down as a sum of two contributions $$\bar{\SGamma}=\SGamma_{<}+\SGamma_{>}=
-\frac{1}{2}\int\limits_{0}^{\infty }
\frac{ds}{s}{\rm Tr}\,K_{<}(s)
-\frac{1}{2}\int\limits_{s_{\ast }}^{\infty }
\frac{ds}{s}\left( {\rm Tr}\,K_{>}(s)
-{\rm Tr}\,K_{<}(s)\right) . \label{ll}$$ The first integral here has already been calculated and is given by the sum of the expressions (\[moddiv\]) and (\[CW\]) with $\tilde{a}_{0}=1$ and $\tilde{a}_{n}=0,\,n\geq 1$, which in our particular case give rise to $$\begin{aligned}
&&\SGamma_{<}=\SGamma_{{\rm div}}
+\SGamma_{{\rm CW}} \nonumber \\
&&\qquad \equiv \frac{1}{64\pi ^{2}}
\int dx\,\left[ \left( -\frac{1}{2-\omega}
+2{\bf C}-3-\ln 4\pi \right) V^{2}
+V^{2}\ln \frac{V}{\mu ^{2}}\,\right],
\,\,\,\omega \rightarrow 2, \label{uv}
\end{aligned}$$ where ${\bf C}=0,577...$ is the Euler’s constant. The first term here is responsible for the renormalization of the original action and the second one is just the Coleman-Weinberg potential. The second integral in (\[ll\]) can also be calculated exactly. Integrating by parts and taking into account (\[eqfors\]), we obtain $$\begin{aligned}
&&\SGamma_{>}\equiv
-\frac{1}{2}\int\limits_{s_{\ast }}^{\infty }
\frac{ds}{s}\,\left( {\rm Tr}\,K_{>}(s)
-{\rm Tr}\,K_{<}(s)\right) \nonumber \\
&&\qquad \qquad \qquad \qquad
=\frac{1}{64\pi ^{2}}\int dx\,
\left[ \,\frac{V\Phi }{s_{\ast }}
-\frac{Ve^{-s_{\ast }V}}{s_{\ast }}
+V^{2}\Gamma (0,s_{\ast}V)\,\right], \label{ir}
\end{aligned}$$ where $\Gamma (0,x)$ is an incomplete gamma function, $\Gamma
(0,x)=\int_{x}^{\infty }\,dt\,t^{-1}e^{-t}$, with the following asymptotics $$\Gamma (0,x)\sim \left\{
\begin{array}{lc}
\ln \displaystyle{\frac{1}{x}}-{\bf C}, &\,\,\,\,x\ll 1, \\
\displaystyle{\frac1x}\,e^{-x}, &\,\,\,\, x\gg 1.
\end{array}\right. \label{gass}$$
Further steps strongly depend on the class of potentials, for which the consistency of the piecewise approximation (\[ass1\]) - (\[lass\]) should be carefully analyzed.
Small potential
---------------
The approximation (\[ass1\]) - (\[lass\]) is efficient only if the ranges of validity of two asymptotic expansions (respectively for small and big $s$) overlap with each other and the point $s_{\ast}$ belongs to this overlap. In this case the corrections due to the deviation of ${\rm Tr}\,\bar{K}(s)$ from the exact ${\rm Tr}\,K(s)$ are uniformly bounded everywhere and one can expect that (\[act1a\]) would give a good zeroth-order approximation to an exact result. Below we show that this necessary requirement can be satisfied at least for two rather wide classes of potentials $V(x)$.
The modified gradient expansion is well applicable in the overlap range of the parameter $s$ if $$s\nabla \nabla V\ll V, \label{5.1}$$ (cf. Eq.(\[parexp\]) with $s$ replaced by effective cutoff $s=1/V$) and the applicability of the large $s$ expansion in the same domain reads as $$s\int dx\,V\Phi \gg \int dx\,
\nabla _{\mu }\Phi \frac{1}{V-\Box }
\nabla^\mu\Phi, \label{5.4}$$ which means that the subleading term (quadratic in $\nabla_\mu\Phi$) of the late time expansion (\[LATETR\]) is much smaller than the leading second term.
To implement these requirements, let us make some simplifying assumptions. Instead of the power law falloff, assume that $V(x)$ has a compact support of finite size $R$ $$V(x)=0,\,\,\,|x|\geq R.$$ Let us also assume that the characteristic magnitude of the potential inside its support is given by $V_{0}$. Then the estimate for the derivatives is obvious $$\nabla \nabla V\sim \frac{V_{0}}{R^{2}}$$ and (\[5.1\]) reads as $sV_{0}/R^{2}\ll V_{0}$, that is $$s\ll R^{2}. \label{a2}$$ To find out what does the criterion (\[5.4\]) mean let us make a simplifying assumption, namely, that the potential $V$ is small. In this case it can be disregarded in the Green’s functions and $1/(\Box-V)$ can be replaced by $1/\Box $. Therefore the following estimates hold $$\begin{aligned}
&&\frac{1}{\Box -V}V(x)\sim \int_{|y|\leq R}dy\,
\frac{1}{|x-y|^{d-2}} V(y)\sim \frac{1}{R^{d-2}}R^{d}V_{0}
\sim V_{0}R^{2}, \nonumber\\
&&\int dx\,V\Phi \sim V_{0}R^{d}, \nonumber\\
&&\int dx\,\nabla_\mu\Phi
\frac{1}{V-\Box }\nabla ^{\mu }\Phi
\simeq V_{0}^{2}R^{d+4}. \label{est}
\end{aligned}$$ Roughly, every Green’s function gives the factor $R^{2}$, every derivative – $1/R$, integration gives the volume of compact support $R^{d}$, etc. Applying these estimates to eq. (\[5.4\]) we get $sV_{0}R^{d}\gg V_{0}^{2}R^{d+4}$, whence $$s\gg V_{0}R^{4}.$$ Combining this with (\[a2\]) one gets the following range of overlap of our asymptotic expansions $$R^{2}\gg s\gg V_{0}R^{4}. \label{a5}$$ It immediately follows from here that this overlap domain is not empty only if $$V_{0}R^{2}\ll 1. \label{sp}$$ Moreover, the assumption of disregarding the potential in the Green’s function is also justified in this case since $V\sim
V_{0}\ll 1/R^{2}\sim \Box $. In other words this bound means that the potential is small in units of the inverse size of its compact support.
Now let us check whether $s_{\ast }$ introduced above belongs to the overlap domain (\[a5\]). Note that if it is really so, then $s_{\ast}V$ in Eq.(\[eqfors\]) is much smaller than unity because in the overlap range one has $sV\sim sV_0\ll R^2 V_0\ll
1$. Hence the exponent in the left hand side of (\[eqfors\]) can be expanded in powers of $s_{\ast }V$, and the resulting equation for $s_{\ast}$ becomes[^8] $$\int dx\,\left( 1-s_{\ast }V
+\frac{s_{\ast }^{2}}{2}V^{2}
+O\left( \left(s_{\ast}V\right)^{3}\right) \right)
=\int dx\,\left( 1-s_{\ast }V\Phi\right)$$Its solutions has the following form: $$s_{\ast }\simeq 2\,\frac{\int dx\,
V(1-\Phi )}{\int dx\,V^{2}}=2\,\frac{\int dx\,
V\frac{1}{V-\Box }V}{\int dx\,V^{2}}.$$ Taking into account the estimates (\[est\]) we see that the point $s_{\ast}\sim R^{2}$ belongs to the upper edge of the interval (\[a5\]). Late time expansions is fairly well satisfied here, but the small $s$ expansion is on the verge of breakdown. At this level of generality it is hard to overstep the uncertainty of this estimate. There is a hope that numerical coefficients in more precise estimates (with concrete potentials) can be large enough to shift $s_{\ast}$ to the interior of the interval (\[a5\]) and, thus, make our approximation completely reliable.
Bearing in mind all these reservations let us proceed with the calculation of the effective action. Using the small $x$ asymptotics (\[gass\]) in the expression (\[ir\]) we get $$\SGamma_{>}\simeq \frac{1}{64\pi ^{2}}\int d^4x\,\left[ \,-V\frac{1-\Phi }{s_{\ast }}+V^{2}\left( \ln \frac{1}{s_{\ast }V}-{\bf C}+1\right) \,\right] .$$It is interesting to note that in the whole action $\bar{\SGamma}=\SGamma_{<}+\SGamma_{>}$ the Coleman-Weinberg term disappears and the final answer reads $$\begin{aligned}
&&\bar{\SGamma}\simeq \frac{1}{64\pi ^{2}}\left( -\frac{1}{2-\omega }+{\bf C}-2-\ln 4\pi \right) \int d^4x\,V^{2} \nonumber \\
&&\quad +\frac{1}{64\pi ^{2}}\int d^4x\,V^{2}\ln \Big(\int dx\,V^{2}\Big)-\frac{1}{64\pi ^{2}}\int d^4x\,V^{2}\ln \Big(\int dx\,V\frac{\mu ^{2}}{V-\Box }V\Big). \label{EAsmallV}\end{aligned}$$The first term here differs from $\SGamma_{{\rm div}}$ in (\[uv\]) by a finite renormalization of the local $V^{2}$-term, while the two other terms have entirely new nonlinear and nonlocal structure advocated in the Introduction. The ultraviolet renormalization mass parameter $\mu ^{2}$ makes the argument of the second logarithm dimensionless – it plays the same role as for the Coleman-Weinberg potential, but now it enters the new essentially nonlocal structure.
It is natural that the original Coleman-Weinberg term for the case of small potentials (\[sp\]) gets replaced by the other qualitatively new nonlocal structure. Potentials which are small in units of the inverse size of their support are qualitatively very different from nearly constant potentials for which the Coleman-Weiberg action was originally derived. In the case of small potentials spacetime gradients dominate over their magnitude and, therefore, one should not expect that the Colman-Weiberg term would survive the inclusion of nonlocal structures.
Big potential
-------------
Remarkably, the case of the small potentials (\[sp\]) is not the only one when one can find a non-empty domain of overlap where both asymptotics for ${\rm Tr}\,K(s)$ are applicable. Namely, the opposite case of big potentials (in units of the inverse size of their support) $$V_{0}R^{2}\gg 1, \label{bp}$$ is equally good. The key observation here is that in this case the kernel of the Green’s function $1/(\Box -V)$ can be replaced within the compact support of $V$ by $-1/V$ ($\Box \sim 1/R^{2}\ll
V_{0}\sim V$) and correspondingly $$\begin{aligned}
&&\frac{1}{\Box -V}V(x)
\sim -\frac{1}{V}V=-1, \label{5.12a} \\
&&\int dx\,\nabla _{\mu }
\Phi \frac{1}{V-\Box }
\nabla ^{\mu }\Phi \simeq
\frac{R^{4}}{V_{0}R^{2}}. \label{5.12}
\end{aligned}$$ Therefore, the criterion of applicability of the late time expansion (\[5.4\]) becomes $s\gg 1/V_{0}^{2}R^{2}$. Together with (\[a2\]) it yields the new overlap range $$R^{2}\gg s\gg \frac{1}{V_{0}^{2}R^{2}} \label{bpint}$$ which is obviously not empty if the potential satisfies (\[bp\]).
To find $s_{\ast}$ in this case we have to solve the equation (\[eqfors\]) for the case when $s_{\ast }V$ is not anymore a small quantity. Since $V$ is big, the exponent in (\[eqfors\]) can be replaced by zero inside the compact support, $\exp(-s_{\ast}V(x))\sim 0$, $|x|\leq R$, and by one outside of it where the potential vanishes, $\exp (-s_{\ast }V(x))\sim 1$, $|x|>R$. Rewriting the integrals in both sides of the equation (\[eqfors\]) as a sum of contributions of $|x|\leq R$ and $|x|>R$, we see that the contribution of the non-compact domain gets cancelled and the equation becomes $$s_{\ast }\int\limits_{|x|\leq R}dx\,
V\Phi \simeq \int\limits_{|x|\leq R}dx.$$ Then it follows that $s_{\ast}$ is approximately given by the inverse of the function $V\Phi (x)$ [*averaged*]{} over the compact support of the potential $$\begin{aligned}
&&s_{\ast }\simeq \frac{1}{\big<V\Phi \big>}, \\
&&\big<V\Phi \big>\equiv
\frac{\int\limits_{|x|\leq R}\!dx\,V\Phi }
{\int\limits_{|x|\leq R}\!dx}.
\end{aligned}$$ A qualitative estimate of $\big<V\Phi \big>\sim V_{0}$ implies that $s_{\ast}\sim 1/V_{0}$ and it belongs to the middle of the interval (\[bpint\]). This makes the case of big potentials fairly consistent. On the other hand, the value of $\Phi (x)$ is close to zero inside the potential support (see (\[5.12a\])), so most likely the estimate for $\big<V\Phi \big>$ is smaller by at least one power of the quantity $1/V_{0}R^{2}$, which is the basic dimensionless small parameter in this case. Therefore the magnitude of $s_{\ast}$ becomes bigger by one power of $V_{0}R^{2}$, $s_{\ast }\simeq R^{2}$, which is again near the upper boundary of the overlap interval (\[bpint\]). Similarly to the small potential case, a more rigorous analyses is needed (maybe for more concretely specified potentials) to account for subtle edge effects at the boundary of compact support, which might shift the value of $s_{\ast }$ to a safe region inside (\[bpint\]).
With the above estimate for $s_{\ast }\sim R^{2}$ the magnitude of $s_{\ast}V$ in the expression for the infrared part of the effective action (\[ir\]) becomes big, $s_{\ast }V\sim
s_{\ast}V_{0}\sim V_{0}R^{2}\gg 1$, and we use the big $x$ asymptotics in (\[gass\]) to get the contribution $$\SGamma_{>}\simeq \frac{1}{64\pi ^{2}s_{\ast }}
\int d^4x\,V\Phi =\frac{1}{64\pi ^{2}}\int\limits_{|x|\leq R}
\!dx\,\big<V\Phi \big>^{2}.$$ In this case the Coleman-Weinberg term is not cancelled in complete agreement with what we would expect for big potentials and the final result reads $$\bar{\SGamma}=\SGamma_{{\rm div}}
+\SGamma_{{\rm CW}}+\frac{1}{64\pi ^{2}}
\int\limits_{|x|\leq R}\!d^4x\,\big<V\Phi \big>^{2}.$$
Comments
========
We developed a new technique for the calculation of late time asymptotics of the heat kernel and its functional trace. Using these asymptotics we found previously unknown essentially nonlocal and nonperturbative contributions to the effective action for two large classes of potentials with compact supports. Therefore, the generalization of these results to potentials with power-law falloff, which would imply subtler analyses, deserves further studies.
Our results in their present form are applicable only in higher dimensions $d\geq 3$. One can easily check that the expression $(1/\Box)V$ is not well defined in low dimensions $\left(
d\leq2\right)$. Note, that the logarithmic kernel of the Green’s function of the massless field is defined in $d=2$ only up to an additive constant. Moreover, the convolution of the Green’s function with the potential makes sense in $d=2$ only if the latter is the total derivative of some other function [@CPTII], as, for instance, the 2-dimensional curvature scalar in the Polyakov action [@Polyakov]. Thus, the extension of our results to low-dimensional models, where other calculational schemes completely fail, is especially important. This will be done in the forthcoming paper [@infrII].
Another possibility is the generalization of our technique to potentials with isolated zeroes in the interior of spacetime. Even more interesting is the situation when the potential becomes negative $V(x)<0$ in some spacetime domains. In this case there is a tachionic instability, and it is worth deriving quantitative criteria describing this instability in terms of the properties of $V(x)$.
Finally, it is important to generalize our results to curved spacetimes and in addition consider fields with non-trivial spin-tensor structure. All these issues are addressed in [@infrII]. The nonlocal effective action can then be applied to study interesting physical problems, like quantum black holes evaporation [@MWZ], quantum cosmology, etc.
It is worth mentioning that the developed technique for late time asymptotics of the heat kernel could also be useful in statistical physics for calculating low temperature partition functions.
Integral equation for $W$
=========================
In this Appendix we derive the integral form of the equation $$\frac{\partial W(s|\,x,y)}{\partial s}
+\frac{(x-y)^{\mu }}{s}
\nabla _{\mu}W(s|\,x,y)
-\Box W(s|\,x,y)=f(s|\,x,y), \label{A1}$$ where $$f(s|\,x,y)\equiv V(x)
-(\nabla W(s|\,x,y))^{2}, \label{A2}$$ and $W(s=0|\,x,y)=0$. With this purpose we first introduce the new function $\tilde{W}$ $$W(s|x,y)=e^{-s\Box }\tilde{W}(s|\,x,y). \label{A3}$$
Using the relation $$e^{s\Box }(x-y)^{\mu }e^{-s\Box }
=(x-y)^{\mu }+2s\nabla ^{\mu } \label{A4}$$ one finds that this function satisfies the following equation $$\frac{\partial \tilde{W}}{\partial s}
+\frac{(x-y)^{\mu }}{s}\nabla _{\mu }
\tilde{W}=e^{s\Box }f, \label{A6}$$ which does not contain anymore $\Box $-term on the left hand side. To write down the formal solution of this equation in terms of the ”source” term $f=f(s|\,x,y)$, let us introduce the characteristic curve $\bar{x}^{\mu }(t)$ of (\[A6\]), which satisfies the equation $$\frac{d\bar{x}^{\mu }(t)}{dt}
=\frac{(\bar{x}(t)-y)^{\mu }}{t}, \label{A7}$$ with the boundary conditions $$\bar{x}^{\mu }(t=0)=y^{\mu },\,\,\,
\bar{x}^{\mu }(t=s)=x^{\mu }. \label{A8}$$ The solution of (\[A7\]) is $$\bar{x}^{\mu }(t)=y^{\mu }
+\frac{(x-y)^{\mu }}{s}t. \label{A9}$$ The total derivative of $\tilde{W}(t|\bar{x}(t),y)$ with respect to $t$ along this characteristic curve is then equal to $$\frac{d}{dt}\tilde{W}(t|\bar{x}(t),y)
=\left[ \,\frac{\partial }{\partial t}+
\frac{(x-y)^{\mu }}{t}
\frac{\partial }{\partial \bar{x}^{\mu }}\right]
\tilde{W}(t|\bar{x},y)
=e^{t\overset{-}{\Box }}f(t|\,\bar{x}(t),y), \label{A10}$$ where $\overset{-}{\Box }\equiv \partial ^{2}/\partial
\bar{x}^{\mu }\partial \bar{x}_{\mu }$. Integrating this equation from $0$ to $s$ with the initial condition $\tilde{W}=0$ at $t=0$ and taking into account the boundary conditions (\[A8\]) for $\bar{x}(t)$, one gets $$\tilde{W}(s|\,x,y)=\int_{0}^{s}dt\,
e^{t\overset{-}{\Box }}\,
f(t|\,\bar{x}\left( t\right) ,y). \label{A11}$$ Returning to the original $W$ which is related to $\tilde{W}$ via (\[A3\]) and taking into account that $\Box=\left(t/s\right)^{2}\overset{-}{\Box}$ we finally obtain: $$W(s|\,x,y)=s\int_{0}^{1}d\alpha \,
e^{s\alpha (1-\alpha )\overset{-}{\Box}}
f(s\alpha |\,\bar{x},y)\,
\Big|_{\,\bar{x}=\alpha x+(1-\alpha )y}, \label{A12}$$ where instead of $t$ the new integration variable $\alpha =t/s$ was introduced. This is exactly the integral form (\[intform\]) of the equation (\[A1\]) that we used in Sect.3.
Covariant perturbation theory and late time behavior of the functional trace of the heat kernel
===============================================================================================
Here we consider the nonlocal covariant perturbation theory (CPT) of [@CPTI; @CPTII; @CPTIII]. In CPT the functional trace of the heat kernel for the covariant second order differential operator is expanded as nonlocal series in powers of the potential $V$ with explicitly calculable coefficients - nonlocal form factors $F_{n}(s|\,x_{1},x_{2},...x_{n})$. Their leading asymptotic behavior at large $s$ was obtained in [@CPTII]. Here we calculate them up to the first subleading order in $1/s$ inclusive for a simple operator $\widehat{F}=\Box-V$. Then we explicitly perform infinite summation of power series in the potential to obtain the nonlocal and nolinear expression (\[sol\]) for late time behavior of ${\rm Tr}\,K(s)$.
According to [@CPTII] the heat kernel trace is local in the first two orders of the perturbation theory in the potential (\[CPT1\]) $$\begin{aligned}
&&{\rm Tr}\,K_{0}(s)=\frac{1}{(4\pi s)^{d/2}}\int dx, \\
&&{\rm Tr}\,K_{1}(s)=-\frac{s}{(4\pi s)^{d/2}}\int dx\,V(x),
\end{aligned}$$ and in higher orders it reads as $${\rm Tr}\,K_{n}(s)=\frac{(-s)^{n}}{(4\pi s)^{d/2}n}
\int dx\,\big<e^{s\Omega_{n}}\big>\,
V(x_{1})V(x_{2})...V(x_{n})
\Big|_{x_{1}=...=x_{n}=x},\,\,\,n\geq 2. \label{B1}$$ Here $\Omega _{n}$ is a differential operator acting on the product of $n$ potentials $$\Omega _{n}=\sum\limits_{i=1}^{n-1}
\nabla_{i+1}^{2}+2\sum\limits_{i=2}^{n-1}
\sum\limits_{k=1}^{i-1}\beta _{i}(1-\beta_{k})
\nabla _{i+1}\nabla _{k+1}, \label{B2}$$ expressed in terms of the partial derivatives labelled by the indices $i$ implying that $\nabla _{i}$ acts on $V(x_{i})$. It is assumed in (\[B1\]) that after the action of all derivatives on the respective terms all $x_{i}$ are set equal to $x$. It is also assumed that the spacetime indices of all derivatives $\nabla
=\nabla ^{\mu }$ are contracted in their bilinear combinations, $\nabla _{i}\nabla _{k}\equiv \nabla _{i}^{\mu }\nabla _{\mu\,k}$. The differential operator (\[B2\]) depends on the parameters $\beta _{i}$, $i=1,...n-1$, which are defined in terms of the parameters $\alpha_{i}$, $i=1,...n$ as $$\beta _{i}=\alpha _{i+1}+\alpha _{i+2}+...+\alpha _{n},$$ and the angular brackets in $\big<e^{s\Omega _{n}}\big>$ imply that this operator exponent is integrated over compact domain in the space of $\alpha $-parameters $$\big<e^{s\Omega _{n}}\big>\equiv
\int\limits_{\alpha _{i}\geq 0}d^{n}\alpha
\,\delta \Big(\sum_{i=1}^{n}\alpha _{i}-1\Big)\,
\exp (s\Omega _{n}).$$
The late time behavior of ${\rm Tr}\,K_{n}(s)$ is thus determined by the asymptotic behavior of this integral at $s\rightarrow\infty$, which can be calculated using the Laplace method. To apply this method, let us note that $\Omega _{n}$ is a [*negative*]{} semidefinite operator (this is shown in the Appendix B of [@CPTII]) which degenerates to zero at $n$ points of the integration domain: $(0,...0,\alpha _{i}=1,0,...0)$, $i=1,...n$. Therefore the asymptotic expansion of this integral is given by the contribution of the corresponding $n$ maxima of the integrand at these points. The integration by parts in (\[B1\]) is justified by the formal identity $\nabla_{1}+\nabla_{2}+...+\nabla _{n}=0$ . Using it one can show that the contributions of all these maxima are equal, so that it is sufficient to calculate only the contribution of the point $\alpha _{1}=1$, $\alpha _{i}=0$, $i=2,...n$. In the vicinity of this point it is convenient to rewrite the expression for $\Omega_{n}$ in terms of the independent $(n-1)$ variables $\alpha_{2},\alpha _{2},...\alpha _{n}$, the remaining $\alpha_{1}=1-\sum_{i=2}^{n}\alpha _{i}$, $$\Omega _{n}=\sum_{i=2}^{n}\alpha _{i}D_{i}^{2}
-\sum_{m,k=2}^{n}\alpha_{m}\alpha _{k}D_{m}D_{k},$$ where the operator $D_{m}$ is defined as $$D_{m}=\nabla _{2}+\nabla _{3}+...+\nabla _{m},
\,\,\,m=2,...n. \label{D}$$
Substituting this expression for $\Omega _{n}$ in (\[B1\]) and expanding in powers of the term bilinear in $\alpha$-parameters one gets $$\begin{aligned}
&&{\rm Tr}\,K_{n}(s)=
\frac{(-s)^{n}}{(4\pi s)^{d/2}}\int
dx\,\int_{0}^{\infty }d^{n-1}\alpha \,
\exp \Big(s\sum_{i=2}^{n}\alpha_{i}D_{i}^{2}\Big) \nonumber \\
&&\qquad \qquad \qquad \qquad \qquad
\times \left( 1-s\sum_{m,k=2}^{n}\alpha_{m}
\alpha _{k}D_{m}D_{k}+...\right) \,V_{1}V_{2}...V_{n}.
\end{aligned}$$ Here $1/n$ factor disappeared due to the contribution of $n$ equal terms and the range of integration over $\alpha_{2},...\alpha_{n}$, $\sum_{i=2}^{n}\alpha _{i}\leq 1$, was extended to all positive values of $\alpha _{i}$. This is justified since the error we make by extending the integration range is exponentially small and goes beyond the the accuracy of asymptotic expansion in inverse powers of $s$. The second term in the round brackets can be rewritten in terms of the derivatives with respect to $D_{m}^{2}$ acting on the exponential, so that $$\begin{aligned}
&&{\rm Tr}\,K_{n}(s)=\frac{(-s)^{n}}{(4\pi s)^{d/2}}
\int dx\,\left( 1-\frac{1}{s}\sum_{m,k=2}^{n}D_{m}D_{k}
\frac{\partial }{\partial D_{m}^{2}}\frac{
\partial }{\partial D_{k}^{2}}+...\right) \nonumber \\
&&\qquad \qquad \qquad \qquad \qquad \times
\int_{0}^{\infty }d^{n-1}\alpha
\,\exp \Big(s\sum_{i=2}^{n}
\alpha _{i}D_{i}^{2}\Big)\,V_{1}V_{2}...V_{n}.
\end{aligned}$$ In this form it is obvious that further terms of expansion in powers of the quadratic in $\alpha $ part of $\Omega _{n}$ bring higher order corrections of the $1/s$-series. Doing the integral over $\alpha$ here and performing differentiations one obtains $$\begin{aligned}
&&{\rm Tr}\,K_{n}(s)=\frac{1}{(4\pi s)^{d/2}}\int dx\,
\left[ -s\frac{1}{D_{2}^{2}...D_{n}^{2}}
+2\sum_{m=2}^{n}\frac{1}{D_{2}^{2}...D_{m-1}^{2}}\,
\frac{1}{(D_{m}^{2})^{2}}\,
\frac{1}{D_{m+1}^{2}...D_{n}^{2}}\right. \nonumber \\
&&\qquad +2\sum_{m=2}^{n-1}\sum_{k=m+1}^{n}
\frac{1}{D_{2}^{2}...D_{m-1}^{2}}
\,\frac{D_{m}^{\mu }}{(D_{m}^{2})^{2}}\,
\frac{1}{D_{m+1}^{2}...D_{k-1}^{2}}\,
\frac{D_{k\mu }}{(D_{k}^{2})^{2}}\,
\frac{1}{D_{k+1}^{2}...D_{n}^{2}} \nonumber \\
&&\qquad \qquad \qquad \qquad \qquad \qquad
\left. +O\left( \frac{1}{s}
\right) \right] \,V_{1}V_{2}...V_{n}. \label{bigequation}
\end{aligned}$$
The first term in the square brackets gives the leading order term of the late time expansion. It can be further transformed by taking into account that any operator $D_{m}$ defined by (\[D\]) acts as a partial derivative only on the group of factors $V_{m}V_{m+1}...V_{n}$ in the full product $V_{1}...V_{n}$, $D_{m}V_{1}...V_{n}=V_{1}...V_{m-1}\nabla (V_{m+1}...V_{n})$. Therefore all the operators understood as [*acting to the right*]{} can be ordered in such a way $${\rm Tr}\,K_{n}(s)=-\frac{s}{(4\pi s)^{d/2}}
\int dx\,V_{1}\frac{1}{D_{n}^{2}}
V_{n}\frac{1}{D_{n-1}^{2}}V_{n-1}...
\frac{1}{D_{2}^{2}}V_{2}+O\left( \frac{1}{s^{d/2}}\right)$$ that the labels of $D_{m}^{2}$’s can be omitted and all $D_{m}^{2}$ can be identified with boxes also acting to the right $${\rm Tr}\,K_{n}(s)=-\frac{s}{(4\pi s)^{d/2}}
\int dx\,\underbrace{V\frac{1}{\Box }V\frac{1}{\Box }...
V\frac{1}{\Box }}_{n-1}\,V(x)
+O\left( \frac{1}{s^{d/2}}\right) . \label{b1}$$ Infinite summation of this series is not difficult to perform because this is the geometric progression in powers of the nonlocal operator $V(1/\Box )$ and $${\rm Tr}\,K(s)={\rm Tr}\,K_{0}(s)
-\frac{s}{(4\pi s)^{d/2}}\int
dx\,\sum_{n=0}^{\infty }
\left( V\frac{1}{\Box }\right) ^{n}V(x)
+O\left(\frac{1}{s^{d/2}}\right),$$ or $${\rm Tr}\,K(s)=\frac{1}{(4\pi s)^{d/2}}
\int dx\,\Big(1-s\Box \frac{1}{\Box-V}\,V(x)
+O(s^{0})\Big). \label{b2}$$ The second term here looks as a total derivative. However, it does not vanish because this is a derivative of the nonlocal expression and the corresponding surface term does not vanish at infinity in view of the Green’s function asymptotics. This term can be rewritten as $$\Box \frac{1}{\Box -V}\,V(x)
=V(x)+V\frac{1}{\Box -V}\,V(x)=V\Phi (x), \label{b3}$$ and, therefore, $${\rm Tr}\,K(s)=\frac{1}{(4\pi s)^{d/2}}
\int dx\,\Big(1-s\,V\Phi (x)+O(s^{0})\Big), \label{b4}$$ where $O(s^{0})$ denotes the subleading in $s$ terms which depend on the potential in a nontrivial way. They are given by infinite resummation over $n$ of the second and third terms in square brackets of Eq.(\[bigequation\]). Remarkably, this summation can again be explicitly done. In this case one has to sum multiple geometric progressions.
Indeed, the second term of (\[bigequation\]) gives rise to the series $$\frac{2}{(4\pi s)^{d/2}}\int dx\,
\sum_{n=2}^{\infty }\sum_{m=2}^{n}V
\underbrace{\frac{1}{\Box }V...
\frac{1}{\Box }V}_{n-m}\frac{1}{\Box ^{2}}
\underbrace{V\frac{1}{\Box }...
V\frac{1}{\Box }}_{m-2}\,V(x). \label{1}$$ By summing the two geometric progressions with respect to independent summation indices $0\leq n-m<\infty $ and $0\leq
m-2<\infty$ one finds that this series reduces to $$\frac{2}{(4\pi s)^{d/2}}\int dx\,V\,
\frac{1}{(\Box -V)^{2}}\,V(x), \label{2}$$ which after the integration by parts amounts to $$\frac{2}{(4\pi s)^{d/2}}\int dx\,
\left( \frac{1}{\Box -V}\,V(x)\right) ^{2}=
\frac{2}{(4\pi s)^{d/2}}\int dx\,
\Big(1-\Phi (x)\Big)^{2}. \label{B1000}$$
Similarly, the third term of (\[bigequation\]) gives rise to the triplicate geometric progression which after summation and integration by parts reduces to $$\begin{aligned}
&&\frac{2}{(4\pi s)^{d/2}}\int dx\,
V\sum_{i=0}^{\infty }\Big(\frac{1}{\Box }V
\Big)^{i}\frac{1}{\Box }\nabla ^{\mu }
\frac{1}{\Box }\,\sum_{j=0}^{\infty }
\Big(V\frac{1}{\Box }\Big)^{j}
\frac{1}{\Box }\nabla ^{\mu }\frac{1}{\Box}
\sum_{l=0}^{\infty }
\Big(V\frac{1}{\Box }\Big)^{l}\,V(x) \nonumber \\
&&\qquad \qquad \qquad \qquad
=-\frac{2}{(4\pi s)^{d/2}}\int dx\,
\Big(\nabla_{\mu }\Phi (x)\Big)
\frac{1}{\Box -V}V\frac{1}{\Box }
\nabla ^{\mu }\Phi (x). \label{B100}
\end{aligned}$$ Taking here into account that $$\frac{1}{\Box -V}V\frac{1}{\Box }
=\frac{1}{\Box -V}-\frac{1}{\Box }$$ one finds that the sum of (\[B1000\]) and (\[B100\]) is equal to $$\begin{aligned}
&&\frac{2}{(4\pi s)^{d/2}}\int dx\,
\left( (1-\Phi )^{2}-\nabla _{\mu }\Phi
\frac{1}{\Box -V}\nabla ^{\mu }\Phi
+\nabla _{\mu }\Phi \frac{1}{\Box }
\nabla ^{\mu }\Phi \right) \nonumber \\
&&\qquad \qquad \qquad \qquad \qquad \qquad
=-\frac{2}{(4\pi s)^{d/2}}\int
dx\,\nabla _{\mu }\Phi
\frac{1}{\Box -V}\nabla ^{\mu }\Phi , \label{B10000}
\end{aligned}$$ where the cancellation of the first and the third terms takes place after rewriting $\nabla _{\mu }\Phi $ in the third term as $\nabla _{\mu }(\Phi -1)$ and integrating it by parts[^9]. Together with (\[b4\]) the contribution (\[B10000\]) forms the nonlinear and nonlocal late time expression for the heat kernel trace (\[sol\]) up to the first subleading order in $1/s$ inclusive.
Acknowledgement {#acknowledgement .unnumbered}
===============
A.O.B. is grateful for hospitality of the Physics Department of LMU, Munich, where a major part of this work has been done. The work of A.O.B. was also supported by the Russian Foundation for Basic Research under the grant No 02-02-17054. V.M. thanks SFB375 for support.
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V.F.Mukhanov, A.Wipf and A.I.Zelnikov, Phys. Lett. [**B332**]{} (1994) 283-291.
A.O.Barvinsky and V.F.Mukhanov, work in progress.
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V.P.Frolov and G.A.Vilkovisky, Phys. Lett. [**B106**]{} (1981) 307; Proceedings of the Second Seminar on Quantum Gravity (Moscow, 1981), ed. by M.A.Markov and P.C.West (Plenum, London, 1983) p.267
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[^1]: Note that this expression for ${\rm Tr}\,K(s)$ cannot be obtained directly by integrating the asymptotics (\[ass1a\]) over the whole spacetime because for a given $s$ this asymptotics fails at $\left|x\right|^{2}>s$. Its derivation is given in Sect. 4 and Appendix B, where we show that the expressions (\[ass1a\]) and (\[latetr\]) are in complete agreement with each other.
[^2]: For the convergence of the integral in $\left(1/\Box \right) V$ the potential $V(x)$ should fall off at least as $1/|x|^{3}$ in any spacetime dimension [@CPTII].
[^3]: This action can be obtained by integrating the conformal anomaly [@FrolVilk; @Polyakov].
[^4]: The conditions of the smallness of the potential are exactly opposite to those of (\[parexp\]), e.g., $V^{2}/\nabla ^{2}V\ll
1$. However, this is true only as a rather rough estimate, because CPT is a nonlocal perturbation theory and its actual smallness “parameters” are some nonlocal functionals of the potential.
[^5]: This graphical interpretation should not be taken too literally because integration over $\alpha $-parameter(s) involves also the argument $\bar{x}\equiv \bar{x}(\alpha |x,y)$ of the potential.
[^6]: The boundary condition $K(s|\,x,y)\rightarrow 0$ at $|x|\rightarrow \infty$ is enforced by the gaussian factor in (\[ansatz\]), even for nonvanishing finite $\Omega(s|\,x,y)=\exp(-W(s|\,x,y)$.
[^7]: This follows from the derivation of (\[hkas\]) above, which is based on discarding the second term of eq.(\[eqw\]) linearly growing in $(x-y)/s$.
[^8]: Note that the quadratic term should be retained in the expansion of $e^{-s_*V}$ if we want to get a nontrivial solution for $s_{\ast}$.
[^9]: Straightforward integration by parts of $\nabla_{\mu }\Phi (1/\Box )\nabla^{\mu }\Phi $ is impossible because $\Phi (x)$ does not vanish at $|x|\rightarrow \infty$, while $\Phi (x)-1$ does.
|
---
address:
- 'Claudio Fuentes is Ph.D. candidate, Department of Statistics, University of Florida, Gainesville, Florida 32611, USA .'
- 'George Casella is Distinguished Professor, Department of Statistics, University of Florida, Gainesville, Florida 32611, USA .'
author:
-
-
title: 'Discussion of “Estimating Random Effects via Adjustment for Density Maximization” by C. Morris and R. Tang'
---
and
We congratulate Morris and Tang for an interesting addition to empirical Bayes methods, and for tackling a difficult and nagging problem in variance estimation. The ADM adjustment appears to bring on interesting properties, not just in variance estimation but also in estimation of the means. In this discussion we want to focus on the latter topic, and see how the ADM-derived estimators of a normal mean perform in a decision-theoretic way. To facilitate this we will stay with the simple model $$\label{eq:model}
y_i \vert \theta_i \sim N(\theta_i, V),\quad \theta_i \sim N(0, A).$$
The James–Stein Estimator as Generalized Bayes (Not!)
=====================================================
$\!\!$We first address the comment of Morris and Tang in Section 2.5, that the prior $ A \sim \operatorname{Unif}(0, \infty)$ is strongly suggested because the James–Stein estimator is the posterior mean if we take $A
\sim \operatorname{Unif}(-V, \infty)$. Professor Morris has noted this before, and in the interest of understanding, we want to show this calculation and comment on its relevance.
Writing ${\mathbf{y}}=(y_1, \ldots, y_k)$ and ${\bolds{\theta}}=(\theta_1, \ldots,
\theta_k)$, the posterior expected loss from model (\[eq:model\]), with the $A \sim \operatorname{Unif}(-V, \infty)$ prior, is $$\begin{aligned}
\label{eq:bayesrisk}
&&\int_{-V}^\infty \int_{\Re^p} \vert {\bolds{\theta}}- \delta({\mathbf{y}})
\vert^2\nonumber\\[-8pt]\\[-8pt]
&&\hphantom{\int_{-V}^\infty \int_{\Re^p}}{}\cdot\frac{e^{-\vert {\mathbf{y}}-\theta \vert^2/(2V)}}{(2 \pi V)^{k/2}}\frac{e^{-\vert \theta \vert^2/(2A)}}{(2 \pi A)^{k/2}}d{\bolds{\theta}}\, dA,\nonumber\end{aligned}$$ and factoring the exponent in (\[eq:bayesrisk\]) and writing $B=V/(V+A)$ shows that $$\begin{aligned}
{\bolds{\theta}}\vert {\mathbf{y}}, A &\sim& N\bigl((1-B){\mathbf{y}}, V(1-B)\bigr), \\
A \vert {\mathbf{y}}&\sim& \biggl(\frac{1}{V+A} \biggr)^{k/2} e^{- ({1}/{(2(V+A))} )\vert {\mathbf{y}}\vert^2}.\end{aligned}$$ The Bayes rules is the posterior mean, which we can calculate as $$\begin{aligned}
\mathrm{E} ({\bolds{\theta}}\vert {\mathbf{y}}) &=& \mathrm{E}[\mathrm{E} ({\bolds{\theta}}\vert {\mathbf{y}}, A)]\\
&=& \mathrm{E}[(1-B){\mathbf{y}}\vert {\mathbf{y}}] = [1-\mathrm{E}(B \vert {\mathbf{y}})]{\mathbf{y}}.\end{aligned}$$
We now, very carefully, calculate $\mathrm{E}(B \vert {\mathbf{y}})$, yielding $$\begin{aligned}
\mathrm{E}(B \vert {\mathbf{y}})&\propto& \int_{-V}^\infty \biggl(\frac{V}{V+A} \biggr) \biggl(\frac{1}{V+A} \biggr)^{k/2}\\
&&\hphantom{\int_{-V}^\infty}{}\cdot e^{- (1/{(2(V+A))} )\vert {\mathbf{y}}\vert^2}\,dA\\
&=& V \int_{1/V}^\infty t^{k/2-1}e^{-t|{\mathbf{y}}|^2/2}\,dt\\
&&{} + V \int_{0}^{1/V} t^{k/2-1}e^{-t|{\mathbf{y}}|^2/2}\,dt,\end{aligned}$$ where we make the transformation $t=1/(V+A)$, with the first integral coming from $A \in (-V,0)$. Noting that the integrand is the kernel of a chi-squared density, we finally have $$\begin{aligned}
\label{eq:twoparts}
\qquad \mathrm{E}(B \vert {\mathbf{y}})&\propto& \frac{V \Gamma(k/2) 2^{k/2}}{(|{\mathbf{y}}|^2)^{k/2}} [P(\chi^2_{k} \ge |{\mathbf{y}}|^2/V)\nonumber\\[-8pt]\\[-8pt]
&&\hphantom{\frac{V \Gamma(k/2) 2^{k/2}}{(|{\mathbf{y}}|^2)^{k/2}} [}
{} + P(\chi^2_{k} \le |{\mathbf{y}}|^2/V) ],\nonumber\end{aligned}$$ where $\chi^2_{k}$ is a chi-squared random variable with $k$ degrees of freedom. Since the chi-squared probabilities sum to 1, normalizing this expectation (dividing by ) results in $\mathrm{E}(B \vert {\mathbf{y}})= V(k-2)/|{\mathbf{y}}|^2$, yielding the James–Stein estimator. There are a number of things to note:
If this were a valid calculation, it would contradict such important papers as Brown ([-@Bro71]) and Strawderman and Cohen ([-@StrCoh71]), which providedcomplete characterizations of admissible generalized Bayes estimators.
In fact, Strawderman and Cohen \[([-@StrCoh71]), Section 4.5\], explicitly tell us that the James–Stein estimator cannot be generalized Bayes.
In fact, the calculation leading to $\mathrm{E}(B \vert {\mathbf{y}})= V(k-2)/|{\mathbf{y}}|^2$ is invalid. To see this note that, starting from (\[eq:model\]), with $A \sim U(-V, \infty)$, the prior on $\theta$ is $$\int_{-V}^\infty \frac{e^{-\vert \theta \vert^2/(2A)}}{(2 \pi A)^{k/2}} \,dA,$$ and, even if we take $k$ to be even to avoid complex integration, it is straightforward to verify that the integral over $(-V, 0)$ is infinite.
What does this tell us about the James–Stein estimator? The “bad” part of the integral, which leads to the piece in (\[eq:twoparts\]) corresponding to $P(\chi^2_{k} \ge |{\mathbf{y}}|^2/V)$, is to be avoided. We can informally interpret this as pointing to the region where $
|{\mathbf{y}}|^2/V$ is small, resulting in shrinkage factors that could be greater than 1 (in absolute value), and result in the James–Stein estimator both changing the sign and expanding ${\mathbf{y}}$. When we lop off this part, we are led to estimators such as the positive-part James–Stein estimator, or admissible estimators, like those based on (32) in Morris and Tang.
Minimaxity of ADM
=================
The lesson from the previous section is to avoid estimators that do not control the shrinker to be between $0$ and $1$. So we turn to ADM and ask if it can do this. We find, interestingly, that the ADM approach will, almost automatically, give us a minimax estimator and, moreover, it controls the shrinker.
$\!$Typically, minimax estimators have been constructed using empirical Bayes arguments and a bit of customizing, or using formal Bayes derivations with priors like $A \sim \operatorname{Unif}(0, \infty)$. The derivation of Morris and Tang in Section 2.7 is a straightforward differentiation, and we can apply the following theorem. \[This is Theorem 5.5, Chapter 5, Lehmann and Casella ([-@LehCas98]), and can be traced back to Baranchik ([-@Bar70]).\]
Under model (\[eq:model\]), the estimator $$\delta({\mathbf{y}}) = \biggl(1-\frac{V g(\vert {\mathbf{y}}\vert)}{\vert {\mathbf{y}}\vert^2} \biggr) {\mathbf{y}}$$ is minimax under the loss $ \vert {\bolds{\theta}}- \delta({\mathbf{y}}) \vert^2$ if
the function $g(\vert {\mathbf{y}}\vert)$ is nondecreasing,
$0 \le g(\vert {\mathbf{y}}\vert) \le 2(k-2)$.
In the notation of Morris and Tang, we are considering the case $r=0$, and writing $T=\vert {\mathbf{y}}\vert^2/(2V)$, the ADM shrinkage factor is $$\label{eq:adm}
\quad \hat B = \frac{1}{T} \biggl[\frac{2(m-c+1)T}{T + m+1 +\sqrt{(T-m-1)^2 +4cT}} \biggr].\hspace*{-10pt}$$ Morris and Tang note that $\hat B$ is monotone decreasing in $T$, but for minimaxity we need the function in square brackets, which corresponds to $g(\cdot)$ of the theorem, to be nondecreasing.
Unfortunately, this is as far as we can go. The estimator based on (\[eq:adm\]), which is reminiscent of a ridge regression estimator, cannot be admissible. Again we can trace this back to Strawderman and Cohen ([-@StrCoh71]), and also Berger and Srinivasan ([-@BerSri78]). The problem is that (being a bit informal here) admissible estimators must be analytic in the complex plane which is not the case with those based on (\[eq:adm\]).
Lastly, we wanted to see the risk performance of ADM. Morris and Tang set $m=(k-2)/2$, but there is actually a range of values of $m$ for which the estimator is minimax. To clarify, denote $(k-2)/2=m^\ast$, the Morris and Tang choice, and consider the $m$ in (\[eq:adm\]) to be a variable. Then, for $c=1$ the estimator is minimax for all $m \le
2(k-2)$. In Figure \[fig:ADM\] we see, in the left panel, the risk of five ADM estimators, along with the risk of the James–Stein estimator for comparison. There we see that the choice of $m$ completely orders the ADM risk, with $m^\ast=(k-2)/2$ being the best choice, resulting in an estimator with risk similar to that of James–Stein. In the right panel we compare the ADM estimator, with $m^\ast=(k-2)/2$, to the James–Stein estimator, its positive-part version, and the admissible estimator with $B$ given in (32). There we see that ADM compares favorably with the James–Stein estimator, is uniformly dominated in risk by the admissible estimator, but not by the positive-part estimator, whose risk crosses that of ADM for large $|\theta|$.
Is ADM Automatic?
=================
The automatic appearance of the ADM minimax estimator gives support to the claim of Morris and Tang that “ADM maintains the spirit of MLE while making small sample improvements.” In fact, examination of the ADM shrinker $B$, and its risk functions, shows that “automatic” ADM produces an estimator that does not shrink as strongly as either the admissible estimator or the positive-part and, hence, can have smaller risk for larger values of the norm of $\theta$. It is not clear to us that such small sample properties as minimaxity will continue to hold for other models, for example for the estimation of a Poisson mean, where many similar minimaxity results hold. However, the results of Morris and Tang are encouraging and certainly deserve further investigation.
Acknowledgment {#acknowledgment .unnumbered}
==============
$\!$This work is supported by NSF Grant MMS 1028329.
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|
---
abstract: |
We present hydrodynamical simulations of the formation of galaxies in the mass range $10^9 - 10^{13} {\rm M}_{\odot}$, with the focus on the efficiency of gas cooling.
The effect of a background UV radiation field, and the effect of metal enrichment of halo gas due to star formation and stellar evolution, are investigated. A background radiation field is found to strongly suppress the formation of galaxies with circular velocities less than $\approx 50$ km/s. The effect is, however, not large enough to reconcile hierarchical clustering models with observations.
Metal enrichment of the halo gas increases the cooling rate at low redshifts. We find that the mass fraction of gas at virial temperatures may be reduced by a factor of two, in simulations with a UV background field added. The decrease in overall efficiency of gas cooling due to the inclusion of a UV background field can be more than compensated for by the increased cooling that follows metal enrichment of halo gas, but the effect may depend strongly on the assumed model of metal enrichment.
author:
- 'Daniel K[ä]{}llander'
- John Hultman
date: Accepted
subtitle: Gas dynamical simulations incorporating a background UV field and metal enrichment
title: Cooling Limits on Galaxy Formation
---
Introduction
============
It is generally believed that galaxies formed by gravitational collapse. Small fluctuations in the primeval density field grew through gravitational instabilities. When the mean density in a proto-galaxy reaches approximately twice the mean density of the Universe, the proto-galaxy breaks away from the general expansion of the Universe and collapses, and forms a gravitationally bound galactic halo. If the gas in a galactic halo is able to cool efficiently, it can dissipate kinetic energy and condense into a galactic object at the center of the halo.
The purpose of this paper is to study under what circumstances the gas in proto-galaxies is able to cool efficiently, and condense into objects with galactic densities. Particularly, we take a close look at the possible effects of a strong background UV radiation field and the dynamical effects of metal enrichment of halo gas. The results are based on three-dimensional hydrodynamical computer simulations, and a standard CDM cosmology is assumed throughout. The results should have some relevance for all hierarchical scenarios of galaxy formation.
In hierarchical models of galaxy formation, the statistical cosmological time of collapse for a proto-galaxy is an increasing function of mass. More massive galactic halos thus form later. This general picture is consistent with observations indicating that clusters are younger than the contained galaxies, and that the Universe is homogeneous on very large scales.
Any theory of galaxy formation must address fundamental properties of the observed galaxy population like characteristic masses and sizes. Rees & Ostriker ([@Rees77a]), and Silk ([@Silk77a]), showed that the characteristic masses and sizes of galaxies are just those to be expected of gas clouds that can cool in a Hubble time. This provides a natural explanation for the sharp upper cut-off in the galaxy luminosity function; more massive proto-galaxies can not cool in a Hubble-time. These calculations were based on simple models of homogeneous gas clouds, very unlike the highly inhomogeneous collapse of a proto-galaxy in a hierarchical model, where objects are formed by merging of smaller ones, where the gas have already had time to cool. If a substantial fraction of the gas is still heated to virial temperatures, these models still shows that limits due to gas cooling are likely to play an important role in the formation of galaxies.
Observations suggest that only a small fraction, around $1-2 \: \%$, of the closure density of the Universe is in the form of stars. It therefore seems likely that most of the baryonic matter has avoided falling into compact objects, where efficient star formation could take place. Hierarchical models (White & Frenk [@White91a], Cole et al. [@Cole94a]), on the other hand, tend to lock up most of the baryonic mass in the universe in small galactic objects, at early times. If the gas in these halos is able to cool and condense into dwarf galaxies, hierarchical models seem to over-predict the current mass fraction of stars in the Universe. One possible explanation is that supernova driven winds are able to suppress the formation of dwarf galaxies (Dekel & Silk [@Dekel86a]). Another process that may be important involves the presence of a photo-ionizing UV background radiation field. A photo-ionizing background field would heat the gas, and, by keeping the gas at a high degree of ionization, it would suppress collisional line cooling from neutral atoms. Evidence for such a field is given by the lack of a Gunn-Peterson effect even at high redshifts $z\approx 5$ (Webb et al. [@Webb92a]), indicating that the Universe was highly ionized at these redshifts.
Based on simple models of the collapse of gas clouds, Efstathiou ([@Efstathiou92a]) argued that a strong background UV radiation field might indeed suppress the formation of dwarf galaxies. The gas is heated by photo-ionization, and the gas is also kept at a high degree of ionization that suppresses collisional line cooling. Efstathiou found that this might prevent the gas in galactic halos with circular velocities less than $50 km/s$ from cooling.
The UV background photo-ionization heating is able to raise the equilibrium gas temperature to only about $10^4 - 10^5$ K, for the typical range of densities in question, ($n_H = 10^{-6} - 10^{-2} \; {\rm cm}^{-3}$). Further, the photo-ionization process is proportional to the density, through relative abundances, but the excitation and collisional cooling processes are proportional to the density squared. Therefore, the effect of photo-ionization is larger for low densities, and becomes less significant for higher densities. Above $10^5-10^6$ K photo-ionization effects also diminishes, due to first hydrogen, later helium, becoming completely ionized. It is therefore only likely to be important for the dynamics of small galaxies. But, large galaxies could also have been affected, since their lower mass progenitors may have been unable to cool.
Typical virial temperatures of galactic halos are $10^4-10^6$ K. In this temperature range the cooling rate of interstellar gas is highly dependent on the metallicity. The cooling rate of a gas with solar metallicity can be more than one order of magnitude larger than for a gas of primordial composition (Sutherland & Dopita [@Sutherland93a]). Observations of intra-cluster gas typically indicates metallicities of $[{\rm Fe/H}]=-0.4$. This metal enrichment can have a strong effect on the amount of gas that is able to cool at low redshifts.
Källander ([@Daniel96a]) has performed simulations similar to the ones presented here, using a similar model for the metal enrichment of halo gas, (described in the next section), but twice as high metallicity yield. The numerical resolution was similar, but no UV field was included, and only galaxies with a total mass of $10^{12} {\rm M}_{\odot}$ were studied. The increase in cooling rate, due to line cooling in metals, was in that case found to be enough to cause the hot halo gas to collapse in a cooling flow, in roughly one Hubble time. Without the increased cooling rate, significantly less cooling flow was present, and essentially no disk formed. Some experimentation seems to indicate that the magnitude of the gas dynamical effects of metal enrichment, are sensitive to the details of how the metal enrichment is modelled.
White and Frenk ([@White91a]) used an intricate semi-analytic framework to compare different models of hierarchical galaxy formation to observations, and found that the results were sensitive to the assumptions made for the metal enrichment of the gas.
Other simulations including the effects of photo-ionization have been presented by Vedel, Hellsten, & Sommer-Larsen ([@Vedel94a]), Thoul & Weinberg ([@Thoul96a]), Quinn, Katz, & Efstathiou ([@Quinn96a]), Weinberg, Hernquist, & Katz ([@Weinberg97a]), Navarro & Steinmetz ([@Navarro97a]). These authors uses different implementations, initial conditions and strengths of the photo-ionizing field. Vedel et al. uses three dimensional SPH simulations of a Milky Way sized object, but cannot draw conclusions on objects of other masses. Thoul & Weinberg uses one dimensional models, and have much higher resolution. The remaining uses 3D SPH simulations and emphasise on resolution effects, which are shown to be of critical importance to the results in these simulation. In spite of that these authors uses different implementations, the main conclusions are very similar. A background photo-ionizing field does have some effects on lower mass objects, but cannot alone be responsible for suppressing the formation of dwarf galaxies, and make the hierarchical CDM models consistent with observations. These authors does not include the effects of metal enrichment, but this does not alter the general result, however.
Hierarchical clustering is a highly inhomogeneous process. Direct three-dimensional simulations are therefore a vital complement to lower dimensional analytical models. Gas dynamics must be included because dissipation is a fundamental part of galaxy formation. In this article we present computer simulations of galaxy formation that provide further insight on hierarchical clustering models, with an emphasis on cooling arguments. Even though a CDM cosmological model is assumed throughout, the results should have some relevance for all hierarchical scenarios. The effects of a UV background is investigated by running otherwise identical simulations, with and without the effects of an ionizing field incorporated. To be able to address questions about the mass dependence of physical processes, simulations covering a wide range of galactic masses are presented.
Simulations
===========
The simulations presented here follow the evolution of two fluids: a collision-less component that comprises $95\%$ of the mass and represents the dark matter, and a second hydrodynamical gas component. Gravitational forces are calculated using a tree code and the hydrodynamical equations are solved for the gas by use of the smooth particle hydrodynamics (SPH) method. Details of the implementation can be found in Hultman & Källander ([@Pohlman97b]).
The gas consists of hydrogen and a $24\%$ mass fraction of helium, typical of estimated primordial values. Furthermore, we assume that the gas is optically thin and in ionization equilibrium at all times. The most important cooling processes are radiative cooling by bound-bound and bound-free transitions. Compton cooling by electron scattering against the Cosmic Microwave Background (CMB) can also have a significant effect at high redshifts, but at low redshifts, $z < 4$, the photon density of the CMB is too low to lead to any significant Compton cooling. In some of the simulations an external strong UV radiation background field is present. The time evolution of this field is fixed already at the start of the simulations, and does not depend on the state of the gas in the simulated region. This field changes the gas cooling rate, and adds a redshift dependence to the cooling function. The gas is also heated by this field.
Radiative molecular cooling is not included, and this limits us when choosing objects to study. Below $\approx 5\times 10^3$ K, atomic radiative cooling is inefficient and molecular radiative cooling may be important. By neglecting molecular cooling we cannot with any certainty simulate systems where the gas pressure at temperatures below $10^4$ K can be dynamically important. This implies a lower mass limit on the applicability of these simulations.
Neutral gas of primordial abundance is stable against collapse in a dark matter halo if the halo mass, $M_{halo}$, is (Rees [@Rees86a], Quinn et al. [@Quinn96a]) $$\begin{aligned}
M_{halo} & < & 1.09\times 10^9 \left( \frac{T}{10^4 {\rm K}} \right)^{3/2}\times
\nonumber\\
& & h_{50}^{-1} (1 + z)^{-3/2} \left( \frac{\mu}{m_p} \right) {\rm M}_{\odot}.\end{aligned}$$ T, $m_p, \mu, z, h_{50}$ are respectively the gas temperature, the proton mass, the gas mean molecular weight, the formation redshift and the Hubble constant in units of km $\rm s^{-1}$ $\rm Mpc^{-1}$. For a neutral gas, $T \sim 10^4$ K, and a formation redshift of $z=3$ this corresponds to a mass of $1.25\times 10^8 {\rm M}_{\odot}$. A further complication at these mass scales is that galactic halos with as low masses as $10^8 {\rm M}_{\odot}$ are very susceptible to disruption by supernovae (Dekel & Silk [@Dekel86a]). The least total mass in the sequence of simulations, was for these reasons chosen to have a total mass of $10^9 {\rm M}_{\odot}$.
Each object was simulated both with and without a photo-ionizing background, using the same initial density field. In this way it is possible to investigate the effects of the photo-ionizing background, without making a statistical study that would require a large number of simulations. The assumed background radiation field is of the form $$\begin{aligned}
J(\nu ) = J_{-21}(z) \times
10^{-21} (\nu_L/\nu) \nonumber\\ {\rm erg \: cm^{-2} sr^{-1} Hz^{-1} s^{-1}},\end{aligned}$$ where $\nu_L$ is the Lyman limit frequency, and the time dependence of the field is contained in the function $J_{-21}(z)$.
Observational determinations of $J_{-21}(z)$ in recent years include the following (Haardt and Madau [@Haardt96a]): $\log J_{-21}$ $\approx$ $0.5$ at $1.6 < z < 4.1$ (Bechtold [@Bechtold94a]), $-1.0 < \log J_{-21} < -0.5$ at $z\approx 4.2$ (Williger et al. [@Williger94a]), $\log J_{-21} = 0.0 \pm 0.5$ at $1.7 < z < 3.8$ (Bajtlik et al. [@Bajtlik88a], Lu et al. [@Lu91a]), and $\log J_{-21} = -0.3 \pm 0.2$ at $1.7 < z < 3.8$ (Espey [@Espey93a]). A decline in the intensity is expected at high redshifts due to a decline in the number density of quasars and increased absorption by Lyman-alpha clouds. Here $J_{-21}(z)$ is taken to have the form $$J_{-21}(z) =
\left\{ \begin{array}{l@{\quad \quad}l}
4/(1 + z) & 3 < z < 6 \\
1 & 2 < z < 3 \\
z - 1 & 1 < z < 2. \\
\end{array} \right .$$
Proto-galactic gas is enriched with metals ejected from heavy stars, through stellar winds and supernovae. This leads to an significant increase in the gas cooling rate at the temperatures that are relevant for galaxy formation, as can be seen from cooling curve by Sutherland & Dopita ([@Sutherland93a]), in Fig \[coolfunc\]. To take cooling effects of metals into account, a description of the star formation rate is necessary. Star formation is, however, not well understood. Since the only purpose here is to roughly estimate the amount of heavy metals ejected, a simple description is adequate.
All gas with a density contrast above 200 is taken to be inside “collapsed structures”, and gas inside such collapsed structures is assumed to be transformed completely into stars. This is only made for the purpose of estimating the average gas metallicity in the proto-galaxy; dynamically, baryonic mass remains gaseous everywhere. Instantaneous recycling is also assumed (heavy stars eject their metals immediately after being formed) and complete mixing of the metals throughout the proto-galaxy (making the gas metallicity homogeneous) which simplifies the model further. The gas metallicity is then given by (Tinsley [@Tinsley80a])
$$Z(t) = y \ln{M/(M - M_s(t))}$$
where $Z$, $y$, $M$, $M_s$ and t are, respectively, gas metallicity, yield, total baryonic mass, total mass in stars and time. That is, the average gas metallicity in the proto-galaxy, is only dependent of the average gas fraction in the proto-galaxy, (the proto-galaxy being the entire simulation volume in this context). Here a yield $y=0.5 {\rm Z}_{\odot}$ is adopted, which was found to give a gas metallicity in accordance with observed values.
Assuming a time evolution and spectral shape for the background field, a four dimensional table in redshift, metallicity, density and temperature was calculated for both the cooling and heating. We used the publicly available program CLOUDY (Ferland [@CLOUDY]) to calculate cooling and heating tables. The tables were stored in a file and subsequently read in at the beginning of each simulation. Linear interpolation was then used to calculate heating and cooling rates during the simulation.
Initial conditions
==================
In setting up initial conditions for a simulation it is necessary to specify the cosmological model. Here a CDM model, with a baryon fraction of $5\%$, a bias parameter of $b=1.5$, $\Omega = 1$ , $\Lambda = 0$ , and a Hubble constant $h=0.5$ in units of 100 km/s/Mpc, is employed. The main significance of the cosmological model is in this case to provide a normalization, and shape, of the power spectrum of density fluctuations. The amplitude of the power spectrum on galactic scales is constrained by observations, choosing $\sigma_8 = 1/b$, (the rms field value, smoothed with a top hat filter on a scale of $8/h$ Mpc). Choosing another hierarchical cosmological model is therefore likely to have only moderate effects on the results of these simulations.
The simulations presented here cover a mass range of objects, $10^{9} - 10^{13} {\rm M}_{\odot}$, in order to address questions about the mass dependence of different physical mechanisms. The gas is initially represented by $8000$ particles and the dark matter by $4000$ particles. Gas particles are under certain conditions allowed to merge with other nearby gas particles, and the number of gas particles is therefore a decreasing function of time, (see Hultman & Källander [@Pohlman97b] for details). The simulations start with a spherical region of the Universe at a redshift of $z = 30$.
We choose the simulation volumes as spheres, containing precisely the mass of the object in question. This minimizes the volume needed to be simulated, reducing costs in computational time, allowing for highest possible resolution, which is of critical importance in these type of simulations. The initial conditions used here are similar to the ones used by Katz & Gunn ([@Katz91a]), but there are some important differences, and improvements as will be explained below.
Selecting a limited spherical region to study the formation of a galaxy leads to two major complications, the neglect of the surroundings and the selection of a proper site.
Galaxies do not form at random locations. Finding good candidates for galactic seeds at high redshifts, using only the density distribution as a guide, is a difficult task. One possible procedure is to make a low resolution simulation of a large region, and thereby identify the sites where galactic halos of the desired type form. After that, such a region can be re-simulated with higher resolution. Although cumbersome, this is feasible and has been done (Navarro & White [@Navarro94b]).
We chose instead to rely on the “peaks formalism” (Bardeen et al. [@BBKS86a]) to identify likely sites of the formation of galactic halos. This enables us to use an explicit scheme where key properties of the proto-galaxy can be specified, “seeding” the formation of the galactic object to form at the desired location. The details of this are described in Appendix \[app\_a\].
By ignoring the surrounding regions outside the proto-galaxy, no account is taken of tidal interactions. Tidal interactions with the surrounding matter is believed to be what gives galaxies their spin. In its early pre-collapse phase, the proto-galaxy acquires angular momentum through gravitational interactions with surrounding matter. Density fluctuations grow by gravitational instabilities and at some time the proto-galaxy collapses. When it collapses, the quadrupole, and higher, moments of its mass distribution diminish as it shrinks in size. This drastically reduces any tidal interactions, and the angular momentum accumulated in the pre-collapse phase is effectively frozen in at this time. It is therefore believed that a galaxy that has not been involved in any major merger events has had an almost constant angular momentum since the main collapse of the proto-galaxy.
Thus, to approximate the neglected effects of the proto-galactic surroundings, the simulated spherical region is started in solid body rotation. The amount of angular momentum added corresponds to a spin parameter of $\lambda = 0.05$ (Barnes & Efstathiou [@Barnes87a]).
[cllllllllllllll]{} Total mass & $r_{init}$ & $V_{circ}$ & $\epsilon_g$ & $\epsilon_{DM}$ & $T_{vir}$\
\[${\rm M}_{\odot}$\] & \[Mpc\] & \[km/s\] & \[kpc\] & \[kpc\] & \[K\]\
$10^{9}$ & 0.15 & 28 & 0.37 & 1.0 & $2.85\cdot 10^4$\
$10^{10}$ & 0.33 & 55 & 0.74 & 2.0 & $1.10\cdot 10^5$\
$10^{11}$ & 0.71 & 100 & 1.1 & 3.0 & $3.63\cdot 10^5$\
$10^{12}$ & 1.52 & 170 & 1.5 & 4.0 & $1.05\cdot 10^6$\
$10^{13}$ & 3.27 & 330 & 2.2 & 6.0 & $3.95\cdot 10^6$\
Primordial gas simulations
==========================
A first series of simulations does not include a UV field, nor metal enrichment of gas. The gas has a primordial composition, with a 24% mass fraction of helium, and the rest of the mass in hydrogen.
All the simulations in this series result in the formation of a single dominant collapsed gas object at $z=0$, which contain more than 90% of the collapsed gas mass. This object is built up through hierarchical merging of smaller objects. The mass of the most massive progenitor as a function of redshift can be seen in Figure \[mainobjmassplain\]. The redshift, at which the mass of the largest progenitor has acquired half of the final mass, is an increasing function of total proto-galactic mass. For the $10^9 {\rm M}_{\odot}$ simulation this redshift is $z \approx 2.2$, and for the $10^{13} {\rm M}_{\odot}$ simulation it is $z \approx 0.2$.
During the collapse of the proto-galaxy the gas acquires kinetic energy, which is then converted into thermal energy by shocks and radiated away. In the low mass galaxies the radiative gas cooling is efficient and keeps the gas cool, as can be seen in Figure \[hotvirfracplain\]. The fraction of gas that can cool depends on the redshift of formation. At high redshifts the density of the Universe is higher, and the cooling therefore stronger.
The shape of the cooling curve also gives a strong implicit redshift dependence of the efficiency of gas cooling. Radiative energy losses are highest in the temperature range $10^4 - 10^5$ K. Most gas that is shock heated to temperatures above $10^5$ K stays hot for more than a Hubble time, whereas gas that is heated to less than $10^5$ K experiences an order of magnitude stronger cooling and cools well within a Hubble time.
It can be seen in Figure \[hotvirfracplain\] and Table \[simpars\] that the galactic halos that contain gas at virial temperatures at $z=0$, are those with a virial temperature exceeding $10^5$ K. These galactic halos form late, and have accumulated less than half the final mass at $z=1$.
The gas core in the $10^{13} {\rm M}_{\odot}$ simulation forms late, at $z\approx 0.2$, see Figure \[mainobjmassplain\], in a series of merging events. However, the hot halo remaining at $z=0$, and containing $30\%$ of the gas mass, forms much earlier, at $z\approx 1$. Between $z=1$ and $z=0.2$, several collapsed gas clumps, none of which has a mass in excess of 20% of the mass of the final gas core, share a common hot pressure supported gas halo. The gas cooling is relatively inefficient at the high temperatures of this hot halo, and there is no significant amount of cooling inflow of gas until the present epoch.
The central gas core that forms in all simulations is very compact, and is not resolved. This over-concentration of mass can be seen from the rotation curves in Figure \[rotcurveplain\], which rise very rapidly in the innermost regions.
The galactic objects that form are very compact because the collapsed gas has lost much of the angular momentum that was injected at the start of the simulations. This effect is well known, and have been previously investigated by Navarro & Benz ([@Navarro91a]), Katz & Gunn ([@Katz91a]), Vedel, Hellsten, & Sommer-Larsen ([@Vedel94a]), Navarro, Frenk, & White ([@Navarro95a]), Navarro & Steinmetz ([@Navarro97a]), but was first found by Lake & Carlberg ([@Lake88a]), (using a “sticky particle” method).
The total angular momentum is in all cases conserved to well within 1%, and if the collapsed gas has lost angular momentum that angular momentum must have been transferred to the dark matter component. Figure \[angmomplain\] shows the total angular momentum of the gas, normalized to the initial value. It is clear that between 10 and 90% of the gas angular momentum has been transferred to the dark matter component. In the $10^{12}$ and $10^{13} {\rm M}_{\odot}$ simulations this angular momentum transfer is, however, not very pronounced. Instead, most of the gas angular momentum is contained in the hot halo of pressure supported gas that surrounds the central collapsed object. (Note that above mentioned authors considered the angular momentum of the disk, and did not include the halo.)
Simulations with an ionizing field
==================================
A background UV field affects the gas in two ways: (i) the gas is heated by photo-ionization; and (ii) it ionizes the gas and thereby reduces its ability to cool by collisional excitation of neutral atoms. In these simulations the background radiation field is non-zero in the range $1<z<6$.
The rate of photo-ionization heating by the background field is proportional to the local gas density, and the rate of cooling by collisional line radiation, the dominant cooling mechanism, is proportional to the square of the local gas density. Heating by photo-ionization is therefore most important in low density regions, where the equilibrium temperature can rise to $T \approx 3\cdot 10^4$ K. This temperature is comparable to the virial temperature of the galactic halos that form in the $10^9 {\rm M}_{\odot}$ and $10^{10} {\rm M}_{\odot}$ simulations, but much less than the corresponding temperature for the largest simulations. The dynamical effects of heating should therefore be small for the largest galaxies.
Figure \[hotvirfracion\] shows the mass fraction of gas within the virial radius that has a temperature exceeding half the virial temperature as a function of redshift. The difference is modest for the more massive galaxies, when comparing with the corresponding curves for the simulations without a background field. In the $10^{12} {\rm M}_{\odot}$ simulation the fraction of hot gas at $z=0$ increases from around 10% to 20%, when the background radiation field is included. On the other hand, the smallest galaxies show dramatic changes. In the $10^9 {\rm M}_{\odot}$ simulation, most of the gas is heated to temperatures above the virial temperature by photo-ionization. This prevents the gas from falling into the galactic halo potential well. When the background field falls in strength, after $z = 2$, about $10\%$ of the gas is able to cool and condense into a galactic object.
The reduced cooling rate in this series of simulations, with a background UV field, also reduces the gas mass of the most massive objects that form. As can be seen when comparing Figure \[mainobjmassion\] with Figure \[mainobjmassplain\], the reduction in object masses is most pronounced in the smallest simulations, with a total mass of $10^9 {\rm M}_{\odot}$ and $10^{10} {\rm M}_{\odot}$.
There is less mass in the clumpy and cold gas component when a background UV field is incorporated. This lowers the angular momentum transfer to the dark matter component, as can be seen in Figure \[angmomion\], to be compared with Figure \[angmomplain\].
In all cases, the cold collapsed gas is concentrated in a compact object with an extent of less than a few smoothing lengths. Around this object is a second gas component in the form of a hot pressure supported halo. In the simulations where most of the gas can cool and condense into an object, most of the angular momentum is transferred to the dark matter component. In the cases where cooling is less efficient, more angular momentum is retained in the gas component, but instead the gas angular momentum is contained in the hot gas halo. The resulting gas cores are still very compact, as can be seen in Figure \[rotcurveion\].
Simulations with an ionizing field and metal enrichment
=======================================================
The cooling rate of interstellar gas depends sensitively on the composition assumed. Metals can increase the cooling rate by an order or two of magnitude in the temperature range relevant here. This increase is mostly due to collisional line cooling by oxygen and carbon. We employ a very simplistic scheme to estimate the metallicity of the gas, as described earlier in this paper. We have included the effects of this modulation of the gas cooling rate, in addition to the inclusion of a background radiation field. To some extent these two processes are expected to counteract each other, since a background field limits the ability of the gas to cool, and a high metal content increases the cooling rate. However, the effects of a background radiation field vanishes at low redshifts, whereas the metal content of the gas is only significant at lower redshifts, after significant star formation has taken place.
Through the amount of collapsed gas it is possible to get a very rough estimate of the star formation rate, and, with the further assumption of instant mixing, consequently an estimate of the metal enrichment. Metallicity as a function of redshift is displayed in Figure \[metallicity\], for the different simulations. The gas in the $10^{9} {\rm M}_{\odot}$ simulation does not reach metallicities high enough to significantly affect the gas cooling rate, and the result for this mass is therefore practically identical to that of the corresponding simulation without metal enrichment.
The early time evolution, for $z>2$, is almost unchanged for all simulated proto-galaxies, when comparing with the simulations that include a background field, but no metal enrichment. At later times, the metallicity of the gas leads to more efficient cooling, as can be seen when comparing Figure \[hotvirfracmet\] with Figure \[hotvirfracion\]. A hot halo of gas, at $z=0$, is only present in the $10^{12} {\rm M}_{\odot}$ and $10^{13} {\rm M}_{\odot}$ simulations, and, in fact, all galactic halos contain less gas at virial temperatures than in the corresponding simulations without a background field and metal enrichment. The gas mass of the most massive progenitor is displayed in Figure \[mainobjmassmet\].
Conclusions
===========
The mass of the most massive progenitor as a function of redshift, for all the simulations presented here, is shown in Figures \[objmass09\], \[objmass10\], \[objmass11\], \[objmass12\], and \[objmass13\].
These simulations clarify some points of previous arguments about galaxy formation that are based on simple analytic models, and estimates of the efficiency of gas cooling. It is clear that the inclusion of a background radiation field, consistent with the observed Gunn-Petersson effect, can strongly suppress the formation of galaxies with total mass less than $\approx 10^9 {\rm M}_{\odot}$ (circular velocity $<30$ km/s). Furthermore, the formation of galaxies with total mass less than $\approx 10^{10} {\rm M}_{\odot}$ (circular velocity $<60$ km/s) may be significantly delayed. These results of course depend on the assumed temporal and spectral form of the background radiation field. Nevertheless, the results are in reasonable agreement with previous analytic estimates (Efstathiou [@Efstathiou92a]), and in good agreement with recently published results based on simulations (Quinn et al. [@Quinn96a], Navarro & Steinmetz [@Navarro97a], Weinberg et al. [@Weinberg97a]).
Hierarchical models of galaxy formation tend to over-produce galaxies with circular velocities less than 100 km/s. Our results indicate that photo-ionization alone is not sufficient to suppress the formation of these galaxies, since the effects on galaxies with circular velocities larger than $\approx$ 60 km/s is very limited.
The galactic objects that form in three-dimensional hydrodynamical simulations, are too compact when compared with observed disk galaxies. The reason for this is that most of the angular momentum in the gas component is transferred to the dark matter. Navarro & Steinmetz ([@Navarro97a]) find that collapsed objects acquire even less angular momentum, when the effects of a UV field is included. Comparing Figure \[angmomplain\] and Figure \[angmomion\], these simulations show that the angular momentum transfer, from the gas to the dark matter, decreases in magnitude when a UV field is included. This is not a contradiction. When a UV field is included, most of the gas angular momentum at $z=0$ is contained in a hot pressure supported halo. The angular momentum content of the collapsed gas cores does in fact decrease also in our simulations, in agreement with the results of Navarro & Steinmetz ([@Navarro97a]).
Metal enrichment of the interstellar gas increases the gas cooling rate at late times, and may have significant effects on the amount of gas that may cool and sink to the center of a galactic halo in a Hubble time. The inclusion of a background radiation field leads to more massive hot halos in large galaxies, $10^{12}$ and $10^{13} {\rm M}_{\odot}$, as seen when comparing Figure \[hotvirfracion\] with Figure \[hotvirfracplain\]. These hot halos contain most of the gas angular momentum. Metal enrichment increases the gas cooling rate at late times, and could potentially lead to the collapse of the hot halo gas, to the center of the galactic dark matter halo, in a cooling flow. However, as seen in Figure \[hotvirfracmet\], the increase in cooling rate is not enough for this to happen. See however Appendix \[two\_body\] for potential two body heating effects. Increased cooling due to metal enrichment does decrease the mass of the hot halo that forms, (Figure \[hotvirfracmet\]), but most of the gas angular momentum still resides in the remaining hot halo.
If inhomogeneities in the gas are smoothed out by the limited resolution used, the average cooling rate in the region will change. This is a problem common to all hydrodynamical simulations of galaxy formation. Some implicit assumption must be made, e.g., that the density field is smooth on unresolved scales due to physical processes not incorporated into the simulation or, that the density in regions where this could have an effect is already so high that the cooling is extremely efficient both with and without unresolved density fluctuations. Previous simulations [*without*]{} UV background ionization and heating, did [*not*]{} suffer from resolution effects, as badly as one might (naively) expect from the squared density dependence of the cooling function. The reason being that the cooling function is divided by the density, i.e. the gas cooling rate, per unit mass, is (roughly) proportional to the density, when thermal energy is integrated. For example, Navarro & White ([@Navarro94b]) varied the gas mass- resolution by a factor of two, and Hultman & Källander ([@Pohlman97b]), by a factor of ten, both showing comparatively small effects.
Then, when a UV background field is included, there is a competition between density vs. density squared dependencies, being explicitly sensitive to resolution. Indeed, this was observed by Weinberg et al. ([@Weinberg97a]). Varying the mass-resolution by a factor of eight, had severe effects on the outcome of their results. Navarro & Steinmetz ([@Navarro97a]) performed similar simulations varying the mass-resolution of identical runs with a factor of six. They found much smaller effects, that in addition decreased with redshift. However, the mass- resolution was [*much*]{} higher than in Weinberg et al. (In fact, even the lower resolution runs of Navarro & Steinmetz had slightly higher resolution as compared to the “high resolution” runs of Weinberg et al.) For the simulations presented here, the absolute resolution varies, since it is proportional to the total mass, but for comparison the $10^{12} {\rm M}_{\odot}$ runs are of comparable resolution to those of Navarro & Steinmetz. All in all, this is reassuring but not conclusive evidence that effects from limited numerical resolution are small. Ultimately, the tenacity of the underlying assumptions must be judged by comparisons with observations, and with other models that make different simplifying assumptions.
Appendix
========
Initial density fields {#app_a}
----------------------
In hierarchical clustering models, structure forms by the gravitational growth of small inhomogeneities in the density field. The fluctuations in the density are conveniently characterised by the density contrast $${\delta (\vec{x})} = \frac{{\rho (\vec{x})} - \rho_b}{\rho_b}.$$ where $\rho_b$ is the mean density of the universe. In an Einstein-de Sitter universe, a region with a mean density contrast of zero is marginally bound by gravity. Regions of space where the density contrast is positive are bound and will eventually collapse. At early times $|\delta| << 1$ and gravitational dynamics can be handled by linear theory. The mathematical formulation for this is laid out in detail by Bardeen et al. ([@BBKS86a]), BBKS henceforth.
To single out sites where objects of mass M are likely to form, the density contrast is first smoothed on the corresponding length scale $${\delta_{s} (\vec{x})} = \int{\delta (\vec{x'})}W_s(|\vec{x'}-\vec{x}|)d^3x'.$$ where $$W_s(|\vec{x}|) = \exp{\left(-\frac{\vec{x}^2}{2 s^2}\right)}/(2 \pi s^2)^{3/2}.$$ s is the smoothing length scale and is given by $s$ $\approx$ $(M/\rho_b)^{1/3}$. If ${\delta (\vec{x})}$ is a Gaussian random field then ${\delta_{s} (\vec{x})}$ is also a Gaussian random field. The power spectrum of ${\delta_{s} (\vec{x})}$ is given by $P_s(k) =
P(k)\exp{(-s^2k^2)}$, where $$P(k)\equiv {\left\langle |{\delta_{k}}|^2 \right\rangle},$$ and ${\delta_{k}}$ is the Fourier transform of the density contrast, and ${\left\langle ... \right\rangle}$ is the mean value. A peak in the field ${\delta_{s} (\vec{x})}$ is a point where the mass inside a spherical region of size s has a maximum. Such points are plausible sites for the formation of objects of mass M.
Hoffman et al. ([@Hoffman92b]) extended the formalism of BBKS to investigate the influence of the background density field on a given smaller scale peak. By proceeding in a similar manner we consider two Gaussian random fields, a peak field $\delta^p$, and a background field $\delta^b$, which are defined by their power spectra $$P^p(k) =
\left\{ \begin{array}{l@{\quad \quad}l}
0 & k < k_{lim} \\
P_s(k) & k > k_{lim} \\
\end{array} \right.$$ and $$P^b(k) =
\left\{ \begin{array}{l@{\quad \quad}l}
P_s(k) & k < k_{lim} \\
0 & k > k_{lim} \\
\end{array} \right.$$ The combined field $\delta^p + \delta^b$ is statistically identical to ${\delta_{s} (\vec{x})}$, the density contrast smoothed on a scale s. By requiring that $\frac{2 \pi}{k_{lim}} >> s$, extremum points in ${\delta_{s} (\vec{x})}$ will also be extrema of $\delta^p$. When smoothing on galactic scales, maxima in the density field are assumed to be progenitors of galaxies.
Dimensionless field values, $\nu$, are conveniently expressed in units of $\sigma$, i.e., $\nu = \delta/\sigma$, where $$\sigma^2 \equiv \int{P(k)\frac{4 \pi k^2dk}{(2\pi)^3}}$$ is the mean square density contrast (BBKS). Galaxies are believed to form around peaks of height $1 < \nu < 3$, assuming the density field has been smoothed on an appropriate galactic scale. Specifying an equivalent value $\nu_b$ for the background field, at the same point, determines the density field in a larger surrounding of the peak. $\nu_s$, $\nu^p$ and $\nu^b$, are related by (Hoffman et al. [@Hoffman92b]) $$\nu_s \sigma_s = \nu^p \sigma^p + \nu^b \sigma^b.$$ Inside radius s, the statistical properties of the density field are primarily determined by the small scale peak constraint. But, further away from the peak, the statistical properties depends on the value of $\nu^b$. The main effect of the background field is to change the overall density in the vicinity of the smaller scale peak.
The approach then, is to set up a density field ${\delta (\vec{x})}$, with the constraint that the corresponding field ${\delta_{s} (\vec{x})}$ should have a maximum at the center of the simulated region. The small scale field $\delta_s$ is constrained by $$\delta_s(0) = \nu^p \sigma^p.$$ To ensure that the field has an extremum at the center, it is also required to have vanishing first derivatives there. These constraints specifies the statistical properties of the field. The second derivative of the field could also have been used as a constraint, to ensure that the central extremum is in fact a maximum. This is neglected since it would be a largely redundant constraint. It is very unlikely that a $\nu^p \approx 2$ extremum, as is the case here, is a minimum, and therefore the statistical properties of the field are only marginally affected by adding the contrary as a constraint.
Hoffmann & Ribak ([@Hoffman92a]) showed how to create a random realization of a Gaussian field, that is subject to constraints that are linear in the field. The method is computationally efficient, and it is easy to handle several constraints. Any quantities, or combinations of, that can be represented within linear theory, can be used as constraints for the field. In addition to the “seeding’ peak itself, we have found it useful to also specify $v(0) = 0$, i.e. requiring that the “peak” is stationary. This increases the likelihood that the forming object stays in the center of the simulation volume.
The field is set up in a cubic region of size $L_{cube}$, where $L_{cube} >> s$. The largest wavelengths that can be represented is therefore $2\pi/L_{cube}$. The values used for s are shown in Table \[simpars\]. The smoothed density field is then split into $\delta^b$ and $\delta^p$, in such a way that $\delta^p$ contains all wavelengths that can be represented in the cube, and $\delta^b$ contains all longer wavelengths.
The background field cannot be accurately represented inside the cubic region, where the density field is constructed, since it consists of modes with wavelengths exceeding the size of the region. These long wavelength modes can still have important dynamical effects, because, by changing the overall density inside the cube, these modes change the collapse times for structures in the region. A constant density contrast of $\delta_{b} =
\nu^b \sigma^b$ was added to the region to approximate the lowest order effects of the background field. All wavelengths longer than the expansion volume will be assumed to give a constant shift of the density.
There is, however, by no means a one-to-one correspondence between such density peaks and the galaxies that later form (Katz et al. [@Katz93a]). Objects that form when peaks collapse at a high redshift can at later times merge with other collapsed objects, and form new, more massive, objects. Peaks in the density field at a high redshift will then not be in a one to one correspondence with the galaxies that later form. This is a nonlinear dynamical process that cannot be handled by linear theory. N-body calculations have shown that a significant number of the galactic halos that form, do in fact correspond to peaks in the early density field. It may be argued that the “peaks formalism” would generate atypical initial conditions. This should not be the case however, as long as the parameters are within statistically reasonable limits, i.e. $\nu \leq 3 - 4$. The generated field would still be a statistically typical one.
The gravitational particle smoothing used in the simulations are shown in Table \[simpars\].
[cllllllllllllll]{} Total mass & $r_{init}$ & $\nu^p$ & $\nu^b$ & $L_{cube}$\
\[${\rm M}_{\odot}$\] & \[Mpc\] & & & \[Mpc\] &\
$10^{9}$ & 0.15 & 2.0 & 2.0 & 2.0\
$10^{10}$ & 0.33 & 2.0 & 2.0 & 4.0\
$10^{11}$ & 0.71 & 2.0 & 2.0 & 8.0\
$10^{12}$ & 1.52 & 2.0 & 2.0 & 16.0\
$10^{13}$ & 3.27 & 3.0 & 3.0 & 32.0\
Two-body heating {#two_body}
----------------
There have been recent results showing that there can be undesired, artificial heating of the gas component, due to two-body relaxation like effects, caused by the dark matter halo (Steinmetz & White [@Steinmetz97a]). This could be the case for our simulations, where the dark matter particles are of higher mass. (i.e. the simulations of $10^{12}$ and $10^{13} {\rm M}_{\odot}$ total mass.) The effect would be that, e.g., the cooling flow might be reduced. In the hot gas halo, the cooling times are on the other hand very long, as can be seen from Fig. \[cool\_time\], and it cannot cool within a Hubble time. Corresponding temperatures are shown in Fig. \[temp\_rad\]. Only in the absolute center, (i.e. the disk), the cooling times are short. A few particles just within 10 kpc seems to be possibly prevented from cooling, so we cannot rule out that there are some two-body heating effects. We doubt, however, that a large fraction hot gas halo, is being held at virial temperatures, (or was formed), solely due to two-body effects. However, we believe these two-body heating effects should be taken seriously, and avoided in future work.
We are grateful to the theoretical astrophysics group, at Uppsala Astronomical observatory, in particular for providing us with the computational resources needed.
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|
---
abstract: |
Gamma-ray bursts (GRBs) are divided into two classes according to their durations. We investigate if the softness of bursts plays a role in the conventional classification of the objects. We employ the BATSE (Burst and Transient Source Experiment) catalog and analyze the duration distributions of different groups of GRBs associated with distinct softness. Our analysis reveals that the conventional classification of GRBs with the duration of bursts is influenced by the softness of the objects. There exits a bimodality in the duration distribution of GRBs for each group of bursts and the time position of the dip in the bimodality histogram shifts with the softness parameter. Our findings suggest that the conventional classification scheme should be modified by separating the two well-known populations in different softness groups, which would be more reasonable than doing so with a single sample. According to the relation between the dip position and the softness parameter, we get an empirical function that can roughly set apart the short-hard and long-soft bursts: $SP
= (0.100 \pm 0.028) T_{90}^{-(0.85 \pm 0.18)}$, where $SP$ is the softness parameter adopted in this paper.
author:
- 'Y.-P. Qin, A. C. Gupta, J. H. Fan, C.-Y. Su, R.-J. Lu'
title: 'Duration distributions for different softness groups of gamma-ray bursts'
---
[**Key words:**]{} gamma-rays, bursts — methods, data analysis — methods, statistical
[**PACS:**]{} 98.70.Rz, 07.05.Kf, 43.60.Cg, 97.10.Ri
Introduction
============
The bimodality in the duration distribution of GRBs suggests that, the GRB events exist in short and long duration classes, which are separated at 2 seconds \[1\]. In general, the short duration bursts are harder and long duration bursts are softer. In the hardness ratio vs. duration plot, the two classes are seen to distribute in distinct domains \[1, 2\], where the hardness ratio $HR_{32}$ concerned is defined as the ratio of the total counts in the $100-300$ keV and $50-100$ keV energy range (later, the hardness ratio is defined as the ratio of the fluence in the $100-300$ keV range to the fluence in the $50-100$ keV range \[3\]). When sources of both classes are combined, the hardness ratio is obviously correlated to the duration, while for each of the two classes alone the two quantities are not correlated at all \[4\]. This in turn strongly suggests the existence of the two classes of GRBs. The study of GRB classification is very important since different classes might have different progenitors. It was proposed that the long duration bursts are caused by the massive star collapsars \[5$-$7\] and shot duration bursts are produced in the event of binary neutron star or neutron star-black hole mergers \[8, 9\].
After the successful launch of the Swift satellite \[10\], a large number of evidences favor the two progenitor proposal for GRBs. Long duration bursts were found to be originated from star-forming regions in galaxies \[11\], and in some of these, supernovae were detected to accompany the bursts \[12, 13\]. On the other hand, short duration bursts were detected in regions with lower star-formation rates, with no evidence of supernovae to accompany them \[14$-$16\].
Although GRBs are known to belong to the two distinct classes, the membership of the classes is hard to establish due to the overlap of the duration distributions, which makes the classification of the bursts an unsettled issue. Authors of Ref. \[17\] showed that, the bimodal distribution of GRBs can be well accounted for by two overlapped lognormal distributions, which suggests that each of the two GRB populations is likely to form a single peak modality and there would be a sufficient number of bursts that are mis-classified by simply applying the criterion $T_{90}=2s$. Authors of Ref. \[18\] used a multivariate analysis to discriminate between distinct classes of GRBs, and from the third BATSE catalog they found that instead of two, there are three classes of GRBs, which triggered a debate lasting until the present time.
In distinguishing the X-ray flashes (XRFs), X-ray rich bursts (XRRs) and conventional gamma-ray bursts (CGRBs), authors of Ref. \[19\] used the ratio between the X-ray fluence $S_X$ and the gamma-ray fluence $S_\gamma$ as a softness parameter to divide them. Their results seem to be quite satisfactory. We wonder, if the softness parameter can play a role in separating the short-hard and long-soft populations of GRBs from the BATSE catalog. This is the motivation for our analysis given below.
The BATSE catalog has the largest sample of GRBs available until today. We employ this sample in the following analysis since the total number of GRBs in the sample is large enough for a meaningful statistical analysis.
In Section 2, we investigate if the duration distribution depends on the softness parameter. Based on this investigation, we shortly explore a possible classification of GRBs with an empirical curve in Section 3. The case of Swift is considered in Section 4. Discussion and conclusions are presented in Section 5.
Duration distributions of GRBs with different softness parameters
=================================================================
Motivated by Ref. \[19\]’s work, we wonder if the softness of BATSE bursts can play a role in separating the short-hard and long-soft populations of GRBs. However, most of the BATSE bursts do not have X-ray fluxes, so, we cannot define the softness of the bursts with $S_X/S_\gamma$. Therefore, we turn to consider the ratio between the fluence in the lowest energy band and that in higher energy bands, as adopted in Ref. \[20\] to distinguish them. This parameter is no longer applicable to distinguish XRFs, XRRs and CGRBs. Here, we define the softness parameter of BATSE bursts by $$SP\equiv\frac{f_1}{f_2+f_3+f_4},$$ where $f_1$, $f_2$, $f_3$, and $f_4$ are the fluences of the first ($20-50$ keV), second ($50-100$ keV), third ($100-300$ keV), and fourth ($>300$ keV) BATSE channels, respectively.
![Duration distributions of the BATSE bursts with different softness parameters. Range of the adopted softness parameter is mentioned at the upper left corner for each panel.[]{data-label="Fig. 1"}](f1.eps){width="5in"}
As hinted by Ref. \[19, 20\], we roughly divide the bursts into several groups according to the softness parameter $SP$. To be comparable to the conventional duration distribution plot, we divide the duration range from 0.01 s to 2000 s in the logarithm format into 30 bins. We require that the maximum count of each group should be no less than 30 to keep a statistical significance. For the softest group, we require that if there exist some short duration bursts in this group, the corresponding count should be noticeable. In this way, we get four groups of bursts. The distributions of durations ($T_{90}$) of these groups are displayed in Figure 1, where $T_{90}$ of a burst is the time interval during which 90% of the total observed counts have been detected. We find that the duration histogram of each group bears a bimodality. The positions of the dip lying between the two peaks of the bimodality of different groups differ significantly, where the softer the group, the smaller value of the duration position of the dip. If separating the bursts into short and long duration classes, then in different ranges of the softness parameter the classes occupy significantly different percentage of the total counts. The figure shows that the long duration bursts occupy a very large percentage of the softest population ($SP \ge 0.1$), and they cover a large range of durations, starting from $\sim 1$s to $\sim
1000$s. In contrast, short duration bursts occupy a very small percentage of the softest population, and they cover a very small range of durations. The situation changes for harder groups: the percentage of long duration bursts becomes smaller and they cover smaller ranges of durations, while the percentage of short duration bursts becomes larger and they cover larger range of durations.
[lllll]{} $ 0.1 \leq SP $ & $0.28 \pm 0.24$ & $0.52^{+0.65}_{-0.42}$ & 41& 895\
$ 0.05 \leq SP < 0.1$ & $0.073 \pm 0.014 $ & $0.82^{+1.52}_{-0.44}$ & 42& 328\
$ 0.02 \leq SP < 0.05$ & $0.0338 \pm 0.0086 $ & $3.3^{+5.2}_{-2.0}$ & 135& 186\
$ SP < 0.02$ & $0.0103 \pm 0.0056 $ & $16.2^{+81.9}_{-3.2}$ & 278& 67\
In the conventional classification of GRBs, a dip around 2 s in the duration histogram plot in Figure 1a of Ref. \[1\] was proposed. For a more precise measurement, the authors fitted a quadratic function between the two peaks in the histogram and determined its minimum to be $T_{90} = (1.2 \pm 0.4) s$. As an empirical analysis, we adopt this quadratic function fitting method to locate the dip position from the plots in Figure 1. In Table 1, dip positions and the numbers of long and short duration bursts divided by this feature for various softness groups are listed. In the first column, the softness parameter range of each group is presented. The mean of the softness parameter calculated with all the individual $SP$ values of the bursts of the corresponding group is presented in the second column. Positions of the dip are given in the third column. The numbers of short and long duration bursts separated by the dip are presented in the fourth and fifth columns respectively.
Empirical curve roughly setting apart two populations of BATSE bursts
=====================================================================
A relation between the softness parameter and the dip position of the bi-modal duration distribution of BATSE bursts hinted by Figure 1 can be evaluated from the data of Table 1. In Figure 2, we display the result of a power-law analysis on the data. Fitting the data with a power-law function yields (by performing a Spearman correlation analysis with the ORIGIN software): $$SP = (0.100 \pm 0.028) T_{90}^{-(0.85 \pm 0.18)}.$$ The following are the statistical results of a linear correlation analysis between $log SP$ and $log T_{90}$: the correlation coefficient, -0.958; the number of data points, 4; the probability of rejecting the null hypotheses, 0.0417. Owing to the limit of the total counts, we have only 4 data points in Figure 2. The statistical analysis performed here is thus not robust (we hence regard this fitting curve as an empirical function — see the statement below). Note that one can divide the BATSE catalog into more groups to have more data points. As a result, one might find difficulties to locate the dip suggested in Figure 1 due to data fluctuations.
![Relation between the softness parameter and the position of the duration distribution dip drawn from the BATSE catalog.[]{data-label="Fig. 2"}](f2.eps){width="5in"}
![Plot of the softness parameter vs. the duration of BATSE bursts, where the solid line is the function of Equation (2).[]{data-label="Fig. 3"}](f3.eps){width="5in"}
In Figure 3, we display the plot of the softness parameter vs. the duration of BATSE bursts, where the function of Equation (2) is also plotted. The figure shows that there exist two populations of bursts which are clustered in two distinct domains in the plot. Between the two clusters there is a “gap” with relatively sparse counts. The function of Equation (2), which is the fitting curve in Figure 2, is seen to pass through the “gap”, roughly separating the two populations. As the figure shows, most sources of the population below the fitting curve are the conventional short bursts (the duration is smaller than 2 s), and most sources of the population above the fitting curve are the conventional long bursts (the duration is larger than 2 s).
![Duration distributions of the two populations divided by the curve of Equation (2), where the thick line represents the short-hard population while the thin line denotes the long-soft population drawn from the BATSE catalogue.[]{data-label="Fig. 1"}](f4.eps){width="5in"}
We find that: a) the number of the population below the fitting curve is 511, and among them there are 433 bursts (84.7% of the population) with their duration satisfying $T_{90}\leq 2 s$; b) the number of the population above the fitting curve is 1461, and among them there are 1418 bursts (97.1% of the population) with their duration satisfying $T_{90} > 2 s$. The duration distributions of the two populations divided by the fitting curve is shown in Figure 4. Remind that the fitting curve represents the relation between the dip positions of the duration distribution of bursts and the softness parameter. According to this analysis (refer to Figures 1, 3 and 4), we regard the fitting curve as an empirical curve roughly setting apart the short-hard and long-soft populations of BATSE GRBs in the softness vs. duration plot.
In the case of Swift
====================
GRBs observed by Swift are becoming more and more important due to the excellent ability of Swift to detect the afterglows of the objects. The number of bursts collected by Swift has been steadily growing up since the launch of the satellite \[10\]. Would the statistical analysis performed above be applicable to the current Swift data set?
To answer this question, we search in literature and collect the Swift data which contain the values of the duration and the four Swift channel fluences. The data are available in Ref. \[21\] where the first Swift BAT GRB catalog is presented. From Ref. \[21\] we get 222 bursts with their $T_{90}$ as well as $f_1$ ($15-25$ keV), $f_2$ ($25-50$ keV), $f_3$ ($50-100$ keV), and $f_4$ ($100-150$ keV) being available. Compared with Table 1 we find that this number of bursts is too small to be used to perform the same analysis. This explains why we focus our attention on the BATSE catalog instead of the Swift catalog.
Even though the number is so small, we manage to divide the bursts into two groups according to their softness parameters to check their duration distributions. The same definition of the softness parameter is adopted, although the energy channels of Swift are not the same as those of BATSE (also, $T_{90}$ would not be exactly the same as that measured in BATSE, since the energy range of bursts would play a role in determining the quantity). Those Swift bursts with $SP \geq 0.1$ are included in the softer group, whilst the others belong to the harder group. Duration distributions of the two groups are shown in Figure 5. We observe that, although the statistical significance is much less than that in the BATSE catalog due to the small number of bursts, the same trend is observed in the figure: the duration histogram of each group is likely to bear a bimodality; the positions of the dip lying between the two peaks of the bimodality of the two groups are different, where the softer the group, the smaller value of the duration position of the dip; when separating the bursts of each group into short and long duration classes according to the apparent dip, then any of the two classes occupies significantly different percentage of the total count in different groups, e.g., the softer the group, the larger percentage of the long duration bursts. Comparing Figure 1 with Figure 5 one can find that the adopted statistical analysis is hard to be applied to the current Swift data due to the limited number of bursts.
![Duration distributions of Swift bursts with different softness parameters. The range of the adopted softness parameter is mentioned at the upper left corner of each panel.[]{data-label="Fig. 1"}](f5.eps){width="5in"}
To check the trend in other ways, let us ignore the statistical significance and fit a quadratic function between the two peaks in the histogram of each panel of Figure 5 and determine the minimum of the curve as the position of the dip. The result is displayed in Figure 6, where as a comparison, the BATSE result shown in Figure 2 is also presented. The same trend of the empirical function is maintained if we rely on the two Swift data points in the figure. However, we would like to emphasize that one should not take the two data points so serious since the numbers adopted in Figure 5 are quite small so that the statistical result is not so robust. In addition, we adopt the same definition of the softness parameter to analyze the BATSE and Swift catalogs but this definition would correspond to at least a slightly different quantities since the energy ranges of the corresponding channels are not the same. While this is not an unsolvable problem currently, but the number of bursts is. We hope that the growing Swift burst number would provide a reliable analysis in the near future.
![Relation between the softness parameter and the position of the duration distribution dip drawn from the Swift catalog (open circles). Other symbols are the same as they are in Figgure 2.[]{data-label="Fig. 2"}](f6.eps){width="5in"}
Discussion and conclusions
==========================
Motivated by Ref. \[19\]’s work, we investigate if the softness of bursts plays a role in the conventional classification of GRBs. We employ the BATSE catalog and define the softness parameter as the ratio of the fluence in the first channel to the sum of the fluences in the second, third and fourth channels. The duration distributions of different groups of BATSE GRBs associated with distinct softness parameters are explored. From the analysis we get an empirical curve that can roughly set apart short and long duration bursts. The analysis shows that the conventional classification of GRBs with the duration of bursts is influenced by the softness of the objects.
![Distributions of the conventional hardness ratio (the thick line) and the softness parameter (the thin line) of the bursts in the BATSE catalog.[]{data-label="Fig. 1"}](f7.eps){width="5in"}
As is known, the hardness ratio is an active factor to tell the spectral difference between the short and long duration bursts of GRBs. Why do we adopt the softness parameter instead of the conventional hardness ratio to investigate the issue? As mentioned above, it is Ref. \[19\]’s work on dividing XRFs, XRRs, and CGRBs with another softness parameter ($S_X / S_\gamma$) that motivates us for the exploration of the softness parameter as a possible factor active in the classification of GRBs. While the role of the conventional hardness ratio in GRB classification has been well studied for a long time, the role of the softness parameter has not been sufficiently explored. Another reason for doing so is illustrated in Figure 7, where distributions of both the conventional hardness ratio and the softness parameter are presented. We find that, while the conventional hardness ratio spans about one magnitude interval, the softness parameter covers approximately two magnitude range. This suggests that, if both the conventional hardness ratio and the softness parameter can be adopted as a parameter to distinguish different types of burst, then the latter must be more sensitive than the former, since the difference of the latter is more easily noticeable.
![Hardness ratio vs. duration plot for the short duration bursts (filled circles in the upper panel; open circles in the lower panel) and long duration bursts (filled circles in the lower panel; open circles in the upper panel) divided by the function of Equation (2) from the BATSE catalog.[]{data-label="Fig. 1"}](f8.eps){width="5in"}
What would one get in the conventional hardness ratio vs. duration plot, if the bursts of the BATSE catalog are divided by the empirical curve, the function of Equation (2)? The result is shown in Figure 8 where we find that two distinct populations are clustered in the conventional domains of the short and long duration classes. The well-known correlation properties between the hardness ratio and duration of the two classical populations are maintained. That is, for any of the two populations, the hardness ratio is not at all correlated with the duration, while the two elements are obviously correlated when the two populations are combined. This is not surprising, since 84.7% bursts of the population below the fitting curve are the conventional short duration bursts ($T_{90}\leq 2 s$), and 97.1% bursts of the population above the fitting curve are the conventional long duration bursts ($T_{90} > 2 s$). This together with Figure 4 show that dividing BATSE bursts with the empirical curve or with the duration criterion $T_{90} = 2 s$ would not give rise to a statistical difference (or, the statistical properties of the two resulting populations would not be significantly different for the two criterions). This in turn suggests that the modification to the conventional method (e.g., by considering different softness groups instead of a single sample, or by adopting the empirical curve instead of the $T_{90}=2s$ criterion) would not lead to a noticeable physical discrepancy between the two populations.
As Figure 8 shows, the two populations divided by the empirical curve heavily overlap in the hardness ratio vs. duration plot. This together with the overlapping in Figures 3 and 4 imply that, the two well-known populations are unlikely to be sharply separated merely by quantities such as the duration, the hardness ratio and the softness parameter, or the relations between them. To distinguish the intrinsic short-hard and long-soft populations, other supplemental criterions are needed. Among them, the most desirable one is the location of bursts in the host galaxies, as observable in some Swift bursts.
One might notice that the softness parameter adopted here is in fact an inverted BATSE gamma-ray hardness ratio $HR_{234/1}\equiv
(f_2+f_3+f_4)/f_1$. Therefore, the fact that the dip existed in the bimodality in the duration distribution of GRBs shifts with the softness parameter (see Figure 1) is equivalent to the fact that the dip position shifts with the hardness ratio $HR_{234/1}$.
What is the difference between our analysis and the conventional one? The conventional investigation considers a single sample from a catalog whilst our study considers different groups of softness from the same catalog. The conventional one yields a single dip position of the histogram of the duration distribution while our analysis produces several dip positions which are found to vary with the softness. The statistical methods adopted to distinguish the long and short duration classes in both cases are exactly the same, but the results are different. Our analysis suggests that the conventional classification should be modified by setting apart the two populations in different softness groups, which would be more reasonable than doing this with a single sample. Another possible modification is to separate the two populations in the softness vs. duration plot with an empirical function as long as the function is reliable enough (when the total number of bursts is much larger than the current available one, the function so obtained would be more robust in terms of statistics, e.g., in that case one would get more data points rather than only 4 in Figure 2 under the same requirements).
![Duration distributions (the thin line) of BATSE bursts fitted by a lognormal distribution curve (the thick line) for the conventional short (the left bottom panel; the reduced chi-square for the fit is 85.6) and long (the right bottom panel; the reduced chi-square for the fit is 305) duration classes, and for the newly classified short-hard (the left top panel; the reduced chi-square for the fit is 35.8) and long-soft (the right top panel; the reduced chi-square for the fit is 227) populations by the empirical function of Equation (2), respectively.[]{data-label="Fig. 1"}](f9.eps){width="5in"}
Assume that we take the empirical function as the criterion to classify GRBs. If so, is the new classification statistically better than the old one? As discussed above, as much as 84.7% of the newly classified short-hard bursts are the conventional short duration bursts and as much as 97.1% of the newly classified long-soft bursts are the conventional long duration bursts. The new method is only a modification to the conventional one but not a totally different one. Therefore we cannot expect an entire difference between bursts identified by the two kinds of classification. However, if we believe that each of the two populations follows a lognormal distribution of the duration of bursts, as suggested in Ref. \[17\], then we can check the statistical difference of the two classifications in this light. Presented in Figure 9 is the fitting result of a lognormal distribution curve to the duration distributions of BATSE bursts for the two classifications. The statistical improvement from the new method is obviously observable.
As Figure 9 shows, compared with the short duration class, the long duration class is more poorly fitted by a lognormal distribution. This indicates that if the empirical function is used as a criterion to separate the BATSE bursts, the resulting long-soft population might contain a “third” subclass. As mentioned above, about one decade ago, authors of Ref. \[18\] used a multivariate analysis to discriminate between distinct classes of GRBs and found that there exist three classes of GRBs instead of only two. The debate of the existence of the “third class” has then been triggered and lasting until the present time \[22$-$28\]. We suspect, if a detailed analysis based on what is shown in Figure 9 is helpful in ending the debate, and hence hope to see such an investigation in the near future (for example, to explore a fit to the duration distribution of the long-soft population with the method of the superposition of two lognormal distributions, as that adopted in Ref. \[25\]).
![Duration distributions of seven groups of the BATSE bursts divided according to the softness parameters. Range of the adopted softness parameter is mentioned at the upper left corner of each panel.[]{data-label="Fig. 1"}](f10.eps){width="5in"}
What result would a different number of intervals of the softness parameter produce? Obviously, larger intervals would lead to a smaller number (smaller than 4) of data points in Figure 2, which will not be considered here. As mentioned previously, smaller intervals would give rise to more data points in the figure, but the statistical significance will be harmed. For example, when we divide the BATSE catalog into 7 instead of 4 groups, the requirement that the maximum count of each group should be no less than 30 would no longer be satisfied (this is the reason why we prefer to divide the BATSE catalog into 4 groups). As a result, more obvious fluctuation of data would be expected. The result is shown in Figure 10. Compared with Figure 1 we find that data fluctuation is indeed more obvious in Figure 10. Following the same method, we get an empirical curve from Figure 10, which is $SP = (0.104 \pm 0.019) T_{90}^{-(0.95 \pm
0.11)}$. Parameters of this curve are in agreement with Equation (2) (within the error bars, values of the fitting parameters obtained in both cases are the same), indicating that our analysis does not rely strongly on the choice of the number of intervals of the softness parameter.
According to the above analysis, we reach the following conclusions. a) When dividing BATSE bursts into several groups according to their softness parameters, the duration distribution of each group bears a bimodality histogram. b) Between the two peaks of the bimodality exhibits a feature of dip that sets apart two subgroups, shorter duration subgroup and longer duration subgroup. c) The dip position shifts with the softness parameter, where the softer the group, the smaller value of the duration position of the dip. d) For groups with different softness parameters, the short duration subgroup occupy significantly different percentage of the total count, where the softer the group, the smaller percentage of the total count it occupies (for the long duration subgroup, the conclusion is the opposite). e) Deduced from the relation between the dip position and the softness parameter we find an empirical function that can roughly separate the short-hard and long-soft populations.
As a primary investigation, we find a statistical difference between the duration distributions of a single sample and multi-group samples from the BATSE catalog. We do not know what causes this difference. Nor are there any models that have ever predicted this. Lacking the knowledge of the causing, the physical interpretation associated with the proposed modification of the classification is currently unavailable. In the same way, the phenomenon of overlapping is currently not able to be explained. Other independent investigations are needed to understand the new findings as well as the overlapping phenomenon.
This work is supported in part by the National Natural Scientific Foundation of China (10633010, 10573005, 10747001) and the 973 project (No. 2007CB815405). We also thank the financial support from the Guangzhou Education Bureau and Guangzhou Science and Technology Bureau.\
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---
abstract: 'The Bayesian approach for the feed-forward neural networks is reviewed. Its potential for usage in hadron physics is discussed. As an example of the application the study of the the two-photon exchange effect is presented. We focus on the model comparison, the estimation of the systematic uncertainties due to the choice of the model, and the over-fitting. As an illustration the predictions of the cross sections ratio $d \sigma(e^+ p\to e^+ p)/d \sigma(e^- p\to e^- p)$ are given together with the estimate of the uncertainty due to the parametrization choice.'
address: |
Institute of Theoretical Physics, University of Wrocław,\
pl. M. Borna 9, 50-204, Wrocław, Poland
author:
- 'Krzysztof M. Graczyk and Cezary Juszczak'
title: Applications of Neural Networks in Hadron Physics
---
[*Keywords*]{}: form-factors, proton structure, neural networks, Bayesian statistical analysis
Introduction
============
One of the goals of physics is to construct models, which describe a part of reality. A promising model should be able to reproduce the experimental data with *reasonable precision* and should also be characterized by good predictive power. The closest to Nature seems to be a theory, which is based on fundamental symmetries or some other beautiful mathematical structure and contains a minimal number of internal parameters. However, in many cases either the fundamental underlying theory is not known yet or the model is not fully solvable yet. Therefore effective approaches are often utilized to describe physical observables and properties. They are defined by a set of internal parameters usually inferred from the measurements. Some of them, like particle masses, have particular physical interpretation but many are just introduced to reproduce the experimental data.
Modelling the internal structure of the nucleon is an example of a situation where the theory, at least in some regions, is unsolvable or very difficult to apply. Because of the asymptotic freedom, the perturbative methods in quantum chromodynamics (QCD) work well at large energies, but they fail in the confinement region. In this low energy range it is more convenient to describe the system in terms of hadronic degrees of freedom (baryons and mesons) rather than the quarks and gluons. Therefore the internal hadronic structure, in the confinement region, is usually investigated within effective approaches. In many of them the information about the static and dynamical internal structure [@Thomas_book] of hadrons is parametrized by the transition form factors (FFs). In the case of the nucleon they describe its electromagnetic (E-M) as well as electroweak properties. Their functional form is not known and they are inferred from the scattering data[^1].
Usually in the particle and nuclear physics the methods of frequentistic rather than Bayesian statistics are used. There are fundamental differences between both methodologies starting from the very definition of probability (for comprehensive review see [@Jeffreys; @De_Agositini]). In the Bayesian approach the probability is *the measure of the degree of belief that an event will occur* [@De_Agositini]. Seemingly this definition is subjective and non-operational in contrast to the frequentistic approach, where the probability is defined by *the ratio of the number of times the event occurs in a test series to the total number of trials in the series*. The latter definition implies an additional assumption that every event occurred/occurs/will occur with the same probability [@De_Agositini]. In the Bayesian statistics, with the use of the Bayes’ rule, one can construct the probability (posterior), which accommodates the initial model assumptions (prior and likelihood) with the data. The posterior should always be updated after new data arrive. The statistical model (see definition in the next section) is defined by the probability distribution of its parameters and the model assumptions contained in the definition of the prior and the likelihood.
Having a set of physical hypotheses (models) it is natural to ask: *which one is the most favourable by given data?* Within the Bayesian statistics the hypothesis can be ranked by the conditional probability $P({\rm hypothesis}|{\rm data})$. Therefore the comparison of different models and the discussion of the impact of the initial assumptions on the results of the analysis can be naturally performed. Moreover the analysis of every possible model brings a valuable contribution. Indeed even negative verification of a particular hypothesis is constructive information which contributes to the posterior needed to classify the hypotheses.
It is believed that the laws of Nature are simple, therefore the desired theory, which aims to approach the *true underlying theory*, should be based on a small set of fundamental assumptions and it should contain a minimal number of internal parameters. Hence it is rather natural to search for simpler rather than more complex descriptions of the physical reality. An instructive example is the extraction of the value of the proton radius from the elastic $ep$ scattering data. This quantity is related with the slope of the electric proton form factor ($G_{E}$) at vanishing four-momentum transfer $Q^2\to 0$. In the typical analysis the parametrizations for the electric $G_{E}$ and magnetic $G_{M}$ proton form-factors are postulated. Usually these are arbitrary functions, which obey some general properties and they are fitted to the experimental data. It turns out that the obtained value of the proton radius depends on the choice of the parametrization [@Hill_et_al]. This difficulty can be approached within the Bayesian statistics [@Graczyk:2014lba], which in natural way embodies the Occam’s razor principle [@Occam_razor] (models with lower number of parameters are preferred).
In this paper we shall introduce a statistical framework, based on the Bayesian statistics, which allows to quantitatively control the model-dependence of predictions of physical quantities, and to estimate the systematic uncertainties caused by a particular choice of the model. The proper estimate of the statistical and systematic uncertainties is of importance in atomic physics [@theeditors_of_PRA], nuclear physics [@Dobaczewski:2014jga] but also in the physics of hadrons.
There are a lot of lepton-hadron and hadron-hadron scattering cross section measurements. The analysis of these data brings information about the internal structure of hadrons and allows for validation of the theoretical models. With the help of the neural network methods one can try to analyse these data in a model-independent way, constructing the statistical model, based on which the predictions about the transition FFs [@Graczyk:2010gw; @Graczyk:2011kh] and the parton distribution functions [@Ball:2013lla] can be made.
In the following sections we shall introduce the Bayesian framework (BF) for feed-forward neural networks and as an instructive example of application we will present results of our studies of the proton structure, E-M FFs and the two-photon exchange (TPE) effect [@Graczyk:2013pca; @Graczyk:2011kh]. Investigation of the proton FFs and related observables (proton radius, two photon exchange effect) is an important topic of the hadron physics [@Thomas:2014zja]. In this paper we discuss the statistical features of the framework concentrating our attention on the quantitative model comparison and the estimate of the systematic uncertainties due to the choice of initial model assumptions.
The paper is organized as follows. Sect. 2 introduces the BF for feed-forward neural networks. In Sect. 3 the application to hadron physics is presented. In Sect. 4 the features of the approach and the results are discussed.
Remarks on the Bayesian Framework
=================================
In this section we recall, following D. MacKay [@MacKay_thesis], some general features of the BF.
Our purpose is to find the optimal model or a set of models having a given set of measurements. By a statistical model we mean:
- the function, $\mathcal{N}$, used to fit the data;
- two conditional probabilities: the distribution of the function parameters $P(\{w_i\}|\mathcal{N})$ and the likelihood $P(\mathcal{D}|\{w_i\}, \mathcal{N})$, where $\mathcal{D}$ denotes the data, while $\{w_i\}$ are the model parameters.
In principle one should consider all possible hypotheses, for each of them find the most optimal set of parameters and rank them by the conditional probability $P(\mathcal{N}|D)$, which estimates how plausible any given hypothesis is according to the measurements.
The optimal configuration of the parameters, $\{w_i \}_{MP}$, of the model maximizes the posterior (obtained from the Bayes’ rule): $$\label{posterior_paraeters_general}
P(\{w_i\}| \mathcal{D},\mathcal{N}) = \frac{P(\mathcal{D}|\{w_i \}, \mathcal{N})P(\{w_i \}|\mathcal{N})}{P(\mathcal{D}| \mathcal{N})},$$ where $P(\mathcal{D}|\{w_i \}, \mathcal{N}) $ is the likelihood and $P(\{w_i \}|\mathcal{N})$ is the prior which contains information about the initial assumptions. The denominator of the right-hand-side of Eq. \[posterior\_paraeters\_general\] is equal to: $$\label{evidence_gen}
P(\mathcal{D}| \mathcal{N}) = \int \prod_k d w_k \, P(\mathcal{D}|\{w_i \}, \mathcal{N}) P(\{w_i \}|\mathcal{N})$$ and it is called the evidence for the model $\mathcal{N}$. On the other hand, from the Bayes’ rule we have: $$\label{Posterior_para_general}
P(\mathcal{N}|\mathcal{D}) = \frac{P(\mathcal{D}| \mathcal{N}) P( \mathcal{N})}{P(\mathcal{D})}.$$ For given data $\mathcal{D}$, the $P(\mathcal{D})$ is fixed. Moreover if one assumes that there are no model preferences at the beginning of the analysis (the prior is uniform over neural networks of certain scheme and then uniform within particular model) i.e. $P( \mathcal{N}_1)=P( \mathcal{N}_2)=...$, then $P(\mathcal{N}|\mathcal{D}) \sim P(\mathcal{D}| \mathcal{N})$ and that the evidence (\[evidence\_gen\]) can be used to rank the models. In practice
In a typical situation the integrated function in the formula (\[evidence\_gen\]) is peaked at some configuration $\{w\}_{MP}$, so the evidence can be computed in the Hessian approximation [@MacKay_thesis], $$P(\mathcal{D}|\mathcal{N}) \approx P(\mathcal{D}|\{w_i\}_{MP},\mathcal{N}) \underbrace{(2\pi)^\frac{p}{2} |A|^{-\frac{1}{2}}}_{Occam\,factor},$$ where $p$ is the number of parameters, $A_{ij}= - \left. \nabla_{w_i}\nabla_{w_j} \ln P(\{w_i \}| \mathcal{D},\mathcal{N})\right|_{\{w_i \}=\{w_i \}_{MP}}$, and $|A|=\det A$ . In this case the evidence is proportional to the likelihood at the maximum multiplied by the Occam factor, which penalizes too complex models.
In a non-Bayesian analysis only the likelihood at the maximum is accessible and the comparison of non-nested models is not straightforward. Moreover when the number of parameters of the model increases then the maximum of the likelihood also grows – the model with larger number of parameters can fit the data better. In the Bayesian approach the contribution from the Occam factor makes models which over-fit the data less likely. Therefore we expect that the best model should be characterized by a good predictive power.
Artificial Neural Networks
==========================
![\[Fig\_Architecture\] Fig. (a): MLP utilized to approximate the electric or magnetic FF of the nucleon [@Graczyk:2010gw]. It contains one input unit, one output unit, and one hidden layer with 7 (plus one bias) units. Fig. (b): MLP used to analyse the elastic $e^\pm p$ data in order to extract the FFs of the proton and TPE correction [@Graczyk:2010gw]. MLP contains two input units, three output units and 5 (plus one bias) units in the hidden layer. The hidden layer is divided into FFs sector (blue vertices), which connects $Q^2$ input only with FFs, and TPE sector (red vertices), which does not connect with FFs. Each line corresponds to one weight parameter. The bias weights are denoted by dashed lines and the bias units by crossed circles.](nnpatternHadronic-eps-converted-to.pdf){width="\textwidth"}
In the particle and nuclear physics the artificial neural networks (ANNs) are used to identify the interaction vertices and particles in the detectors [@NN_in_Physics]. Recently they are also exploited to interpolate the parton distribution functions (PDFs) [@Ball:2013lla] and the nucleon form factors [@Graczyk:2010gw].
In the following two subsections we will review the foundations of the BF for feed forward neural networks [@MacKay_thesis; @Bishop_book].
Multi-layer perceptron
----------------------
In order to construct a statistical model for the function $\mathcal{N}$ we consider feed-forward neural networks in multilayer perceptron (MLP) configurations[^2]. A neural network is a non-linear map $\mathcal{N} : \mathbb{R}^{n_{in}} \to \mathbb{R}^{n_{out}}$, where $n_{in/out}$ is the dimension of the input/output vector space, which is usually represented as a graph with several layers of units (vertices) such that only the vertices from the consecutive layers can be connected. The first layer is the input, the last one is the output, and all other layers are hidden. Each unit (see Fig. \[Fig\_Unit\]) contains a real-valued function (called the activation function $f^{act}$) depending on one argument which is the weighted sum of the values obtained from the connected units from the previous layer, $$y=f^{act}\left(\sum_{i\in \,{\rm previous\, layer}} w_{i} y_{i}\right).$$ The weights $\{w_{i}\}$ (located at the edges of the graph) are real numbers, which are the parameters established during the training (learning) process, so as to maximize the posterior probability (\[posterior\_w\]). A simple example of MLP, which was used to fit the electromagnetic proton, neutron FFs data is shown in Fig. \[Fig\_Architecture\] (a).
![\[Fig\_Unit\] Single unit connected with $n$ units (solid lines) and one bias (dashed line). ](unit-eps-converted-to.pdf){width="40.00000%"}
According to the Cybenko theorem [@Cybenko_Theorem] the class of networks with only one hidden layer containing sigmoid-like activation functions and the output layer with linear activation functions is dense in the space of continuous functions, $\mathbb{R}^{n_{in}} \to \mathbb{R}^{n_{out}}$, defined on the unit hypercube. This means that any continuous function can be approximated, with arbitrary precision, by such a network, if it has sufficient number of units in the hidden layer. Therefore we restrict our analysis to networks of this class.
In the hidden layer we use sigmoid activation function $ f_{act}(x) = {1}/{(1 + \exp(-x))} $, which has effective support limited to a close neighbourhood of $x=0$. This property is used when the prior for the weights is postulated and when the weights are randomly initialized at the beginning of each training process.
Bayesian framework for MLP
--------------------------
To construct a statistical model the prior for the model parameters, weights, must be postulated. We consider a Gaussian distribution centred at zero. $$\begin{aligned}
\label{prior_BF}
\mathcal{P}\left(\{w_i\}\right|\left. \alpha, \mathcal{N}\right)
&=&
\frac{e^{- \alpha E_w}}{N} ,\; E_w=\frac{1}{2}\sum_{k=1}^{p}w_k^2, \; N = \int d^{p} w\, e^{- \alpha E_w}.\end{aligned}$$
The parameter $\alpha$ (or rather the square root of its inverse), called later regularizer, defines the width of the Gaussian. If $\alpha$ is large then the prior (\[prior\_BF\]) dominates in the posterior (\[posterior\_paraeters\_general\]), which forces the optimal weights $\{w_i\}_{MP}$ to be small. For low $\alpha$ the weights are unconstrained and the maximum of the likelihood dominates. As a result the optimal model may over-fit the data. The properly adjusted $\alpha$ prevents the over-fitting but does not affect strongly the obtained results, asserting that the model will have a good predictive power. Hence the prior plays also the role of a penalty contribution, which allows to prefer smaller rather than more complex networks.
In general one could introduce a separate regularizer for every weight. However, because of the internal symmetry of the MLP (hidden units from the same layer can be interchanged without affecting the value of the output) the set of weights can be divided in several distinct classes of parameters with one $\alpha$ parameter shared by all weights of a given class. For an instructive example see Sect. 3.2 of Ref. [@Graczyk:2010gw]). However, to simplify our numerical calculations we consider only one common regularizer.
In principle $\alpha$ is one of the model parameters. Hence there should exist a configuration $\{\alpha, w_{i}\}_{MP}$, which maximizes expression (\[Posterior\_para\_general\]). In order to obtain their optimal values together with the value of the evidence we use the so-called [*evidence approximation*]{} [@evidence].
The idea of this approach is to calculate the posterior for weights, $$\begin{aligned}
\label{posterior_w}
P(\{w_i\}| \mathcal{D}, \alpha, \mathcal{N}) &=&
\frac{\mathcal{P}\left(\mathcal{D}\right|\left.\{w_i\}, \alpha, \mathcal{N} \right)
\mathcal{P}\left(\{w_i\}\right|\left. \alpha, \mathcal{N} \right)}{\mathcal{P}\left(\mathcal{D}\right|\left. \alpha, \mathcal{N} \right)},\end{aligned}$$ assuming a fixed $\alpha$. It is done by taking the Gaussian (\[prior\_BF\]) for the prior and assuming that the likelihood is given by $$\begin{aligned}
\label{likelihood_BF}
\mathcal{P}\left(\mathcal{D}\right|\left.\{w_i\}, \alpha, \mathcal{N} \right)&=&
\frac{e^{-\chi_{ex}(\mathcal{D},\{w_i\})}}{n},\end{aligned}$$ where $\chi_{ex}(\mathcal{D},\{w_i\})$ is the $\chi$-square distribution for given data $\mathcal{D}$, and $n$ – the normalization factor calculated in the Hessian approximation, see Eq. (3.8) of Ref. [@Graczyk:2010gw].
In this approximation the maximum of the posterior (\[posterior\_w\]) (for fixed $\alpha$) corresponds to the minimum of the error function: $$\label{minimum_of_error}
S(\mathcal{D},\alpha,\{w_i\}) = \chi^2_{ex}(\mathcal{D},\{w_i\}) + \alpha E_w.$$ Hence the optimal configuration of weights minimizes $S(\mathcal{D},\alpha_{MP},\vec{w})$.
On the other hand, it can be shown [@Bishop_book] that the necessary condition for the optimal $\alpha$ reads $$\label{alpha_iter}
\left.\frac{\partial }{\partial \alpha} \mathcal{P}\left(\mathcal{D}\right|\left. \alpha, \mathcal{N} \right)\right|_{\alpha=\alpha_{MP}} =0.$$ (notice that $ \mathcal{P}\left(\mathcal{D}\right|\left. \alpha, \mathcal{N} \right) = \int d^p w \mathcal{P}\left(\mathcal{D}\right|\left.\{w_i\}, \alpha, \mathcal{N} \right)
\mathcal{P}\left(\{w_i\}\right|\left. \alpha, \mathcal{N}\right) $).
Both conditions (\[minimum\_of\_error\]) and (\[alpha\_iter\]) are used to find (iteratively) the optimal configuration $\{\alpha, w_{i}\}_{MP}$. Then the evidence for the model $P(\mathcal{D}|\mathcal{N})$ is computed. It is done by using the results of the previous step, for the details see [@Bishop_book] (Chap. 10).
Application: Investigation of the Proton Structure
==================================================
The feed-forward neural networks can be used to approximate the FFs [@Graczyk:2010gw]. The methods of neural networks allow one to reduce the model-dependence of the results of the analysis and to make predictions in kinematic regions where there are no measurements.
As an example of application to hadron physics we present the extraction of the E-M proton FFs and the TPE correction from the elastic $e^\pm p$ scattering data. Our aim is to discuss the statistical features of the approach like dealing with the bias-variance trade-off and the problem of estimating of the systematic uncertainty due the model-dependence.
Form-factors and two-photon exchange correction
-----------------------------------------------
A part of the information about the structure of the nucleon is hidden in the E-M FFs [@Belitsky:2003nz]. They are the functions which parametrize the E-M proton vertex [@Thomas_book]: $$\Diagram{ & \vertexlabel^{q} \\
& ![ulft]{gv}{} & \\
{fA} & ![bot]{fA}{} \\
} = \Gamma^\mu(q) = F_1(Q^2) \gamma^\mu + \frac{\mathrm{i}\sigma^{\mu\nu}q^\nu }{2M_p} F_2(Q^2),$$ where $M_p$ is the proton mass, $q$ is the four-momentum of the virtual photon, and $F_{1,2}$ denote the form-factors – Lorentz invariant scalars depending only on the four-momentum transfer $Q^2=-q^2$.
It is convenient to consider the electric, $G_E = F_1 - (Q^2/4M^2_p)F_2$, and the magnetic, $G_M = F_1+F_2$, FFs which at low $Q^2$ and in the Breit frame can be related with the Fourier transform of the charge and the magnetic distributions inside the proton [@Ernst:1960zza].
The E-M FFs are extracted from the elastic $e^- p$ scattering data within two methods [@form_factors_review]. In one method the Rosenbluth separation of the unpolarized cross section data is performed, and the values of $G_E$ and $G_M$ are simultaneously obtained. In the other method (denoted later as PT) the ratio $G_E/G_M$ is extracted from the measurements of the polarization transfer observables. In both types of the analyses the radiative corrections are subtracted from the scattering data in order to get the cross sections, polarizabilities, in *the one-photon exchange approximation*.
It turned out that the $G_E/G_M$ ratios obtained from the Rosenbluth separation and from the PT measurements differ at larger $Q^2$ values (for the review see [@Arrington:2011dn]). This inconsistency can be partially removed if the Rosenbluth data are corrected by the TPE contribution (hard photon part)[@tpe_wyjasnienie; @Arrington:2007ux], which was neglected in the old analyses. It is a small correction but its inclusion into the Rosenbluth analysis changes the results of the separation (affecting mainly the resulting value of $G_E$), while the form factor ratio obtained from the PT data analysis is affected to much lesser extent.
The TPE correction is given by the interference of the Born diagram and the diagrams describing the exchange of two virtual photons between the electron and the proton (Fig. \[Fig\_tpe\]).
1 ![Two-photon exchange contribution to elastic $ep$ scattering. \[Fig\_tpe\]](tpe_diagrams-eps-converted-to.pdf "fig:")
The hard photon contribution of the TPE is induced by the hadronic proton structure. Many efforts have been made to calculate this contribution based on the theoretical and phenomenological models (see references 6-18 in [@Graczyk:2011kh]). However, for larger $Q^2$ values the predictions of TPE are model-dependent. On the other hand, there are attempts to get the TPE contribution directly from the scattering data, by using constructively the inconsistency of Rosenbluth-PT data. Obviously such analysis requires model assumptions about the functional form of the FFs and the TPE term. However, the extracted value of the TPE correction depends also on the choice of the model i.e. the FFs and the TPE parametrizations. The BF for MLP turned out to be a useful methodology for dealing with the model-dependence and finding the optimal statistical model [@Graczyk:2011kh]. In the next subsections we review the main features of this approach.
The TPE correction is particularly well suited to discuss model-dependence because both the extraction of the TPE correction from the data and also its theoretical predictions are affected by the choice of the model. On the other hand recently the ratio $R_{+/-} = \sigma(e^+p)/\sigma(e^-p)$ of the cross sections for elastic positron-proton to the electron-proton scattering is measured in two dedicated experiments [@Gramolin:2011tr; @Bennett:2012zza]. From (\[ratio\_R+-\]) it is evident that the TPE correction contributes to the deviations of $R_{+/-}$ from unity.
Analysis of the elastic $e^\pm p$ scattering data
-------------------------------------------------
For the purpose of this presentation we made a revision of our previous analysis [@Graczyk:2011kh]. The present results are obtained by an improved version of the neural network program with more efficient learning algorithm (Levenberg-Marquardt [@LMalogorithm]). The structure of the program has also been changed. It is now possible to consider larger number of models in shorter time.
The idea of the analysis is to assume that the missing correction, responsible for the Rosenbluth-PT data disagreement, affects mainly the cross section data and the PT observables to much lesser extent [@tpe_wyjasnienie]. Hence performing a combined analysis of both types of the data should allow one to get ”missing” contribution responsible for the disagreement. In practice we consider three types of the data:
1. unpolarized cross sections (27 independent data sets);
2. the $G_E/G_M$ ratio (from PT measurements);
3. the $R_{+/-}$ ratio.
All three types of data depend on $(G_E,G_M, \Delta \tilde{C}_{2\gamma})$, where $\Delta \tilde{C}_{2\gamma}$ denotes the ”missing” correction responsible for the disagreement. $\Delta \tilde{C}_{2\gamma}$ contributes to the reduced unpolarized cross section: $$\label{sigma_reduced}
\sigma_{R}(Q^2, \varepsilon,s) = \frac{Q^2}{4M^2_p} G_M^2(Q^2) + \varepsilon G_E^2(Q^2) + s\Delta \tilde{C}_{2\gamma}(Q^2, \varepsilon),$$ and it is interpreted as TPE contribution. In (\[sigma\_reduced\]) $s=\pm 1$ corresponds to the cross-section for $e^\pm p $ scattering and $\varepsilon = \left[1+2\left(1+{Q^2}/{4M_p^2}\right)\tan^2\!\left({\theta}/{2}\right)\right]^{-1}$ is the photon polarizability ($\theta$ is the scattering angle between incoming and outgoing electrons). It is easy to see that the ratio of the positron/electron cross-sections has the form: $$\label{ratio_R+-}
R_{+/-} = \sigma_{R}(Q^2, \varepsilon,-1)/\sigma_{R}(Q^2, \varepsilon,+1).$$
In the combined analysis of the data we consider networks with two inputs $(Q^2,\varepsilon)$ and three outputs $(G_E,G_M, \Delta \tilde{C}_{2\gamma})$, see Fig. \[Fig\_Architecture\]. Because the FFs do not depend on $\varepsilon$ some of the connections in the network are erased. As a result the hidden layer of the network is divided into two parts: one (called FF sector) contains $g$ units connected only with $Q^2$ input and both outputs, and the other (called TPE sector) with units disconnected only from the TPE output.
The likelihood (\[likelihood\_BF\]) and prior (\[prior\_BF\]) are defined in the same way as in our previous analysis: Eqs. 10, 11 A1-3 and 12, 13 of Ref. [@Graczyk:2011kh] respectively. The selection of the data sets is the same as well. The $\chi^2_{ex}$ in (\[likelihood\_BF\]) is the sum of the $\chi^2$s’ for the cross section, FFs ratio and $R_{\pm}$ data. The experimental data points are characterized by statistical and point to point systematic uncertainties. In the case of the cross section data, for every separate set of measurements, the systematic normalization uncertainty is taken into account and corresponding normalization parameter is introduced into the fit, for more details see Appendixes A and B of Ref. [@Graczyk:2011kh].
Numerical Algorithm
-------------------
The scheme of the numerical analysis is the following:
1. consider MLP with definite number of hidden units;
2. find the optimal configuration of weights $\{w_i\}_{MP}$ and $\alpha_{MP}$ by the use of a learning algorithm:
1. randomly initialize the weights;
2. perform the learning trial, iterating on-line $\alpha$ parameter (as described above);
3. compute the evidence for the model (for the analytic expression see Eqs. 33-35 of [@Graczyk:2013pca]);
4. for a given network type (scheme) choose the best model (one with the highest evidence);
5. change the network type (by increasing by one the number of units either in form-factor or TPE part of the hidden layer) and repeat the steps (i-v).
We considered 102 different MLP schemes. The maximal number of units in the hidden layer was 14. For each type of the network about 2300 learning processes were performed reaching the total number of about 226 000 considered networks. After that we collected distribution of 102 models ranked by the evidence $P(\mathcal{D}|\mathcal{N})$. The model with the highest evidence contains 6 hidden units: 2 in the FF sector and 4 in the TPE sector (see Fig. \[Fig\_evidence\]).
Searching for the optimal model
-------------------------------
The optimal model, which we search for, should be rather simple (low number of parameters) to have ability for generalization (making the predictions about new data). On the other hand the number of parameters should be large enough so that model be able to reproduce the current data with reasonable precision. These two requirements are opposite, which is called the bias-variance trade-off. The optimal solution is a compromise between both tendencies [@Bishop_book].
In the BF finding the optimal solution is achieved in a very natural way. As mentioned before, the Bayesian statistics embodies Occam’s razor. In our approach this appears in two ways: first, by adjusting value of the regularizer $\alpha$, second, by using the evidence to choose the best model.
The most spectacular differences between the predictions of the model which is optimal (maximizes the evidence) and the one, which minimizes the error function and over-fits the data are seen in the plot of the $R_{+/-}$ (Fig. \[Fig\_R\_+-\_best\_evidence\_error\]). Similar comparison for the $\mu_p G_E/G_M$ ratio is shown in Fig. \[Fig\_FF\_ratio\]. The overfitted parametrization is characterized by high curvatures, while the optimal parametrization is given by much smoother curves. In this case the example of over-fitted parametrization is rather spectacular, it minimizes the error function, and from the point of view of $\chi^2/NDF$ is highly acceptable. The only reason (in qualitative sense) for rejecting this model from the typical non-Bayesian analysis is the presence of unacceptably high curvatures. But in a less spectacular case such model would be accepted. In the BF the best model is indicated by a mathematical objective algorithm i.e. the way of learning the networks and the evidence hence the ”human” decision is reduced to a minimum.
Systematic uncertainty induced by choice of the parametrization
---------------------------------------------------------------
The main result of the analysis is the set (denoted by $\mathcal{M}$) of the neural networks, parametrizations, which are ranked by the evidence. Each of them has a different connection graph and maximizes the evidence in its class of functions. Having such distribution of models one can estimate the systematic uncertainty due to the choice of the functional parametrization (the network shape).
Let us introduce the mean value, according to the space of *the best parametrizations* of the observable $\mathcal{F}$, which is a function of outputs of the network $\mathcal{N}$, $$\begin{aligned}
\label{discrete_series}
\overline{\mathcal{F}(G_E,G_M, \Delta
\tilde{C}_{2\gamma}) } &=&
\sum_{\mathcal{N}\in \mathcal{M}}
\mathcal{F}(G_E^{\mathcal{N}},G_M^{\mathcal{N}}, \Delta \tilde{C}_{2\gamma}^{\mathcal{N}}) \mathcal{P}_{nor}(\mathcal{D}|\mathcal{N}),\end{aligned}$$ where, $$\mathcal{P}_{nor}(\mathcal{D}|\mathcal{N}) = \frac{\mathcal{P}(\mathcal{D}|\mathcal{N})}{\sum_{\mathcal{N}\in \mathcal{M}} \mathcal{P}(\mathcal{D}|\mathcal{N})}$$ and the relation $\mathcal{P}(\mathcal{D}|\mathcal{N}) \approx \mathcal{P}(\mathcal{N}|\mathcal{D})$ is also imposed.
Then the systematic uncertainty due to the choice of the functional parametrization is given by the square root of the variance $\Delta \mathcal{F}(G_E,G_M, \Delta
\tilde{C}_{2\gamma})$.
In Fig. \[Fig\_systematic\_3\_poins\] we plot the estimates of the $R_{+/-}$ (for three points) against the logarithm of evidence. Each point denotes the value predicted by a neural network with the highest evidence in its class. The prediction is plotted together with the error bounds ($1\sigma$ error due the distribution of weights) see Eqs. 3.15 and 3.16 of [@Graczyk:2010gw]. The horizontal lines denote the mean value (weighted by the evidence) see (\[discrete\_series\]), while the shaded areas denote the $1\sigma$ uncertainty given by $\Delta R_{+/-}$, calculated as described above. Notice that the best model predictions agree well with the new data from [@Gramolin:2011tr], which were not included in the analysis. On the other hand it is clear that many models give negligible contribution to the mean value and dispersion because they have very low values of the evidence.
In Fig. \[Fig\_systematic\_plot\] we present another example of estimating the systematic uncertainty due to the model-dependence. It is the $Q^2$ dependence of ratio $R_{+/-}$ calculated for a fixed value of $\varepsilon$ . The systematic uncertainty is very small on a large $Q^2$ range. In the same figure for the qualitative comparison we plot also the best 20 fits, due to the evidence.
Obviously the above estimate of the systematic model-dependence uncertainty does not include all the model dependence of the approach. Indeed, an important assumption to perform the analysis was to neglect the TPE correction to the PT data. This model-assumption is difficult to quantitatively account for. One can only estimate the systematic bias with respect to the theoretical model calculations. In Fig. \[Fig\_NN\_th\] we compare the predictions of $R_{+/-}$ obtained by neural network with hadronic model predictions [@Graczyk:2013pca]. In the latter the TPE is calculated within the quantum field theory approach, in which it was assumed that the hadronic intermediate state is given by either a proton or the $P_{33}(1232)$ resonance [@Graczyk:2011kh]. It is expected that this model should work well in the low and intermediate $Q^2$ range. It can be seen that in the low $Q^2$ there is a systematic deviation between the neural network response and the theory. It seems that it is the result of the theoretical assumption mentioned above. On the other hand it can also be caused by the inaccuracy of the theoretical model.
Summary
=======
We have shown that the approach based on neural networks can be used to extract important information about the structure of the proton from the scattering data. The approach offers tools which make it possible to control and reduce the model-dependence. The result of the analysis is a statistical model with good predictive power which can be used to make predictions about the FFs and TPE correction in the kinematic region where there are no measurements.
It seems that one can try to introduce an analogical Bayesian framework also in the case of non-neural network analyses, especially, for the set of theoretical models. However, it seems that in that case the construction of an objective prior distribution is a challenge. Additional difficulty is the over-fitting problem and construction of a suitable penalty term.
Acknowledgements {#acknowledgements .unnumbered}
================
The calculations have been carried out at the Wroclaw Centre for Networking and Supercomputing (<http://www.wcss.wroc.pl>), grant No. 268.
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[^1]: We notice that many efforts have been made to calculate the FFs within the lattice QCD [@latice].
[^2]: More detailed description of the MLP properties can be found in our previous paper [@Graczyk:2010gw] (Sect. 2).
|
---
abstract: 'The leading-order hadronic vacuum polarization contribution to the hyperfine splitting of true muonium is reevaluated in two ways. The first considers a more complex pionic form factor and better estimates of the perturbative QCD contributions. The second, more accurate method directly integrates the Drell ratio $R(s)$ to obtain $C_{1,\rm hvp}=-0.0489(3)$. This corresponds to an energy shift in the hyperfine splitting of $\Delta E^\mu_{hfs,\rm hvp}=276196(51)$ MHz.'
author:
- Henry Lamm
bibliography:
- '/home/hlamm/wise.bib'
title: Hadronic Vacuum Polarization in True Muonium
---
Introduction
============
True muonium is the yet unidentified $({\mu\bar{\mu}})$ bound state. The bound states have lifetimes between ps to ns [@Brodsky:2009gx]. QED dominates the characteristics of true muonium, while QCD effects appear at $\mathcal{O}(m_\mu\alpha^5)$ [@Jentschura:1997tv; @Jentschura:1997ma]. Electroweak effects appear at $\mathcal{O}(m_\mu\alpha^7)$ [@PhysRevD.91.073008]. Measurements of Lamb shift, $1s-2s$ splitting, and the hyperfine splitting (hfs) will occur in the future. These experiments are motivated by the existing discrepancies in muon physics [@PhysRevD.73.072003; @Antognini:1900ns; @Aaij:2014ora; @Aaij:2015yra; @Pohl1:2016xoo]. Numerous new physics models have been suggested to explain these discrepancies [@TuckerSmith:2010ra; @Jaeckel:2010xx; @Batell:2011qq; @Barger:2011mt; @Karshenboim:2010cm; @Karshenboim:2010cg; @Karshenboim:2010cj; @Karshenboim:2010ck; @Karshenboim:2011dx; @Karshenboim:2014tka; @Carlson:2012pc; @Carlson:2015poa; @Izaguirre:2014cza; @Kopp:2014tsa; @Martens:2016zzx; @Liu:2016qwd; @Onofrio:2013fea; @Wang:2013fma; @Gomes:2014kaa; @Brax:2014vva; @Lamm:2015gka; @Lamm:2016jim]. True muonium can produce competitive constraints on most models if standard model predictions are known to the 100 MHz level, corresponding to $\mathcal{O}(m_\mu\alpha^7)$.
Beyond new physics, a further motivation for considering true muonium comes from the anomalous magnetic moment of the muon ($a_\mu$). There exists a discrepancy between the measurement at BNL and theory, $\Delta a_\mu=a_{\mu,exp}-a_{\mu,th}=288(80)\times10^{-11}$ [@PhysRevD.73.072003; @Aoyama:2014sxa]. Hadronic contributions dominate the theoretical uncertainty, and hadronic vacuum polarization (hvp) is the largest term. One way to reduce the theoretical uncertainty would be consistency checks from other systems. By its particle/antiparticle nature, the annihilation channel contributes to true muonium, leading to an enhancement of hvp contributions to the hfs. These contributions are measurable in true muonium unlike positronium were they are mass-suppressed.
The theoretical expression for the hfs corrections to true muonium from QED can be written $$\begin{aligned}
\Delta E_{\rm hfs}=m_\mu\alpha^4\bigg[&C_0+C_1\frac{\alpha}{\pi}+C_{21}\alpha^2\ln\left(\frac{1}{\alpha}\right)+C_{20}\left(\frac{\alpha}{\pi}\right)^2\nonumber\\
&+C_{32}\frac{\alpha^3}{\pi}\ln^2\left(\frac{1}{\alpha}\right)+C_{31}\frac{\alpha^3}{\pi}\ln\left(\frac{1}{\alpha}\right)\nonumber\\
&+C_{30}\left(\frac{\alpha}{\pi}\right)^3+\cdots\bigg],\end{aligned}$$ where $C_{ij}$ indicate the coefficient of the term proportional to $(\alpha)^i\ln^j(1/\alpha)$. All dependence of the hfs to mass scales other than $m_\mu$ is in the $C_{ij}$. The coefficients of single flavor QED bound states, used in positronium, are known up to $\mathcal{O}(m_e\alpha^6)$ and some partial results for $\mathcal{O}(m_e\alpha^7)$(For an updated review of the coefficients see [@Adkins:2014dva; @PhysRevA.94.032507]). The exchange $m_e\rightarrow m_\mu$ translates these results to true muonium.
True muonium has extra contributions that must be considered. The lighter electron allows for large loop contributions. The relative smallness of $m_\tau/m_\mu\approx 17$ and $m_{\pi}/m_{\mu}\approx1.3$ produce contributions to true muonium much larger than analogous contributions to positronium. Of these true muonium specific contributions, which we denote by $C_{ij}^\mu$, only a few terms are known. The $\mathcal{O}(m_\mu\alpha^5)$ contributions from electron loops were found to be $C_{1,e}^\mu=1.684$ [@Jentschura:1997tv]. The $\mathcal{O}(m_\mu\alpha^6)$ contribution from leptonic loops to the two-photon annihilation channel $C^\mu_{20,2\gamma}=-2.031092873$ was recently computed exactly [@PhysRevA.94.032507], and the electron loop in three-photon annihilation at $\mathcal{O}(m_\mu\alpha^7)$ is $C_{30,3\gamma}^\mu=-5.86510(20)$ [@Adkins:2015jia]. For a $\mathcal{O}(m_\mu\alpha^7)$ prediction of the hfs, contributions from $Z$-bosons must be considered [@PhysRevD.91.073008].
The hadronic vacuum polarization contributes at $\mathcal{O}(m_\mu\alpha^5)$ and was previously calculated to be $C_{1,\rm hvp}=-0.047(5)$ in [@Jentschura:1997tv] where the error is an estimate of the model-dependence. We will refer to this result as JSIK throughout, after the authors of that paper. This result mixed a Gounaris-Sakurai form factor for the $\pi$ and $\rho$ contributions, a simple pole approximation for the $\omega$ and $\phi$, and a two-constant perturbative contribution above 1 GeV.
Together, these contributions predict the hfs of true muonium to be $\Delta E_{\rm hfs}^{1s}=42329730(800)(700)$ MHz where the first, dominant, uncertainty is from the hadronic model-dependence and the second is an estimate of uncalculated $\mathcal{O}(m_\mu\alpha^6)$ contributions. The goal of this work is to recalculate $C_{1,\rm hvp}$ such that we can both reduce the model dependence and better estimate the uncertainty.
The hadronic vacuum polarization contribution is given by $$\begin{aligned}
\label{eq:energy}
\Delta E_{1,\rm hvp}=&\left[m_\mu^2\int_{4m_\pi^2}^{\infty}\mathrm{d}s\frac{\rho(s)}{4m^2_{\mu}-s}\right]\frac{m_\mu\alpha^5}{n^3\pi}\nonumber\\
=&C_{1,\rm hvp}\frac{m_\mu\alpha^5}{n^3\pi}\end{aligned}$$ where $\rho(s)$ is the spectral function that must be specified.
JSIK Calculation
================
The calculation of $C^{\rm JSIK}_{1,\rm hvp}$ in [@Jentschura:1997tv] is given by the sum of four terms, $$C^{\rm JSIK}_{1,\rm hvp}=C_{1,\pi}+C_{1,\omega}+C_{1,\phi}+C_{1,>}$$ where $C_{1,\pi}$ is the contribution from the pion form factor, $C_{1,\omega}$ and $C_{1,\phi}$ are the simple-pole terms for these two meson, and $C_{1,>}$ is the contribution from the regime above 1 GeV were perturbative QCD was applied.
The main contribution is from the pionic loop, where the result is given by [@Sapirstein:1983xr; @Jentschura:1997tv] $$\rho(s) = \frac{(s - 4\,m_{\pi}^2)^{3/2}}{12\,s^{5/2}}\,
|F_{\pi}(s)|^2 \,.$$ JSIK chose to use the simple Gounaris-Sakurai form factor [@Gounaris:1968mw]. For brevity, this choice of $F_{\pi}(s)$ is often written as $$\label{eq:fpi}
F_{\pi}(s)=F_{\rho,\rm GS}(s) = \frac{N}{D_1 + D_2 - i\,D_3}\,.$$ In this decomposition, $N$, $D_1$, $D_2$ and $D_3$ are given by $$\begin{aligned}
N = m_\rho^2 + d m_\rho \, \Gamma_\rho\,,&\phantom{xxx}D_1 = m_\rho^2 - s,\nonumber\\
D_2 = \Gamma_\rho \frac{m_\rho^2}{k_\rho^3}
\bigg[k(s)^2(h(s) &- h_\rho) + k_\rho^2 h'_\rho
(m_\rho^2 - s) \bigg],\nonumber\\
\quad D_3 = \frac{m_\rho^2\Gamma_\rho}{\sqrt{s}} &
\left(\frac{k(s)}{k_\rho}\right)^3 \,,\end{aligned}$$ with the parameter $d$ defined via $$d = \frac{3}{\pi} \frac{m_\pi^2}{k_\rho^2}
\ln\frac{m_\rho + 2 \, k_\rho}{2\,m_\pi} +
\frac{m_\rho}{2\,\pi\,k_\rho} -
\frac{m_\pi^2\, m_\rho}{\pi\,k_\rho^3} \approx 0.48.$$
The functions $k(s)$ and $h(s)$ are defined as $$k(s)=\frac{1}{2}\sqrt{s - 4m_\pi^2}, \quad h(s) = \frac{2}{\pi} \,
\frac{k(s)}{\sqrt{s}} \,
\ln \left(\frac{\sqrt{s} + 2\, k(s)}{2\,m_\pi}\right)\,.$$ Where $h'$ denoted the derivative of $h(s)$ with respect to $s$ and the subscript $\rho$ indicated evaluation of the function at $m^2_\rho$. In this form factor, only the contributions for the $\rho$ mesons are included. The physical values used by JSIK were $\Gamma_\rho = 150.7(1.2) \, {\rm MeV},$ and $m_\rho = 768.5(6) \, {\rm MeV}$. Integrating these expressions, the value for the pionic loop was found to be $C_{1,\pi}=-0.032$.
To include other meson resonances, a simple pole approximation is taken. The spectral function contribution from a vector meson is given by $\rho(s)=4\pi^2/f_V^2\delta(s-m_V^2)$ [@Bauer:1977iq] where $f_V$ are coupling constants. These were estimated in [@Bauer:1977iq] to be $f_\omega^2/4\pi=18(2)$ and $f_\phi^2/4\pi=11(2)$. The masses of the vector mesons are $m_\omega=782.71(8)$ MeV and $m_\phi=1019.461(19)$ MeV. JSIK obtained $C_{1,\omega}=-0.004$ and $C_{1,\phi}=-0.003.$
The final contribution, $C_{1,>}$ was obtained by applying the relation between the spectral function and the Drell ratio, $$\label{eq:rhors}
\rho(s)=\frac{R}{3s}, \text{ where } R=\frac{\sigma(e^+e^-\rightarrow h)}{\sigma(e^+e^-\rightarrow\mu^+\mu^-)}.$$ In perturbative QCD, $R$ at leading order is given by $R_{LO}=N_c\sum q_i^2$ where $N_c$ is the number of colors and $q_i$ is the charge of quark $i$. Below the $c$ threshold at $\approx4$ GeV, $R_{LO}=2$. Between 4 GeV and 10 GeV, the $c$ quark goes on shell and $R_{LO}=10/3$. Above 10 GeV the $b$ quark changes the ratio to $R_{LO}=11/3$. At present, perturbative calculations exist up to $\mathcal{O}(\alpha_s^4)$ that better account for the experimental results. JSIK estimated from the experimental results in [@Barnett:1996hr] that $R_{\rm 2 GeV<s<4 GeV}\approx2$ and $R_{s>4 GeV}\approx4$ (See Fig. \[fig:rs\]). With these values, they obtained $C_{1,>}=-0.008.$
Putting all of these together, and including a 11% estimate of the model-dependent uncertainties, their final result was $C_{1,\rm hvp}^{\rm JSIK}=-0.047(5)$
{width="\linewidth"}
Investigating the Pieces
========================
One way to reduce the uncertainty in $C_{1,\rm hvp}$ would be to improve the calculations of the pieces of the JSIK value. Since that work, improvements in experimental measurements of the pion form factor have lead to the development of an improved Gounaris-Sakurai parameterization that more properly accounts for the $\rho-\omega$ mixing as well as the $\rho'$ and $\rho''$ states. $C_{1,>}$ can also be better estimated by computing the numerical averages of $R(s)$ in the regimes and accounting for non-constant terms.
Improved Gounaris-Sakurai Parameterization
------------------------------------------
Instead of the simple Gounaris-Sakurai form factor, a more complex form exists that features two improvements. The improved form [@Schael:2005am] includes $\rho-\omega$ mixing and the $\rho'$ and $\rho''$ resonances. The form factor is given by $$\begin{aligned}
F_{\pi,\rm IGS}=\frac{1}{1+\beta+\gamma}\big[F_{\rho,\rm GS}&(s)\left(1+\delta \frac{s}{m^2_\omega}F_{\omega,\rm BW}(s)\right)\nonumber\\&+\beta F_{\rho',\rm GS}(s)+\gamma F_{\rho'',\rm GS}(s)\big]\end{aligned}$$ where $F_{i,\rm GS}(s)$ are given by Eq.(\[eq:fpi\]) with the additional masses and decay constants: $m_{\rho'}=1409(12)$ MeV, $\Gamma_{\rho'}=501(37)$ MeV, $m_{\rho''}=1740(21)$ MeV, $\Gamma_{\rho''}=235(1)$ MeV, and $\Gamma_\omega=8.68$ MeV [@Schael:2005am]. Further the parameters $\delta=2.03(10)e^{0.2269(401)i}$, $\beta=-0.166(6)$, and $\gamma=0.071(6)$ determine the mixing and relative strengths [@Schael:2005am]. For the $\omega$ meson, a Breit-Wigner form factor is used $$F_{\omega,\rm BW}(s)=\frac{m^2_{\omega}}{m^2-s+i\Gamma_{\omega}m_\omega}.$$ Integrating, we compute a coefficient $C_{1,\rm IGS}=-0.0377(5)$ which should include the same physics as $C_{1,\pi}+C_{1,\omega}$ as the well previously uncalculated higher-order terms from $\rho',\rho''$. The error on $C_{1,\rm IGS}$ is estimated from parameter variation.
Perturbative QCD Regime
-----------------------
We use the data for R(s) compiled for the software packages alphaQED [@Jegerlehner:2003qp; @Jegerlehner:2003rx; @Jegerlehner:2003ip; @Jegerlehner:2015stw] and rhad [@HARLANDER2003244]. These packages may be found at [@alphaqed]. It can be seen in Fig. \[fig:rs\] that the JSIK value of $C_{1,>}$ can be improved. For $s\approx1$ GeV$^2$ to $s\approx2$ GeV$^2$, the JSIK estimate overestimates the contribution, and ignores the $s$-dependence. We find that in the region $s=[1.2,2.3]$ GeV$^2$ is well fit to $R(s)=0.0895(9)s^{3.43(11)}+0.63(2)$. Above this, we take the $R$ to be a constant fit to the average value without resonances. Between $s=2.3$ GeV$^2$ and the $s=16$ GeV$^2$ $R\approx2.15(1)$. In the region $s=[16,120]$ GeV$^2$, we find $R\approx3.71(1)$ and above this we take $R\approx3.95(1)$. Together, these choices give a value of $C_{1,>}=-0.00574(4)$
Adding our values of $C_{1,\rm IGS}$ and $C_{1,>}$ to the previously obtained value of $C_{1,\phi}$, our final results for the improved piecewise coefficient is $C^{\rm imp}_{1,\rm hvp}=-0.0467(5)$. This value represents an improvement on the JSIK value, but we note that it still has a large model-dependence which is difficult to estimate, and doesn’t fully encapsulate the effect of resonances.
Numerical Integration of $R(s)$
===============================
Another method to obtain $C_{1,\rm hvp}$ is numerically integrating $R(s)$. This method has negligible model-dependence and theoretical uncertainties. We numerically integrate the full $R(s)$ data from Ref. [@HARLANDER2003244; @Jegerlehner:2003qp; @Jegerlehner:2003rx; @Jegerlehner:2003ip; @Jegerlehner:2015stw; @alphaqed] seen in Fig. \[fig:rs\] using Eq.(\[eq:energy\]) and Eq.(\[eq:rhors\]). Without interpolation, the values of $R(s)$ tend to be overestimated due to the step behavior from binning in the data, especially around resonances. To avoid this, we interpolate the data with $n-$order polynomials before numerical integration. The results for $C_{1,\rm hvp}$ are found in Table \[tab:rs\]. We average these values to yield our final result, $C^R_{1,\rm hvp}=-0.0489(3)$. The error is the standard deviation of the interpolated results.
By replacement of $m_\mu\rightarrow m_e$ we can also compute the correction to positronium. We find that value to be $C^{R,e}_{1,\rm hvp}=-1.030(6)\times10^{-6}$, which is too small to be relevant in the near-future.
Order $C^{R,\mu}_{1,\rm hvp}$ $C^{R,e}_{1,\rm hvp}$
------- ------------------------- --------------------------
0 -0.05001 -1.051$\times10^{-6}$
1 -0.04879 -1.028$\times10^{-6}$
2 -0.04874 -1.027$\times10^{-6}$
3 -0.04869 -1.026$\times10^{-6}$
4 -0.04846 -1.021$\times10^{-6}$
Avg. -0.0489(3) -1.030(6)$\times10^{-6}$
: \[tab:rs\]$C^R_{1,\rm hvp}$ from directly integrating $R(s)$ for both true muonium and positronium. The order indicates the polynomial order used to fit the experimental data.
Summary and Conclusion
======================
In this work, we have recomputed the coefficient $C_{1,\rm hvp}$ in two ways. The first was improving upon the work of [@Jentschura:1997tv] through the use of a more complex pionic form factor and better modeling of the perturbative regime. The final calculation in this technique was $C^{\rm imp}_{1,\rm hvp}=-0.0467(5)$, where the error was only estimated by parameter variation and would miss systematic errors. While more precisely accounting for some of the features of the full spectral function, it has some drawbacks. It treats the $\phi$ meson as a simple pole, which will underestimate its contribution. Further, the treatment of all physics above 1 GeV$^2$ by the perturbative background neglects resonances and other features.
In order to avoid these problems, we computed $C_{1,\rm hvp}$ directly from experimental $R(s)$. To account for the binning of the data, we used $n-$order polynomial interpolators. The final value for this method was $C^R_{1,\rm hvp}=-0.0489(3)$. This value is larger than both JSIK and our improved method by more than estimated error of $C^{\rm imp}_{1,\rm hvp}$. We attribute this to the inadequate treatment of resonances in the these methods. Further, the model-dependence inherent in fitting the form factor has been avoided in this method. Therefore, we take this as our final value for calculations of the hfs.
With this contribution found, we can reevaluate the prediction for hfs. Our result reduces the hadronic error estimate of JSIK[@Jentschura:1997tv] from $800$ MHz to $51$ MHz. The current value is $\Delta E^{1s}_{\rm hfs}=42329355(51)_{\rm had}(700)_{\rm miss}\text{ MHz}$, where the first uncertainty is from hadronic contribution, and the second an estimate of missing $\mathcal{O}(m_\mu\alpha^6)$ terms. With this reduction in hadronic uncertainty, the missing QED corrections now dominates and is the only remaining step to obtaining $\mathcal{O}(100$ MHz) predictions for use in new physics searches.
HL would like to thank E Lee for her assistance in locating an error in the numerical integration. HL is supported by the National Science Foundation under Grant Nos. PHY-1068286 and PHY-1403891.
|
---
abstract: 'In this paper we intend to study implications in their most general form, generalizing different classes of implications including the Heyting implication, sub-structural implications and weak strict implications. Following the topological interpretation of the intuitionistic logic, we will introduce non-commutative spacetimes to provide a more dynamic and subjective interpretation of an intuitionistic proposition. These combinations of space and time are natural sources for well-behaved implications and we will show that their spatio-temporal implications represent any other reasonable abstract implication. Then to provide a faithful well-behaved syntax for abstract implications, we will develop a logical system for the non-commutative spacetimes for which we will present both topological and Kripke semantics. These logics unify sub-structural and sub-intuionistic logics by embracing them as their special fragments.'
author:
- 'Amirhossein Akbar Tabatabai[^1]'
date:
title: Implication via Spacetime
---
Introduction
============
> *I remember that back in 1980, as an undergraduate, I was disappointed in logic, and was thinking of shifting to topology. Then Van Dalen came along and gave a course at the University of Amsterdam on sheaves and their relation to logic (the first such course in Holland), and subsequently organised a stimulating seminar on the subject. A course of lectures on Kripke-Joyal semantics by Michael Fourman formed part of this seminar. I was immediately fascinated by the subject, and still am.* [@Moer]
Replacing topology with algebraic geometry and categorical logic with Brouwer’s liberating revolution, I can hardly imagine a more vivid explanation of Mohammad Ardeshir’s eye-opening influence on my life, both academic and personal, than what Ieke Moerdijk is drawing in Dirk van Dalen festschrift. Through Ardehsir’s fascinating explanation of the intutionistic philosophy and its huge impact on the everyday practice of mathematics, I found the realm of constructive mathematics and its *implications* haunting and hence decided to leave not only my possible future in algebraic geometry but the whole discipline of everyday mathematics, altogether. However, a true revolution knows no border and Brouwer’s was no exception. Starting from the second half of the last century, the anti-realistic interpretation of mathematics has emerged unexpectedly and as a technical inevitable necessity in the mainstream mathematics, first in algebraic geometry through Alexander Grothendieck’s inexhaustible quest for the generalized space and then in higher geometry, homotopy theory and the so-called homotopical mathematics. Following this historical thread, my fascination for intuitionism and more specifically the intuitionistic implication, is now slowly bringing me back to algebraic geometry again, where intuitionism might play its most deserved technical role. In this introduction I intend to explain how such a seemingly unrelated notion of space can be useful to understand intuitionism and hence intuitionistic implications. Far better, I will explain how intuitionism and geometry, interpreted in its most general sense, are nothing but the two sides of the same coin.\
To establish this connection, we have to first understand the spatial interpretation of the notion of construction. For that purpose, let us start with the easier notion of constructibility rather than the explicit constructions, themselves. This means that we are interested in propositions and the provability relation between them rather than the actual proofs. Let us start with the creative subject’s mind that may have many possible *states*. These states may encode many different data including the knowledge that she possesses in that mental state. It generally consists of all the constructions to which she has some reasonable access. For an intuitionist, a proposition is simply an entity that in every state of the creative subject’s mind, it possesses a truth value and if the proposition happens to be true at some point, it must be possible to verify this truth in a finite number of steps. The truth value checks whether the proposition is derivable from the knowledge in a given state or not. Interpreting the knowledge as the story that has been told, a true proposition is exactly what the story can imply. Note that the finite verifiability condition is different from the decidability of a proposition in a mental state. For instance, let the knowledge content of a mental state be the axioms of Peano arithmetic. Then if something is not derivable from this theory, there is no a priory way to verify that.\
The key point in the connection between intuitionism and topology is the set of these finitely verifiable propositions. This set has exactly the structure of the open subsets of a topological space and conversely, for any topological space, the set of its open subsets can be interpreted as the set of finitely verifiable propositions in a given theory.[^2] To explain how to interpret the set of all finitely verifiable propositions as the open subsets of a topological space, let us explain the three main structures that this set possess. Let $S$ be the set of all possible mental states. Then a proposition can be identified by a subset of $S$, consisting of all the mental states for which the proposition holds. First, note that these subsets are ordered by the partial order $A \vdash B$ that encodes the situation that the truth of $A$ in any state implies the truth of $B$ in the same state. The second structure is the finite meets of the poset, called conjunctions. The reason is that if both $A$ and $B$ are finitely verifiable propositions, then so is $A \wedge B$. Because, if $A \wedge B$ holds in a state, there are finite verifications for both of them and the combination of these verifications is also finite. Note that the same claim is not necessarily true for infinite conjunctions, because, if the infinite conjunction is true, we need possibly infinite number of verifications that may exceed any possible finite memory. The last and the third structure is the arbitrary joins called disjunctions. For some set $I$, if $A_i$ is finitely verifiable for any $i \in I$, then so is $\bigvee_{i \in I} A_i$. Because, if $\bigvee_{i \in I} A_i$ holds in a state, then one of them must hold and since it has a finite verification, the verification also works for the whole disjunction. Note that the semi-decidability condition and the existential nature of validity allows arbitrary disjunctions while it prohibits infinite conjunctions.[^3] These ingredients are nothing but the conditions on a topology of a topological space. Therefore, the set of all finitely verifiable propositions is actually the set of opens of the space of the mental states. Therefore, it should not be surprising that intuitionistic propositional logic is sound and complete with respect to its topological interpretation that reads a proposition as an open subset of a given topological space; see [@Mc]. In this sense, intuitionism may be interpreted as the logic of space as opposed to the classical logic that corresponds to the logic of sets or discrete spaces. Compare the set of all opens of a space to the opens of a discrete space, namely the Boolean algebra of all subsets.\
Now let us leave the truncated constructibility to address the actual explicit constructions. In this move, for any state we need a $\mathbf{Set}$-like world to encode the constructions of the propositions and not just their truth values. In this setting, the three structures that we have explained transform to the following higher order notions: First, a poset transforms into a category whose objects and morphisms are propositions and the constructions between them. Secondly, for conjunctions we need the categorical version of finite meets, i.e., finite limits. And finally, for disjunctions we have to bring categorical joins, i.e., small colimits. Together with some technical conditions, this *new space* is nothing but a Grothendieck topos. In this sense, the generalized notion of space is canonically conceivable from the pure intuitionistic conception of a proposition - a truly borderless revolution, indeed! Moreover, it implies that we should not be surprised that Grothendieck topoi or their elementary version can serve as the models for intuitionistic set theories or type theories, since the latter is simply the syntactic axiomatization of the constructions that the former formalizes model-theoretically. Unfortunately, this paper does not have enough space to explain all the details of this interpretation. However, we strongly encourage the reader to pursue this logical/philosophical path to geometry and read any geometrical construction by keeping an eye on the foregoing interpretation. This briefly explained connection between constructivism and the different incarnations of the notion of space is a very well-established tradition and here we only had time to see the tip of the iceberg. To see how this connection may lead to some useful interpretations in topos theory, higher geometry and even computer science, see [@JoyalTierney], [@Anel], [@Ab0], [@Ab1], [@Ab] and [@TopViaLogic].\
Now, considering propositions as the open subsets of a (new) space, we are ready to address the complex, ubiquitous and hard to comprehend notion of implication. First note that any sophisticated anti-realistic philosophy needs an act of internalization; the way by which the creative subject internalizes her own notion of construction to be able to bring them to her consideration as the object of the study and not just its instrument. This internalization is actually what the implication is developed for. It transforms the provability order between propositions, $A \vdash B$, a meta-mathematical property, into the validity of another proposition, i.e., $A \to B$. In the case we also care about the explicit constructions, the implication or in this case the function space, implements the same idea to transform the set of constructions from $A$ to $B$ to the constructions of $A \to B$.\
What is an internalizer? For the sake of simplicity, let us limit ourselves only to the constructibility case. Therefore, we have the provability order which we intend to internalize. There are many different structures that we can expect an implication to internalize. For instance, the order is reflexive, i.e., $A \vdash A$ for any proposition $A$ and it is transitive, i.e., “$A \vdash B$ *and* $B \vdash C$ *implies* $A \vdash C$" for any propositions $A$, $B$, and $C$. The internalizations for these basic properties are $\vdash A \to A$ and $$(A \to B) \wedge (B \to C) \vdash (A \to C),$$ for any propositions $A$, $B$, and $C$. The order has also all finite conjunctions meaning that for any two propositions $B$ and $C$, there exists a proposition $B \wedge C$ such that for any $A$ we have “$A \vdash B \wedge C$ *iff* “$A \vdash B$ *and* $A \vdash C$“” whose internalization is: $$A \to (B \wedge C)=(A \to B) \wedge (A \to C),$$ and for all finite disjunctions it means the existence of $A \vee B$ such that for any $C$ we have “$A \vee B \vdash C$ *iff* “$A \vdash C$ *and* $B \vdash C$“” whose internalization is: $$(A \vee B) \to C=(A \to C) \wedge (B \to C)$$ As we can observe by the foregoing instances, there can be many structures or properties that we may want to internalize and depending on that, there can be many different possible implications. The usual Heyting implications in posets, exponential objects in categories, the many-valued, the relevant and the linear implications and the monoidal internal hom structures in monoidal categories are only some of these many implications. See [@Mac], [@Bor1] and [@Substructural]. There are also some non-substructural internalizations. One of the early examples that also motivated the present work was introduced first by Visser [@Vi], [@Vi2] and re-emerged in a more philosophically motivated form by Ruitenburg [@Ru2] to address the impredicativity problem of the implication. This implication is morally the Heyting implication without its modus ponens rule; see [@Ard2], [@Ard3], [@BasicPropLogic], [@CJ1]. The emergence of these weak implications then set the scene for a plethora of other and sometimes even weaker implications emerging philosophically [@Ru]; algebraically [@Restall], [@CJ2], [@Aliz1], [@Aliz2], [@Aliz3], [@Aliz4], [@Aliz5], [@Lat]; proof theoretically [@Corsi], [@Dosen], [@Suz], [@Sas]; arithmetically [@Vi3], [@Iem1], [@Iem2] and via relational semantics [@Ard], [@LitViss], almost everywhere in the logical realm. Apart from the philosophically oriented reasons, the weak implications raise also some independent mathematical interests. In their propositional form, they appear in different logical disciplines including provability logic [@Vi] and preservability logic [@Vi3], [@Iem1], [@Iem2], [@LitViss]. In their higher categorical form, they capture some type constructors called arrows by the functional programming community. Arrows were first introduced by Hughes [@Hughes] to encode some natural types of function-like entities that are not really functions. For instance, the type of all partial functions from $A$ to $B$, for the given types $A$ and $B$ is such an arrow type. Categorically speaking, they generalize monads, used elegantly to formalize the computational effects in [@Moggi]. For the categorical formalizations of arrows see [@Jacobs] and for more information on their role in programming and type theory see [@Pat] and [@Lin].\
Coming back to the spatial interpretation, we are facing a question: If the notion of space is powerful enough to formalize constructions, why not using them to also understand implications and exponentials? For this purpose, we have to bring in another important intuitionistic notion, different from the usual constructions. This notion is time. Assume that the mental states encode not only the current knowledge of the mind, but also the relevant temporal data including the actual moment that the mental state occupies in the time line. To encode this temporal structure, we add a temporal modality, $\nabla$, to construct a proposition $\nabla A$ from a proposition $A$, meaning “*$A$ holds at some point in the past*". First note that $\nabla A$ is a proposition itself. Since, if $\nabla A$ holds in a mental state, there is some point in the past in which $A$ holds. But $A$ is a proposition and hence has a finite verification at that point. Therefore, it is easy to bring that verification to the current mental state and save it as some temporal information of the past. Secondly, $\nabla$ is clearly monotone and union preserving. The reason for the latter is the existential nature of $\nabla$. More precisely, if $\nabla (\bigvee_{i \in I} A_i)$ holds at some state, then there exists some point in the past in which $\bigvee_{i \in I} A_i$ holds. Hence, one of $A_i$’s must hold in that point which implies $\nabla A_i$ holds at the current state. The converse is similar and easy. This completes the data we need for the temporal modality.\
Back to the implications, using $\nabla$ as the temporal modality, it is possible to design an implication that brings the temporal structure to the scene. Define the implication by $$A \to_{\nabla} B= \bigcup \{C | \; \nabla C \wedge A \vdash B\}.$$ By this definition and the fact that $\nabla$ preserves all disjunctions, it is not hard to prove $$\nabla C \wedge A \vdash B \;\;\; \text{iff} \;\;\; C \vdash A \to_{\nabla} B,$$ which can be read as a pair of the introduction-elimination rules that defines the implication. These rules state that $A \to_{\nabla} B$ is a consequence of $C$ if the fact that $C$ constructed before plus the truth of $A$ at this moment implies the truth of $B$. Note that the only role that $\nabla$ plays is delaying the implication. Philosophically speaking, it is the machinery to ensure a delay between constructing an implication and using it. For instance, based on the introduction-elimination rules, we know that $\nabla (A \to_{\nabla} B) \wedge A \vdash B$ while there is no reason to have $(A \to_{\nabla} B) \wedge A \vdash B$. The former means that $A \to_{\nabla} B$ holds (constructed) before and hence, at his moment we can argue that in the presence of $A$, we can use the implication to show $B$. While in the latter case, $A \to_{\nabla} B$ is just constructed and it can not be applicable at the moment. Now, identifying the set of propositions by the opens of a topological space, we have a mathematical formalization of the foregoing discussion. It is enough to have a topological space and a monotone and union preserving map $\nabla: \mathcal{O}(X) \to \mathcal{O}(X)$ encoding the temporal modality. Calling such a data a spacetime, we can ensure that all spacetimes have their canonical implications, as defined above. Admittedly, these implications define a special class of all possible implications. However, we will show that any reasonable implication is actually representable by these temporal implications. The advantage of a temporal implication is the full introduction-elimination rules that it possesses. These rules make a natural machinery for internalization and leads to a very well-behaved implication as opposed to the arbitrary selection of structures that an implication may randomly internalize. In sum, our motto is that the study of the notion of time can almost be the study of the notion of implication. In this paper and in its sequel, we intend to follow this motto to investigate the general notion of implication via its incarnations in the above-mentioned spacetimes. Here, we will focus on the algebraic side of the story and leave the full general categorical setting and its categorical spacetimes as the more structured Grothendieck topoi to the forthcoming work.\
The structure of the present paper is as follows. In Section \[Pre\], we will present a rather intense section on preliminaries to make the paper self-contained and hence accessible for a wider range of audience. In Section \[QuantalesAndIntuitionism\] quantales will be presented as the natural generalization of the notion of space. We will also discuss how to capture a more subjective formalization of finitely verifiable propositions in which even observing the truth of a proposition changes the mental state. In Section \[Imp\], we will define an abstract implication as an order internalizing operation. Then in Section \[Non-ComSpacetime\], we will develop a generalized version of spacetimes via quantales as developed in Section \[QuantalesAndIntuitionism\]. Section \[Rep\] is devoted to the representation theorems to show that a considerable class of abstract implications are essentially the implications of the generalized spacetimes. In Section \[LogicsofSpcaeTimes\], we will continue by developing a series of sub-structural logics for spacetimes and we will study their topological semantics. Their Kripke semantics will be introduced in Section \[KripkeModels\]. And finally, in Section \[Sub-int\], we will show how to embed the sub-intuitionistic logics, the logics of weak implications into these more well-behaved logics of spacetime.
Preliminaries {#Pre}
=============
In this section we will review some basic facts and some useful constructions, including the notions of poset, adjunction, the monoidal posets, quantales and some completion techniques. These are very well-known facts and constructions. However, for the sake of completeness and being accessible to a wider range of audience, we prefer to briefly explain some necessary parts here. For more information, see [@StoneSpaces], [@TopViaLogic] and [@Bor3] on locales and completions and [@Ros2] on quantales.
\[DefMonoid\] By a monoid $\mathcal{M}=(M, \otimes, e)$, we mean a set $M$ equipped with a binary multiplication function $\otimes: M \times M \to M$ and an element $e \in M$ such that the multiplication is associative, i.e., for all $m, n, k \in M$ we have $(m \otimes n) \otimes k= m \otimes (n \otimes k)$ and $e$ is the identity element, i.e., for all $m \in M$ we have $e \otimes m=m = m \otimes e$. If $\mathcal{M}=(M, \otimes_M, e_M)$ and $\mathcal{N}=(N, \otimes_N, e_N)$ are two monoids, by a homomorphism $f: \mathcal{M} \to \mathcal{N}$ we mean a structure preserving function $f: M \to N$, i.e., $f(e_M)=e_N$ and for any $m, n \in M$, $f(m \otimes_M n)=f(m) \otimes_N f(n)$.
\[DefPoset\] By a poset we mean a pair $\mathcal{A}=(A, \leq)$, where $A$ is a set and $\leq$ is a reflexive, anti-symmetric and transitive binary relation over $A$. By $\mathcal{A}^{op}$ we mean the opposite poset of $\mathcal{A}$, consisting of $A$ with the opposite order. When there is no risk of confusion, we denote $\mathcal{A}^{op}$ simply by $A^{op}$. By a downset of $\mathcal{A}$, we mean a subset of $A$ that is $\leq$-downward closed, i.e., a subset $S$ such that if $a \leq b$ and $b \in S$, then $a \in S$. By an upset we mean a $\leq$-upward closed subset, i.e., a subset $S$ such that if $a \leq b$ and $a \in S$, then $b \in S$.\
By the join (the meet) of a subset $S \subseteq A$, we mean the greatest lower bound (the least upper bound) of $S$ in $A$, if it exists. We denote it by $\bigvee S$ ($\bigwedge S$). If $S$ has at most two elements $a, b \in A$, we use the notation $a \vee b$ for the join ($a \wedge b$ for the meet) and we denote the join of the empty set by $0$ (the meet of the empty set by $1$). A poset is called join semi-lattice or finitely cocomplete (meet-semilattice or finitely complete) if the join (meet) of all finite subsets of $A$ exist. It is called cocomplete (complete) if the join (meet) of all subsets of $A$ exist. And finally by a map between two posets $\mathcal{A}=(A, \leq_A)$ and $\mathcal{B}=(B, \leq_B)$, denoted by $f: \mathcal{A} \to \mathcal{B}$, we simply mean an order preserving function $f: A \to B$ meaning $f(a) \leq_B f(b)$ for any $a \leq_A b$. An order-preserving map is called an embedding if for any $a, b \in A$, the inequality $f(a) \leq_B f(b)$ implies $a \leq_A b$.
Note that any cocomplete poset is also complete and vice verse. It is easy to see that if $(A, \leq)$ is cocomplete and $S \subseteq A$ then $\bigvee \{x \in A | \forall s \in S \; (x \leq s)\}$ exists and serves as the meet $\bigwedge S$. The converse is similar.
\[DefAdjunction\] Let $\mathcal{A}=(A, \leq_A)$ and $\mathcal{B}=(B, \leq_B)$ be two posets and $f: \mathcal{A} \to \mathcal{B}$ and $g: \mathcal{B} \to \mathcal{A}$ be two maps. The map $f$ is called a left adjoint for $g$ (or equivalently $g$ is a right adjoint for $f$), if for all $a \in A$ and $b \in B$, $$f(a) \leq_B b \;\;\;\; \text{iff} \;\; \; \; a \leq_A g(b)$$ In such situation the pair $(f, g)$ is called an adjunction and it is denoted by $f \dashv g: \mathcal{B} \to \mathcal{A}$ or simply $f \dashv g$.
\[SemiInverse\] Note that given $f \dashv g: \mathcal{B} \to \mathcal{A}$, we have $fg(b) \leq_B b$, for all $b \in B$ because $g(b) \leq_B g(b)$. Similarly, $a \leq_A gf(a)$, for all $a \in A$. Moreover, $fgf=f$ since, for any $a$, $a \leq_A gf(a)$ from which $f(a) \leq_B fgf(a)$ and $fg(f(a)) \leq_B f(a)$. Similarly, $gfg=g$.
(Adjoint Functor Theorem for Posets)\[AFT\] Let $\mathcal{A}=(A, \leq_A)$ be a complete poset and $\mathcal{B}=(B, \leq_B)$ be a poset. Then an order preserving map $f: \mathcal{A} \to \mathcal{B}$ has a right (left) adjoint iff it preserves all joins (meets).
See [@Bor1].
\[DefMonoidalPoset\] A monoidal poset is a structure $\mathcal{A}=(A, \leq, \otimes, e)$ where $(A, \leq)$ is a poset and $(A, \otimes, e)$ is a monoid whose multiplication is compatible with the order, i.e., $\otimes$ is order-preserving in each of its arguments. A monoidal poset is called distributive if its poset is a join-semilattice and its multiplication distributes over all finite joins in each of its arguments.
\[MonoidalMap\] Let $\mathcal{A}=(A, \leq_A, \otimes_A, e_A)$ and $\mathcal{B}=(B, \leq_B, \otimes_B, e_B)$ be two monoidal posets. By a lax monoidal map $f: \mathcal{A} \to \mathcal{B}$ we mean an order preserving function $f: A \to B$ such that $f(e_A) \geq e_B$ and for any $a, b \in A$ we have $f(a \otimes_A b) \geq f(a) \otimes_B f(b)$. A map is called oplax monoidal if it is order preserving and the last two inequalities are in the reverse order, i.e., $f(e_A) \leq e_B$ and for any $a, b \in A$ we have $f(a \otimes_A b) \leq f(a) \otimes_B f(b)$. A map is called strict monoidal if it is both lax monoidal and oplax monoidal.
\[MonoidalAdjoint\] Let $\mathcal{A}=(A, \leq_A, \otimes_A, e_A)$ and $\mathcal{B}=(B, \leq_B, \otimes_B, e_B)$ be two monoidal posets, $f: \mathcal{A} \to \mathcal{B}$ be an oplax monoidal (lax monoidal) map and $g:\mathcal{B} \to \mathcal{A}$ be its right (left) adjoint. Then $g$ is lax monoidal (oplax monoidal).
We prove the case when $f$ is oplax and $f \dashv g$. The other case is similar. Since $f$ is oplax we have $f(e_A) \leq_B e_B$ from which and by using the adjunction we have $e_A \leq_A g(e_B)$. For the other condition, note that by the Remark \[SemiInverse\], the adjunction implies $f(g(a)) \leq_B a$ and $f(g(b)) \leq_B b$. By the fact that $f$ is oplax, we have $$f (g(a) \otimes g(b)) \leq_B f(g(a)) \otimes f(g(b)) \leq_B a \otimes b$$ and by the adjunction again, we have $g(a) \otimes g(a) \leq_B g (a \otimes b)$, which completes the proof.
\[Quantale\] A monoidal poset $\mathscr{X}$ is called a quantale if its order is cocomplete and its multiplication distributes over all joins on both sides. A quantale is a locale if its monoidal structure is the meet structure of the poset. In other words, a locale is a cocomplete poset whose meet distributes over all of its joins.
Note that quantales are also complete. This provides the enough structure to interpret conjunctions in a quantale, as we will see later.
Here are some prototypical examples of locales and quantales that we will need in the future.
Let $\mathcal{S}=(S, \leq)$ be a cocomplete poset and define $X$ as the set of all join preserving functions $f: \mathcal{S} \to \mathcal{S}$ with the pointwise order $\leq_X$. Then $\mathscr{X}=(X, \leq_X, \circ, id)$ is a quantale where $\circ$ is the usual composition and $id: \mathcal{S} \to \mathcal{S}$ is the identity map.
Let $X$ be a set and $\mathcal{R}$ be a set of binary relations over $X$ that includes the equality and is closed under composition and arbitrary union. Then $\mathscr{X}=(\mathcal{R}, \subseteq, \circ, =)$ is a quantale where $\circ$ is the relation composition.
Let $X$ be a topological space. Then $\mathscr{X}=(\mathcal{O}(X), \subseteq, \cap, X)$ is a locale where $\mathcal{O}(X)$ is the set of all open subsets of $X$.
Let $\mathcal{M}=(M, \otimes, e)$ be a monoid. Consider $I(\mathcal{M})$ as the set of all left ideals of $\mathcal{M}$, i.e., the subsets of $M$ closed under arbitrary left multiplication. Then $(I(\mathcal{M}), \subseteq, \cdot, M)$ is a quantale where $$I \cdot J=\{ij | i \in I, j \in J \}$$ The reason is that the union of any set of ideals is an ideal again and the multiplication clearly distributes over the union.
Note that if $\mathscr{X}$ is a quantale, then for any fixed $a \in \mathscr{X}$, the functions $l_a, r_a : \mathscr{X} \to \mathscr{X}$ mapping $x$ into $a \otimes x$ and $x \otimes a$, respectively, preserve all joins and since the poset is cocomplete, by the adjoint functor theorem, Theorem \[AFT\], they both have right adjoints. Because of some technical reasons, we are only interested in $l_a$. Therefore, it will be useful to have a name and a notation for $l_a$’s right adjoint. We denote it by $a \Rightarrow (-)$ and we call the binary operator $\Rightarrow$, *the canonical implication* of the qunatale $\mathscr{X}$. Spelling out the adjunction conditions, it means that for any $a, b, c \in \mathscr{X}$, we have $a \otimes b \leq c$ iff $b \leq a \Rightarrow c$. Note that if $\mathscr{X}$ is a locale, its canonical implication is just the usual Heyting implication of $\mathscr{X}$.
\[GeometricMap\] Let $\mathscr{X}, \mathscr{Y}$ be two quantales. Then by a lax/oplax/strict geometric morphism $f: \mathscr{X} \to \mathscr{Y}$, we mean a lax/oplax/strict monoidal join preserving map $f: \mathscr{X} \to \mathscr{Y}$.
Let $X$ and $Y$ be two topological spaces, $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ be the poset of all open subsets of $X$ and $Y$, respectively and $f: X \to Y$ be a continuous function. Then $f^{-1}: \mathcal{O}(Y) \to \mathcal{O}(X)$ is a strict geometric morphism.
It is worth mentioning that over locales, any join preserving map $f: \mathscr{X} \to \mathscr{X}$ is an oplax geometric morphism because $f$ is order preserving which implies $f(a \wedge b) \leq f(a) \wedge f(b)$.
\[LiftingFromSetsToQuantales\] Let $X$ and $Y$ be two sets and $f: X \to Y$ be a function. Then $f$ induces a lax geometric morphism $f^*: P(Y \times Y) \to P(X \times X)$ by $f^*(R)=f^{-1}(R)$. It is clearly union preserving. Moreover, for any two relations $R, S \subseteq Y \times Y$, we have $f^{-1}(R) \circ f^{-1}(S) \subseteq f^{-1}(R \circ S)$, because, if $(x, y) \in f^{-1}(R) \circ f^{-1}(S)$ then there is $z \in X$ such that $(x, z) \in f^{-1}(S)$ and $(z, y) \in f^{-1}(R)$. Therefore, $(f(x), f(z)) \in S$ and $(f(z), f(y)) \in R$ which implies $(f(x), f(y)) \in R \circ S$ from which $(x, y) \in f^{-1}(R \circ S)$.\
The function $f$ also induces an oplax geometric morphism. Define $f_*: P(X \times X) \to P(Y \times Y)$ by $f_*(R)=f[R]$ as the image of $R$. This is also union preserving. Moreover, we have $f(R \circ S) \subseteq f(R) \circ f(S)$, because, if $(x, y) \in f(R \circ S)$ then there is $x', y', z' \in X$ such that $x=f(x')$, $y=f(y')$, $(x', z') \in S$ and $(z', y') \in R$. Therefore, $(f(x'), f(z')) \in f_*(S)$ and $(f(z'), f(y')) \in f_*(R)$. Hence, $(x, y)=(f(x'), f(y')) \in f_*(S) \circ f_*(R)$.
\[LiftingFromMonToQuantales\] Let $\mathcal{M}=(M, \otimes_M, e_M)$ and $\mathcal{N}=(N, \otimes_N, e_N)$ be two monoids and $f: \mathcal{M} \to \mathcal{N}$ be a homomorphism. Then $f$ induces a lax geometric morphism $f^*: I(\mathcal{N}) \to I(\mathcal{M})$ by $f^*(I)=f^{-1}(I)$. It is clearly union preserving. Moreover, we have $f^*(I) f^*(J) \subseteq f^*(IJ)$ because if $x \in f^*(I) f^*(J)$ then there are $y \in f^*(I)$ and $z \in f^*(J)$ such that $x=yz$. Since $f$ is a homomorphism we have $f(x)=f(y)f(z) \in IJ$. Therefore, $x \in f^*(IJ)$. The homomorphism $f$ also induces an oplax geometric morphism defined by $f_*: I(\mathcal{M}) \to I(\mathcal{N})$ by $f_*(I)=Mf[I]$, where $f[I]$ is the image of $I$ and $Mf[I]$ is the generated left ideal of the image of $I$. This map clearly preserves union. Moreover, $f_*(IJ) \subseteq f_*(I) f_*(J)$, because if $x \in f_*(IJ)$, then there are $m \in M$, $i \in I$ and $j \in J$ such that $x=mf(ij)$. Since $f$ is a homomorphism we have $x=mf(i)f(j)=(mf(i))f(j) \in f_*(I) f_*(J)$.
In the rest of this section, we will recall some of the main completion techniques for the monoidal posets. We will address the details of constructions as we need them later in some other constructions of the paper.
\[Completions\](Downset and Ideal Completions) Let $\mathcal{A}=(A, \leq, \otimes, e)$ be a monoidal poset. Then there exists a quantale $D(\mathcal{A})$, called the downset completion of $\mathcal{A}$ and a strict monoidal embedding $i: \mathcal{A} \to D(\mathcal{A})$. If $\mathcal{A}$ has all finite joins and distributive, then there exists another quantale $I(\mathcal{A})$, called the ideal completion of $\mathcal{A}$ and a finite join-preserving strict monoidal embedding $i: \mathcal{A} \to I(\mathcal{A})$. If $\mathcal{A}$ has all finite meets, then in both cases $i$ preserves all finite meets.
First let us explain the downset completion that works for monoidal posets that do not necessarily have the join structure. Later we will also address the joins and the distributive case. Define $\mathscr{X}=D(\mathcal{A})$ as the set of all downsets of $A$ with the inclusion as its order. Since downsets are closed under arbitrary union and intersection, they are the joins and the meets of the poset, respectively. Define the map $i: A \to \mathscr{X}$ by $i(a)=\{x \in A | x \leq a\}$ and the monoidal structure of $\mathscr{X}$ by $e_{\mathscr{X}}=i(e)$ and $$I \otimes_{\mathscr{X}} J = \{x \in A | \; \exists i \in I \exists j \in J \; (x \leq i \otimes j)\},$$ for any downsets $I$ and $J$. Note that $I \otimes_{\mathscr{X}} J $ is also a downset. Moreover, it is not hard to prove that this multiplication is associative with the identity element $e_{\mathscr{X}}$ and it distributes over all unions. Therefore, $(\mathscr{X}, \otimes_{\mathscr{X}}, e_{\mathscr{X}})$ is actually a quantale. Moreover, $i$ is a strict monoidal map because by definition, $e_{\mathscr{X}}=i(e)$ and $$i(a) \otimes_{\mathscr{X}} i(b)=\{x \in A| \exists i \leq a \exists j \in b \; (x \leq i \otimes j)\} = \{x \in A | x \leq a \otimes b\}.$$ Finally, note that $i$ is clearly an embedding, because, $$i(a) \subseteq i(b) \;\;\; \text{iff} \;\;\; \{x \in A | x \leq a\} \subseteq \{x \in A | x \leq b\} \;\;\; \text{iff} \;\;\; a \leq b,$$ and if $\mathcal{A}$ has all finite meets, $i$ preserves them because, $i(1)=\{x \in A | x \leq 1\}=A$ and $$x \in i(a) \cap i(b) \;\;\; \text{iff} \;\;\; (x \in a \;\; \text{and} \;\; x \in b) \;\;\; \text{iff} \;\;\; x \leq a \wedge b \;\;\; \text{iff} \;\;\; x \in i(a \wedge b),$$ which implies $i(a) \cap i(b)=i(a \wedge b)$.\
Now, let us move to the distributive case, where $\mathcal{A}=(A, \leq, \otimes, e)$ has all finite joins. Then the foregoing function $i$ does not necessarily preserve the join structure of $\mathcal{A}$. To handle this issue, we have to change $\mathscr{X}$ a little bit: Define $\mathscr{Y}=I(\mathcal{A})$ as the poset of all ideals of $A$, i.e., all downsets $I \subseteq A$ such that $0 \in I$ and $a \vee b \in I$, for any $a, b \in I$. We want to show that $\mathscr{Y}$ with the join $$\bigvee_{i \in N} I_i= \{x \in A | \exists x_1, \ldots x_n \in \bigcup_{i \in N} I_i \; (x \leq \bigvee_{j=1}^n x_j) \}$$ and the same monoidal structure as of $\mathscr{X}$’s is a quantale and the previous function $i$ is again an embedding that also preserves all finite joins. First, it is not hard to prove that $\bigvee$ maps ideals to ideals and is actually the join of the family $\{I_i\}_{i \in N}$ in the inclusion order over ideals. Secondly, note that the original $i: A \to \mathscr{X}$ actually lands into the set of ideals $\mathscr{Y}$, because, $\{x \in A | x \leq a\}$ is closed under all finite joins. Note also that $i$ preserves all finite joins because, $$i(a) \vee i(b)=\{x \in A| \exists i \leq a \exists j \leq b \; (x \leq i \vee j)\} = \{x \in A | x \leq a \vee b\}.$$ Since the intersection of ideals is also an ideal, the meet structure for ideals is also the intersection. Hence, the same argument for meet preservation by $i$ works here, as well. Thirdly, note that the defined $\otimes$ on $\mathscr{X}$ maps ideal to ideals, meaning that if $I$ and $J$ are ideals then so is $I \otimes J$. To prove this claim, first note that $0 \otimes 0 \leq 0 \otimes e=0$ from which $0 \otimes 0=0$ and hence $0 \in I \otimes J$. Secondly, assume that $x, y \in I \otimes J$. We want to show that $x \vee y \in I \otimes J$. By definition, there exist $i, i' \in I$ and $j, j' \in J$ such that $x \leq i \otimes j$ and $y \leq i' \otimes j'$. By monotonicity of $\otimes$ we have $x \leq (i \vee i') \otimes (j\vee j')$ and $y \leq (i \vee i') \otimes (j \vee j')$ and hence $ x \vee y \leq [(i \vee i') \otimes (j \vee j')]$. Since both $I$ and $J$ are closed under finite joins, $i \vee i' \in I$ and $j \vee j' \in J$ and hence, $x \vee y \in I \otimes J$.\
\
Finally, we show that the multiplication distributes over joins, i.e., $$\bigvee_{n \in N} (I_n \otimes J)=(\bigvee_{n \in N} I_n) \otimes J
\;\;\;\; \text{and} \;\;\;\;
I \otimes (\bigvee_{n \in N} J_n)=\bigvee_{n \in N} (I \otimes J_n).$$ We will prove the left equality. The right one is similar. There are two directions to prove. $\bigvee_{n \in N} (I_n \otimes J) \subseteq (\bigvee_{n \in N} I_n) \otimes J $ is clear by monotonicity. For the other direction, assume $x \in (\bigvee_{n \in N} I_n) \otimes J$. By definition, there exist $y \in \bigvee_{n \in N} I_n$ and $j \in J$ such that $x \leq y \otimes j$. Again by definition, there exist $i_1, i_2, \ldots, i_k \in \bigcup_{n \in N} I_n$ such that $y \leq i_1 \vee \ldots \vee i_k$. By distributivity, we have $$x \leq (i_1 \otimes j) \vee (i_2 \otimes j) \vee \ldots \vee (i_k \otimes j).$$ But since each $i_r$ is in at least one $I_{m_r}$, we have $$i_r \otimes j \in (I_{m_r} \otimes J) \subseteq \bigvee_{n \in N} (I_n \otimes J).$$ Since $\bigvee_{n \in N} (I_n \otimes J)$ is closed under finite joins, we have $x \in \bigvee_{n \in N} (I_n \otimes J)$.
\[2\] Note that in the both downset and ideal completions, if the monoidal structure of $\mathcal{A}$ is just the meet structure, i.e., $\otimes=\wedge$ and $e=1$, then $\otimes_{\mathscr{X}}$ is the intersection because $$I \otimes_{\mathscr{X}} J=\{x \in A| \; \exists i \in I \exists j \in J \; (x \leq i \wedge j)\}= I \cap J,$$ which is the meet of $\mathscr{X}$ and also $e_{\mathscr{X}}$ is $i(1)$ which is the top element $1_{\mathscr{X}}=i(1)=A$.
(Lifting Monoidal Maps)\[LiftingMonoidalMaps\] Let $\mathcal{A}=(A, \leq_A, \otimes_A, e_A)$ and $\mathcal{B}=(B, \leq_B, \otimes_B, e_B)$ be two monoidal posets and $f: \mathcal{A} \to \mathcal{B}$ be a lax (oplax) monoidal map. Then there exists a lax (oplax) geometric map $f_!: D(\mathcal{A}) \to D(\mathcal{B})$ such that $f_! i_A=i_Bf$, where $i_A$ and $i_B$ are the canonical embeddings of the downset completions of $\mathcal{A}$ and $\mathcal{B}$, respectively. Moreover, if both $\mathcal{A}$ and $\mathcal{B}$ have all finite joins and are distributive, and if $f: \mathcal{A} \to \mathcal{B}$ is finite join preserving, then the same holds for some map $f_!: I(\mathcal{A}) \to I(\mathcal{B})$.
First let us prove the downset case. We will address the ideal case later. Define $$f_!(I)=\{x \in A | \; \exists i \in I \; (x \leq_B f(i))\}.$$ This set is clearly a downset, hence $f_!$ is well-defined. Moreover, note that $$f_!(i_A(a))=\{x \in A | \; \exists i \leq_A a \; (x \leq_B f(i))\}=\{x \in A | (x \leq_B f(a))\}=i_B(f(a)).$$ The map $f_!$ obviously preserves all unions. We have to prove that if $f$ is lax (oplax), then so is $f_!$. Assume $f$ is lax monoidal. The other case is similar. We have to prove that $i_B(e_B) \subseteq f_!(i_A(e_A))$ and $f_!(I) \otimes f_!(J) \subseteq f_!(I \otimes J)$, for any downsets $I$ and $J$ of $\mathcal{A}$. For the first, assume $x \in i_B(e_B)$, then $x \leq_B e_B \leq_B f(e_A)$. Hence, $x \in f_!(i_A(e_A))$. For the second, if $x \in f_!(I) \otimes f_!(J)$, then there are $y \in f_!(I)$ and $z \in f_!(J)$ such that $x \leq y \otimes z$. Since $y \in f_!(I)$ and $z \in f_!(J)$ there are $i \in I$ and $j \in J$ such that $y \leq f(i)$ and $z \leq f(j)$. Hence, $x \leq f(i) \otimes f(j) \leq f(i \otimes j)$, which implies $x \in f_!(I \otimes J)$.\
For the ideal completion case, we define the same $f_!$. However, we have to check whether it is ideal and join preserving. It is an ideal because, $0 \leq f(0)$ and since $0 \in I$ we have $0 \in f_!(I)$. Moreover, if $x, y \in f_!(I)$ then there are $i, j \in I$ such that $x \leq f(i)$ and $y \leq f(j)$. Since $f$ is monotone, we have $x \vee y \leq f(i \vee j)$. Since $I$ is an ideal we have $i \vee j \in I$ and hence $x \vee y \in f_!(I)$. Furthermore, we have to check that $f_!$ is join preserving. For that matter, we have to show $f_!(\bigvee_{n \in N} I_n) = \bigvee_{n \in N} f_!(I_n)$. From right to left is easy by monotonicity of $f_!$. For the left to right, assume $x \in f_!(\bigvee_{n \in N} I_n)$. Hence, there are $i_1, \ldots, i_k \in \bigcup_{n \in N} I_n$ such that $x \leq f(i_1 \vee \ldots \vee i_k)$. Since $f$ is join preserving we have $x \leq f(i_1) \vee \ldots \vee f(i_k)$ which implies that $x \in \bigvee_{n \in N} f_!(I_n)$.
**Upset and Filter Completions.** Using two ideal completions in an appropriate way leads to a very useful construction that we call the upset construction. The details follow. Let $\mathcal{A}=(A, \leq, \otimes, e)$ be a monoidal poset and denote the downset qunatale of $\mathcal{A}$ by $D(\mathcal{A})$ and the opposite of $\mathcal{A}$, the same structure with the reverse order, by $\mathcal{A}^{op}$. Then by the downset completion for $\mathcal{A}^{op}$, there exists a strict monoidal embedding $i : \mathcal{A}^{op} \to D(\mathcal{A}^{op})$ or equivalently $i: \mathcal{A} \to D(\mathcal{A}^{op})^{op}$. It is useful to observe that $D(\mathcal{A}^{op})$ is nothing but the poset of all upsets of $\mathcal{A}$ with the multiplication: $$P \otimes Q=\{x \in A | \exists y \in P \exists z \in Q \; (x \geq y \otimes z)\}.$$ Denote this poset by $U(\mathcal{A})$. Now we use the same operation again to embed $D(\mathcal{A}^{op})^{op}$ into $D(D(\mathcal{A}^{op})^{op})$. Combining these two embeddings, we reach a strict monoidal embedding of $\mathcal{A}$ into $D(D(\mathcal{A}^{op})^{op})$ which we call the upset completion of $\mathcal{A}$. Spelling out the construction of the upset completion, the set consists of all the upsets of the upsets of $\mathcal{A}$ with the inclusion as its order and the following multiplication for any upsets of upsets $X$ and $Y$: $$X \otimes Y=\{P \in U(\mathcal{A}) | \exists Q \in X \exists R \in Y \; (P \supseteq Q \otimes R)\}.$$ Moreover, the embedding is simply expressible by $i(a)=\{P \in U(\mathcal{A}) | a \in P\}$.\
In the case that the monoidal poset is a meet semi-lattice $\mathcal{A}=(A, \leq, \wedge, 1)$, there is another construction that is called the canonical construction $\mathcal{C}(\mathcal{A})$ and an embedding $i: \mathcal{A} \to \mathcal{C}(\mathcal{A})$ that respects all finite meets. A non-empty upset of $A$ is called a filter if it is closed under all finite meets. Denote the class of all filters of $\mathcal{A}$ by $F(\mathcal{A})$ and then define $ \mathcal{C}(\mathcal{A})$ as the poset of all upsets of filters and use the same $i$ as defined before. The embedding $i: \mathcal{A} \to \mathcal{C}(\mathcal{A})$ preserves all finite meets. First note that all filters include $1$, thus $$i(1)=\{P \in F(\mathcal{A}) | 1 \in P\}=F(\mathcal{A}).$$ Secondly, note that the filters are closed under meets. Hence, $$i(a \wedge b)=\{P \in F(\mathcal{A}) | a \wedge b \in P\}=\{P \in F(\mathcal{A})| a \in P \; \text{and} \; b \in P\}=i(a) \cap i(b).$$ In case $\mathcal{A}$ has all finite joins and it is distributive, it is also possible to change the canonical construction so that $i$ also preserves the finite joins. The construction is as follows: A filter is called prime if it is proper and for any $a, b \in A$, the assumption $a \vee b \in P$ implies either $a \in P$ or $b \in P$. Denote the set of all prime filters by $P(\mathcal{A})$. If we change $C(\mathcal{A})$ to the poset of all upsets of $P(\mathcal{A})$ with the same $i$, then $i$ preserves both finite joins and finite meets. The reasoning for the meet is the same as before. For the joins, since prime filters are proper, we have $0 \notin P$, which implies $ i(0)=\{P \in P(\mathcal{A}) | 0 \in P\}=\emptyset$ and $$i(a \vee b)=\{P \in P(\mathcal{A}) | a \vee b \in P\}=\{P \in P(\mathcal{A})| a \in P \; \text{or} \; b \in P\}=i(a) \cup i(b).$$
Intuitionism via Quantales {#QuantalesAndIntuitionism}
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In the Introduction, we have seen that any finitely verifiable proposition can be interpreted as an open subset of a topological space. In this interpretation, the corresponding open subset captures the set of all the mental states in which the proposition actually holds. More operationally, a finitely verifiable proposition $A$ is just an *observation* that reads a mental state and finds the truth value of $A$ in that state, in the same way that a physical quantity like the speed or the temperature can be seen as an observation that reads a physical state to find the value of the quantity.\
Reading propositions as observations suggests that we silently believe in some sort of an independent objective mind. Let us assume that the creative subject observes her mental state to check the validity of a proposition. It seems that this introspection only observes a mental state and extracts some needed information from it but it does not affect the mental state at all. The situation is similar to the classical assumption that the physical observations do not affect the physical phenomenon that they are observing. It measures a quantity ideally without distorting the picture or interfering with any other observation. This may be the case when we interpret the knowledge content of a mental state as a set of propositions and the validity of a proposition as its provability. Then it is just a real factual situation and it is not important what, when and in what order we are observing the validity of the propositions. However, it is totally possible to imagine a more subjective, more dynamic and more interactive formalization of knowledge. One possible scenario to show how natural such a situation could be is the following: Interpret the knowledge content of a mental state as a set of propositions as before but change the validity from provability to immediate provability. It means that a valid proposition is either in the set or provable in one step via some given proving methods from the set. Observing a proposition in this scenario clearly affects the mental state. If a proposition holds in a mental state, it is provable in at most one step. Then since the creative subject thinks about the proposition and finds out the proof, it is totally reasonable to assume that she then modifies her knowledge to add this new proposition to the set she had before. The observation process is also interactive. In each step, there could be many one-step provable propositions and hence it could be important to choose which way she wants to proceed. This choice may change her path forever. It is also non-commutative because $A$ may be immediately provable and its presence makes $B$ also immediately provable while the proposition $B$ is not immediately provable without using $A$. Hence, proving $A$ after $B$ may not be even possible.\
This is only one possible scenario. Now let us find a more formal way to express not only this scenario but its essential dynamic, interactive and non-commutative nature. We will begin by a toy example to be prepared to find the algebraic abstract formalization later. Let $S$ be the set of all the mental states and identify a proposition not by a subset of $S$ but by a binary relation $A \subseteq S \times S$ that includes $(s, t)$ if the proposition $A$ holds in the state $s$, its truth is verifiable in a finite number of steps and this verification changes the mental state $s$ to $t$. Using this example, we can also identify the previous static interpretation of knowledge as the non-state-changing relations, i.e., the relations like $A$ with the property that if $(s, t) \in A$ then $s=t$. These $A$’s are simply identifiable by the subset $\{s \in S | (s, s) \in A\}$ of the mental states where they are valid. This is simply our previous proposition-as-subset formalization.\
To formalize the calculus of this new interpretation of finitely verifiable propositions, we try to provide an algebraic axiomatization reflecting the main intuitive properties of this toy example. Here again we have three main structures. The first obvious structure is the order between the propositions encoding how a proposition implies another one. This order in our toy example is the inclusion order between the binary relations. Secondly, propositions has a natural notion of composition. Philosophically speaking, for any two propositions $A$ and $B$, we can imagine $A \otimes B$ as the composition of observations, first applying $B$ and then $A$. $A \otimes B$ changes the state $s$ to $t$ if $B$ holds in $s$ and maps $s$ to some state $r$ where $A$ holds. This $r$ then must be mapped to $t$ by $A$. In our toy example composition is simply the composition of relations. Note that this composition is clearly associative and has an identity element. The identity element is simply the do-nothing observation. In our toy example it is the equality relation over $S$. Moreover, note that in the static interpretation of propositions when $A$ and $B$ are encoded by subsets $\{s \in S | (s, s) \in A\}$ and $\{s \in S | (s, s) \in B\}$, their composition $A \otimes B$ will be $\{s \in S | (s, s) \in A\} \cap \{s \in S | (s, s) \in B\}$ which is nothing but the intersection. This shows how this dynamic approach really generalizes the static topological interpretation of the Introduction.\
Finally, let us address the finiteness condition. Note that the poset of propositions is cocomplete as we explained in the Introduction, simply because for any set $I$, if all $A_i$’s are all finitely verifiable, then their disjunction $\bigvee_{i \in I} A_i $ is also finitely verifiable. The main point is that for verifying a disjunction it is enough to verify one of them. How does a disjunction act on the states? It just combines the actions of all $A_i$’s, since observing the validity of $\bigvee_{i \in I} A_i $ is just observing one of $A_i$’s and hence it changes a states $s$ to one of the states that one of $A_i$’s may dictate. The disjunction in our toy example is just the union of relations. Moreover, note that the composition distributes over all joins because doing the observation $B$ after “at least one of $A_i$’s" is nothing different than doing one of “$A_i$’s before $B$". The same also goes for the other argument of the multiplication. Therefore, to make a calculus for finitely verifiable propositions in its dynamic interactive sense, we need a cocomplete monoidal poset whose multiplication distributes over all joins on both sides. This is nothing but a quantale. Note that if we collapse the monoidal structure to the meet structure as in the non-state-changing-observation interpretation dictates, then the quantale turns into a locale, the point-free version of a topological space. Interpreting locales as the calculus of non-state-changing observations were developed by Abramsky in [@Ab0] and [@Ab1] and Vickers in [@TopViaLogic]. This generalization to quantales has its roots even in [@Mul] where quantales first appeared to provide an algebraic formalization for non-commutative $C^*$-algebras. However, in its explicit form, the state-changing interpretation is developed in [@Ab] and has been important in the connection between the quantales and their categorical monoidal versions on the one hand and the formalization of processes and observations in computer science and quantum physics on the other.
Abstract Implications {#Imp}
=====================
Philosophically speaking, an implication is a conditional proposition internalizing the provability order of the poset of all propositions. Traditionally, the internalization has been implemented via Heyting implications or in a more general setting of monoidal posets via residuations for right and left multiplications. We argue that this tradition is far more restricting than what a basic internalization task demands. As we have seen already in Introduction, internalizations can take place in many different levels to internalize many different structures. For instance, if we have a meet-semilatice, the implication may internalize the basic structures of reflexivity and transitivity via the axioms $a \to a=1$ and $$(a \to b) \wedge (b \to c) \leq (a \to c),$$ or it can go one step further to also internalize the finite meet structure via $$a \to (b \wedge c)=(a \to b) \wedge (a \to c),$$ or in the case that the meet-semilatice has all finite joins, the join structure via $$(a \vee b) \to c=(a \to c) \wedge (b \to c).$$ We propose that the minimum reasonable conditions for any internalization is the inernalization of reflexivity of the order and its transitivity. However, it does not need to be over meet-semilattices. We can use a more general setting where we only have a monoidal poset:
\[DefImplication\] Let $\mathcal{A}=(A, \leq, \otimes, e)$ be a monoidal poset. By an implication $\to : A^{op} \times A \rightarrow A$, we mean any order preserving map with the following properties:
- $e \leq a \to a$,
- $(a \to b) \otimes (b \to c) \leq (a \to c)$,
The structure $\mathcal{A}=(A, \leq, \otimes, e, \to)$ is called a strong algebra if $\to$ is an implication. And if $\mathcal{A}=(A, \leq_A, \otimes_A, e_A, \to_A)$ and $\mathcal{B}=(B, \leq_B, \otimes_B, e_B, \to_B)$ are two strong algebras, by a strong algebra morphism we mean a strict monoidal map $f: A \to B$ that also preserves $\to$, i.e., $f((-) \to_A (-))=f(-) \to_B f(-)$.
Based on the order preservability of the implications in their second arguments, it is possible to strengthen the axiom $(i)$ by the following more general axiom: $(i')$: If $a \leq b$ then $e \leq a \to b$.
Different versions of strong algebras are defined in the literature under many different names. Usually, the definitions use lattices and the meet structure as the monoidal structure, i.e., $\otimes = \wedge$ and $e=1$. They also start with relatively more internalization axioms, including the internalization of finite meets and finite joins, as mentioned above. These algebras are the natural algebraic models for sub-intuitionistic logics. See for instance [@Restall], [@CJ2], [@Lat], [@Aliz1], [@Aliz2], [@Aliz3] and [@Aliz4] for the algebraic notions and [@Ard2], [@BasicPropLogic], [@Aliz5] and [@Restall] for their role in sub-intuitionistic logics.
By a left residuated algebra we mean a monoidal poset $\mathcal{A}=(A, \leq, \otimes, e)$ with a binary operation $\Rightarrow$ such that $x \otimes y \leq z$ is equivalent to $y \leq x \Rightarrow z$, for all $x, y, z \in A$. More specifically, a finitely complete and finitely cocomplete left residuated algebra with the meet structure as its monoidal structure is called a Heyting algebra. Spelling out, a Heyting algebra is a finitely complete and finitely cocomplete poset $\mathcal{H}=(H, \leq, \wedge, \vee ,1, 0)$ with a binary operation $\Rightarrow$ such that $x \wedge y \leq z$ is equivalent to $y \leq x \Rightarrow z$, for all $x, y, z \in H$. It is clear that $\Rightarrow$ in any left residuated algebra is an implication. Note that if $\mathscr{X}$ is a quantale, then $(\mathscr{X}, \Rightarrow_{\mathscr{X}})$ is a left residuated algebra where $\Rightarrow_{\mathscr{X}}$ is the canonical implication of $\mathscr{X}$. Therefore, $\Rightarrow_{\mathscr{X}}$ is also an implication.
### Constructing New Implications from the Old {#Constructing .unnumbered}
There are some simple methods to make new implications from the old. Two of these methods play an important role in our future investigations. Here we will explain them. See also [@Lat].\
*The First Method*. For the first method, let $\mathcal{A}=(A, \leq, \otimes, e, \to)$ be a strong algebra and $F: A \to A$ be a monotone function (not necessarily lax or oplax). Then $\mathcal{A}=(A, \leq, \otimes, e, \to_F)$ where $a \to_F b=F(a) \to F(b)$ is a strong algebra. The reason is that $F(a) \leq F(a)$ and since $\to$ is an implication, we have $e \leq F(a) \to F(a)$. The other axiom is trivial, because $$(F(a) \to F(b)) \otimes (F(b) \to F(c)) \leq (F(a) \to F(c))$$
*The Second Method*. Let $\mathcal{A}=(A, \leq, \otimes, e, \to)$ be a strong algebra and let $G: A \to A$ be a lax monoidal map. Then the structure $\mathcal{A}=(A, \leq, \otimes, e, \to_G)$ where $a \to_G b=G(a \to b)$ is also a strong algebra. The reason is the following. Since $\to$ is an implication, then $e \leq a \to a$. Since $G$ is monotone $G(e) \leq G(a \to a)$. Since $G$ is lax we have $e \leq G(e)$ which implies $e \leq G(a \to a)$. For the second axiom, since $G$ is lax and $\to$ is an implication, we have $$G(a \to b) \otimes G(b \to c) \leq G((a \to b) \otimes (b \to c)) \leq G(a \to c)$$ Later in Section \[Rep\], we will prove some representation theorems to show that any implication is essentially the result of applying these two methods on the canonical implication of a quantale.
Let $\mathcal{H}=(H, \leq, \wedge, \vee ,1, 0, \Rightarrow)$ be a Heyting algebra. Then for some $a \in H$, consider $C_a: H \to H$ as the constant function with the value $a$ i.e., $C_a(x)=a$; $M_a: H \to H$ as $M_a(x)=a \wedge x$ and $J_a: H \to H$ as $J_a(x)=a \vee x$. Then since $C_a$ is meet-preserving and $M_a$ and $J_a$ are monotone, all the following operations are implications: $[x \to_{C_a} y=a]$, $[x \to_{M_a} y=(x \wedge a \Rightarrow y \wedge a)]$ and $[x \to_{J_a} y=(x \vee a \Rightarrow y \vee a)]$.
Let $X$ be a topological space, $f: X \to X$ be a continuous function and $\mathcal{O}(X)$ be the locale of all open subsets of $X$. Since $f^{-1}: \mathcal{O}(X) \to \mathcal{O}(X)$ preserves all unions, by the adjoint functor theorem, Theorem \[AFT\], it has a right adjoint. Call it $g: \mathcal{O}(X) \to \mathcal{O}(X)$. Since $g$ is a right adjoint, it preserves all meets. Therefore, it is lax monoidal. Therefore, the operation $U \to V=g(U \Rightarrow V)$, where $\Rightarrow$ is the Heyting implication on $\mathcal{O}(X)$ is an implication by the second construction.
Let $\mathcal{A}=(A, \leq, \otimes, e, \to)$ be a strong algebra. It internalizes its monoidal structure if for all $a, b, c \in A$: $$a \to b \leq c \otimes a \to c \otimes b$$ $\mathcal{A}$ is called closed if it has the left residuation, i.e., the operation $\Rightarrow$ such that $a \otimes b \leq c$ iff $b \leq a \Rightarrow c$, for any $a, b, c \in A$. A strong algebra internalizes the closed monoidal structure if it is closed, it internalizes the monoidal structure and for all $a, b, c \in A$: $$a \otimes b \to c \leq b \to (a \Rightarrow b)$$
\[MonoidalInternalForMeet\] For strong algebras for which the monoidal structure is the meet structure, internalizing the monoidal structure simply means $a \to (b \wedge c)=(a \to b) \wedge (a \to c)$, for all $a, b, c \in A$. First note that we always have $a \to (b \wedge c) \leq (a \to b) \wedge (a \to c)$ because, $\to$ is order preserving in its second argument. Now, assume that $\mathcal{A}$ internalizes its monoidal structure, then we have $$(a \to b) \leq (a \wedge a \to a \wedge b)
\;\;\;\; \text{and} \;\;\;\;
(a \to c) \leq (b \wedge a \to b \wedge c)$$ implying $$(a \to b) \wedge (a \to c) \leq (a \wedge a \to a \wedge b) \wedge (b \wedge a \to b \wedge c) \leq (a \to b \wedge c)$$ Therefore, $ (a \to b) \wedge (a \to c) \leq a \to (b \wedge c)$ and hence $$a \to (b \wedge c)=(a \to b) \wedge (a \to c)$$ Conversely, since $c \wedge a \leq c$ we have $c \wedge a \to c=1$. Moreover, $c \wedge a \leq a$ implies $(a \to b) \leq (c \wedge a) \to b$. Hence, $$(a \to b) \leq [(c \wedge a) \to c] \wedge [(c \wedge a) \to b]=(c \wedge a \to c \wedge b)$$
Let $X$ be a set and $f: X \to X$ be a function. Consider $P(X)$, the poset of all subsets of $X$ and $F: P(X) \to P(X)$ defined by $F(A)=f[A]$, where $f[A]$ is the image of $A$. Since $F$ is monotone, $A \to B=F(A) \Rightarrow F(B)$ is an implication, where $\Rightarrow$ is the usual Boolean implication on $P(X)$. In a special case, if we choose $X$ and $f$ such that $f$ is surjective and for some subsets of $X$ such as $A, B$ we have $f[A\cap B] \neq f[A] \cap f[B]$, then $\to_F$ does not internalize the monoidal structure (the meet) because, $$[1 \to_F (A \cap B)]=[F(1) \Rightarrow F(A \cap B)]= F(A \cap B)$$ $$[(1 \to_F A) \cap (1 \to_F B)]= [(F(1) \Rightarrow F(A)) \cap (F(1) \Rightarrow F(B))]=[F(A) \cap F(B)]$$ are not equal. There are many such arrangements. For instance, take $X=\mathbb{N}$, $f(n)=\lfloor \frac{n}{2} \rfloor$ and $A=2\mathbb{N}$ and $B=2\mathbb{N}+1$ as the set of even and odd natural numbers, respectively. Then $A \cap B=\emptyset$ and hence $f[A \cap B]=\emptyset$, while $0 \in f[A] \cap f[B]$. This example provides an implication that does not internalize the monoidal structure.
Non-Commutative Spacetimes {#Non-ComSpacetime}
==========================
As we have discussed in Section \[QuantalesAndIntuitionism\], quantales provide a natural formalization for a more subjective notion of intuitionistic proposition. However, to address the full intuitionistic picture, along the constructibility formalized by the order, we also need to formalize the independent notion of time. How can we formalize such a temporal structure? The answer is the modality $\nabla$ that we introduced in the Introduction. Recall that $\nabla a$ must be read as the proposition “$a$ hold at some point in the past".
\[Spacetime\] A pair $\mathcal{S}=(\mathscr{X}, \nabla)$ is called a non-commutative spacetime if $\mathscr{X}$ is a quantale and $\nabla: \mathscr{X} \to \mathscr{X}$ is an oplax geometric map, i.e., a monotone and join preserving map such that $\nabla e \leq e$ and $\nabla (a \otimes b) \leq \nabla a \otimes \nabla b$, for all $a, b \in \mathscr{X}$. A non-commutative spacetime is called a spacetime if its monoidal structure is a meet structure. Spelling out, $\mathcal{S}=(\mathscr{X}, \nabla)$ is a spacetime if $\mathscr{X}$ is a locale and $\nabla :\mathscr{X} \to \mathscr{X}$ is just a join preserving map. Note that the oplax condition is a consequence of monotonicity of $\nabla$ and the fact that $1$ is the greatest element.
Our notion of spacetime is similar to dynamic topological spaces studied in [@DTL]. However, in spacetimes, we are interested in the combination of both adjoints rather than the $\Box$ as the right adjoint of $\nabla$, alone. Moreover, we depart from topological spaces and the inverse image of continuous functions to quantales and oplax join preserving maps. The latter is extremely more general than the former.
\[t6-6\] Assume that $X$ is a topological space and $f: X \to X$ is a continuous function. Then $\mathcal{S}=(\mathcal{O}(X), f^{-1})$ is a spacetime where $\mathcal{O}(X)$ is the locale of the open subsets of $X$.
\[KripkeToSpacetime\] By a *Kripke frame*, we mean a tuple $\mathcal{K}=(W, \leq, R)$ where $(W, \leq)$ is a poset and $R \subseteq W \times W$ is a relation compatible with the order $\leq$, meaning that for all $u, v, u', v' \in W$, if $(u, v) \in R$, $u' \leq u$ and $v \leq v'$ then $(u', v') \in R$. For any Kripke frame $\mathcal{K}$, define $\nabla_{\mathcal{K}} : U(W, \leq) \to U(W, \leq)$ as $\nabla_{\mathcal{K}}(U)=\{v \in W | \exists u \in U \; R(u, v) \}$. The map $\nabla_{\mathcal{K}}$ is trivially monotone and join preserving. For the latter, note that $w \in \bigcup_{i \in I} \nabla U_i$ iff $\exists i \in I \; (w \in \nabla U_i)$ iff $$\exists i \in I \exists u \in W \; ((u, w) \in R \; \wedge (u \in U_i))$$ $$\text{iff} \;\; \exists u \in W \;(u \in \bigcup_{i \in I} U_i \wedge (u, w) \in R) \;\; \text{iff} \;\; w \in \nabla (\bigcup_{i \in I} U_i).$$ Therefore, $\mathcal{S}_{\mathcal{K}}=(U(W, \leq), \nabla_{\mathcal{K}})$ is a spacetime. Note that if we take equality $=_W$ for $\leq$, it transform any usual Kripke frame $(W, R)$ with arbitrary $R$ to a spacetime. Philosophically speaking, in an arbitrary Kripke frame, $W$ can be interpreted as the set of the creative subject’s mental states, $\leq$ as the encoding of the order on the knowledge content of states and $R$ as the encoding of the order of time of the states.
Let $X$ be a set, $f: X \to X$ be a function and $P(X \times X)$ be the quantale of all binary relations over $X$. Consider $f_!: P(X \times X) \to P(X \times X)$ defined as $f_!(R)=\{(f(x), f(y)) | x, y \in X \; \text{and} \; (x, y) \in R\}$. By Example \[LiftingFromSetsToQuantales\], the map $f_!$ is an oplax geometric morphism and hence $(P(X, X), f_!)$ is a non-commutative spacetime.
Let $\mathcal{M}=(M, \otimes, e)$ be a monoid, $I(\mathcal{M})$ be the quantale of its left ideals and $f: M \to M$ be an endomorphism. Consider $f_!: I(\mathcal{M}) \to I(\mathcal{M})$ defined as $f_!(I)=M f[I]=\{mf(i) | i \in I , m \in M\}$. By Example \[LiftingFromMonToQuantales\], the map $f_!$ is an oplax geometric morphism and hence $(I(\mathcal{M}), f_!)$ is a non-commutative spacetime.
Any non-commutative spacetime has its own canonical implication. It is constructible via the second method we have explained before. This implication is nothing but the usual implication, delayed by the passage of time. The main point of these canonical implications is the full adjunctions that they present. This means that the structure is complete enough to fully capture the behaviour of the implication. We will explain these matters more later.
\[t6-8\] Let $\mathcal{S}=(\mathscr{X}, \nabla)$ be a non-commutative spacetime. Then there exists an implication $\to_{\mathcal{S}}: \mathscr{X}^{op} \times \mathscr{X} \to \mathscr{X}$ such that $$a \otimes \nabla b \leq c \;\; \text{iff} \;\; b \leq a \to c$$
Since $\mathscr{X}$ is a quantale and $\nabla: \mathscr{X} \to \mathscr{X}$ is a join preserving monotone map, by the adjoint functor theorem, Theorem \[AFT\], it has a right adjoint $\Box: \mathscr{X} \to \mathscr{X}$. Now, define $a \to_{\mathcal{S}} b=\Box (a \Rightarrow b)$ where $\Rightarrow$ is the canonical implication of $\mathscr{X}$. This map has the desired property since $$a \otimes \nabla b \leq c \;\; \text{iff} \;\; \nabla b \leq a \Rightarrow c \;\; \text{iff} \;\; b \leq \Box(a \Rightarrow c)$$ Moreover, note that $\Box$ is the right adjoint of $\nabla$. Therefore, since $\nabla$ is oplax, by Theorem \[MonoidalAdjoint\], $\Box$ must be lax monoidal and hence by the second construction method for implications, the operation $\to_{\mathcal{S}}$ must be an implication.
It is worth defining an elementary version of the previous adjunction situation. This is similar to how Heyting algebras provide an elementary version of locales:
Let $(A, \leq, \otimes, e)$ be a monoidal poset and $\nabla: A \to A$ and $\to : A^{op} \times A \to A$ be two order preserving functions. Then the structure $\mathcal{A}=(A, \leq, \otimes, e, \nabla, \to)$ is called a temporal algebra if for any $a, b, c \in A$, we have $a \otimes \nabla b \leq c$ iff $b \to a \to c$. A temporal algebra is called distributive if $(A, \leq, \otimes, e)$ is a distributive monoidal poset. A temporal algebra is called a left residuated algebra if $\nabla=id$. And finally, if $\mathcal{A}=(A, \leq_A, \otimes_A, e_A, \nabla_A, \to_A)$ and $\mathcal{B}=(B, \leq_B, \otimes_B, e_B, \nabla_B, \to_B)$ are two temporal algebras, by a temporal algebra morphism we mean a strict monoidal map $f: A \to B$ that also preserves $\nabla$ and $\to$, i.e., $f\nabla_A=\nabla_B f$ and $f((-) \to_A (-))=f(-) \to_B f(-)$.
Interpreting a temporal algebra $\mathcal{A}=(A, \leq, \otimes, e, \nabla, \to)$ as the world of propositions and $\nabla a$ as “$a$ happened at some point before", we claim that moving from the set $A$ to $\nabla [A]=\{\nabla a | a \in A\}$ and changing $\to$ to $\nabla( \to )$, forgets the time structure and results in a usual left residuated algebra. The reason is that the only role that the time structure plays is delaying the implication. If we apply $\nabla$ on the implication, it almost brings it back to the original timeless implication. It is almost, because moving in time and coming back, $\nabla \Box$, is not a real identity, but it is its best approximation. However, if we move from $A$ to $\nabla [A]$, this approximation becomes the real equality. In this sense we can claim that a temporal algebra (with meet structure for the monoidal part) is a refined version of the usual left residual algebra (Heyting algebra).
\[AlgebraicTranslation\] Let $\mathcal{A}=(A, \leq, \otimes, e, \nabla, \to)$ be a temporal algebra and $\nabla$ preserves all finite multiplications. Then, the structure $\nabla \mathcal{A}=(\nabla [A], \leq, \otimes, e, \nabla, \to')$ is a left residuated algebra where $\nabla[A]=\{\nabla x | x \in A\}$ and $a \to' b=\nabla (a \to b)$, for any $a, b \in \nabla [A]$. Moreover, if $\mathcal{A}$ is finitely complete (finitely cocomplete), so is $\nabla \mathcal{A}$. The same is also true for completeness. Finally, if the monoidal structure of $\mathcal{A}$ is the meet structure, $\nabla \mathcal{A}$ is a Heyting algebra.
Since $\nabla$ preserves the monoidal structure, the set $\nabla [A]$ is closed under all finite multiplications. Therefore, the only thing to prove is the adjunction $a \otimes (-) \dashv (a \to' (-))$, for any $a \in \nabla [A]$. It means $a \otimes b \leq c$ iff $b \leq \nabla (a \to c)$, for all $a, b, c \in \nabla [A]$. From left to right, since $b \in \nabla [A]$, there exists $b' \in A$ such that $b=\nabla b'$. Since $a \otimes \nabla b' \leq c$ we have $b' \leq a \to c$ which implies $b=\nabla b' \leq \nabla (a \to c)$. From right to left, if $b \leq \nabla (a \to c)$ then $a \otimes b \leq a \otimes \nabla (a \to c) \leq c$.\
Note that if $A$ has also all (finite) joins or all (finite) meets, so does $\nabla [A]$. For joins, since $\nabla$ has a right adjoint and preserves all joins, $\nabla [A]$ is closed under all (finite) joins. Therefore, $\nabla [A]$ has also all (finite) joins. For meet, the situation is a bit more complex. We will address the binary meet. The rest is similar. For any $a, b \in \nabla [A]$, we claim that $\nabla \Box (a \wedge b) \in \nabla [A]$ is the meet of $a$ and $b$ in $\nabla \mathcal{A}$. Because, $\nabla \Box (a \wedge b) \leq (a \wedge b) \leq a$ and similarly for $b$ we also have $\nabla \Box (a \wedge b) \leq b$ . If for some $c \in \nabla [A]$ we have $c \leq a$ and $c \leq b$, then $c \leq a \wedge b$ and hence $\nabla \Box c \leq \nabla \Box (a \wedge b)$. Since $c \in \nabla [A]$, there exists $c' \in A$ such that $c=\nabla c'$. Hence, $\nabla \Box c=\nabla \Box \nabla c'=\nabla c'=c$. Thus, $c \leq \nabla \Box (a \wedge b)$.
Let $\mathcal{S}=(\mathscr{X}, \nabla_{\mathcal{S}})$ and $\mathcal{T}=(\mathscr{Y}, \nabla_{\mathcal{T}})$ be two non-commutative spacetimes. By a geometric map $f: \mathcal{S} \to \mathcal{T} $, we mean a strict geometric morphism $f: \mathscr{X} \to \mathscr{Y}$ such that $f \nabla_{\mathcal{S}} = \nabla_{\mathcal{T}} f$. A geometric map is called logical if it also preserves the implication, i.e., $f [(-) \to_{\mathcal{S}} (-)]=f(-) \to_{\mathcal{T}} f(-)$.
Let $\mathcal{K}=(W, =_W, R)$ and $\mathcal{L}=(V, =_V, S)$ be two Kripke frames. A map $p:W \to V$ is called a p-morphism if $(u, v) \in R$ implies $(p(u), p(v)) \in S$, for any $u, v \in W$ and for any $w \in W$ and $s, t \in V$ if $p(w)=s$ and $(s, t) \in S$, then there exists $u \in W$ such that $p(u)=t$ and $(w, u) \in R$. A map $p:W \to V$ is a p-morphism iff $p^{-1}: \mathcal{S}_{\mathcal{L}^{op}} \to \mathcal{S}_{\mathcal{K}^{op}}$ is a geometric morphism, where $\mathcal{K}^{op}=(W, R^{op})$ and $(v, u) \in R^{op}$ iff $(u, v) \in R$ and similarly for $\mathcal{L}$. We only prove the left to right direction. The other direction is similar. First note that $p^{-1}$ preserves all unions and all finite intersections. Therefore, the only thing we have to prove is the preservability of $\nabla$, i.e., $p^{-1} \nabla_{\mathcal{L}^{op}}=\nabla_{\mathcal{K}^{op}} p^{-1}$. Let $U$ be a subset of $V$. Then if $u \in \nabla_{\mathcal{K}^{op}} p^{-1}(U)$, then there exists $w \in W$ such that $(w, u) \in R^{op}$ or equivalently $(u, w) \in R$ and $p(w) \in U$. Since $p$ is a p-morphism we have $(p(u), p(w)) \in S$ which means $(p(w), p(u)) \in S^{op}$. Hence, $p(u) \in \nabla_{\mathcal{L}^{op}} (U)$ which implies $u \in p^{-1} \nabla_{\mathcal{L}^{op}}(U)$. Conversely, if $u \in p^{-1} \nabla_{\mathcal{L}^{op}}(U)$, we have $p(u) \in \nabla_{\mathcal{L}^{op}}(U)$ from which, there exists $v \in U$ such that $(v, p(u)) \in S^{op}$ or equivalently $(p(u), v) \in S$. Since $p$ is a p-morphism, there exists $w \in W$ such that $p(w)=v$ and $(u, w) \in R$. Hence, $(w, u) \in R^{op}$ from which, $u \in \nabla_{\mathcal{K}^{op}}(p^{-1}(U))$.
\[Logical\] Let $\mathcal{S}$ and $\mathcal{T}$ be two non-commutative spacetimes and $f: \mathcal{S} \to \mathcal{T}$ be a geometric morphism with a left adjoint $f_!$. Then $f$ is logical iff $f_!(f b \otimes \nabla_{\mathcal{T}} a)=b \otimes \nabla_{\mathcal{S}} f_! a$.
Using the adjunctions $x \otimes \nabla_{\mathcal{T}}(-) \dashv x \to_{\mathcal{T}} (-)$, $y \otimes \nabla_{\mathcal{S}} (-) \dashv y \to_{\mathcal{S}} (-)$ and $f_! \dashv f$ we have $$f_!(fb \otimes \nabla_{\mathcal{T}} a) \leq c \;\; \text{iff} \;\; fb \otimes \nabla_{\mathcal{T}} a \leq fc \;\; \text{iff} \;\; a \leq fb \to_{\mathcal{T}} fc$$ and $$b \otimes \nabla_{\mathcal{S}} f_! a \leq c \;\; \text{iff} \;\; f_!a \leq b \to_{\mathcal{S}} c \;\; \text{iff} \;\; a \leq f(b \to_{\mathcal{S}} c)$$ These equivalences imply exactly what we wanted. Because, if $f_!(f b \otimes \nabla_{\mathcal{T}} a)=b \otimes \nabla_{\mathcal{S}} f_! a$, then the left hand sides of the above lines are equivalent which implies the equivalence of the right hand sides from which $fb \to_{\mathcal{T}} fc=f(b \to_{\mathcal{S}} c)$. The converse is similar.
Sometimes, it would be reasonable to investigate the pure spatial behaviour of a non-commutative space, meaning the properties that hold for *all* possible time structures or more formally *all* possible $\nabla$’s over a fixed space. The following theorem provides a method to transfer these properties along certain geometric morphisms. We will use this theorem when we have a suitable syntax for non-commutative spacetimes to formally address what we mean by a “property".
\[TransferringNabla\] Let $\mathcal{S}=(\mathscr{X}, \nabla_{\mathcal{S}})$ be a non-commutative spacetime, $\mathscr{Y}$ be a quantale and $f: \mathscr{X} \to \mathscr{Y}$ be a strict geometric embedding with a left adjoint $f_!$. Then there exists $\nabla$ on $\mathscr{Y}$ such that $\mathcal{T}=(\mathscr{Y}, \nabla)$ is a non-commutative spacetime and $f: \mathcal{S} \to \mathcal{T}$ is a logical morphism.
Define $\nabla=f \nabla_{\mathcal{S}} f_!$. Since $f_!$ is a left adjoints and both $f$ and $\nabla_{\mathcal{S}}$ preserves all joins, the operator $\nabla$ is also join preserving. Moreover, since $f$ is strict monoidal, its left adjoint, $f_!$ is oplax, by Theorem \[MonoidalAdjoint\]. Therefore, $\nabla$ as a composition of three oplax monoidal maps is also oplax. To prove the geometricity of $f: (\mathscr{X}, \nabla_{\mathcal{S}}) \to (\mathscr{Y}, \nabla)$, since $f_! \dashv f$, by Remark \[SemiInverse\], $ff_!f=f$. Since $f$ is an embedding we have $f_! f= id$. Therefore, $ \nabla f = f \nabla_{\mathcal{S}} f_! f = f \nabla_{\mathcal{S}}$. Hence $f: (\mathscr{X}, \nabla_{\mathcal{S}}) \to (\mathscr{Y}, \nabla)$ is geometric. To prove it is logical, by Theorem \[Logical\] we have to show that $f_!(\nabla a \otimes f b)=f_!(f \nabla_{\mathcal{S}} a \otimes f b)$. Since $f$ is strict monoidal, the right hand side is equivalent to $f_!f (\nabla_{\mathcal{S}} a \otimes b)$. Since $f_!f= id$, the latter is equivalent to $\nabla_{\mathcal{S}} f_! a \otimes b$. Hence, the geometric map $f: (\mathscr{X}, \nabla_{\mathcal{S}}) \to (\mathscr{Y}, \nabla)$ is logical.
Corollary \[TransferringNabla\] is useful in case the quantales are the open posets of topological spaces and the space for $\mathscr{X}$ is an Alexandroff space. Recall that a topological space is Alexandroff if any arbitrary intersection of its open subsets is also open.
\[TransferringNablaAlex\] Let $X$ be a topological space, $Y$ be an Alexandroff space, $f:X \to Y$ be a continuous surjection and $\mathcal{S}=(\mathcal{O}(Y), \nabla_{Y})$ be a spacetime. Then there exists $\nabla_{X}: \mathcal{O}(X) \to \mathcal{O}(X)$ such that $\mathcal{T}=(\mathcal{O}(X), \nabla_{X})$ is a spacetime and $f^{-1}: \mathcal{S} \to \mathcal{T}$ is a logical morphism.
Since the space $Y$ is Alexandroff, $\mathcal{O}(Y)$ is closed under all intersections. Therefore, since $f^{-1}$ preserves arbitrary intersections it also preserves arbitrary meets. Hence, by the adjoint functor theorem, Theorem \[AFT\], it has a left adjoint $f_!$. Moreover, note that $f: X \to Y$ is surjective which means that $f^{-1}: \mathcal{O}(Y) \to \mathcal{O}(X)$ is an embedding. Hence, it is enough to use Corollary \[TransferringNabla\].
Representation Theorems {#Rep}
=======================
In this section we will present some quantale-based representations for different classes of strong algebras. The main motive is embedding an abstract strong algebra in a quantale in a way that the implication presents a possible well-behaved left adjunction. We call this process *resolving the implication*. In a technical sense, these left adjoints make the implications easier to handle as it is usual all over mathematics. However, resolutions have a very philosophical role, as well. We know that adjunctions are the algebraic term for the usual proof theoretical situation in which we have a pair of introduction and elimination rules for a logical connective that we try to capture. For instance, think about the intuitionistic implication and its natural deduction rules. Following Gentzen, a connective is fully captured if it enjoys a pair of introduction and elimination rules. In this sense, resolving an implication is an attempt to fully identify an abstract implication as a logical connective.\
Having all said, resolving all the implications is unfortunately impossible. We will explain the reason later. We will also see some necessary and partially sufficient conditions to make resolutions possible. But first let us begin by a general yet weak resolution-type result. We will prove that any strong algebra is embeddable in a quantale equipped with an implication. The implication is not necessarily a non-commutative spacetime implication but it is a substitution of it. We can think of the implication as the result of the application of the two construction methods that we explained before, applied on the canonical implication of the quantale.\
Let $\mathcal{A}=(A, \leq, \otimes, e, \to)$ be a strong algebra. A priory, there is no reason to assume that the structure $\mathcal{A}$ has the power (enough elements or structure) to resolve the implication and find an adjunction-type situation. However, if we extend the domain to also include the *relative elements*, meaning the monotone functions $A^{op} \to A$, then we can provide the following characterization for the implication: $$c \leq a \to b \;\;\; \text{iff} \;\;\; (x \to a) \otimes c \leq (x \to b)$$ where $x$ is a variable and the right-hand side consists of the functions for which the order and the monoidal structure are both defined pointwise. The reason is simple. From left to right, note that $$c \leq a \to b \;\;\; \text{implies} \;\;\; (x \to a) \otimes c \leq (x \to b) \otimes (a \to b) \leq (x \to b)$$ and from right to left, it is enough to put $x=a$ to have $$c =e \otimes c \leq (a \to a) \otimes c \leq a \to b$$ Note that while this adjunction-type characterization handles all the elements of $A$, it can not handle the functions that it adds. To solve this problem we simply need infinitely many of such variables:
For any strong algebra $\mathcal{A}$ there exists a non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla)$, a monotone map $F: \mathscr{X} \to \mathscr{X}$ and a strict monoidal embedding $i: \mathcal{A} \to \mathscr{X}$ such that $i(a \to_{\mathcal{A}} b)= F(i(a)) \to_{\mathcal{S}} F(i(b))$.
Define $E$ as the set of all monotone functions $f: (\Pi_{n \in \mathbb{N}} A^{op}) \to A$ with finite support, i.e., all order preserving maps that depend only on some finitely many of their arguments. Define $\leq_E$ as the pointwise order on $E$ and use $\otimes_E$ and $e_E$ to represent the pointwise monoidal structure of $E$. Then the structure $\mathcal{E}=(E, \leq_E, \otimes_E, e_E)$ is clearly a monoidal poset. Define $j: \mathcal{A} \to \mathcal{E}$ by mapping any elemnt $a \in A$ to the constant function with the value $a$. Since the structure of $\mathcal{E}$ is defined pointwise, it is clear that $j$ is a strict monoidal embedding.\
Define $s: E \to E$ as the coordinate shift map by $f(\vec{x}_n) \mapsto f(\vec{x}_{n+1})$. This map is clearly strictly monoidal. Moreover, define $l: E \to E$ mapping $f(\vec{x}_n) \mapsto (x_0 \to f(\vec{x}_{n+1}))$. Now use the downset completion on $\mathcal{E}=(E, \leq_E, \otimes_E, e_E)$ to construct our $\mathscr{X}$. Let $k: \mathcal{E} \to \mathscr{X}$ be the canonical strict monoidal embedding from the downset completion. Define $$\nabla I = s_!=\{f \in E | \exists g \in I \; (f \leq_{E} s(g))\}$$ and $$F(I) = \{f \in E | \exists g \in I \; [f \leq_{E} l(s(g))]\}$$ The map $F$ is clearly monotone and mapping downsets to downsets. By Theorem \[LiftingMonoidalMaps\], $\nabla$ preserves the joins and it is oplax because $s$ is oplax. Therefore, $\mathcal{S}=(\mathscr{X}, \nabla)$ is a non-commutative spacetime. We claim that $i=kj: \mathcal{A} \to \mathscr{X}$ is the strict monoidal embedding that we are looking for. The only thing to prove is that $i(a \to_{\mathcal{A}} b)= F(i(a)) \to_{\mathcal{S}} F(i(b))$. To prove that, we show $$J \subseteq i(a \to b) \;\; \text{iff} \;\; F(i(a)) \otimes \nabla J \subseteq F(i(b)), \;\;\;\;\; (*)$$ for any downset $J$ of $E$. First to simplify the proof, note that for any $c \in A$ $$f \in F(i(c)) \;\;\; \text{iff} \;\;\; f \leq (x_0 \to c)$$ The reason is that $f \in F(i(c))$ iff there exists a function $f'\in i(c)$ such that $f \leq (x_0 \to s(f'))$. This is equivalent to $f \leq (x_0 \to c)$.\
Now to prove $(*)$, for left to right, if $J \subseteq i(a \to b)$ and $f \in F(i(a)) \otimes \nabla J$ by the definition of the multiplication on downsets, there exist $g \in F(i(a))$ and $h \in \nabla J$ such that $f \leq g \otimes_E h$. By the above point, since $g \in F(i(a))$ we have $g \leq (x_0 \to a)$. By definition of $\nabla$ there exists $h' \in J $ such that $h \leq s(h')$. Since $h' \in J \subseteq i(a \to b)$ we have $h' \leq a \to b$ and hence $h \leq s(a \to b)= a \to b$. Therefore, $g \otimes_E h \leq (x_0 \to a) \otimes_E (a \to b) \leq x_0 \to b$. Therefore, by the above-mentioned point we have $g \otimes_E h \in F(i(b))$ and since $f \leq g \otimes_E h$ and $F(i(b))$ is a downset, we have $f \in F(i(b))$.\
For the converse, assume $F(i(a)) \otimes \nabla J \subseteq F(i(b))$ and we want to show that $J \subseteq i(a \to b)$. Assume $f \in J$. Then by the definition of $\nabla$, we have $s(f) \in \nabla J$. Moreover, by the above mentioned point we have $(x_0 \to a) \in F(i(a))$. Hence, $(x_0 \to a) \otimes_E s(f) \in F(i(a)) \otimes_{\mathscr{X}} \nabla J$. Therefore, $(x_0 \to a) \otimes_E s(f) \in F(i(b))$. Hence, $(x_0 \to a) \otimes_E s(f) \leq (x_0 \to b)$. Since the order of $E$ is pointwise, put $x_0=a$ and keep the other variables intact. Since $s(f)$ does not depend on $x_0$, it does not change after the substitution. Hence, $(a \to a) \otimes_E s(f) \leq (a \to b)$. Since $e \leq a \to a$, we have $s(f) \leq a \to b=s(a \to b)$. Since $s$ is an embedding, $f \leq a \to b$ and hence $f \in i(a \to b)$.
Although, the previous theorem provides a weak resolution for any abstract implication, it can only resolve it up to a factor $F$ which breaks the full adjunction situation. This $F$ is inevitable, simply because it is impossible to embed any implication into a non-commutative spacetime. The reason is that for any non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla)$, its implication, $\to_{\mathcal{S}}$, internalizes the closed monoidal structure of $\mathcal{S}$, i.e., for all $a, b, c \in \mathscr{X}$ we have $$a \to_{\mathcal{S}} b \leq c \otimes a \to_{\mathcal{S}} c \otimes b$$ because by the associativity and the adjunction $$c \otimes a \otimes \nabla (a \to_{\mathcal{S}} b) \leq c \otimes b$$ Therefore, if we seek an embedding into a non-commutative spacetime we have to restrict our domain to the implications that internalize their monoidal structure. Unfortunately, we do not know if this necessary condition is also sufficient. However, if the multiplication has left residuation and the implication internalizes the closed monoidal structure, we will have the following representation. Here our main ingredient is the ternary frames introduced first in [@Meyer] as the Kripke models for the relevant logics.
For any strong algebra $\mathcal{A}$ whose multiplication has left residual and $\mathcal{A}$ internalizes its closed monoidal structure, there exists a non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla)$ and a strong algebra embedding $i:\mathcal{A} \to \mathcal{S}$.
Recall that $U(\mathcal{A})$ is the poset of all upsets of $\mathcal{A}$ with inclusion. Define $\mathcal{R}$ as a ternary relation over $U(\mathcal{A})$ as: $(P, Q, R) \in \mathcal{R}$ iff for all $a, b \in A$ if $a \to b \in P$ and $a \in Q$ then $b \in R$. Note that the relation $\mathcal{R}$ is order-reversing in its first two arguments while it is order preserving in its third argument. Consider $\mathscr{X}=U(U(\mathcal{A}))$ and $i: \mathcal{A} \to \mathscr{X}$ by defining $i(a)=\{P \in U(\mathcal{A}) | \ a \in P\}$. As we have observed in Preliminaries, this $i$ is clearly a strict monoidal embedding. Our strategy is first defining an implication on $\mathscr{X}$ and showing how $i$ maps the implication of $\mathcal{A}$ to this implication and then finding an oplax $\nabla$ such that $X \otimes \nabla (-) \dashv (X \to (-))$ for any $X \in \mathscr{X}$.\
For any upsets of $U(\mathcal{A})$ such as $X$ and $Y$ define $X \to Y$ as: $$\{P \in U(\mathcal{A}) | \; \forall Q, R \in U(\mathcal{A}), \; \text{if} \; (P, Q, R) \in \mathcal{R} \; \text{and} \; Q \in X \; \text{then} \; R \in Y\}$$ Since $\mathcal{R}$ is order-reversing in its first argument, $X \to Y$ is an upset. To prove that $i$ maps the implication of $\mathcal{A}$ into this implication, i.e., $i(a \to b)=i(a) \to i(b)$, we need to address the following two directions:\
For $i(a \to b) \subseteq i(a) \to i(b)$, if $P \in i(a \to b)$ then $a \to b \in P$. To show that $P \in i(a) \to i(b)$, assume for some $Q, R \in U(\mathcal{A)}$ we have $(P, Q, R) \in \mathcal{R}$ and $Q \in i(a)$. Then by the definition of $i$ we have $a \in Q$ and since $a \to b \in P$, by the definition of $\mathcal{R}$ we have $b \in R$ implying $R \in i(b)$. Conversely, for $i(a) \to i(b) \subseteq i(a \to b)$, if $P \in i(a) \to i(b)$ define $Q=\{x \in A| x \geq a\}$ and $R=\{y \in A | a \to y \in P\}$. We have $(P, Q, R) \in \mathcal{R}$ because if $x \to y \in P$ and $x \in Q$ then $x \geq a$ and hence $x \to y \leq a \to y$ which implies $a \to y \in P$. Therefore, by definition $y \in R$. Finally, Since $a \in Q$ we have $Q \in i(a)$ and since $(P, Q, R) \in \mathcal{R}$ we have $R \in i(b)$ which implies $b \in R$. By definition of $R$ it means that $a \to b \in R$.\
To complete the proof, we have to introduce an oplax $\nabla$ and show that for any upsets of $U(\mathcal{A})$ such as $X, Y, Z$ we have $X \subseteq Y \to Z$ iff $Y \otimes \nabla X \subseteq Z$. Define $\nabla$ as: $$\nabla X=\{R \in U(\mathcal{A})| \; \exists P, Q \in U(\mathcal{A}) \; [(P, Q, R) \in \mathcal{R}, (P \in X) \; \text{and} \; (e \in Q)] \}$$ Since $\mathcal{R}$ is order-preserving in its third argument, $\nabla$ is an upset. To prove the adjunction condition and the fact that it is oplax, we need a claim first:
- For any upsets $P, Q, R, S \in U(\mathcal{A})$, if $(P, Q, R) \in \mathcal{R}$ then $(P, S \otimes Q, S \otimes R) \in \mathcal{R}$.
- For any upsets $P, Q, R \in U(\mathcal{A})$, if $(P, Q, R) \in \mathcal{R}$ then $(P, E, Q \Rightarrow R) \in \mathcal{R}$ where $E=\{x \in A | x \geq e\}$ and $ \Rightarrow $ is the canonical implication of the quantale $U(\mathcal{A})$.
- For any upsets $P_1, P_2, Q, R \in U(\mathcal{A})$, if $(P_1 \otimes P_2, Q, R) \in \mathcal{R}$ and $e \in Q$, then there are upsets $Q_1, Q_2, R_1, R_2 \in U(\mathcal{A})$ such that $e \in Q_1$, $e \in Q_2$, $(P_1, Q_1, R_1) \in \mathcal{R}$, $(P_2, Q_2, R_2) \in \mathcal{R}$ and $R_1 \otimes R_2 \subseteq R$.
*Proof of the Claim.* For $(i)$, if $x \to y \in P$ and $x \in S \otimes Q $ then there are $z \in S$, $w \in Q$ such that $x \geq z \otimes w $. Since $x \geq z \otimes w $ and $x \to y \in P$ we have $ z \otimes w \to y \in P$. Since, $\mathcal{A}$ internalizes its closed monoidal structure we have $$z \otimes w \to x \leq w \to (z \Rightarrow_A x)$$ where $\Rightarrow_A$ is the left residual of multiplication in $\mathcal{A}$. Since $P$ is an upset, $w \to (z \Rightarrow_A x) \in P$. Since $(P, Q, R) \in \mathcal{R}$ and $w \in Q$ we have $z \Rightarrow_A x \in R$. Since $z \in S$ and $z \otimes (z \Rightarrow_A x) \leq x$ we have $x \in S \otimes R$.\
For $(ii)$, assume $x \to y \in P$ and $x \geq e$ then we have to show that $y \in Q \Rightarrow R$. Equivalently, it means $Y \subseteq Q \Rightarrow R$ where $Y=\{x \in A | x \geq y\}$. The latter is equivalent to $Q \otimes Y \subseteq R$ because $\Rightarrow$ is the left residual in $U(\mathcal{A})$. Assume $z \in Q \otimes Y$. Therefore, there exist $w \in Q$ and $u \geq y$ such that $z \geq w \otimes u$ implying $z \geq w \otimes y$. Since $\mathcal{A}$ internalizes its monoidal structure, we have $$x \to y \leq w \otimes x \to w \otimes y$$ Hence, $w \otimes x \to w \otimes y \in P$. Since $e \leq x$ we have $w= w \otimes e \leq w \otimes x$. By $w \in Q$ we have $w \otimes x \in Q$. Since $(P, Q, R) \in \mathcal{R}$ we have $w \otimes y \in R$. Since $z \geq w \otimes y$ we conclude $z \in R$ that completes the proof.\
To prove $(iii)$, if $(P_1 \otimes P_2, Q, R) \in \mathcal{R}$ and $e \in Q$, then define $Q_1=Q_2=\{x \in A | x \geq e\}$ and $R_i=\{x \in A | e \to x \in P_i\}$ for $i \in \{1, 2\}$. By definition it is clear that $(P_1, Q_1, R_1) \in \mathcal{R}$ and $(P_2, Q_2, R_2) \in \mathcal{R}$, because if $u \to v \in P_i$ and $u \geq e$ then $e \to v \in P_i$ which by definition means $v \in R_i$. Finally, to prove $R_1 \otimes R_2 \subseteq R$, assume $z \in R_1 \otimes R_2$. Therefore, there are $x \in R_1$ and $y \in R_2$ such that $z \geq x \otimes y$. Since $x \in R_1$ and $y \in R_2$ we have $e \to x \in P_1$ and $e \to y \in P_2$. Therefore, $(e \to x) \otimes (e \to y) \in P_1 \otimes P_2$. Since $\mathcal{A}$ internalizes its monoidal structure we have $$e \to y \leq (x \otimes e \to x \otimes y)=(x \to x \otimes y)$$ Therefore, $$(e \to x) \otimes (e \to y) \leq (e \to x) \otimes (x \to x \otimes y) \leq e \to x \otimes y$$ Hence, $e \to x \otimes y \in P_1 \otimes P_2$. Since $e \in Q$ and $(P, Q, R) \in \mathcal{R}$ we have $x \otimes y \in R$ and since $z \geq x \otimes y$ we have $z \in R$.\
\
Now let us come back to prove that $\nabla$ is a join preserving oplax map. We have to show that $\nabla i(e) \subseteq i(e)$ and for any upsets of $U(\mathcal{A})$ such as $X, Y$ we have $\nabla (X \otimes Y) \subseteq \nabla X \otimes \nabla Y $. For the first one, if $R \in \nabla i(e)$, by definition there exist upsets $P$ and $Q$ such that $(P, Q, R) \in \mathcal{R}$, $e \in Q$ and $P \in i(e)$. Therefore, $e \in P$. Since $e \leq e$, we have $e \leq e \to e$. Since $P$ is an upset we have $e \to e \in P$. Then since $e \in Q$ and $(P, Q, R) \in \mathcal{R}$ we have $e \in R$ which means that $R \in i(e)$. For $\nabla (X \otimes Y) \subseteq \nabla X \otimes \nabla Y $, assume $R \in \nabla (X \otimes Y)$ then again by definition there exist upsets $P \in X \otimes Y$ and $Q$ such that $e \in Q$ and $(P, Q, R) \in \mathcal{R}$. Since $P \in X \otimes Y$, there are $P_1 \in X$ and $P_2 \in Y$ such that $P_1 \otimes P_2 \subseteq P$. Since $\mathcal{R}$ is order reversing in its first argument and $(P, Q, R) \in \mathcal{R}$ we have $(P_1 \otimes P_2, Q, R) \in \mathcal{R}$. By the part $(iii)$ of the claim, there are upsets $Q_1, Q_2, R_1, R_2$ such that $e \in Q_1$, $e \in Q_2$, $(P_1, Q_1, R_1) \in \mathcal{R}$ and $(P_2, Q_2, R_2) \in \mathcal{R}$ and $R_1 \otimes R_2 \subseteq R$. Hence, by definition $R_1 \in \nabla X$ and $R_2 \in \nabla Y$ and since $R_1 \otimes R_2 \subseteq R$ we have $R \in \nabla X \otimes \nabla Y$. Therefore, $\nabla (X \otimes Y) \subseteq \nabla X \otimes \nabla Y $.\
For the adjunction conditions, i.e., $X \subseteq Y \to Z$ iff $Y \otimes \nabla X \subseteq Z$, we need to address the following two directions. For left to right, if $X \subseteq Y \to Z$ and $P \in Y \otimes \nabla X$ we have to show that $P \in Z$. Since $P \in Y \otimes \nabla X$, by definition there exist $Q, R$ such that $Q \otimes R \subseteq P$ and $Q \in Y$ and $R \in \nabla X$. Again by definition since $R \in \nabla X$ there exist $P', Q'$ such that $(P', Q', R) \in \mathcal{R}$, $P' \in X$ and $e \in Q'$. Since $e \in Q'$ for any $q \in Q$ we have $q=q \otimes e \in Q \otimes Q'$. Therefore, $Q \subseteq Q \otimes Q'$. Since $Q \in Y$ we have $Q \otimes Q' \in Y$. Since $(P', Q', R) \in \mathcal{R}$ by the part $(i)$ of the Claim, we have $(P', Q \otimes Q', Q \otimes R) \in \mathcal{R}$ and since $P' \in X \subseteq Y \to Z$ and $Q \otimes Q' \in Y$, we have $Q \otimes R \in Z$. Finally since $Z$ is an upset and $Q \otimes R \subseteq P$ we have $P \in Z$.\
For right to left, if $Y \otimes \nabla X \subseteq Z$ and $P \in X$ we want to show that $P \in Y \to Z$. Pick $Q$ and $R$ such that $(P, Q, R) \in \mathcal{R}$ and $Q \in Y$. We have to show that $R \in Z$. By the part $(ii)$ of the Claim, since $(P, Q, R) \in \mathcal{R}$ we have $(P, E, Q \Rightarrow R) \in \mathcal{R}$ where $e \in E$. Hence, by definition of $\nabla$, we have $Q \Rightarrow R \in \nabla X$ and hence $Q \otimes (Q \Rightarrow R) \in Y \otimes \nabla X$. Since $Y \otimes \nabla X \subseteq Z$ we have $Q \otimes (Q \Rightarrow R) \in Z$. Finally, since $Q \otimes (Q \Rightarrow R) \subseteq R$ and $Z$ is an upset we have $R \in Z$.
Fortunately, if the monoidal structure is just the meet structure, it is possible to show that the internalization of the monoidal structure is sufficient for resolution. Moreover, it is possible to show that the quantale is actually a locale or even better an Alexandroff space:
For any (distributive) strong algebra $\mathcal{A}=(A, \leq, \wedge, 1, \to)$ that internalizes its monoidal structure \[not necessarily its closed structure if it has any\] (and its join structure), there exists a Kripke frame $\mathcal{K}$ and a (join preserving) strong algebra embedding $i: \mathcal{A} \to \mathcal{S}_{\mathcal{K}}$. Moreover, if $\mathcal{A}$ is a reduct of a (distributive) temporal algebra, $i$ also preserves $\nabla$.
See Section \[KripkeModels\].
And finally, in case that we already have a nice left adjoint for the implication, it is possible to make the algebra cocomplete, preserving the temporal structure. This will be useful later in topological completeness theorem.
Let $\mathcal{A}=(A, \leq, \otimes, e, \nabla, \to)$ be a (distributive) temporal algebra. Then there exists a non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla)$ and a (join preserving) temporal algebra embedding $i: \mathcal{A} \to \mathcal{S}$. Moreover, if $\mathcal{A}$ has all finite meets, then $i$ also preserves them.
See Section \[LogicsofSpcaeTimes\].
Logics of Spacetime {#LogicsofSpcaeTimes}
===================
In the previous section we presented some methods to represent some classes of implications via a diamond-type modality $\nabla$, encoding the abstract notion of time. In this section we bring the adjunction into the syntax of logic to provide a more expressible language to address non-standard weak implications. Later, we will see how this new language provides a conservative extension for some weak implication logics including Visser-Ruitenburg’s basic logic. However, the fully captured implications of these new logics make the non-standard implications more suitable for foundational studies. We will present an embedding of a fragment of full Lambek calculus (without element $0$ and right implication) into our logic and full intuitionistic logic into our logic equipped with the structural rules. Therefore, the logics of spacetime can be interpreted as a unification of sub-structural and sub-intuionistic logics.\
Let $\mathcal{L}_{\nabla}$ be the usual language of propositional logic equipped with a new unary modal operator $\nabla$. To introduce some formal systems in this language, consider the following set of sequent-style rules in which the left side of a sequent is a sequence of formulas:
**Axioms:**
--------- -- -- -- --
\[3ex\]
--------- -- -- -- --
**Cut:**
[c]{}
**Conjunction Rules:**
---------- -- --
\[3 ex\]
---------- -- --
**Disjunction Rules:**
--------- -- --
\[3ex\]
--------- -- --
**Rule for 1:**
[c]{}
**Multiplication Rules:**
-- --
-- --
**Modal Rules:**
-- --
-- --
**Implication Rules:**
--------- --
\[3ex\]
--------- --
Now define the logic of spacetime, $\mathbf{STL}$, as the logic of the proof system consisting of all the axioms, cut and propositional rules. The provability of a sequent $\Gamma \Rightarrow A$ in $\mathbf{STL}$ is denoted by $\mathbf{STL} \vdash \Gamma \Rightarrow A$ or $\Gamma \vdash_{\mathbf{STL}} A$.\
By the basic rule schemes $ \{N, H, P, F, wF\}$, we mean one of the following schemes:
**Rule Schemes:**
---------- -- -- --
\[3 ex\]
---------- -- -- --
[c c]{}
Also consider the structural rules:
**Structural Rules:**
-- -- --
-- -- --
For any $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, by the logic $\mathbf{STL}(\mathcal{R})$ we mean the logic of all rules of $\mathbf{STL}$ plus the rules of $\mathcal{R}$. By $i\mathbf{STL}(\mathcal{R})$ we mean $\mathbf{STL}(\mathcal{R})$ with all structural rules. And finally we denote $\mathbf{STL}(\{P, F\})$ by $\mathbf{FL}_l$ and $i\mathbf{STL}(\{P, F\})$ by $\mathbf{IPC}$.
\[3\] Note that in the presence of all the structural rules, the connective $\otimes$ collapses to $\wedge$ and the constant $1$ is reduced to $\top$. Therefore, it is possible to axiomatize the structural logics of spacetime by eliminating the connective $\otimes$ and $1$ from the language and the axiom $\Rightarrow 1$ and the rules $L\otimes$, $R \otimes$, $L1$ and $Oplax$ from the system.
Note that in the presence of both $(F)$ and $(P)$, the connective $\nabla$ trivializes to identity. Therefore, in such logics and more specifically in $\mathbf{FL}_l$ and $\mathbf{IPC}$, it is possible to formalize the logics without the axiom $\nabla 1 \Rightarrow 1$ and the rules $\nabla$ and $Oplax$, by eliminating $\nabla$ in the implication rules. In such a situation, the implication rules become the usual left implication rules in $\mathbf{FL}$. This explains our terminology. In fact, our logic is exactly the fragment of $\mathbf{FL}$ excluding the right implication and $0$ from both the language and the rules. For $\mathbf{IPC}$, it is easy to see that the system becomes the original system $\mathbf{LJ}$ for intuitionistic propositional logic if we forget the collapsed $\otimes$. See Remark \[3\].
\[PropertiesOgfLogic\] Note that the following sequents are provable in the system. First $\nabla (A \otimes B) \Rightarrow \nabla A \otimes \nabla B$ stating the oplax condition for $\nabla$:
[c]{}
Secondly, $\mathbf{STL}$ proves the distributivity of multiplication over disjunction, on both sides, i.e., $(A \otimes B) \vee (A \otimes C) \Rightarrow A \otimes (B \vee C) $ and $A \otimes (B \vee C) \Rightarrow (A \otimes B) \vee (A \otimes C)$. The first is a simple result of monotonicity of $\otimes$. For the second:
[c]{}
Thirdly, the system proves the sequent $A \otimes \nabla (A \to B) \Rightarrow B$:
[c]{}
Therefore, the sequents $A, \nabla B \Rightarrow C$ and $B \Rightarrow A \to C$ are equivalent. From left to right is just one application of the rule $R \to$. From right to left, by the rule $\nabla$, we have $\nabla B \Rightarrow \nabla (A \to C)$. Using cut with $A, \nabla (A \to C) \Rightarrow C$ we reach what we wanted. Note that this adjunction situation simply implies that $\nabla$ preserves all disjunctions, i.e., $\nabla \bot \Rightarrow \bot$, $\nabla (A \vee B) \Rightarrow \nabla A \vee \nabla B$ and $\nabla A \vee \nabla B \Rightarrow \nabla (A \vee B)$. Fourthly, the system proves the sequent $A \to B \Rightarrow C \otimes A \to C \otimes B$:
[c]{}
Note that the defined extensions of the system $\mathbf{STL}$ can be also axiomatized with some axioms instead of rules. For $(N)$ the axioms are $\Rightarrow \nabla 1$ and $\nabla A \otimes \nabla B \Rightarrow \nabla (A \otimes B)$. These are provable by the rule $(N)$ because:
-- --
-- --
The converse is also true. For the empty $\Gamma$, if $\Rightarrow A$, then by $(L1)$, we have $1 \Rightarrow A$. By $\nabla$ we have $\nabla 1 \Rightarrow \nabla A$. Hence, by $\Rightarrow \nabla 1$ we have $\Rightarrow \nabla A$. For $\Gamma$ with at least one element, by induction, it is possible to use the axiom to prove that $\bigotimes (\nabla \Gamma) \Rightarrow \nabla (\bigotimes \Gamma) $. Hence,
[c]{}
where the double line means the existence of an easy omitted proof tree there. Therefore, since $\nabla 1 \Rightarrow 1$ and $\nabla (A \otimes B) \Rightarrow \nabla A \otimes \nabla B$ are already provable in $\mathbf{STL}$ without $(N)$, the rule $(N)$ just states the strictness of $\nabla$, i.e., for any sequence $\Gamma$, the sequents $\bigotimes (\nabla \Gamma)$ and $\nabla (\bigotimes \Gamma)$ are equivalent. This justifies the name of the rule, $(N)$, that stands for normality, reflecting the normality condition of the usual conjunction-preserving modalities. For $(H)$, note that this rule implies the rule $(N)$ for $I=\emptyset$. It also implies that $\nabla A \to \nabla B \Rightarrow \nabla (A \to B)$ because:
[c c]{}
Therefore, $H$ implies $(\Rightarrow \nabla 1$), ($\nabla A \otimes \nabla B \Rightarrow \nabla (A \otimes B)$) and ($\nabla A \to \nabla B \Rightarrow \nabla (A \to B) $). These are enough to prove $(H)$ because the first part implies the rule $(N)$ and then
[c]{}
Moreover, in the presence of $(H)$ we also have:
[c]{}
Therefore, the rule $(H)$ is equivalent to the strictness of $\nabla$ and the equivalence between $\nabla (A \to B)$ and $\nabla A \to \nabla B$. We will see that these conditions when applied on a locale of the open subsets of a topological space is equivalent to the condition that $\nabla$ be the inverse image of a homeomorphism. This justifies the name of the rule, $(H)$. For $(P)$ and $(F)$, they are equivalent to $\nabla A \Rightarrow A$ and $A \Rightarrow \nabla A$, respectively. $(P)$ stands for past and $(F)$ for future, reflecting the temporal nature of the modality $\nabla$. We will see the details in Section \[KripkeModels\]. Finally, $(wF)$ is equivalent to $1 \to \bot \Rightarrow \bot$. It is provable via $(wF)$ because
[c c]{}
Conversely, if we have the axiom $1 \to \bot \Rightarrow \bot$, then
[c]{}
In this rule, $(wF)$ stands for “weak future", since the rule $(F)$ clearly implies $(wF)$.
\[t4-1\](Topological Semantics) Let $\mathcal{S}=(\mathscr{X}, \nabla_{\mathcal{S}})$ be a non-commutative spacetime and $V:\mathcal{L}_{\nabla} \to\mathscr{X}$ an assignment. A tuple $(\mathcal{S}, V)$ is called a topological model for the language $\mathcal{L}_{\nabla}$ if:
- $V(1)=e$, $V(\bot)=0$ and $V(\top)=1$,
- $V(A \wedge B)=V(A) \wedge V(B)$,
- $V(A \vee B)=V(A) \vee V(B)$,
- $V(A \otimes B)=V(A) \otimes V(B)$,
- $V(\nabla A)=\nabla_{\mathcal{S}} V(A)$,
- $V(A \rightarrow B)= V(A)\to_{\mathcal{S}} V(B)$.
We say $(\mathcal{S}, V) \vDash \Gamma \Rightarrow A$ when $\bigotimes_{\gamma \in \Gamma} V(\gamma) \leq V(A)$ and $\mathcal{S} \vDash \Gamma \Rightarrow A$ when for all $V$, $(\mathcal{S}, V) \vDash \Gamma \Rightarrow A$. For a class $\mathcal{C}$ of non-commutative spacetimes, we write $\mathcal{C} \vDash \Gamma \Rightarrow A$ if for any $\mathcal{S} \in \mathcal{C}$ we have $\mathcal{S} \vDash \Gamma \Rightarrow A$. Moreover, if for some fixed $\mathscr{X}$ and for all $(\mathscr{X}, \nabla)$ in some class $\mathcal{C}$ we have $(\mathscr{X}, \nabla) \vDash \Gamma \Rightarrow A$, we write $\mathscr{X} \vDash_{\mathcal{C}} \Gamma \Rightarrow A$. If $\mathscr{X}$ is $\mathcal{O}(X)$ for some topological space, we simplify it more to $X \vDash_{\mathcal{C}} \Gamma \Rightarrow A$. Furthermore, we omit the symbol $\Rightarrow$ whenever $\Gamma$ is empty.
Let $\mathcal{A}=(A, \leq, \otimes, e, \to, \nabla)$ be atemporal algebra. Then for any rule scheme $R \in \{N, H, P, F, wF\}$, we say $\mathcal{A}$ satisfies $R$ if:
- $\nabla$ preserves all finite multiplications,
- $\nabla$ preserves all the structure including the implication,
- For any $a \in A$ we have $\nabla a \leq a$,
- For any $a \in A$ we have $a \leq \nabla a$,
- (If $\mathcal{A}$ has zero) for any $a \in A$, if $\nabla a =0$ then $a=0$.
For any set of rule schemes $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, by the class $\mathbf{ST}(\mathcal{R})$ we mean the class of all non-commutative sapcetimes $(\mathscr{X}, \nabla)$ that satisfies all the rule schemes in $\mathcal{R}$. The class $i\mathbf{ST}(\mathcal{R})$ is defined similarly for spacetimes, i.e., non-commutative spacetimes with locales.
\[HforAlgebra\] Note that the condition $(H)$ implies that $\nabla$ is an isomorphism with the inverse $\Box=e \to (-)$. The proof is the following. Since $\nabla e=e$ we have $$e \to \nabla a=\nabla e \to \nabla a=\nabla (e \to a)$$ but since $\nabla \dashv e \to (-)$, we have $\nabla (e \to a) \leq a \leq e \to \nabla a$. Hence, $\nabla (e \to a)=a=e \to \nabla a$. This means that $\nabla$ and $\Box$ are inverses of each other over $A$.
\[HforSpace\] Note that for non-commutative spacetimes, the conditions $(N)$ and $(H)$ are equivalent to “$\nabla$ is a strict geometric morphism" and “$\nabla$ is a strict geometric isomorphism", respectively. The reason for the first one is that $\nabla$ has a right adjoint and hence preserves all joins. Hence, the only geometricity condition is the preservation of multiplications. For the second, we have to show that if $\nabla$ is a geometric isomorphism, then it also preserves the implication. Let $\mathcal{S}=(\mathscr{X}, \nabla)$ be a non-commutative spacetime where $\nabla$ is a strict geometric isomorphism. Then, to reduce the risk of confusion, let us denote $\nabla$ by $f$. We know that $f$ has an inverse. Call it $g$. Since they are inverses, we have $g \dashv f$. Then since $f$ preserves $\nabla$, it can be seen as a geometric map between non-commutative spaces, i.e., $f: \mathcal{S} \to \mathcal{S}$. Finally, by Theorem \[Logical\], to prove it is logical meaning that it respects the implication, it is enough to check that $
g(f b \otimes \nabla a)=b \otimes \nabla g a
$. Since $f=\nabla$ is strict and $gf =id=fg$ we have $
g (f b \otimes f a)=gf (b \otimes a)=b \otimes fg a
$. Therefore, $f=\nabla$ preserves the implication.
\[t4-2\](Soundness) For any set of rule schemes $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, if $\Gamma \vdash_{\mathbf{STL}(\mathcal{R})} A$ then $\mathbf{ST}(\mathcal{R}) \vDash \Gamma \Rightarrow A$. Specially, if $\Gamma \vdash_{i\mathbf{STL}(\mathcal{R})} A$ then $i\mathbf{ST}(\mathcal{R}) \vDash \Gamma \Rightarrow A$.
Since the logics are just the syntactical elementary representations of the structure of the non-commutative spacetimes, the soundness theorem is clear and we will leave the details to the reader. There are only four points to make. First about the rule $Oplax$ and the axiom $\nabla 1 \Rightarrow 1$. They are clearly valid whenever the interpretation of $\nabla$ is oplax. Hence, they are valid in our topological interpretation. Secondly, consider the rule $R \to$. If $\Gamma \Rightarrow A \to B$ is proved by $A, \nabla \Gamma \Rightarrow B$, then by IH, for any non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla_{\mathcal{S}})$ and any $V: At(\mathcal{L}_{\nabla}) \to \mathscr{X}$ we have: $V(A) \otimes \bigotimes_{\gamma \in \Gamma} \nabla_{\mathcal{S}} V(\gamma) \leq V(B)$. Since $\nabla_{\mathcal{S}}$ is oplax, we have $V(A) \otimes \nabla_{\mathcal{S}} (\bigotimes_{\gamma \in \Gamma} V(\gamma)) \leq V(B)$. By adjunction, we have $\bigotimes_{\gamma \in \Gamma} V(\gamma) \leq V(A) \to_{\mathcal{S}} V(B)$. Therefore, the rule $R \to$ is also valid. Thirdly, note that all the rule schemes are equivalent to some axioms and those axioms are exactly the corresponding conditions on the non-commutative spacetimes. Hence, their validity is evident. Finally, note that for the spacetimes $\otimes=\wedge$ and $e=1$. Therefore, it is clear that all the structural rules are valid.
To prove the completeness theorem, we need the Lindenbaum construction together with a completion technique. For the former, set $L=\mathbf{ST}(\mathcal{R})$. Define $\mathcal{B}(L)$ to be the set of all formulas of the language $\mathcal{L}_{\nabla}$ with the equivalence relation $\equiv$ as $A \equiv B$ iff $L \vdash A \Rightarrow B$ and $L \vdash B \Rightarrow A$. It is clear that $(\mathcal{B}(L)/\equiv, \vdash)$ is a monoidal poset with all finite meets and all finite joins. Moreover it is also a distributive temporal algebra with its canonical $\nabla$ and $\rightarrow$ such that $[A] \otimes \nabla(-)$ is a left adjoint to $[A] \rightarrow (-)$. See Remark \[PropertiesOgfLogic\]. For the completion technique we have the following representation theorem, presented in Section \[Rep\]. Here we present it in a slightly stronger form to also address the rule schemes.
\[RepFromTemp\] Let $\mathcal{A}=(A, \leq_A, \otimes_A, e_A, \nabla_A, \to_A)$ be a (distributive) temporal algebra. Then there exists a non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla)$ and a (join preserving) temporal algebra embedding $i: \mathcal{A} \to \mathcal{S}$. Moreover, if the algebra has all finite meets, $i$ preserves them and if $\mathcal{A}$ satisfies a rule scheme $\mathcal{R} \subseteq \{N, H, P, F\}$, then so does $\mathcal{S}$. The same is also true for $(wF)$ if $\mathcal{A}$ is distributive.
First let us address the case in which the temporal algebra does not necessarily have all the joins. Let $\mathscr{X}=D(\mathcal{A})$ be the downset completion of $\mathcal{A}$ and define $$\nabla I = (\nabla_A)_{!}= \{x \in A| \exists i \in I \; (x \leq \nabla_A i)\}$$ First observe that $\nabla$ maps downsets to downsets. Secondly, note that by Theorem \[LiftingMonoidalMaps\], $\nabla$ is join preserving and since $\nabla_A$ is oplax, $(\nabla_A)_{!}$ is also oplax. Therefore, $\nabla$ has a right adjoint by adjoint functor theorem, Theorem \[AFT\]. Now let us provide the explicit adjoint. Define and $$I \to J=\{x \in A| \; \forall i \in I \; (i \otimes \nabla_A x \in J)\}$$ Again observe that $\to$ maps downsets to downsets. Then note that for any $I \in \mathscr{X}$, the map $I \otimes \nabla (-) \dashv (I \to (-))$ because for any $I, J, K \in \mathscr{X}$ we have $$I \otimes \nabla J \subseteq K \; \; \text{iff} \; \; I \subseteq J \to K$$ For the left to right, note that if $i \in I$, then for any $j \in J$, we have $i \otimes \nabla j \in I \otimes \nabla J \subseteq K$ and hence $i \otimes \nabla j \in K$. Hence, $I \subseteq J \to K$. Conversely, if $I \subseteq J \to K$ and $x \in I \otimes \nabla J$, then there exist $i \in I$ and $j \in J$ such that $x \leq i \otimes \nabla j$. Since $i \in I \subseteq J \to K$, by the definition of the implication we have $i \otimes \nabla j \in K$. Hence, $x \in K$.\
Finally, define $i(a)= \{x \in A | x \leq a\}$. Then by Theorem \[Completions\], the map $i$ is a monoidal poset’s embedding that preserves finite meets (if they exist). Moreover, by Theorem \[LiftingMonoidalMaps\], $i$ also preserves $\nabla$ i.e., $i\nabla_{A} a=\nabla i(a)$, for any $a \in A$. For implication: $$i(a \to b)=\{x \in A | x \leq a \to b\}=\{x \in A | a \otimes \nabla x \leq b\}=$$ $$\{x \in A | \forall y \leq a \; (y \otimes \nabla x \leq b)\}=i(a) \to i(b)$$ Now, let us move to the distributive case. In this case, we have to move from the downset completion to the ideal completion with the same monoidal structure. By Theorem \[LiftingMonoidalMaps\], since $\nabla$ is join preserving so does $(\nabla_A)_!$. Moreover, the same $i$ as before is a join preserving monoidal embedding that respects $\nabla$ and finite meets (if they exist). The only thing we have to check is the stability of the ideals under the implication. This implies that the previous proofs for adjunction $I \otimes \nabla (-) \dashv (I \to (-))$, for any ideal $I$ and preservability of implication under $i$ work again. First note that $0 \in I \to J$ because for any $i \in I$, we have $i \otimes \nabla 0=i \otimes 0=0 \in J$. The last equality is the consequence of distributivity of $\mathcal{A}$. And secondly, note that if $x, y \in I \to J$, then for all $i \in I$, we have $i \otimes \nabla x \in J$ and $i \otimes \nabla y \in J$. Since $J$ is an ideal, $\nabla$ preserves joins and $\mathcal{A}$ is distributive, we have $$[i \otimes \nabla x] \vee [i \otimes \nabla y]=[i \otimes \nabla (x \vee y)] \in J$$ which proves that $I \to J$ is an ideal.\
Finally, for the rule schemes, we have to show that the previous downset or ideal construction respects the rule schemes. For all schemes, except $(wF)$, it is enough to prove the scheme for all downsets. The scheme for the ideals is just its special case.\
For $(N)$, note that $\nabla_A$ is lax and hence by Theorem \[LiftingMonoidalMaps\], $(\nabla_A)_!$ is also lax. Being lax is nothing but satisfying $(N)$.\
For $(H)$, note that if $\mathcal{A}$ satisfies $(H)$, by Remark \[HforAlgebra\], $\nabla$ and $\Box$ are inverses of each other over $\mathcal{A}$. This fact lifts also to $\mathcal{S}$. It is enough to prove that for any ideal $I$, we have $\nabla \Box I=I=\Box \nabla I$. We prove $I \subseteq \nabla \Box I$. The rest is similar. Assume $i \in I$, then $i=\nabla \Box i$. For the sake of readability, let $j=\Box i$. Then $\nabla j=i$. We have $j \in \Box I$ because $e \otimes \nabla j=i \in I$. Therefore, $i=\nabla j \in \nabla \Box I$. Finally, since $\nabla$ has an inverse and is join preserving and strict, it will be a strict geometric isomorphism. The claim follows from Remark \[HforSpace\].\
For $(P)$, we have $\nabla I \subseteq I$ because if $x \in \nabla I$, then there exists $i \in I$ such that $x \leq \nabla i$. Since $\nabla i \leq i$, we have $x \leq i \in I$ which implies $x \in I$. For $(F)$, we have $I \subseteq \nabla I$, because for any $i \in I$ we have $i \leq \nabla i$ which implies $i \in \nabla I$. Finally, for $(wF)$, if $\nabla I=\{0\}$ and $i \in I$, we have $\nabla i \in \nabla I=\{0\}$ which implies $\nabla i=0$. Since $\mathcal{A}$ satisfies $(wF)$, we have $i=0$ that proves $I=\{0\}$.
\[t4-4\](Completeness) For any rule scheme $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, there exists a non-commutative spacetime $\mathcal{S} \in \mathbf{ST}(\mathcal{R})$ such that if $\mathcal{S} \vDash \Gamma \Rightarrow A$ then $\Gamma \vdash_{\mathbf{STL}(\mathcal{R})} A$. The same is also true for $i\mathbf{ST}(\mathcal{R})$ and $i\mathbf{STL}(\mathcal{R})$.
Since the Lindenbaum algebra for $\mathbf{STL}(\mathcal{R})$ is clearly a finitely complete distributive temporal algebra, by Theorem \[RepFromTemp\], there exists a non-commutative spacetime $\mathcal{S}=(\mathscr{X}, \nabla)$ and a finite meet and finite join preserving temporal embedding $i : \mathcal{B}(L) \to \mathcal{S}$. Define $V(p)=i([p])$. It is easy to check that for all formula $C \in \mathcal{L}_{\nabla}$, we have $V(C)=i([C])$. Since $(\mathcal{S}, V) \vDash \Gamma \Rightarrow A$ we have $\bigotimes_{\gamma \in \Gamma} V(\gamma) \leq V(A)$. Hence, $\bigotimes_{\gamma \in \Gamma} i([\gamma]) \leq i([A])$. Since $i$ preserves the monoidal structure and is an embedding, $\bigotimes_{\gamma \in \Gamma}[\gamma] \leq [A]$ or equivalently $\Gamma \vdash_{\mathbf{STL}\mathcal{R})} A$. For the structural version, note that by Remark \[2\], the ideal construction in Theorem \[RepFromTemp\], applied on a monoidal poset with meet structure as its monoidal structure, produces a locale for $\mathscr{X}$.
One of the main advantages of the spacetime logics over the usual sub-intuitionistic logics is their complete pairs of introduction-elimination rules. This well-behaved nature may find some evidence by the following translation that interprets the seemingly more powerful logics into the weaker ones. The translation is the syntactical version of Theorem \[AlgebraicTranslation\]. It helps to import what we have in sub-structural and intuitionistic tradition to the spacetime logics. It also shows that $\mathbf{STL}$ and $i\mathbf{STL}$ are in some sense more powerful that the usual $\mathbf{FL}_l$ and $\mathbf{IPC}$, respectively. In this sense the former refine the timeless spatial structure of the latter by bringing the more temporal and hence more expressive power.
Define the translation $(-)^{\nabla}:\mathcal{L} \to \mathcal{L}_{\nabla}$ as the following, where $\mathcal{L}=\{\wedge, \vee, \top, \bot, 1, \otimes, \to\}$:
- $p^{\nabla}=\nabla \Box p$, $\bot^{\nabla}=\bot$, $\top^{\nabla}=\nabla \Box \top$ and $1^{\nabla}=1$.
- $(A \wedge B)^{\nabla}=\nabla \Box (A^{\nabla} \wedge B^{\nabla})$.
- $(A \vee B)^{\nabla}= A^{\nabla} \vee B^{\nabla}$.
- $(A \otimes B)^{\nabla}=A^{\nabla} \otimes B^{\nabla}$.
- $(A \to B)^{\nabla}=\nabla (A^{\nabla} \to B^{\nabla})$.
For any $\Gamma \cup \{A\} \subseteq \mathcal{L}$,
- $\Gamma \vdash_{\mathbf{FL}_l} A$ iff $ \Gamma^{\nabla} \vdash_{\mathbf{STL}(N)} A^{\nabla}$.
- $\Gamma \vdash_{\mathbf{IPC}} A$ iff $ \Gamma^{\nabla} \vdash_{i\mathbf{STL}(N)} A^{\nabla}$. (Originally proved in [@AMM].)
We will prove $(i)$, the proof for $(ii)$ is the same. For that matter, we will first prove a claim that for any formula $A \in \mathcal{L}$, there exists a formula $A' \in \mathcal{L}_{\nabla}$ such that $A^{\nabla} \vdash_{\mathbf{STL}(N)} \nabla A'$ and $\nabla A' \vdash_{\mathbf{STL}(N)} A^{\nabla}$. The proof for the claim is by induction on the structure of $A$. For atomic cases, considering the fact that $\nabla \bot$ is equivalent to $\bot$, there is nothing to prove. The claim for conjunction and implication is clear by definition of the translation. Finally, for $\otimes$ and $\vee$, note that the translation $(-)^{\nabla}$ commutes with these connectives. Therefore, if there exist $A'$ and $B'$ for $A^{\nabla}$ and $B^{\nabla}$, respectively, for $A \otimes B$ it is enough to consider $A' \otimes B'$. The reason is that $\nabla$ commutes with $\otimes$ because of $(N)$. For $\vee$ the same strategy works. Therefore, the existence of $A'$ is proved. This property implies the following useful fact: For any $B \in \mathcal{L}_{\nabla}$, if $\Gamma^{\nabla} \vdash_{\mathbf{STL}(N)} B$, then $\Gamma^{\nabla} \vdash_{\mathbf{STL}(N)} \nabla \Box B$. The reason is the following. Since the formula $\bigotimes \Gamma^{\nabla}$ is equivalent to $(\bigotimes \Gamma)^{\nabla}$ and the latter is also equivalent to $\nabla C$ for some $C \in \mathcal{L}_{\nabla}$, it is enough to prove the claim for $\nabla C$. Now, since $\nabla C \vdash_{\mathbf{STL}(N)} B$, by $(L1)$ we have $1, \nabla C \vdash_{\mathbf{STL}(N)} B$. By implication introduction we have $C \vdash_{\mathbf{STL}(N)} \Box B$ and hence by the rule $(\nabla)$, we have $\nabla C \vdash_{\mathbf{STL}(N)} \nabla \Box B$.\
Coming back to the proof of the theorem, for the soundness part it is enough to use an induction on the $\mathbf{FL}_l$-proof length of $\Gamma \Rightarrow A$. For axioms, all cases are clear, except $\Gamma \Rightarrow \top$. For this case we have to prove $\Gamma^{\nabla} \vdash \nabla \Box \top$ which is clear from what we observed above.\
For the conjunction rule $(R\wedge)$, assume $\Gamma \Rightarrow A \wedge B$ is proved via $\Gamma \Rightarrow A$ and $\Gamma \Rightarrow B$. Then by IH, we have $\Gamma^{\nabla} \Rightarrow A^{\nabla}$ and $\Gamma^{\nabla} \Rightarrow B^{\nabla}$. Then $\Gamma^{\nabla} \Rightarrow A^{\nabla} \wedge B^{\nabla}$. Therefore, by what we have above $\Gamma^{\nabla} \Rightarrow \nabla \Box (A^{\nabla} \wedge B^{\nabla})$. For the conjunction rule $(L\wedge)$, assume $\Gamma, A \wedge B, \Sigma \Rightarrow C$ is proved via $\Gamma, A, \Sigma \Rightarrow C$. Then by IH, $\Gamma^{\nabla}, A^{\nabla}, \Sigma^{\nabla} \Rightarrow C^{\nabla}$. Then $\Gamma^{\nabla}, A^{\nabla} \wedge B^{\nabla}, \Sigma^{\nabla} \Rightarrow C^{\nabla}$. Since $\nabla \Box (A^{\nabla} \wedge B^{\nabla}) \Rightarrow A^{\nabla} \wedge B^{\nabla}$, we have $\Gamma^{\nabla}, (A \wedge B)^{\nabla}, \Sigma^{\nabla} \Rightarrow C^{\nabla}$.\
For implication rule $(R\to)$, assume $\Gamma \Rightarrow A \to B$ is proved via $A, \Gamma \Rightarrow B$. Then by IH, we have $A^{\nabla}, \Gamma^{\nabla} \Rightarrow B^{\nabla}$. Since $\Gamma^{\nabla}$ is equivalent to $(\bigotimes \Gamma)^{\nabla}$, it is also equivalent to $\nabla C$ for some $C$. We have $A^{\nabla}, \nabla C \Rightarrow B^{\nabla}$. Hence, $ C \Rightarrow (A^{\nabla} \to B^{\nabla})$. Hence, by $(\nabla)$ we have $ \nabla C \Rightarrow \nabla (A^{\nabla} \to B^{\nabla})$. Since $\bigotimes \Gamma^{\nabla}$ is equivalent to $\nabla C$, we have $ \Gamma^{\nabla} \Rightarrow \nabla (A^{\nabla} \to B^{\nabla})$. For implication rule $(L \to)$, assume $\Pi, \Gamma, (A \to B), \Sigma \Rightarrow C$ is proved via $\Gamma \Rightarrow A$ and $\Pi, B, \Sigma \Rightarrow C$. Then by IH, $\Gamma^{\nabla} \Rightarrow A^{\nabla}$ and $\Pi^{\nabla}, B^{\nabla}, \Sigma^{\nabla} \Rightarrow C^{\nabla}$. Hence, $\Pi^{\nabla}, \Gamma^{\nabla}, \nabla (A^{\nabla} \to B^{\nabla}), \Sigma^{\nabla} \Rightarrow C^{\nabla}$.\
For completeness, note that if $\Gamma^{\nabla} \Rightarrow A^{\nabla}$ is provable in $\mathbf{STL}(N)$, then it is also provable in the greater logic $\mathbf{FL}_l=\mathbf{STL}(N, P, F)$. Since for any $B \in \mathcal{L}$, the formulas $B^{\nabla}$ and $B$ are equivalent in $\mathbf{STL}(N, P, F)$, the sequent $\Gamma \Rightarrow A$ is also provable in $\mathbf{FL}_l$.
Kripke Models {#KripkeModels}
=============
In this section we will focus on the structural logics of spacetime and their Kripke semantics. This semantics is essentially the usual Kripke semantics for the intuitionistic modal and implication logics [@Sim], [@Servi] and [@LitViss]. However, to also address $\nabla$, we will add a natural forcing condition using the same accessibility relation that the model uses for $\Box$. In this sense, the structural logics of spacetime are actually the result of a faithful extension of the language and logics to have a better reflection of the Kripke models into the pure syntax.
By a Kripke model for the language $\mathcal{L}_{\nabla}$, we mean a tuple $\mathcal{K}=(W, \leq, R, V)$ where $(W, \leq)$ is a poset, $R \subseteq W \times W$ is a relation over $W$ (not necessarily transitive or reflexive) compatible with $\leq$, i.e., for all $u, u', v, v' \in W$ if $(u, v) \in R$ and $u' \leq u$ and $v \leq v'$ then $(u', v') \in R$ and $V: At(\mathcal{L}_{\nabla}) \to U((W, \leq))$ where $At(\mathcal{L}_{\nabla})$ is the set of atomic formulas of $\mathcal{L}_{\nabla}$ and $U((W, \leq))$ is the set of all upsets of $(W, \leq)$. Define the forcing relation as usual using the relation $R$ and for the $\nabla$ let $u \Vdash \nabla A$ if there exists $v \in W$ such that $(v, u) \in R$ and $v \Vdash A$. A Kripke model is called normal if there exists an order preserving function $\pi : W \to W$ such that $(u, v) \in R$ iff $u \leq \pi(v)$. It is clear that if this $\pi$ exists, it would be unique. Finally, a sequent $\Gamma \Rightarrow A$ is valid in a Kripke model if for all $w \in W$, $\forall B \in \Gamma \; (w \Vdash B)$ implies $w \Vdash A$.
(Monotonicity Lemma) For any formula $A \in \mathcal{L}_{\nabla}$, any Kripke model $\mathcal{K}=(W, \leq, R, V)$ and any $u, v \in W$, if $u \leq v$ and $u \Vdash A$ then $v \Vdash A$.
The proof is a routine induction on the structure of $A$. The only case to mention is when $A=\nabla B$. Then if $u \Vdash \nabla B$, there exist $u' \in W$ such that $(u', u) \in R$ and $u' \Vdash B$. Since $u \leq v$ and $R$ is compatible with $\leq$, we have $(u', v) \in R$. Therefore, $v \Vdash \nabla A$.
Note that in a normal Kripke model $w \Vdash \nabla A$ iff $\pi(w) \Vdash A$. One direction is clear, for the other, if there exists $u \in W$ such that $(u, w) \in R$ and $u \Vdash A$, then since $u \leq \pi(w)$, by the monotonicity lemma we have $\pi(w) \Vdash A$. This means that the normal Kripke models are the models in which we have a canonical way to witness the existential quantifier in the forcing condition of $\nabla$.
For any rule scheme in the set $\{N, H, P, F, wF\}$, we define a corresponding condition on a Kripke model as:
- The model is normal.
- The model is normal and its $\pi$ is a poset isomorphism.
- $R \subseteq \; \leq$. For a normal model, it is equivalent to $\forall w \in W \; (\pi(w) \leq w)$.
- $R$ is reflexive, i.e., for all $w \in W$ we have $(w, w) \in R$. For a normal model, it is equivalent to $\forall w \in W \; (w \leq \pi(w))$.
- $R$ is serial, i.e., for all $u \in W$ there exists $v \in W$ such that $(u, v) \in R$. For a normal model, it is equivalent to $\forall u \in W \exists v \in W \; (u \leq \pi(v))$.
Moreover, if $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, by a $\mathbf{K}(\mathcal{R})$-Kripke model we mean a model satisfying the conditions corresponding to all the schemes in $\mathcal{R}$.
(Soundness) For any rule scheme $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, the logic $i\mathbf{STL}(\mathcal{R})$ is sound for all $\mathbf{K}(\mathcal{R})$-Kripke models.
Our strategy is reducing the soundness for Kripke models to soundness for topological models. It is also possible to prove it directly. However, we follow this strategy to also show how Kripke models must be considered as the special case of the topological models. For that purpose, we show how to assign a topological model to a Kripke model with the same valid sequents. Moreover, we will show that this construction respects the schema conditions. Let $\mathcal{K}=(W, \leq, R, V)$ be a Kripke model. Define the spacetime $\mathcal{S}_{\mathcal{K}}=(U(W, \leq), \nabla_{\mathcal{K}})$ as in Example \[KripkeToSpacetime\] by $$\nabla_{\mathcal{K}} P=\{w \in W | \exists u \in W \; [(u, w) \in R \; \text{and} \; u \in P]\}$$ For any formula $B \in \mathcal{L}_{\nabla}$ define $[B]$ as the set $\{w \in W | w \Vdash B\}$. By the monotonicity lemma, $[B]$ is an upset of $W$. If we define the topological valuation $\bar{V}(p)=V(p)$, it is easy to see that $\bar{V}(B)=[B]$ for any formula $B \in \mathcal{L}_{\nabla}$. Hence, for any sequent $\Gamma \Rightarrow A$, it is valid in $(U(W, \leq), \nabla_{\mathcal{K}}, \bar{V})$ iff $\bigwedge_{\gamma \in \Gamma} \bar{V}(\gamma) \subseteq \bar{V}(A)$ iff $\bigcap_{\gamma \in \Gamma} [\gamma] \subseteq [A]$ which is nothing but the validity of $\Gamma \Rightarrow A$ in $\mathcal{K}$.\
It is remaining to prove the preservation of the schema conditions. First note that for $(N)$, the existence of $\pi$ means that $\nabla_{\mathcal{K}}=\pi^{-1}$. Therefore, $\nabla$ preserves all intersections and hence is a strict geometric map. For $(H)$, since $\pi$ is an order isomorphism, it has an inverse $\rho$. Then $\pi^{-1}, \rho^{-1}: U(W, \leq) \to U(W, \leq)$ are each other’s inverses. Hence, $\nabla_{\mathcal{K}}=\pi^{-1}: U(W, \leq) \to U(W, \leq)$ is a strict geometric isomorphism. For $(P)$, we have $\nabla_{\mathcal{K}} P \subseteq P$. The reason is that if $w \in \nabla_{\mathcal{K}} P$, there exist $u \in W$ such that $(u, w) \in R$ and $u \in P$. Since $R \subseteq \; \leq$, we have $u \leq w$. Since $P$ is an upset we have $w \in P$. For $(F)$, we have $P \subseteq \nabla_{\mathcal{K}} P$ because if $w \in P$ then since $(w, w) \in R$ we have $w \in \nabla_{\mathcal{K}} P$. And finally, for $(wF)$, if $\nabla_{\mathcal{K}} P=\emptyset$, then $P=\emptyset$ because if $w \in P$ then since $R$ is serial, there exists $u \in W$ such that $(w, u) \in R$ which means that $u \in \nabla_{\mathcal{K}} P=\emptyset$. This is a contradiction and hence $P=\emptyset$.
Let $\mathcal{A}=(A, \leq, \wedge, 1, \to)$ be a strong algebra where $(A, \leq)$ is finitely cocomplete. Then $\mathcal{A}$ is called join internalizing if $(a \vee b \to c)=(a \to c) \wedge (b \to c)$, for every $a, b, c \in A$.
For completeness, we need the following representation theorem that we stated in Section \[Rep\]. The proof is essentially the canonical extension construction in [@CJ1] expanded to also cover both weaker and stronger cases. In fact, in [@AMM], we modified this construction to also address the operator $\nabla$. Since [@AMM] is not accessible yet, we restate the full details and we add the proofs for some other cases that are absent in [@AMM].
\[KripkeRepresentation\] For any (distributive) strong algebra $\mathcal{A}=(A, \leq, \wedge, 1, \to)$ that internalizes its monoidal structure \[not necessarily its closed monoidal structure if it has any\] (and internalizes its join structure), there exists a Kripke frame $\mathcal{K}$ and a (join preserving) strong algebra embedding $i: \mathcal{A} \to \mathcal{S}_{\mathcal{K}}$. Moreover, if $\mathcal{A}$ is a reduct of a (distributive) temporal algebra, $i$ also preserves $\nabla$ and for any rule scheme $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, if $\mathcal{A}$ satisfies $\mathcal{R}$, then so does $\mathcal{K}$.
We split the proof to four cases depending on the presence of joins and $\nabla$. For all cases, we need the following constructions. Recall that $F(\mathcal{A})$ is the poset of all filters of $\mathcal{A}$ and define $\mathcal{R}$ as a binary relation over $F(\mathcal{A})$ as: $(P, Q) \in \mathcal{R}$ iff for all $a, b \in A$ if $a \to b \in P$ and $a \in Q$ then $b \in Q$.\
*Case I*. In this case both joins and $\nabla$ are not necessarily present. Define $W=F(\mathcal{A})$ and its order as the equality on $W$. Then it is clear that $\mathcal{K}_1=(W, =, \mathcal{R})$ is a Kripke frame in the sense of Example \[KripkeToSpacetime\]. Consider $i: \mathcal{A} \to U(W, =_W)$ defined by $i(a)=\{P \in F(\mathcal{A}) | \; a \in P\}$. As we observed in the Preliminaries, $i$ is clearly a meet-semilattice embedding. Note that for any $X$ and $Y$ as the upsets of $(W,=_W)$, the implication in $\mathcal{S}_{\mathcal{K}_1}$ is defined by: $$X \to Y=\{P \in F(\mathcal{A}) | \forall Q \in F(\mathcal{A}) \; \text{if} \; (P, Q) \in \mathcal{R} \; \text{and} \; Q \in X \; \text{then} \; Q \in Y\}$$ To prove that $i$ preserves the implication, i.e., $i(a \to b)= i(a) \to i(b)$, we have to check the following two directions:\
To prove $i(a \to b) \subseteq i(a) \to i(b)$, if $P \in i(a \to b)$ then $a \to b \in P$. Then assume $(P, Q) \in \mathcal{R}$ and $Q \in i(a)$. Hence, $a \in Q$ and since $a \to b \in P$, by the definition of $\mathcal{R}$, we have $b \in Q$, meaning $Q \in i(b)$. Therefore, $P \in i(a) \to i(b)$. Conversely, if $P \in i(a) \to i(b)$, then consider $Q=\{x \in A | a \to x \in P\}$. Since $\mathcal{A}$ internalizes its meet structure, by Remark \[MonoidalInternalForMeet\], we have $$(a \to x) \wedge (a \to y)= (a \to x \wedge y)$$ which means that $Q$ is a filter. Moreover, $a \in Q$ because $a \to a=1 \in P$. Note that $(P, Q) \in \mathcal{R}$ because if $x \to y \in P$ and $x \in Q$, then $a \to x \in P$ and since $$(a \to x) \wedge (x \to y) \leq (a \to y)$$ and $P$ is a filter, we have $a \to y \in P$ which means $y \in Q$. Therefore, $(P, Q) \in \mathcal{R}$. Now, since $a \in Q$ we have $Q \in i(a)$. Since $P \in i(a) \to i(b)$ and $(P, Q) \in \mathcal{R}$ we have $Q \in i(b)$, meaning $b \in Q$ which by the definition of $Q$ means $a \to b \in P$.\
*Case II.* In this case, again joins are not necessarily present. However, the algebra $\mathcal{A}$ is a reduct of a temporal algebra. Therefore, there exists $\nabla : A \to A$ such that for any $a \in A$, $a \wedge \nabla (-) \dashv (a \to -)$. First note that the relation $\mathcal{R}$ on $F(\mathcal{A})$ is also definable by $\nabla$ as $(P, Q) \in \mathcal{R}$ iff $\nabla [P]=\{\nabla x | x \in P\} \subseteq Q$. The reason is the following: If $(P, Q) \in \mathcal{R}$ and $x \in P$, since $x \leq 1 \to \nabla x$ and $P$ is a filter, $1 \to \nabla x \in P$. Therefore, by $1 \in Q$ and $(P, Q) \in \mathcal{R}$ we have $\nabla x \in Q$. Hence, $\nabla [P] \subseteq Q$. Conversely, if $\nabla [P] \subseteq Q$, given $a \to b \in P$ and $a \in Q$ we have $\nabla (a \to b) \in \nabla [P] \subseteq Q$ and since $a \wedge \nabla (a \to b) \leq b$ we have $b \in Q$. Therefore, $(P, Q) \in \mathcal{R}$.\
Defining $\mathcal{R}$ in terms of $\nabla$ has the advantage to make $\mathcal{R}$ monotone also in its second argument, i.e, if $(P, Q) \in \mathcal{R}$ and $Q \subseteq Q'$, then $(P, Q') \in \mathcal{R}$. For this part, pick $W=F(\mathcal{A})$ as before and change the order on $W$ to $\subseteq$. Since $\mathcal{R}$ is compatible with $\subseteq$, the tuple $\mathcal{K}_2=(W, \subseteq, \mathcal{R})$ is a Kripke frame. Moreover, note that $i(a)$ for any $a \in A$ is an upset with respect to $\subseteq$. For the preservation of the implication, since it does not depend on the order on $W$, the argument for the previous case also works here. Therefore, we only have to show that $i$ preserves $\nabla$, i.e., $i(\nabla a)=\nabla i(a)$. If $P \in i(\nabla a)$, then $\nabla a \in P$. Pick $Q=\{x\in A | x \geq a\}$. This is clearly a filter and $(Q, P) \in \mathcal{R}$ because $\nabla [Q] \subseteq \nabla\{x \in A | x \geq a\} \subseteq P$ because $\nabla a \in P$. Therefore, there exists $Q$ that includes $a$ and $(Q, P) \in \mathcal{R}$. Therefore, $P \in \nabla i(a)$. Conversely, if $P \in \nabla i(a)$, then there exists $Q$ such that $a \in Q$ and $(Q, P) \in \mathcal{R}$. Therefore, $\nabla a \in \nabla [Q] \subseteq P$ and hence $\nabla a \in P$. Therefore, $P \in i(\nabla a)$\
*Case III.* Now, we move to the case where $\mathcal{A}$ is distributive and the implication internalizes the finite joins while $\nabla$ is not necessarily present. Here, we want to construct a Kripke frame and a join preserving map $i$. For that matter, as we observe in Preliminaries, it is sufficient to change $W$ from the set of filters of $\mathcal{A}$ to the set of all prime filters of $\mathcal{A}$, denoted by $P(\mathcal{A})$. The same $i$ works as an embedding and it preserves both finite meets and finite joins. Define $\mathcal{R}$ as before and $\mathcal{K}_3=(P(\mathcal{A}), =_W, \mathcal{R})$. The only thing to check is whether $i$ preserves both implication and $\nabla$, again.\
For the implication, by the definition of $\mathcal{R}$ and as we had in Case I, $i(a \to b) \subseteq i(a) \to i(b)$ is clear. For the converse, assume $a \to b \notin P$ but $P \in i(a) \to i(b)$. Define $Q=\{x \in A | a \to x \in P \}$. The problem is that this $Q$ is not necessarily prime. The strategy is extending it to a suitable prime filter. Since $a \to b \notin P$ then $b \notin Q$. Define $$\Sigma=\{S \in F(\mathcal{A}) \; | \; (P, S) \in \mathcal{R}, a \in S \; \text{and} \; b \notin S \}.$$ The set $\Sigma$ is non-empty because $Q \in \Sigma$, as we have checked in Case I. Moreover, in $\Sigma$ any chain has an upper bound because if for all $i \in I$ we have $(P, S_i) \in \mathcal{R}$ then $(P, \bigcup_{i \in I} S_i)$. The reason is the following: If $x \to y \in P$ and $x \in \bigcup_{i \in I} S_i$ then for some $i \in I$ we have $x \in S_i$. Since $(P, S_i) \in \mathcal{R}$, we have $y \in S_i$ from which $y \in \bigcup_{i \in I} S_i$. Therefore, by Zorn’s lemma, $\Sigma$ has a maximal element $M$. We will prove that $M$ is prime. First note that $0 \notin M$ because if so, $M=A$ which contradicts with $b \notin M$. Now for the sake of contradiction, let us assume that $x \vee y \in M$ and $x, y \notin M$. Then we claim that either for all $m \in M$ we have $(m \wedge x \to b) \notin P$ or for all $m \in M$ we have $(m \wedge y \to b) \notin P$. The reason is that if for some $m, n \in M$ both $(m \wedge x \to b) \in P$ and $(n \wedge y \to b) \in P$ happen, we would have $(m \wedge n \wedge x \to b) \in P$ and $(m \wedge n \wedge y \to b) \in P$. Then by distributivity and the fact that the implication internalizes the finite joins, we reach $$[m \wedge n \wedge (x \vee y) \to b ]=[(m \wedge n \wedge x \to b) \wedge (m \wedge n \wedge y \to b)] \in P$$ and since $[m \wedge n \wedge (x \vee y)] \in M$ and $(P, M) \in \mathcal{R}$ we have $b \in M$ which is a contradiction. Hence, w.l.o.g. we can assume that for all $m \in M$ we have $(m \wedge x \to b) \notin P$. Then define $$N=\{z \in A| \; \exists m \in M \; (m \wedge x \to z \in P)\}$$ First note that $M \subseteq N$, because for any $m \in M$ we have $m \wedge x \to m=1 \in P$. Similarly, we have $x \in N$. Therefore, $N$ is a proper extension of $M$ because $x \notin M$. Secondly, note that $N$ is a filter because $1=[(1 \wedge 1) \to 1] \in P$ which implies $1 \in N$ and if $z, w \in N$ then there are $m, n \in M$ such that $(m \wedge x \to z) \in P$ and $(n \wedge x \to w) \in P$. Therefore, $(m \wedge n \wedge x \to z) \in P$ and $(m \wedge n \wedge x \to w) \in P$. Since $P$ is a filter and $\mathcal{A}$ internalizes its monoidal structure, by Remark \[MonoidalInternalForMeet\], we have $$(m \wedge n \wedge x) \to (z \wedge w) \in P$$ Since $M$ is a filter we have $m \wedge n \in M$ which implies $z \wedge w \in N$. Thirdly, note that we have $(P, N) \in \mathcal{R}$ because if $z \to w \in P$ and $z \in N$ there exists $m \in M$ such that $m \wedge x \to z \in P$ which implies $m \wedge x \to w \in P$ meaning that $w \in N$. And finally, note that $b \notin N$, because for all $m \in M$ we have $m \wedge x \to b \notin P$. Hence, $N \in \Sigma$ while it is a proper extension of $M$. This contradicts with the maximality of $M$ which implies that $M$ is prime. Finally, since $a \in M$ and $b \notin M$, we have $M \in i(a)$ and $M \notin i(b)$. Since $(P, M) \in \mathcal{R}$, this contradicts with $P \in i(a) \to i(b)$.\
*Case IV.* In this case, the algebra $\mathcal{A}$ is assumed to be a reduct of a distributive temporal algebra and we have to show that $i$ also preserves the $\nabla$ operator, i.e., $i(\nabla a)=\nabla i(a)$. Define $\mathcal{R}$ as before and $\mathcal{K}_4=(P(\mathcal{A}), \subseteq, \mathcal{R})$. As we have seen in Case II, $\mathcal{R}$ is compatible with $\subseteq$ and hence $\mathcal{K}_4$ is a Kripke frame. Again, since the implication does not depend on the order, the proof of preservability of implication in the Case III works here, as well. The only thing to check is whether $i$ preserves $\nabla$. As we have observed in Case II, $\nabla i(a) \subseteq i(\nabla a)$ is an easy consequence of the definition of $\mathcal{R}$. To show $i(\nabla a) \subseteq \nabla i(a)$, if $Q \in i(\nabla a)$ then $\nabla a \in Q$. Define $$\Sigma=\{S \in F(\mathcal{A}) | \; (S, Q) \in \mathcal{R}, a \in S \; \text{and} \; 0 \notin S\}$$ It is clear that $P=\{x \in A| x \geq a\} \in \Sigma$, because $a \in P$, since $\nabla a \in Q$, we have $$\nabla [P]=\{\nabla x | x \geq a\} \subseteq Q$$ and $0 \notin P$ because if $0 \in P$ then $0 \geq a$ which implies $a=0$ and hence $\nabla a=0 \in Q$ which is impossible since $Q$ is proper. Since $P \in \Sigma$, the set $\Sigma$ is non-empty. Any chain in $\Sigma$ has an upper bound because if for all $i \in I$ we have $(S_i, Q) \in \mathcal{R}$ then $\nabla [S_i] \subseteq Q$ from which $\nabla [\bigcup_{i \in I} S_i]=\bigcup_{i \in I} \nabla [S_i] \subseteq Q $ and hence $(\bigcup_{i \in I} S_i, Q) \in \mathcal{R}$. By Zorn’s lemma, $\Sigma$ has a maximal element. Call it $M$. We will prove that $M$ is prime which completes the proof. First note that $M \in \Sigma$ which implies that $0 \notin M$. Hence, $M$ is proper. Now assume $x \vee y \in M$ and $x \notin M$, $y \notin M$. The filters $M_x$ and $M_y$ generated by $M \cup \{x\}$ and $M \cup \{y\}$ are proper extensions of $M$. Therefore, they are not in $\Sigma$ which means that either one of them includes zero or we have both $\nabla M_x \nsubseteq Q$ and $\nabla M_y \nsubseteq Q$. The first is impossible because if $0 \in M_x$, then there is $m \in M$ such that $m \wedge x \leq 0$. Since $\mathcal{A}$ is distributive, we have $$m \wedge (x \vee y)=(m \wedge x) \vee (m \wedge y)=m \wedge y$$ Since $x \vee y \in M$ and $M$ is a filter, we have $m \wedge (x \vee y) \in M$ which implies $m \wedge y \in M$. Therefore, since $m \wedge y \leq y$ we have $y \in M$ which is a contradiction. A similar argument also works for the case $0 \in M_y$. Hence, we are in the case that $\nabla M_x \nsubseteq Q$ and $\nabla M_y \nsubseteq Q$.\
Therefore, there are $z, w \in A$ such that $\nabla z, \nabla w \notin Q$ and $z \in M_x$ and $w \in M_y$. Hence, there are $m, n \in M$ such that $m \wedge x \leq z$ and $n \wedge y \leq w$. Therefore, $\nabla (m \wedge x) \notin Q$ and $\nabla (n \wedge y) \notin Q$. Since $M$ is a filter, $m \wedge n \in M$ and since $x \vee y \in M$, we have $$m \wedge n \wedge (x \vee y)=(m \wedge n \wedge x) \vee (m \wedge n \wedge y) \in M$$ which by $(M, Q) \in \mathcal{R}$ implies $\nabla [(m \wedge n \wedge x) \vee (m \wedge n \wedge y)] \in Q$ and hence $$\nabla (m \wedge n \wedge x) \vee \nabla (m \wedge n \wedge y) \in Q$$ and since $Q$ is prime, either $\nabla (m \wedge n \wedge x) \in Q$ or $ \nabla (m \wedge n \wedge y) \in Q $. If $\nabla (m \wedge n \wedge x) \in Q$ then since $\nabla (m \wedge n \wedge x) \leq \nabla (m \wedge x)$ we have $\nabla (m \wedge x) \in Q$ which is a contradiction. A similar argument also works for the other case. Hence, $M$ is prime. Finally, since $a \in M$ and $(M, Q) \in \mathcal{R}$ we have $Q \in \nabla i(a)$ which completes the proof.\
Finally, we have to address the preservability of the validity of the rule schemes. For $(N)$, if $\mathcal{A}$ satisfies the scheme $(N)$, $\nabla$ commutes with all finite meets. We want to find an order preserving function $\pi(P)$ such that $(P, Q) \in \mathcal{R}$ iff $P \subseteq \pi(Q)$. Define $\pi(P)=\nabla^{-1}[P]$. It is clearly order preserving. Note that $(P, Q) \in \mathcal{R}$ is equivalent to $\nabla [P] \subseteq Q$ which is equivalent to $P \subseteq \pi(Q)$. The only thing to show is that $\pi(P)$ is actually a filter if we are in Case II and it is a prime filter if we are in Case IV. First, $\nabla^{-1}[P]$ is clearly an upset. Since $1=\nabla 1$ and $P$ is a filter, we have $1 \in \nabla^{-1}[P]$. Moreover, if $x, y \in \nabla^{-1}[P]$ then $\nabla x, \nabla y \in P$. Since $P$ is a filter and $\nabla x \wedge \nabla y = \nabla (x \wedge y)$, we have $\nabla (x \wedge y) \in P$ and hence $x \wedge y \in \nabla^{-1}[P]$. Moreover, if $\mathcal{A}$ has all finite joins, then $\nabla^{-1}[P]$ is prime because if $x \vee y \in \nabla^{-1}[P]$, then $\nabla (x \vee y) \in P$. Since $\nabla$ has a right adjoint, it commutes with all joins and hence $\nabla x \vee \nabla y \in P$. Since $P$ is prime, either $\nabla x \in P$ or $\nabla y \in P$. Therefore, either $x \in \nabla^{-1}[P]$ or $y \in \nabla^{-1}[P]$. Moreover, $0 \notin \nabla^{-1}[P]$ because otherwise, $\nabla 0=0 \in P$ which is impossible, since $P$ is prime.\
For $(H)$, note that $\nabla$ and $\Box$ as two operators over the Lindenbaum algebra are inverse of each other. Hence, $\nabla^{-1}$ as an operation over all filters or prime filters is an isomorphism. For $(P)$, given $(P, Q) \in \mathcal{R}$, we have $P \subseteq Q$. Because, given $a \in P$ and the fact that $(P, Q)$, we have $\nabla [P] \subseteq Q$ which implies $\nabla a \in \nabla [P] \subseteq Q$. Hence, $\nabla a \in Q$. Finally, we have $a \in Q$ Since $\nabla a \leq a$. For $(F)$, we have to show $(P, P) \in \mathcal{R}$. The reason is that we have $\nabla [P] \subseteq P$, because for any $a \in P$, we have $a \leq \nabla a$ which implies $\nabla a \in P$.\
$(*)$ \[We denote this part by $(*)$ for the future reference.\] Finally, for $(wF)$, first note that this rule scheme is also expressible by implication via $1 \to 0=0$. The reason is that if we have $(wF)$, then by adjunction $\nabla (1 \to 0) \leq 0$ from which $\nabla (1 \to 0) = 0$ and by $(wF)$ we have $1 \to 0=0$. Conversely, if $1 \to 0=0$ and $\nabla a=0$, then $a \leq 1 \to \nabla a= 1 \to 0=0$ from which $a=0$. Now let us prove that even in the Case III where $\nabla$ is not present, and we have a distributive join internalizing strong algebra $\mathcal{A}=(A, \leq, \wedge, 1, \to)$ if $1 \to 0=0$, then the defined $\mathcal{R}$ is serial. This generality will be useful later in the last section. For the proof, let $P$ be a prime filter. We have to find a prime filter $M$ such that $(P, M) \in \mathcal{R}$. Define $Q=\{x \in A | 1 \to x \in P\}$. Similar to what we had in the four cases above, $Q$ is a filter and $(P, Q) \in \mathcal{R}$. Note that $0 \notin Q$, because otherwise, $1 \to 0=0 \in P$ which is impossible. Define $$\Sigma=\{S \in F(\mathcal{A}) \; | \; (P, S) \in \mathcal{R}\; \text{and} \; 0 \notin S \}.$$ The set $\Sigma$ is non-empty because $Q \in \Sigma$. Moreover, in $\Sigma$ any chain has an upper bound. The proof is similar to the Case III. Hence, by Zorn’s lemma, $\Sigma$ has a maximal element. Similar to the proof of the Case III, this $M$ is prime which completes the proof.
(Completeness) For any rule scheme $\mathcal{R} \subseteq \{N, H, P, F, wF\}$, the logic $i\mathbf{STL}(\mathcal{R})$ is complete with respect to the class of all $\mathbf{K}(\mathcal{R})$-Kripke models.
For completeness, since the Lindenbaum algebra of the logic $i\mathbf{STL}(\mathcal{R})$ is clearly a distributive join internalizing temporal algebra, then if we apply Theorem \[KripkeRepresentation\] on it, it produces a Kripke frame $\mathcal{K}=(W, \leq, R)$ and an embedding $i$. Then define $V: At(\mathcal{L}_{\nabla}) \to U(W, \leq)$ by $V(p)=i([p])$ where $[p]$ is the equivalence class of $p$ in the Lindenbaum algebra. It is routine to check that $\{w \in W| \; w \Vdash B\}=i([B])$ for any formula $B \in \mathcal{L}_{\nabla}$. Therefore, if $\Gamma \Rightarrow A$ is valid in all $\mathbf{K}(\mathcal{R})$-Kripke models including $(W, \leq, R, V)$, we will have $i([\Gamma]) \subseteq i([A])$. Since $i$ is an embedding, it implies $[\Gamma] \leq [A]$ which simply means that $i\mathbf{STL}(\mathcal{R}) \vdash \Gamma \Rightarrow A$.
\[1\] In Corollary \[TransferringNabla\], if $\mathcal{S}$ satisfies any rule scheme in $\{F, wF\}$, then so does $\mathcal{T}$.
Note that we defined $\nabla=f\nabla_{\mathcal{S}}f_!$. If $\mathcal{S} \in \mathbf{ST}(wF)$, then $\mathcal{T} \in \mathbf{ST}(wF)$ because for any $a \in \mathscr{Y}$, if $\nabla a=0$, then $f \nabla_{\mathcal{S}} f_!a=0$. Since $f$ is an embedding, $\nabla_{\mathcal{S}} f_!a=0$. Since $\mathcal{S} \in \mathbf{ST}(wF)$, we have $f_! a=0$. Then $f_! a \leq 0$ implies $a \leq f(0)$. But $f(0)=0$ because $f$ is join preserving. Hence, $a=0$. For $(F)$, if $\mathcal{S} \in \mathbf{ST}(F)$, then we have $\nabla_{\mathcal{S}} f_! a \geq f_! a$ from which $\nabla a=f \nabla_{\mathcal{S}} f_! a \geq ff_!a \geq a$. The last inequality is from the adjunction $f_! \dashv f$. Hence, $\mathcal{T} \in \mathbf{ST}(F)$.
\[TruthTransformation\] Let $X$ be a topological space, $Y$ be an Alexandroff space and $f: X \to Y$ be a continuous surjection. Then for any $\nabla_Y$ over $\mathcal{O}(Y)$ and any valuation $V: At(\mathcal{L}_{\nabla}) \to \mathcal{O}(Y)$, there exist $\nabla_X$ over $\mathcal{O}(X)$ and a valuation $U: At(\mathcal{L}_{\nabla}) \to \mathcal{O}(X)$ such that for any sequent $\Gamma \Rightarrow A$, $(\mathcal{O}(X), \nabla_X, U) \vDash \Gamma \Rightarrow A$ iff $(\mathcal{O}(Y), \nabla_Y, V) \vDash \Gamma \Rightarrow A$. Moreover, for any class $\mathcal{C}$ from $i\mathbf{ST}(F)$ and $i\mathbf{ST}(wF)$, if $(\mathcal{O}(Y), \nabla_Y) \in \mathcal{C} $ then $(\mathcal{O}(X), \nabla_X) \in \mathcal{C} $. Hence, if $X \vDash_\mathcal{C} \Gamma \Rightarrow A$ then $Y \vDash_\mathcal{C} \Gamma \Rightarrow A$.
Let $\nabla_Y: \mathcal{O}(Y) \to \mathcal{O}(Y)$ be a join preserving map and $V: At(\mathcal{L}_{\nabla}) \to \mathcal{O}(Y)$. By Corollary \[TransferringNablaAlex\], since $f$ is a continuous surjection and $Y$ is Alexandroff, there exists a join preserving map $\nabla_X: \mathcal{O}(X) \to \mathcal{O}(X)$ such that $f^{-1}: (\mathcal{O}(Y), \nabla_Y) \to (\mathcal{O}(X), \nabla_X)$ becomes a logical morphism. Therefore, $f^{-1}$ commutes with all connectives of the language $\mathcal{L}_{\nabla}$. Define $U(p)=f^{-1}(V(p))$. For any formula $B \in \mathcal{L}_{\nabla}$, it is evident that $U(B)=f^{-1}(V(B))$. Now note that $(\mathcal{O}(X), \nabla_X, U) \vDash \Gamma \Rightarrow A$ iff $U(\Gamma) \subseteq U(A)$ iff $f^{-1}(V(\Gamma)) \subseteq f^{-1}(V(A))$. Since $f$ is surjective, $f^{-1}$ is an embedding. Thus, the last is equivalent to $V(\Gamma) \subseteq V(A)$ iff $(\mathcal{O}(Y), \nabla_Y, V) \vDash \Gamma \Rightarrow A$. Finally, note that if for any class $\mathcal{C}$ from the classes $i\mathbf{ST}(F)$ and $i\mathbf{ST}(wF)$, if $(\mathcal{O}(Y), \nabla_Y) \in \mathcal{C} $ then $(\mathcal{O}(X), \nabla_X) \in \mathcal{C} $, from Lemma \[1\].
The following theorem uses the Kripke completeness to show that for the for the topological completeness theorem and for the logics $i\mathbf{ST}$, $i\mathbf{ST}(F)$ and $i\mathbf{ST}(wF)$, even one fixed and large enough discrete space is sufficient. This means that despite the intuitionistic logic, $\mathbf{IPC}$, these logics can not understand the difference between discrete sets (complete for classical logic) and topological spaces (complete for intuitionistic logic).
\[StrongTopForNabla\](Topological Completeness Theorem, Strong version) Let $X$ be a set with cardinality at least $2^{\aleph_0}$. Consider $X$ as a discrete space. Then:
- If $X \vDash_{i\mathbf{ST}} A$ then $i\mathbf{STL} \vdash A$.
- If $X \vDash_{i\mathbf{ST}(F)} A$ then $i\mathbf{STL}(F) \vdash A$.
- If $X \vDash_{i\mathbf{ST}(wF)} A$ then $i\mathbf{STL}(wF) \vdash A$.
For $(i)$, let $\mathcal{K}=(W, \leq, R, V)$ be the Kripke model in the proof of Kripke completeness theorem. Note that $U(W, \leq)$ is Alexandroff. The cardinality of this space is at most $2^{\aleph_0}$, since the Lindenbaum algebra is countable. Hence, there exists a surjective function $f: X \to Y$. Since $X$ is discrete, $f$ is also continuous. Therefore, the claim follows from the last part of Theorem \[TruthTransformation\]. The proofs for the other parts are similar.
Note that the Theorem \[StrongTopForNabla\] is not true without the size condition. Interestingly, it is not true for a singleton set $X=\{0\}$. The reason is that in this space we always have $p \vee \neg p$. There are only two possibilities for $\nabla: \{0, 1\} \to \{0, 1\}$. Since $\nabla 0=0$, we have either $\nabla 1=0$ or $\nabla 1=1$. In the second case, $\nabla$ collapses to identity and hence $p \vee \neg p$ is valid because validity is just the boolean validity. In the first case, since $\nabla 1=0$, we have $(\nabla 1 \cap V(p))=0 \leq 0$ which implies $1 \leq (V(p) \to 0)$. Hence, $(V(p) \to 0)=1$ from which $[(V(p) \to 0) \cup V(p)]=1$. However, $p \vee \neg p$ is not provable in neither of the logics $i\mathbf{ST}$, $i\mathbf{ST}(F)$ and $i\mathbf{ST}(wF)$, because all of them are sub-logics of $\mathbf{IPC}$.
Sub-intuitionistic Logics {#Sub-int}
=========================
Sub-intuitionistic logics are the propositional logics of the weak implications. They are usually defined by weakening certain axioms and rules for the intuitionistic implication including the modus ponens rule and the implication introduction rule in the natural deduction system. As we have mentioned before, the logics of spacetime are also designed for the same purpose. In this section we will show how the structural logics of spacetime provide a well-behaved conservative extension for sub-intuitionistic logics. Moreover, we will also use spacetimes to provide a topological semantics for these logics.\
First let us review some important sub-intuitionistic logics, introduced in [@Vi], [@Vi2], [@Restall], [@CJ1], [@Ard], [@Ru2], [@Corsi], and [@Dosen] and investigated extensively in [@BasicPropLogic], [@Ard2], [@Aliz1], [@Aliz2], [@Aliz3], [@Aliz4], [@Aliz5], [@CJ2], [@Sas], and [@Suz]. To complete the list we also define one new logic, $\mathbf{EKPC}$ and we will explain its behaviour later. Consider the following rules of the usual natural deduction system:
**Propositional Rules:**
[c c]{}
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&
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**Formalized Rules:**
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\[4 ex\]
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**Additional Rules:**
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\[3 ex\]
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The logic $\mathbf{KPC}$ is defined as the logic of the system of all the propositional and formalized rules. $\mathbf{BPC}$ is defined as $\mathbf{KPC} + Cur$; $\mathbf{EKPC}$ as $\mathbf{KPC}$ plus the rule $E$; $\mathbf{EBPC}$ as $\mathbf{BPC}$ plus the rule $E$; $\mathbf{KTPC}$ as $\mathbf{KPC}$ plus the rule $MP$ and finally $\mathbf{IPC}$ is defined as $\mathbf{BPC}$ plus the rule $MP$.
\[t4-7\] First note that in the algebraic terminology, the rules state that the connective $\to$ is an implication that internalizes both the monoidal structure, i.e., the meet and the finite joins. Secondly, note that in defining the consequence relation $\vdash$ for sub-intuitionistic logics, we mostly follow [@CJ1], where $\mathbf{KPC}$ and $\mathbf{KTPC}$ are called $wK_{\sigma}$ and $wK_{\sigma}(MP)$. Here, we follow the modal naming tradition to call them $\mathbf{KPC}$ and $\mathbf{KTPC}$ since, they are sound and complete with respect to the class of all and reflexive Kripke models, respectively. The final point to make is on the axiomatization of $\mathbf{BPC}$. This logic can be also defined as $\mathbf{KPC}$ plus the relaxed version of $\to I$ as defined in [@Ard2]:
[c]{}
To prove the equivalence, it is clear that the rule $Cur$ is provable by this more strong version of $\rightarrow I$. For the converse, first we will show that using the rule $Cur$, $C \vdash D \rightarrow C$ is provable, for all the formulas $C$ and $D$. First use $Cur$ on $C$ to prove $C \vdash \top \rightarrow C$ and since $D \vdash \top$, we have $\vdash D \to \top$. By formalized $tr$, we have $C \vdash D \rightarrow C$. Coming back to the proof of the converse part, assume $\Gamma, A \vdash B$. It is easy to see that $\bigwedge \Gamma \wedge A \vdash B$ and then $\vdash \bigwedge \Gamma \wedge A \to B$, by the original version of $\rightarrow I$. By the foregoing point and the formalized $\wedge I$, we can prove $\bigwedge \Gamma \vdash A \to \bigwedge \Gamma \wedge A$, which implies $\bigwedge \Gamma \vdash A \to B$, by $tr_f$. Therefore, $\Gamma \vdash A \to B$.
By a propositional Kripke model for the usual propositional language $\mathcal{L}_p$, we mean a tuple $\mathcal{K}=(W, R, V)$, where $W$ is a set, $R \subseteq W \times W$ is a binary relation over $W$ (not necessarily transitive or reflexive) and $V: At(\mathcal{L}_{p}) \to P(W)$, where $At(\mathcal{L}_{p})$ is the set of atomic formulas of $\mathcal{L}_{p}$ and $P(W)$ is the powerset of $W$. A propositional Kripke model is called persistent if $V(p)$ is $R$-upward closed, i.e., if $u \in V(p)$ and $(u, v) \in R$ then $v \in V(p)$. The model is called serial if $R$ is serial, i.e., for all $u \in W$ there exists $v \in W$ such that $(u, v) \in R$. It is called reflexive if $R$ is reflexive, i.e., $(w, w) \in R$, for all $w \in W$. It is called transitive if $R$ is transitive, i.e., for all $u, v, w \in W$ if $(u, v) \in R$ and $(v, w) \in R$ then $(u, w) \in R$. It is called a rooted tree if it has an element $r$ such that for any $w \neq r$ we have $(r, w) \in R$, it is transitive and for any $u, v, w \in W$, if $(u, w), (v, w) \in R$ and $u \neq v$ then exactly one of the cases $(u, v) \in R$ or $(v, u) \in R$ happens. The forcing relation for a propositional Kripke model is defined as usual using the relation $R$. A sequent $\Gamma \Rightarrow A$ is valid in a propositional Kripke model if for all $w \in W$, $\forall B \in \Gamma \; (w \Vdash B)$ implies $w \Vdash A$.
\[KripkeforPropositional\](Soundness-Completeness for Sub-intuitionistic Logics)
- $\mathbf{KPC}$ is sound and complete with respect to the class of all propositional Kripke models. [@CJ1]
- $\mathbf{EKPC}$ is sound and complete with respect to the class of all serial propositional Kripke models.
- $\mathbf{KTPC}$ is sound and complete with respect to the class of all reflexive propositional Kripke models. [@CJ1]
- $\mathbf{BPC}$ is sound and complete with respect to the class of all transitive persistent propositional rooted Kripke trees. If $\Gamma=\emptyset$, the finite rooted transitive trees are sufficient. [@BasicPropLogic]
- $\mathbf{EBPC}$ is sound and complete with respect to the class of all transitive serial persistent propositional rooted Kripke trees. If $\Gamma=\emptyset$, the finite rooted transitive serial trees are sufficient. [@Ard]
- $\mathbf{IPC}$ is sound and complete with respect to the class of all transitive reflexive persistent propositional rooted Kripke trees. If $\Gamma=\emptyset$, the finite rooted transitive reflexive persistent trees are sufficient.
We have to prove the case of $\mathbf{EKPC}$. For soundness, note that the rule $E$ is valid in all serial Kripke models. Let $(W, R, V)$ be such a model. If
[c]{}
and for some $u \in W$, $u \Vdash \Gamma$, then by the validity of the premise, $u \Vdash \top \to \bot$. Since $R$ is serial, there exists $v \in W$ such that $(u, v) \in R$. Hence, $v \Vdash \bot$, which is impossible. Hence, $u \nVdash \Gamma$ from which $u \Vdash \Gamma \Rightarrow \bot$. For completeness, use the Lindenbaum algebra for $\mathbf{EKPC}$. This algebra is clearly a distributive join internalizing strong algebra that satisfies $1 \to 0=0$. Therefore, by part $(*)$ in the proof of Theorem \[KripkeRepresentation\], it is possible to embed the algebra into its canonical Kripke model with a serial relation $R$. Note that the Kripke frame from the proof of Theorem \[KripkeRepresentation\] is in the form $(W, =_W, R)$. Therefore, since the validity for $\nabla$-free sequents in any model of the form $(W, =_W, R, V)$ is equivalent to its validity in the propositional Kripke model $(W, R, V)$, the completeness follows.
Note that the language $\mathcal{L}_p$ is a fragment of the full language $\mathcal{L}_{\nabla}$. Therefore, it is meaningful to use spacetimes and Kripke models (not propositional Kripke models we have just defined) as models for sub-intuitionistic logics.
\[Embedding\](Embedding Theorem) Assume $\Gamma \cup \{A\} \subseteq \mathcal{L}_p$, where $\mathcal{L}_p$ is the usual language of propositional logic. Then:
- $ \Gamma \vdash_{\mathbf{KPC}} A$ iff $\Gamma \vdash_{i\mathbf{STL}} A$ iff $i\mathbf{ST} \vDash \Gamma \Rightarrow A$ iff $\mathbf{K} \vDash \Gamma \Rightarrow A$.
- $ \Gamma \vdash_{\mathbf{EKPC}} A$ iff $\Gamma \vdash_{i\mathbf{STL}(wF)} A$ iff $i\mathbf{ST}(wF) \vDash \Gamma \Rightarrow A$ iff $\mathbf{K}(wF) \vDash \Gamma \Rightarrow A$.
- $ \Gamma \vdash_{\mathbf{KTPC}} A$ iff $\Gamma \vdash_{i\mathbf{STL}(F)} A$ iff $i\mathbf{ST}(F) \vDash \Gamma \Rightarrow A$ iff $\mathbf{K}(F) \vDash \Gamma \Rightarrow A$.
- $ \Gamma \vdash_{\mathbf{BPC}} A$ iff $\Gamma \vdash_{i\mathbf{STL}(P)} A$ iff $i\mathbf{ST}(P) \vDash \Gamma \Rightarrow A$ iff $\mathbf{K}(P) \vDash \Gamma \Rightarrow A$.
- $ \Gamma \vdash_{\mathbf{EBPC}} A$ iff $\Gamma \vdash_{i\mathbf{STL}(P, wF)} A$ iff $i\mathbf{ST}(P, wF) \vDash \Gamma \Rightarrow A$ iff $\mathbf{K}(P, wF) \vDash \Gamma \Rightarrow A$.
- $ \Gamma \vdash_{\mathbf{IPC}} A$ iff $\Gamma \vdash_{i\mathbf{STL}(P, F)} A$ iff $i\mathbf{ST}(P, F) \vDash \Gamma \Rightarrow A$ iff $\mathbf{K}(P, F) \vDash \Gamma \Rightarrow A$.
Let us start with the embedding of the sub-intuionistic logics into the logics of spacetime. This part is just the syntactical version of the algebraic fact that the connective $\to$ in a temporal algebra is really an implication which internalizes both the monoidal structure and the finite joins. However, to show the proof theoretical flavour of the system, let us present the proof trees for all sub-intuitionistic rules. This hopefully shows the more natural adjoint-based approach to implication compared to the sub-intuitionistic proposal.\
To prove the embedding, we use induction on the length of the sub-intuitionistic proof. Note that all the axioms and the propositional rules except $\to I$ are available in the basic system $i\mathbf{STL}$. Therefore, it remains to prove the formalized rules and the rule $\rightarrow I$. This is what we will do in the following proof trees. Note that by a double line rule, we mean the existence of an easy omitted proof tree between the upper part and the lower part of the double line and by the label $S$ together with a double line, we mean that the omitted tree is a simple combination of the structural rules. For the formalized $\wedge I$, we have:
[c]{}
and for the formalized $\vee I$, we have:
[c]{}
for the formalized $tr$, we have:
[c]{}
And finally for $\rightarrow I$ we have:
[c]{}
Now we have to show that the additional rules are provable by their corresponding additional rules in the logics of spacetime. For $Cur$, we will use its characterization based on $\rightarrow I$ as mentioned in the Remark \[t4-7\].
[c]{}
For $MP$ and $E$ we have:
[c c]{}
&
This completes the embedding part of the theorem. To complete the equivalences, it is enough to close the circle by coming back from the validity in the Kripke models to provability in the sub-intuitionistic logics. For $\mathbf{KPC}$, by Theorem \[KripkeforPropositional\], it is sufficient to prove $\Gamma \Rightarrow A$ is valid in all propositional Kripke models. Let $(W, R, V)$ be a propositional Kripke model. Consider the tuple $(W, =, R, V)$, where the order is just equality. This tuple is a Kripke model, since $R$ is compatible with the equality and $V$ maps atomic formulas to $=$-upward closed subsets of $W$ that are just all subsets. Since $\Gamma \Rightarrow A$ is valid in all Kripke models, it is valid in $(W, =, R, V)$. However, the forcing in this model and the original propositional model is the same for $\nabla$-free formulas. Therefore, $\Gamma \Rightarrow A$ is also valid in $(W, R, V)$. For $(ii)$ and $(iii)$ the argument is similar. For $(iv)$, again by Theorem \[KripkeforPropositional\], it is sufficient to prove the validity of $\Gamma \Rightarrow A$ in all transitive persistent Kripke trees. Let $(W, R, V)$ be such a tree. Define $\leq_R$ as the reflexive extension of $R$, i.e., $R \cup \{(w, w) \in W^2 | w \in W\}$. Since the model is a tree, $\leq_R$ is a partial order. Since, $R$ is transitive, $R$ is also compatible with $\leq_R$ and hence $(W, \leq_R, R, V)$ is a Kripke frame. Moreover, note that $R \subseteq \; \leq_R$ and if a set is $R$-upward closed, it is also $\leq_R$ upward closed. Therefore, $(W, \leq_R, R, V)$ is a $\mathbf{K}(P)$-Kripke model and hence $\Gamma \Rightarrow A$ is valid in $(W, \leq_R, R, V)$. Again since the validity of $\Gamma \Rightarrow A$ in $(W, R, V)$ is the same as validity in $(W, \leq_R, R, V)$ for $\nabla$-free formulas, the theorem follows. The remained cases are similar to $(iv)$.
In the presence of the rule $Cur$, it is also possible to strenghten the topological completeness to capture the logics via one arbitrary infinite fixed Hausdorff space. For that matter, we need the following topological lemma:
\[Hausdorff\] Let $X$ be an infinite Hausdorff space. Then every finite rooted tree is a surjective continuous image of $X$.
Let us first prove the following claims:\
**Claim I.** For any natural numbers $N$ and $K$, there exists a natural number $M=M_{N, K}$ such that for any Hausdorff space $X$ with cardinality greater than or equal to $M$, there are $K$ many open mutually disjoint subspaces of $X$ each of which has at least $N$ elements.\
*Proof of the Claim I*. We prove the claim by induction on $N$. For $N=1$, pick $M_{1, K}=K$ and prove the claim by induction on $K$. For $K=1$, it is enough to pick the whole space as the open subset. To prove the claim for $K+1$, by IH, since $M_{1, K+1}=K+1 \geq K$, it is possible to find at least $K$ non-empty mutually disjoint open subsets $\{U_i\}_{i=1}^K$. Pick $\{x_i\}_{i=1}^K$ as some elements such that $x_i \in U_i$. It is possible because they are not empty. Since the space has at least $K+1$ elements, there should be some point $x \notin \{x_i\}_{i=1}^K$. Now, use the Hausdorff condition to find a sequence $\{V_i\}_{i=1}^{K+1}$ of non-empty mutually disjoint open subsets. The argument is as follows. For any $1 \leq i \leq K$, there exist disjoint open subsets $A_i$ and $B_i$ such that $x \in A_i$ and $x_i \in B_i$. For any $1 \leq i \leq K$, take $V_i=U_i \cap B_i$ and also define $V_{K+1}=\bigcap_{i=1}^K A_i$. They are clearly open non-empty subsets that are mutually disjoint.\
Now, if we have the claim for $N$, we want to prove it for $N+1$. By IH we know that there exists $M'$ that works for $N$ and $K'=2K$. We claim that $M=M'$ works for $N+1$ and $K$. If $X$ has at least $M'$ elements, then there are at lest $2K$ mutually disjoint opens such that each of them has at least $N$ elements. If we arrange these $2K$, to $K$ pairs and compute their unions, then we have $K$ opens, each of which contains at least $2N$ elements, which is greater than or equal to $N+1$.\
\
**Claim II.** For any natural number $n$, there exists a natural number $m$ such that for any Hausdorff space with at least $m$ elements and any finite rooted tree with at most $n$ elements, there exists a continuous surjection from the space to the tree.\
*Proof of the Claim II*. We will prove the claim by induction on $n$. For $n=1$ pick $m=1$ and use the constant function. For $n+1$, by IH, we know that for $n$ there exists an $m'$. Pick $m$ as the number in the claim 1, for $N=m'$ and $K=n$. Therefore, the space $X$ has at least $n$ opens each of which contains at least $m'$ elements. Call them $\{U_i\}_{i=1}^n$. Since the tree has $n+1$ elements, there are at most $n$ branches for the root such that each of them has at most $n$ nodes. Call these branches $\{T_j\}_{j=1}^r$ for some $r \leq n$. By IH, we can find a surjective continuous function $f_i: U_i \to T_i$ for any $1 \leq i \leq r$. Now define $f: X \to T$ as the extension of the union of $f_i$’s such that it sends any $x \notin \bigcup_{i=1}^r U_i$ to the root of the tree. The function is clearly surjective. For continuity, note that any open subset of the tree is an upward-closed subset which means that it is either equal to $T$ or is a union of the upward-closed subsets of the $T_i$’s. For the first case, $f^{-1}(T)=X$ which is open. For the second case, it is implied from the continuity of $f_i$ and the condition that $U_i$ is open.\
\
To prove the theorem, let $T$ be a rooted tree with $n$ elements. Then by Claim II, there exists a bound $m$ such that for any Hausdorff space $X$ with at least $m$ elements, there exists a continuous surjection from $X$ to the tree. The theorem follows from the fact that $X$ is infinite and hence has at least $m$ elements.
Let $\mathcal{R} \subseteq \{P, F, wF\}$ and $X$ be a topological space. By $X \vDash^g_{\mathcal{R}} A$, we mean that for any spacetime $(\mathcal{O}(X), \nabla)$ and any $V: At(\mathcal{L}_p) \to \mathcal{O}(X)$, if $(\mathcal{O}(X), \nabla, V) \vDash i\mathbf{STL}(\mathcal{R})$ then $(\mathcal{O}(X), \nabla, V) \vDash A$.
(Topological Completeness Theorem, Strong version) Let $X$ be an infinite Hausdorff space. Then:
- If $X \vDash^g_{P} A$ then $\mathbf{BPC} \vdash A$.
- If $X \vDash^g_{P, wF} A$ then $\mathbf{EBPC} \vdash A$.
- If $X \vDash^g_{P, F} A$ then $\mathbf{IPC} \vdash A$.
The proof is a truth transformation sequence starting from a propositional Kripke tree, going to an appropriate Kripke model and then to a suitable spacetime to finally land in a spacetime over $X$, using Theorem \[TruthTransformation\]. More precisely, for $(i)$, let $(W, R, V)$ be a finite transitive rooted tree. To prove $\mathbf{BPC} \vdash A$, by Theorem \[KripkeforPropositional\], it is enough to show that $(W, R, V) \Vdash A$. As we have seen in the proof of Theorem \[Embedding\], it is possible to define the Kripke model $\mathcal{K}=(W, \leq_R, R, V)$ such that $\mathcal{K} \vDash i\mathbf{STL}(P)$ and the validity of $\nabla$-free formulas in $(W, R, V)$ and $\mathcal{K}$ are equivalent. Therefore, it is enough to prove $\mathcal{K} \vDash A$. By Example \[KripkeToSpacetime\], it is possible to turn the Kripke model $\mathcal{K}$ to the spacetime $ \mathcal{S}_{\mathcal{K}}$ equipped with a valuation $\bar{V}$, again with the same validity for every sequents. Hence, we will show that $(\mathcal{S}_{\mathcal{K}}, \bar{V}) \vDash A$. By Lemma \[Hausdorff\], there exists a surjective continuous function $f: X \to W$ where $W$ is considered with the upset topology by the order $\leq_R$. By Theorem \[TruthTransformation\] and the fact that the order topology is Alexandroff, there are $\nabla: \mathcal{O}(X) \to \mathcal{O}(X)$ and $U: At(\mathcal{L}_{\nabla}) \to \mathcal{O}(X)$ such that the validity of any sequent in $(\mathcal{S}_{\mathcal{K}}, V)$ and $(\mathcal{O}(X), \nabla, U)$ are the same. Hence, it is enough to prove $(\mathcal{O}(X), \nabla, U) \vDash A$. Since, $(\mathcal{O}(X), \nabla, U)$, the topological model $(\mathcal{S}_{\mathcal{K}}, \bar{V})$ and the Kripke model $\mathcal{K}$ have the same validity and $\mathcal{K} \vDash i\mathbf{STL}(P)$, we have $(\mathcal{O}(X), \nabla, U) \vDash i\mathbf{STL}(P)$. Finally, since, $X \vDash^g_{P} A$, we have $(\mathcal{O}(X), \nabla, U) \vDash A$. The proofs for $(ii)$ and $(iii)$ are exactly the same.
**Acknowledgment.** We would like to thank Majid Alizadeh, Raheleh Jalali and Masoud Memarzadeh for the invaluable discussions and remarks.
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[^1]: Support by the Netherlands Organisation for Scientific Research under grant 639.073.807 is gratefully acknowledged.
[^2]: Technically, this holds for a pointfree version of topological spaces that are called locales. However, for the sake of simplicity, in this introduction we limit ourselves only to topological spaces.
[^3]: The reader may argue that using infinite sets and sequences may be somewhat problematic in the intuitionistic tradition. That is a very reasonable objection but at the same time it is also worth noting that the real meaning of the set $I$ and the sequence of propositions $\{A_i\}_{i \in I}$ is somehow open to meta-mathematical interpretations and therefore they can be chosen completely constructively. For instance, the set $I$ can be just the set of natural numbers and the sequence $\{A_i\}_{i \in I}$ can be a computable sequence of finite subsets. More mathematically, it means that everything in the argument is internalized in an elementary topos that formalizes what the intuitionist means by a set. For instance, the effective topos for the Russian school may be a reasonable choice for the universe. Having all said, the main point here is that while the conjunctions must be finite, the disjunctions can be arbitrary and this arbitrariness is something up to interpretation.
|
---
abstract: '[$\alpha$-RuCl$_3$]{}has attracted enormous attention since it has been proposed as a prime candidate to study fractionalized magnetic excitations akin to Kitaev’s honeycomb-lattice spin liquid. We have performed a detailed specific-heat investigation at temperatures down to $0.4$ K in applied magnetic fields up to $9$ T for fields parallel to the $ab$ plane. We find a suppression of the zero-field antiferromagnetic order, together with an increase of the low-temperature specific heat, with increasing field up to $\mu_0{H_\text{c}}\approx 6.9$ T. Above ${H_\text{c}}$, the magnetic contribution to the low-temperature specific heat is strongly suppressed, implying the opening of a spin-excitation gap. Our data point toward a field-induced quantum critical point (QCP) at ${H_\text{c}}$; this is supported by universal scaling behavior near ${H_\text{c}}$. Remarkably, the data also reveal the existence of a small characteristic energy scale well below $1$ meV above which the excitation spectrum changes qualitatively. We relate the data to theoretical calculations based on a [[$J_1$–$K_1$–$\Gamma_1$–$J_3$]{}]{} honeycomb model.'
author:
- 'A. U. B. Wolter'
- 'L. T. Corredor\*'
- 'L. Janssen\*'
- 'K. Nenkov'
- 'S. Schönecker'
- 'S.-H. Do'
- 'K.-Y. Choi'
- 'R. Albrecht'
- 'J. Hunger'
- 'T. Doert'
- 'M. Vojta'
- 'B. Büchner'
bibliography:
- 'aRuCl3\_Paper\_v1\_LT.bib'
title: 'Field-induced quantum criticality in the Kitaev system $\alpha$-RuCl$_3$'
---
[^1]
[$\alpha$-RuCl$_3$]{}is a $J_{\rm eff}\!=\!1/2$ Mott insulator with a layered structure of edge-sharing RuCl$_6$ octahedra arranged in a honeycomb lattice [@Plumb2014; @Sears2015; @Johnson2015; @Majumder2015; @Kubota2015; @Sinn2016; @Ziatdinov2016; @Weber2016]. It has been suggested [@Jackeli2009; @Chaloupka2010] that strongly spin-orbit-coupled Mott insulators with that lattice geometry realize bond-dependent magnetic “compass” interactions [@Nussinov_RMP] which, if dominant, would lead to a quantum spin liquid (QSL) ground state as discussed by Kitaev [@Kitaev2006]. This exotic spin-disordered state displays an emergent $Z_2$ gauge field and fractionalized Majorana-fermion excitations relevant for topological quantum computation [@Kitaev2006; @Sandilands2015; @Nasu2015; @Trebst2017].
While [$\alpha$-RuCl$_3$]{}displays magnetic long-range order (LRO) of so-called zigzag type, it has been proposed to be proximate to the Kitaev spin liquid based on its small ordering temperature and its unusual magnetic excitation spectrum [@Singh2012; @Banerjee2016; @Cao2016]. The magnetic interactions between the Ru$^{3+}$ magnetic moments are believed to be described by a variant of the Heisenberg-Kitaev model [@Chaloupka2010]: Electronic-structure calculations indicate that the Kitaev interaction in [$\alpha$-RuCl$_3$]{}is ferromagnetic and indeed defines the largest exchange energy scale [@Yadav2016; @Winter2016]. However, the debate about the spin model most appropriate for [$\alpha$-RuCl$_3$]{}– likely to include Heisenberg and off-diagonal exchange interactions, possibly also beyond nearest neighbors – has not been settled [@Kimchi2011; @Rau2014; @Yamaji2014; @Reuther2014; @Perkins2014; @Rousso2015; @Yadav2016; @Winter2016; @Winter2017; @Laubach2017].
The physics of [$\alpha$-RuCl$_3$]{}in an external magnetic field promises to be particulary interesting: It has been reported [@Leahy2016; @Baek2017] that magnetic ordering disappears for fields of the order of $10$ T (depending on the field direction), and NMR measurements performed down to $4$ K have indicated the formation of a sizeable spin gap at high fields [@Baek2017]. Additionally, numerical exact-diagonalization studies of an extended Heisenberg-Kitaev model found hints for a transition from zigzag magnetic ordering to a spin-liquid state when applying a magnetic field [@Yadav2016].
In this Letter, we report a careful heat-capacity study of [$\alpha$-RuCl$_3$]{}down to low temperature $T$ of $0.4$ K in in-plane fields up to $9$ T. We confirm the field-induced suppression of LRO at a critical field of $\mu_0{H_\text{c}}\approx 6.9$ T and provide a detailed account of the field evolution of the spin gap: This is small below $H_c$, closes at $H_c$, and progressively grows above ${H_\text{c}}$. The specific-heat data displays universal scaling consistent with the existence of a quantum critical point (QCP) at ${H_\text{c}}$. The scaling analysis yields critical exponents $d/z = 2.1\pm0.1$ and $\nu z = 0.7\pm0.1$ where $d$ is the space dimension and $\nu$ and $z$ are the correlation-length and dynamic critical exponents, respectively. Based on explicit calculations for a [[$J_1$–$K_1$–$\Gamma_1$–$J_3$]{}]{} spin model we argue that the specific-heat behavior near ${H_\text{c}}$ implies a mode softening at ${H_\text{c}}$ that accompanies the disappearance of magnetic order. The observed violations of scaling for $T \ge 3$ K indicate the presence of an intrinsic sub-meV energy scale near the QCP which we interpret as signature of Kitaev physics.
#### Experimental:
High-quality single crystals of [$\alpha$-RuCl$_3$]{}were grown by a vacuum sublimation method. A commercial RuCl$_3$ powder (Alfa-Aesar) was thoroughly grounded, and dehydrated in a quartz ampoule at $250$C for two days. The ampoule was sealed in vacuum and placed in a temperature-gradient furnace. The temperature of the RuCl$_3$ powder was set at $1080$C. After five hours the furnace was cooled to $600$C at a rate of $-2$C/h. The magnetic properties of the crystal were checked through measurements as a function of $T$ and $H$ using a Vibrating Sample Magnetometer (Quantum Design) with SQUID detection (SQUID-VSM), see supplement [@suppl] for the magnetic characterization. Specific-heat measurements were performed on a single crystal ($m
\sim 7$ mg) between $0.4$ K and $20$ K using a heat-pulse relaxation method in a Physical Properties Measurement System (PPMS, Quantum Design), in magnetic fields up to $9$ T parallel to the $ab$ plane.
#### Results:
The low-$T$ specific heat $C_p/T$ as a function of temperature in different applied fields is shown in Fig. \[Cp\](a). The zero-field curve reveals the good quality of the sample, with a single magnetic transition at $T_N = 6.5$ K determined from the peak position. By applying a magnetic field the peak becomes broader and the transition temperature is gradually suppressed. Finally no thermal phase transition is detected for fields higher than $6.9$ T, i.e., magnetic LRO disappears.
 (a) Temperature dependence of the specific heat, plotted as $C_p/T$, of [$\alpha$-RuCl$_3$]{}for different magnetic fields up to 9 T $\parallel ab$. (b) As before, but showing the magnetic contribution to the specific heat after phonon subtraction on a log-log scale, for details see text. ](Figure1_end)
In order to extract the magnetic contribution to the low-$T$ specific heat, the data were analyzed by subtracting the lattice contribution from the experimental $C_p(T)$ data by measuring the non-magnetic structural analog compound RhCl$_3$ in pressed polycrystalline form. The difference of mass and volume between the Rh and Ru compounds was accounted for by scaling the experimental specific heat curve by the Lindemann factor [@Lindemann1910], which was found to be 0.98. With the aim of ruling out possible errors due to non-perfect sample coupling during the measurements, the phononic contribution was also calculated for RhCl$_3$ by density-functional theory, see supplement [@suppl]. This approach confirmed that the phonon subtraction based on the experimental data is consistent with the theoretical calculations for $T \ge 1$ K.
The temperature dependence of the calculated magnetic contribution to the specific heat is shown in Fig. \[Cp\](b). In the lowest-$T$ region, $T \leq 3$ K, an increase of ${C_\text{mag}}/T$ with the applied field could be observed up to $\mu_0 H = 6.8$ T. Increasing the field even further, the opposite behavior is revealed: the magnetic contribution starts to decrease with field up to the highest field of $9$ T. Hence, low-$T$ entropy accumulates around $6.8-7$ T. Remarkably, around $6.9$ T the magnetic specific heat displays an approximate power-law behavior between $0.4$ and $2.5$ K, with ${C_\text{mag}}\propto T^x$ with $x\approx2.5$. Together, these observations imply the existence of a field-induced QCP [@ssbook; @Vojta2003] at $\mu_0{H_\text{c}}\approx 6.9$ T.
 Exponential fit of ${C_\text{mag}}T$ in order to extract the excitation gap for magnetic fields (a) $5$ T $\leq \mu_0 H \leq 6.8$ T and (b) $7$ T $\leq \mu_0H \leq
9$ T. The data at $6.8$ T cannot be meaningfully fit by an exponential, i.e., the gap is too small. ](Figure2n_end)
#### Excitation gap:
The lowest-temperature data away from the QCP, with a gradual suppression of ${C_\text{mag}}(T)$, indicate the opening of a magnetic excitation gap, Fig. \[Cp\](b). The simplest model of a bosonic mode with gap $\Delta$ and parabolic dispersion in $d=2$ predicts that ${C_\text{mag}}\propto \exp[-\Delta/(k_BT)]/T$, see supplement [@suppl]. According to this, the experimental ${C_\text{mag}}T$ data were fitted to a pure exponential behavior in order to extract the energy gap. The results are shown in Fig. \[Cmagfits\].
Two key observations are apparent: First, the data below about $1.5$ K indeed show an exponential suppression of ${C_\text{mag}}$, and the corresponding gap is minimal near the putative QCP at $\mu_0{H_\text{c}}\approx 6.9$ T. It varies monotonically on both sides of the QCP, consistent with theoretical expectations [@ssbook; @Vojta2003]. (Note that a symmetry-broken phase below ${H_\text{c}}$ should also display a gap, as no Goldstone modes are expected due to the presence of strong spin-orbit coupling.) Second, the data above $\sim$ $1.5$ K do [*not*]{} follow an exponential behavior (at least not in the field range studied here); in fact ${C_\text{mag}}/T$ between $1.5$ and $5$ K appears more consistent with a power law, Fig. \[Cp\](b). This indicates that the density of states of magnetic excitations changes its character at a small energy scale of a few tenths of a meV [@suppl].
 Scaling plot of ${C_\text{mag}}(T,H)$, showing ${C_\text{mag}}/T^{d/z}$ versus $T/(H - {H_\text{c}})^{\nu
z}$, here for $\mu_0{H_\text{c}}=7$ T, $d/z=2.1$, and $\nu z=0.7$. The two panels shows fields (a) slightly below and (b) slightly above ${H_\text{c}}$. The universal piece in the upper (lower) panel corresponds to the scaling function $f_-$ ($f_+$) in Eq. .](Figure3n_end)
#### Scaling analysis:
In order to further substantiate the QCP hypothesis, we have performed a scaling analysis of ${C_\text{mag}}(T,H)$. Provided that hyperscaling holds, the critical contribution to the specific heat is expected [@ssbook; @Vojta2003] to scale as $$\label{eq:scale} C = T^{d/z} f_\pm(T/|H-{H_\text{c}}|^{\nu z}),$$ where $f_\pm$ are universal functions describing the scaling for $H>{H_\text{c}}$ and $H<{H_\text{c}}$, respectively, and the argument $T/|H-{H_\text{c}}|^{\nu z}$ is made dimensionless by using suitable units. Plotting the specific heat as $C/T^{d/z}$ as a function of $T/|H-{H_\text{c}}|^{\nu z}$, separately for $H\lesssim {H_\text{c}}$ and $H\gtrsim
{H_\text{c}}$, we find an approximate data collapse for $d/z = 2.1\pm0.1$, $\nu z = 0.7\pm0.1$, and $\mu_0H_c=6.9\pm0.1$ T, see Fig. \[scaling\] for an example. (Note that the data cannot be collapsed with $d/z=2.5$.) For comparison, the supplemental Fig. S5(c) shows the scaling collapse of specific-heat data obtained from a spin-wave-based model calculation for a field-driven QCP in a [[$J_1$–$K_1$–$\Gamma_1$–$J_3$]{}]{} model, for details see Ref. . The agreement reinforces the notion of a field-induced QCP in [$\alpha$-RuCl$_3$]{}.
It is instructive to analyze deviations from scaling in Fig. \[scaling\]: (i) None of the data sets realizes the critical power law ${C_\text{mag}}\propto T^{d/z}$, indicating that the critical point has not been reached precisely. The most likely reason is sample inhomogeneities, e.g., caused by crystallographic domains with different in-plane orientation. These would lead to a distribution of $|H-{H_\text{c}}|$ values due to anisotropic $g$ factors and hence to a smearing of the QCP. (ii) Only data below $3$ K follow the approximate scaling; this is particularly clear from Fig. \[scaling\](a) where the specific-heat peaks corresponding to $T_N$ do not scale. This again implies the existence of a small energy scale, only below which standard quantum critical scaling applies.
 $T$-$H$ phase diagram for [$\alpha$-RuCl$_3$]{}: magnetic ordering temperature and energy gap as a function of the applied magnetic field $\parallel ab$. The dashed line corresponds to the fit of the gap function to $\Delta \propto |H -
{H_\text{c}}|^{0.7}$. Additionally, the magnetic entropy ${S_\text{mag}}(T,H)$ is shown in color scale.](Figure4n_end)
#### Phase diagram:
Our findings are summarized in the phase diagram, Fig. \[PD\], which displays the Néel temperature (from the peak position in $C_p/T$ as a function of $T$) and the gap values extracted as in Fig. \[Cmagfits\]. The loss of magnetic order at ${H_\text{c}}$ is accompanied by the closing of the magnetic excitation gap $\Delta$. Figure \[PD\] also shows the magnetic entropy ${S_\text{mag}}$, obtained from integrating the specific-heat data from Fig. \[Cp\](b). Focussing on ${S_\text{mag}}(H)$ at fixed $T$, the entropy accumulation near ${H_\text{c}}$ is clearly visible, as is the gap formation at elevated fields.
According to standard scaling, the gap values should follow a power law $\Delta \propto |H - {H_\text{c}}|^{\nu z}$. This is approximately obeyed by the experimental data with $\nu z=0.7$, but deviations are visible very close to ${H_\text{c}}$. These deviations could in principle arise from the transition being weakly first order (in which case the gap would not vanish at ${H_\text{c}}$). We have checked this possibility by performing field sweeps at $1.8$ K searching for hysteresis [@suppl]. However, the detected hysteresis in $M(H)$ is tiny, presumably arising from defects, such that we can exclude intrinsic first-order behavior. Hence, the deviations from power laws likely originate from sample inhomogeneities as discussed above. Alternatively, the formation of an additional narrow low-$T$ phase near ${H_\text{c}}$ appears possible, as theoretically predicted in Ref. for the classical Heisenberg-Kitaev model; this requires more detailed low-$T$ measurements as a function of continuous $H$.
#### Mode softening and nature of the high-field phase:
We now return to the specific-heat data and discuss them in the context of theoretical scenarios for the quantum phase transition (QPT) at ${H_\text{c}}$. The data show that LRO is lost above ${H_\text{c}}$. If the QPT at ${H_\text{c}}$ is continuous then this should be accompanied with a soft mode, i.e., the high-field phase should display a gapped mode with gap $\Delta\to0$ as $H\to {H_\text{c}}^+$, with this mode condensation establishing zigzag LRO below ${H_\text{c}}$. The specific-heat data above ${H_\text{c}}$ is consistent with these considerations.
An exciting possibility is that the phase above ${H_\text{c}}$ is a field-induced spin liquid, accompanied by topological order. Then, the mode which softens at ${H_\text{c}}$ would presumably correspond to an excitation of the emergent gauge field (dubbed vison for a $Z_2$ spin liquid). The field-induced spin liquid cannot exist up to arbitrarily high fields, i.e., a second QPT at a higher field ${H_{\text{c}2}}$ should exist where the spin liquid is destroyed in favor of the high-field phase; this has not been experimentally tested to date. While indications for a field-induced spin liquid in Heisenberg-Kitaev models were found in numerical simulations in Ref. , a full theory is not available.
Alternatively, the phase above ${H_\text{c}}$ could be adiabatically connected to the high-field limit, and the soft mode would then correspond to a high-field magnon. We note that such a magnon condensation is rather different from that in an SU(2)-symmetric Heisenberg magnet due to spin-orbit coupling: First, the zero-temperature magnetisation above ${H_\text{c}}$ can be far below saturation. Second, due to the low symmetry the QPT is not of BEC type ($z=2$, $\nu=1/2$) but generically in the Ising universality class ($z=1$, $\nu=0.630$ in $d=2$) .
We have studied this type of magnon-condensation transition in the framework of an appropriate [[$J_1$–$K_1$–$\Gamma_1$–$J_3$]{}]{} model [@Winter2016] in some detail, see supplement [@suppl]. Within our semiclassical approach, the critical exponents of the transition are $\nu=1/2$ and $z=1$. The results [@suppl], including the value of ${H_\text{c}}$, appear in semiquantitative agreement with the experimental data. This lends further credit to the presence of a field-induced QCP in [$\alpha$-RuCl$_3$]{}, but does not allow us to conclusively identify the nature of the high-field phase. We also note that the theoretical calculation shows the presence of an additional energy scale arising from strong van-Hove singularities in the magnon band structure at high fields. This energy scale varies approximately linearly with field above ${H_\text{c}}$ but does not vanish at ${H_\text{c}}$, see Fig. S6. Beyond the semiclassical limit these elevated-energy features are likely to loose their sharp-mode character, possibly due to fractionalization, as has been found in related models at zero field [@Gohlke2017].
#### Summary:
Via low-temperature specific heat measurements we have demonstrated that the frustrated magnet [$\alpha$-RuCl$_3$]{}displays field-induced quantum criticality at $\mu_0{H_\text{c}}\approx 6.9$ T applied in the $ab$ plane. The high-field phase is characterized by a field-induced gap to magnetic excitations which is clearly visible below $\sim 2$ K. Our scaling analysis of the low-$T$ specific-heat data yields estimates for the critical exponents $d/z = 2.1\pm0.1$ and $\nu z = 0.7\pm0.1$, consistent with Ising universality. While we cannot draw conclusions about the nature of the high-field phase, we believe that the hypothesis of a field-induced spin liquid deserves further studies.
Importantly, the data also reveal the existence of a sub-meV energy scale near the QCP above which the nature of the excitation spectrum changes. It is conceivable that this scale corresponds to a crossover from more conventional dispersive modes at low energies to exotic fractionalized excitations driven by Kitaev interactions. Studying the evolution of these excitations at higher fields is an exciting task for the future.
We acknowledge insightful discussions with C. Hess, A. Isaeva, R. Moessner, S. Nagler, F. Pollmann, S. Rachel, M. Richter, and J. van den Brink. The phonon simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the supercomputer centers in Linköping and Stockholm. This research has been supported by the DFG via SFB 1143.
#### Note added:
While this paper was being written, parallel work [@Sears2017; @Zheng2017] appeared on arXiv documenting related studies of [$\alpha$-RuCl$_3$]{}in a magnetic field. While Ref. reported gapped magnetic excitations at fields above ${H_\text{c}}$, the results of Ref. were interpreted in terms of gapless excitations in this regime. Interestingly, Ref. quotes the order-parameter exponent at ${H_\text{c}}$ to be $\beta=0.28$, in reasonable agreement with the Ising value $0.326$, suggesting a conventional Ising transition. However, in both Refs. the measurements were restricted to temperatures above $2$ K. Our data show that lower temperatures are required to reach the asymptotic regime.
[^1]: These authors contributed equally to this work.
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---
abstract: 'We briefly review the security of the ping-pong protocol in light of several attack scenarios suggested by various authors since the proposal of the protocol. We refute one recent attack on an ideal quantum channel, and show that a recent claim of falseness of our original security proof is erroneous.'
address:
- 'Psychologisches Institut II, Universität Münster, 48149 Münster, Germany'
- 'Institut für Physik, Universität Potsdam, 14469 Potsdam, Germany'
author:
- Kim Boström
- Timo Felbinger
title: 'On the Security of the Ping-Pong Protocol'
---
,
Introduction
============
It is now five years that we proposed a quantum cryptographic protocol [@BostroemFelbinger2002] whose novel feature was that the bits are transmitted in a deterministic manner: Alice, the sender, determines the bit value decoded by Bob, the receiver. Thereby, the transmission efficiency is doubled as compared to non-deterministic protocols like the BB84 [@BennettBrassard1984], where only 50% of the transmitted bits can be used for communication purposes. We were able to rigorously prove that in the case of a perfect quantum channel any effective eavesdropping attack can be detected. Specifically, when Eve, the eavesdropper, tries to gain full information, that is $I_0=1$ bit per message bit, then the detection probability reads $d=1/2$ per control bit. For comparison, in a similar scenario with Eve fully attacking all transmitted bits, the BB84 protocol provides a detection probability of $d=1/4$ per control bit.
Due to its deterministic nature, the protocol allows two parties to communicate directly in a secure manner. More precisely, the direct communication is *quasi-secure*, which means that the probability for an eavesdropper to remain undetected declines exponentially with the length of the message transmitted. Alternatively, when perfect asymptotic security is required, the protocol can be used as a quantum key distribution scheme: The sender transmits a meaningless stream of random bits upon which the usual techniques of error correction and privacy amplification can be applied so that a shared secret key is established for later use in classical encryption.
One peculiarity of the protocol is that the carrier of information, a single qubit, is travelling forth and back between sender and receiver, which motivated the name *ping-pong protocol*. Another peculiarity is that sender and receiver randomly switch between two modes: *message mode* and *control mode*. Only in message mode a message bit is transferred and only in control mode an eavesdropper can be detected with a certain probability. The fact that the mode can only be noticed by the eavesdropper when it is too late to escape detection, is an important ingredient to the security of the protocol and requires careful attention in experimental implementations.
One experimental implementation of the ping-pong protocol using entangled photons has been accomplished at the quantum optics lab in Potsdam, Germany, where the random switching between message mode and control mode is realized in an elegant way [@OstermeyerWalenta2007].
Also meanwhile, several alternative quantum cryptographic protocols have been proposed that are tailored along the scheme of the original ping-pong protocol, providing improved efficiency or experimental feasibility [@CaiLi2004; @CaiLi2004a; @DegiovanniRuo-Berchera2004; @LucamariniMancini2005; @DengLong2004]. Since the security of these protocols is based on the security of the original one, we would like to give a brief review on the security situation of the ping-pong protocol as it appears nowadays, after several attempts to attack the protocol in the case of either a perfect or an imperfect quantum channel. Furthermore, we will show that some recent claims that the protocol is insecure and that the security proof is wrong, are erroneous.
The protocol in short
=====================
Bob, the receiver of information, prepares two qubits in the Bell state $|\Psi^+\rangle_{ht}=\frac1{\sqrt2}(|01\rangle_{ht}+|10\rangle_{ht})$, where $h$ and $t$ refer to the home and travel qubit, respectively. He sends the travel qubit to Alice who randomly selects either message mode or control mode. In message mode, Alice applies the encoding operation $Z_t(j)=\sigma_z^{j}$ to the travel qubit, where $j\in\{0,1\}$ represents the message bit. For $j=1$, the encoding operation transforms $|\Psi^+\rangle_{ht}$ into $|\Psi^-\rangle_{ht}$ and for $j=0$ the state is left unchanged. After encoding, Alice sends the qubit back to Bob, who then applies a Bell measurement on both qubits, yielding either $|\Psi^+\rangle_{ht}$ or $|\Psi^-\rangle_{ht}$, thereby revealing the message bit $j$. In control mode, Alice measures the travel qubit in the $z$-basis and announces the result via the public channel. Upon receiving the announcement, Bob measures his home qubit in the $z$-basis and compares both results. If they differ, the protocol is continued, otherwise aborted.
Cai’s DOS-attack
================
Qing-yu Cai [@Cai2003] has proposed a simple attack scheme that disturbs the information transmission without being detectable and without revealing any message information, that is, a *denial-of-service* (DoS) attack: Eve measures every qubit travelling from Alice to Bob in the $z$-basis. According to the protocol, there is no security check for qubits travelling back from Alice to Bob, hence the attack remains undetected. As the attack destroys the entanglement between home and travel qubit, the bit read out by Bob is completely uncorrelated with the bit encoded by Alice, so the message is scrambled. Since Eve’s measurement result is completely random, she does not gain any message information.
Let us point out that a trivial modification of the attack even allows Eve to derministically change the information transmitted on the channel by flipping message bits of her choice: Instead of measuring the qubit, Eve applies a $\sigma_z$-operation. This way, Eve is able to alter message information, albeit in a “blind” way. Altogether, the protocol in its original form protects the *confidentiality* of the message but not the *integrity* (just like the classical one-time pad).
Cai himself has proposed fairly simple ways to protect the protocol against this type of attack: either quantum mechanically, by slightly modifying the control mechanism, or classically, by performing one of the standard methods of message authentification [@Schneier1996].
Wojcik’s attack on a lossy quantum channel
==========================================
So far, the ping-pong protocol is provably secure only for the case of a perfect quantum channel. Any imperfection of the channel potentially opens the door to effective and undetectable eavesdropping. Since all quantum cryptographic protocols are confronted with such a problem, the standard procedure is to introduce additional steps of error correction and privacy amplification using the public channel to distill an asymptotically perfectly secure key. These procedures can be successful exactly if the mutual information between sender and receiver is greater than that between sender and eavesdropper [@CsiszarKorner1978].
Antoni Wojcik [@Wojcik2003] proposed a smart eavesdropping attack scheme that works on a lossy quantum channel and enables the eavesdropper to gain message information without being detected.
The basic idea is the following. After receiving the travel qubit from Bob, Eve appends an ancilla system ${\cal H}_{xy}$ in some initial state $|e_0\rangle_{xy}$ to the system ${\cal H}_{ht}$ of travel qubit and home qubit in its initial state, yielding the state $$|\text{init}\rangle=\frac1{\sqrt2}(|01\rangle+|10\rangle)_{ht}|e_0\rangle_{xy}. \label{W1}$$ She then unitarily transforms this initial state into $$\begin{split}
|\text{B--A}\rangle=&\frac1{2}|0\rangle_h\big(|1\rangle_t|e_1\rangle_{xy}
+|{\rm vac}\rangle_t|e_2\rangle_{xy}\big)\\
&+\frac1{2}|1\rangle_h\big(|0\rangle_t|e_3\rangle_{xy}
+|{\rm vac}\rangle_t|e_4\rangle_{xy}\big),
\end{split}$$ where $|e_1\rangle$,…,$|e_4\rangle$ are mutually orthogonal. Then Eve resends the travel qubit to Alice, whose encoding operation $\sigma_z^j$ on the travel qubit yields $$\begin{split}
|\text{B--A}'\rangle=&\frac1{2}|0\rangle_h\big((-1)^j|1\rangle_t|e_1\rangle_{xy}
+|{\rm vac}\rangle_t|e_2\rangle_{xy}\big)\\
&+\frac1{2}|1\rangle_h\big(|0\rangle_t|e_3\rangle_{xy}
+|{\rm vac}\rangle_t|e_4\rangle_{xy}\big).
\end{split}$$ It can directly be seen from the above terms that upon capturing the travel qubit being sent from Alice to Bob, Eve is able to find the message bit $j$ with probability $1/2$ by measuring the $txy$ system in a suitable basis or, equivalently, by performing some unitary operation on the $txy$ system and then measuring in the computational basis. In a control run the travel mode is found to be in the vacuum state with probability 1/2. Hence, Eve’s attack introduces 50% channel losses if she attacks all the time.
If the efficiency of the channel is $\eta<0.5$ then Eve replaces the lossy channel with a better one, so that the channel exactly mimics the losses expected by Alice and Bob. That way, she can attack all transmissions while staying undetectable, and she creates mutual information between herself and Alice that exceeds the mutual information between Alice and Bob. Hence, even with error correction and privacy amplification, the protocol would not be secure. If $0.5<\eta\leq0.6$, she attacks the fraction $\mu=2(1-\eta)$ of qubits. For efficiencies above 0.6, the mutual information between Alice and Eve falls below the mutual information between Alice and Bob, so that error correction and privacy amplification can establish the security of the protocol.
Fortunately, Wojcik himself proposed in the same paper two solutions to protect the protocol against his attack. One solution is to estimate the qubit error rate (QBER) which, however, forces Alice and Bob to sacrifice some of their message bits. The other solution is to delay Alice’s announcement of the transmission mode (message or control), until Bob has checked if there is an additional photon in the travel mode. This way, the attack can be detected because the particular attack operation not only produces channel losses, but also (with probability 1/2) inserts a photon into the travel mode which in control mode should be in vacuum state after Alice’s measurement. Such an “illegal” photon travelling to Bob would, if detected, immediately reveal the presence of the eavesdropper.
In summary, Wojcik’s attack exploits a security hole of the protocol in the realistic case of an imperfect quantum channel using photons. A fairly simple modification of the protocol closes the hole and restores the security of the protocol.
The ZML-attack on an imperfect quantum channel
==============================================
In [@ZhangMan2004], Zhan-jun Zhang, Zhong-xiao Man, and Yong Li improved Wójcik’s attack by expanding the domain of effective eavesdropping from nearly 60% to nearly 80% channel efficiency. The basic idea is fully analoguous to Wójcik’s scheme [@Wojcik2003]. Eve appends an ancilla system ${\cal H}_{xy}$ and unitarily transforms the state into the state $$\begin{split}
|\text{B--A}\rangle=&\frac1{2}|0\rangle_h\big(|1\rangle_t|e_1\rangle_{xy}
+|{\rm vac}\rangle_t|e_2\rangle_{xy}\big)\\
&+\frac1{\sqrt2}|1\rangle_h\big(|0\rangle_t|e_3\rangle_{xy}\big),
\end{split}$$ where $|e_1\rangle$,…,$|e_4\rangle$ are mutually orthogonal. Alice’s encoding operation on the travel qubit yields $$\begin{split}
|\text{B--A}'\rangle=&\frac1{2}|0\rangle_h\big((-1)^j|1\rangle_t|e_1\rangle_{xy}
+|{\rm vac}\rangle_t|e_2\rangle_{xy}\big)\\
&+\frac1{\sqrt2}|1\rangle_h\big(|0\rangle_t|e_3\rangle_{xy}\big).
\end{split}$$ Again, by measuring the $txy$ system Eve finds the message bit $j$ with probability $1/2$. In a control run, the travel mode is found to be in the vaccum state with probability 1/4. Hence, Eve’s attack introduces 25% channel losses if she attacks all the time. Reasoning analoguous to that in Wójcik’s publication reveals that the ping-pong protocol can be successfully attacked for channel efficiencies up to nearly 80%. However, also here the security of the protocol can be re-established by introducing the same countermeasures suggested by Wójcik.
The ZLM-attack on a perfect quantum channel
===========================================
Recently, the same three authors proposed an attack scheme against the ping-pong protocol, enabling the eavesdropper to read out message information without being detected even in the case of a perfect quantum channel, in contradiction to our rigorous security proof for this case [@ZhangLi2005]. However, their attack scheme is faulty:
According to their attack scheme, Eve prepares an ancilla state $|\chi\rangle=|{\rm vac},0\rangle_{xy}$ in two additional modes $x$ and $y$, and applies a unitary operation $W_{txy}$ (Eq. (2) in [@ZhangLi2005]) on the compound system $txy$ of the travel qubit and the ancilla modes during the B-A-transmission. Afterwards, the total system is in the state $$\begin{split}
|B-A\rangle
&=\frac12|0,1\rangle_{ht}(|{\rm vac}, 0\rangle_{xy}+|1,{\rm vac}\rangle_{xy})\\
&\quad+\frac12|1,0\rangle_{ht}(|{\rm vac}, 1\rangle_{xy}+|0,{\rm vac}\rangle_{xy}).
\end{split}$$ It is clear that this attack operation cannot be detected by the control measurements of the ping-pong protocol: $z$-basis measurements on $h$ and $t$ will still be strictly anticorrelated.
In message mode, Alice applies the encoding operation $Z_t^{j}=\sigma_z^{j}$ to the travel photon, where $j\in\{0,1\}$ represents the message bit, and sends the photon back to Bob. Eve intercepts the travel photon, applies the inverse operation $W_{txy}^{-1}$ on the compound system $txy$, resends the travel photon to Alice and keeps her ancilla system. The authors claim that a measurement on the ancilla system reveals information about the message bit $j$ encoded by Alice. Indeed, the authors’ Eq. (7), which supposedly shows the state $|A-B\rangle$ after Eve’s (A-B)-attack operation $W_{txy}^{-1}$, indicates that the message bit $j$ is partly encoded in the state of the $y$ photon: $$\begin{split}\label{Eq7}
|A-B\rangle_j
&= \frac12\Big[(-1)^j(\Psi_{ht}^++\Psi_{ht}^-)|j\rangle_y\\
&\quad+(\Psi_{ht}^+-\Psi_{ht}^-)|0\rangle_y\Big]|{\rm vac}\rangle_x.
\end{split}$$
Obvously, a computational-basis measurement by Eve on the $y$-mode reveals the message bit $j$ with probability 1/2, otherwise it yields 0. However, this crucial equation is wrong, which can be seen as follows. When Alice applies her encoding operation to the travel photon $t$, the total system is in the state $$\begin{split}
Z_t^{j}|B-A\rangle
&=\frac1{\sqrt2}\Big[(-1)^j|0,1\rangle_{ht}|\chi_1\rangle_{xy}
+|1,0\rangle_{ht}|\chi_0\rangle_{xy}\Big],
\end{split}$$ where we have set $$\begin{aligned}
|\chi_1\rangle_{xy}&=&\frac1{\sqrt2}(|{\rm vac},0\rangle_{xy}+|1,{\rm vac}\rangle_{xy})\\
|\chi_0\rangle_{xy}&=&\frac1{\sqrt2}(|{\rm vac},1\rangle_{xy}+|0,{\rm vac}\rangle_{xy}).\end{aligned}$$ As can be seen from above, the message bit $j$ is encoded in the relative phase between the two components of the superposition. Since Eve has no access to the home photon $h$, she can in no way read out the relative phase. Generally, consider a state in the space ${\cal H}_{h}\otimes {\cal H}_E$ of the form $$|\Psi\rangle = \alpha|h_1\rangle_h|e_1\rangle_E+\beta e^{i\phi}|h_2\rangle_h|e_2\rangle_E,$$ with $\langle h_1|h_2\rangle=0$, and $\alpha,\beta>0$, and where Eve has only access to the system in ${\cal H}_E$. Then for Eve the state of the entire system is indistinguishable from the reduced density matrix $$\begin{aligned}
\rho_E&=&{\rm Tr}_h\{|\psi\rangle\langle\psi|\}\\
&=&\alpha^2|e_1\rangle\langle e_1|_E+\beta^2|e_2\rangle\langle e_2|_E,\end{aligned}$$ where the relative phase $\phi$ is no longer available.
In the present case, the density matrix of the total state after Alice’s encoding operation is $$\begin{aligned}
\begin{aligned}
\rho_j & = Z_t^j|B-A\rangle\langle B-A|Z_t^{j\dagger}
\cr & = \frac12\Big[|01\rangle_{ht}|\chi_1\rangle_{xy}\langle01|_{ht}\langle\chi_1|_{xy}\\
&\quad+(-1)^j|01\rangle_{ht}|\chi_1\rangle_{xy}\langle10|_{ht}\langle\chi_0|_{xy}\\
&\quad+(-1)^j|10\rangle_{ht}|\chi_0\rangle_{xy}\langle01|_{ht}\langle\chi_1|_{xy}\\
&\quad+|10\rangle_{ht}|\chi_0\rangle_{xy}\langle10|_{ht}\langle\chi_0|_{xy}\Big].
\end{aligned}\end{aligned}$$ The state of the system accessible to Eve is given by partial-tracing over the home photon $h$, $$\rho_{j}^{({\rm Eve})}={\rm Tr}_h\{\rho_j\},$$ which yields $$\begin{split}
\rho_{j}^{({\rm Eve})}
&= \frac12\Big[|1\rangle_{t}|\chi_1\rangle_{xy}\langle1|_{t}\langle\chi_1|_{xy}\\
&\quad+|0\rangle_{t}|\chi_0\rangle_{xy}\langle0|_{t}\langle\chi_0|_{xy}\Big].
\end{split}$$
Since $\rho_j^{\rm (Eve)}$ is independent of $j$, there is no message information available to Eve. Consequently, the authors’ calculation of the state $|A-B\rangle_j$, which results from the application of $W_{txy}^{-1}$ to $Z_t^j|B-A\rangle$ must be faulty. In fact, our own calculations show that the state after Eve’s (A-B)-attack $W_{txy}^{-1}$ reads $$\begin{split}
|A-B\rangle_j
&= \frac12\Big[(-1)^j(\Psi_{ht}^++\Psi_{ht}^-)\\
&\quad+(\Psi_{ht}^+-\Psi_{ht}^-)\Big]|0,{\rm vac}\rangle_{xy},
\end{split}$$ which is different from the authors’ Eq. (7) (Eq. above) in that it is impossible for Eve to read out the message bit by any measurement performed on the ancilla system $xy$.
In summary, we find that the conclusions drawn in the commented paper [@ZhangLi2005] are based on a miscalculation; this attack scheme is not effective and does not impair the security of the ping-pong protocol.
Cai’s invisible photon attack
=============================
Qing-Yu Cai [@Cai2006] adapted the Trojan horse attack introduced by Gisin et al [@GisinFasel2005] to the ping-pong protocol: Eve feeds in an additional photon which is invisible to Alice and Bob’s detectors, but which is affected by Alice’s encoding operation. The illegal photon is inserted into the travel mode on the way from Bob to Alice, and it is filtered out during the transmission from Alice to Bob. Eve detects the state change of the illegal photon which is caused by Alice’s encoding operation, and thereby obtains the message bit without being detected. Choosing a wavelength outside the range of Alice’s detectors is one possible way to make the illegal photon invisible to the control measurements. As Cai has pointed out, the attack does not exploit a weakness of the protocol itself but rather of certain imperfect implementations of the protocol. He also suggests a feasible solution to re-establish the security of the communication: Alice and Bob add filters to their setup whose bandwidth matches the sensitivity range of the detectors.
The generalization is straightforward: The experimental setup should block any quantum carriers of information which are invisible to the detectors but which are affected by the encoding operation.
Zhang’s claim that the security proof is wrong
==============================================
Zhan-jun Zhang challenges the validity of our security proof altogether [@Zhang2006]. We will show that the claim is based 1) on a misunderstanding of the security proof and 2) on a miscalculation at a crucial point in the argument.
The author emphasizes that in our security proof the eavesdropping information $I_0$ is extracted from the travel qubit only. This is not the case. The maximal amount $I_0$ of information that can be extracted from the system available to Eve, is equal to the von-Neumann information $S$ of the state $\rho''$ given by Eq. (8) in our original paper. The state $\rho''$ results from Alice’s encoding operation on the state $\rho'$ which is given by our Eq. (7) as a matrix representation in the orthogonal basis $\{|0,\chi_0\rangle, |1,\chi_1\rangle\}$. As we have pointed out in our paper, the states $|\chi_0\rangle$ and $|\chi_1\rangle$ are states of Eve’s ancilla system ${\cal H}_E$. It is therefore not true that the information $I_0$ is derived only from the state of the travel qubit. Unfortunately, though, we ourselves have made such misunderstanding easy because right before our Eq. (8) we denote $\rho''$ as “the state of the travel qubit after Eve’s attack operation and after Alice’s encoding operation”. This is a misnomer which we apologize for; it should however be clear from the context that $\rho''$ refers to the state $\rho'$ after Alice’s encoding operation, and that the state $\rho'$ explicitely includes Eve’s ancilla system.
Based on this misunderstanding, the author constructs a counterexample against the security proof, where he then miscalculates the information contents $I_{0t}$ and $I_{0c}$ that can be extracted from the travel qubit and the composite system, respectively. He claims that “as can easily be worked out”, the values read $I_{0t}=1$ and $I_{0c}=2$, which would be in contradiction to the prepositions of our security proof.
Let us explicitely perform the calculation. According to the author’s counterexample, Eve captures the travel qubit $t$ in the state $|0\rangle$, attaches an ancilla system $x$ in the state $|\chi\rangle_x = {1\over\sqrt{2}}(|0\rangle_x+|1\rangle_x)$. We find that the state $|\Psi'\rangle$ of the composite system $tx$ after Eve’s attack operation $\hat E$ given by the author reads $$\begin{aligned}
|\Psi'\rangle &=&\hat E (|0\rangle_t |\chi\rangle_x) \\
&=& \hat E\frac1{\sqrt2}(|00\rangle_{tx}+|01\rangle_{tx})\\
&=&\frac1{\sqrt2}|0\rangle_t|\chi_0\rangle_x
+\frac1{\sqrt2}|1\rangle_t|\chi_1\rangle_x,\end{aligned}$$ where we have set $$\begin{aligned}
|\chi_0\rangle_x&=&\frac1{\sqrt2}(|0\rangle_x+|1\rangle_x)\\
|\chi_1\rangle_x&=&\frac1{\sqrt2}(|1\rangle_x-|0\rangle_x).\end{aligned}$$
The state $|\Psi'\rangle$ has exactly the form given in Eq. (4) of our security proof, with $\alpha=\beta=\frac1{\sqrt2}$. When Alice encodes “0” she applies the unity operation which gives $|\Psi'_0\rangle=|\Psi'\rangle$; when she encodes “1” she applies $\sigma_z$ to the travel qubit which gives $$|\Psi'_1\rangle=\frac1{\sqrt2}|0\rangle_t|\chi_0\rangle_x
-\frac1{\sqrt2}|1\rangle_t|\chi_1\rangle_x.$$
Note that $|\Psi'_1\rangle$ is orthogonal to $|\Psi'_0\rangle$. Assuming that she encodes “0” or “1” with equal probability (which is tacitly assumed by the author), the state of the composite system $tx$ reads $$\rho''=\frac12|\Psi'_0\rangle\langle\Psi'_0|
+\frac12|\Psi'_1\rangle\langle\Psi'_1|,$$ which has the entropy $I_{0c}=S(\rho'')=1$, in contrast to the authors’ result. In fact, the author’s value of $I_{0c}=2$ would be very surprising: Alice encoded at most one classical bit by a unitary operation, so the entropy of the resulting state cannot be higher than one bit. For security reasons, our protocol makes use only of a 2-dimensional subspace of the full 4-dimensional Hilbert space spanned by two qubits, and the entropy of a state in a 2-dimensional subspace is at most 1 bit.
Conclusion
==========
So far, the ping-pong protocol has resisted all serious attacks brought forward in the last five years, albeit with slight modifications of the scheme. For the ideal case of a perfect quantum channel, the initial security proof holds and is both rigorous and general. For the realistic case of an imperfect quantum channel, there is no general security proof but the protocol seems to retain its security.
We suggest that future efforts should go into either figuring out more attack scenarios exploiting channel imperfections under realistic circumstances or into finding a general proof for the unconditional security of (a suitable extension of) the ping-pong protocol in the case of an imperfect quantum channel.
[10]{} url \#1[`#1`]{}urlprefix
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|
---
author:
- 'Jeremiah Blocki, Mike Cinkoske'
bibliography:
- 'bounded-parallel-mhf.bib'
- 'password.bib'
- 'cryptobib/abbrev3.bib'
- 'cryptobib/crypto.bib'
- 'some-bib.bib'
title: A New Connection Between Node and Edge Depth Robust Graphs
---
|
---
abstract: 'The Brazilian court system is currently the most clogged up judiciary system in the world. Thousands of lawsuit cases reach the supreme court every day. These cases need to be analyzed in order to be associated to relevant tags and allocated to the right team. Most of the cases reach the court as raster scanned documents with widely variable levels of quality. One of the first steps for the analysis is to classify these documents. In this paper we present a Bidirectional Long Short-Term Memory network (Bi-LSTM) to classify these pieces of legal document.'
author:
- |
F.A. Braz, N.C. Silva, T.E. de Campos[^1], F.B.S. Chaves, M.H.P. Ferreira,\
[**P.H.G. Inazawa, V.H.D. Coelho, B.P. Sukiennik, A.P.G.S. Almeida, F.B. Vidal, D. Alves Bezerra,**]{}\
[**D.B. Gusmão, G.G. Ziegler, R.V.C. Fernandes, R. Zumblick, F. Hartmann Peixoto**]{}\
Universidade de Brasília (UnB), DF, Brazil\
`{fabraz,niltoncs,teodecampos}@unb.br`\
title: 'Document classification using a Bi-LSTM to unclog Brazil’s supreme court'
---
Introduction
============
Ensuring equal access to justice for all is a real challenge in Brazil due to the sluggishness of judicial process and the high volume of new cases entering in the justice system every year. In the Brazilian higher courts, the average time to reach the sentence is 11 months in the Superior Court of Justice (STJ), 1 year and 2 months in the Superior Court of Labor (TST) and 8 months in the Superior Electoral Court (TSE). This scenario is worsening each year: around 28.8 million new cases reach the Brazilian Judiciary and there are already approximately 79.6 million other cases in stock in Brazilian courts, related to 2016. Adding up these figures, there are approximately 108.4 million legal processes being carried on by Brazilian judiciary [@lucia_JusticaEmNumeros_2017]. This overload has a cost of USD 22 billion per year, according to National Justice Council (CNJ), and is one of the main causes of legal insecurity and criminal impunity in the country. The congestion rate was 73.0% in 2016. That means that only 27% of all the cases processed were solved in that year.
Our work aims to contribute towards goal 16.3 of the UN agenda 2030: “ensure equal access to justice for all” [@un_Agenda2030_2015]. To overcome the obstacles that prevent the Brazilian judicial system from democratically guaranteeing the right to due process of law, we propose to apply machine learning methods to classify legal processes that reach the Brazilian Supreme Court (in Portuguese, *Supremo Tribunal Federal - STF*) [@stf_press_30_05_2018; @stf_press_30_08_2018]. According to the STF, it would take 22,000 man-hours by its employees and trainees to analyze the approximately 42,000 processes received per semester [@lucia_JusticaEmNumeros_2017]. The court also points out that the time its employees spend on classifying these processes could be better applied at more complex stages of the judicial work flow.
Most of the cases reach the court are in the form of PDF files with raster scanned documents. About 10% of them are completely unstructured volumes which encloses several documents per file without any digital index. In Brazil’s judiciary, documents that compose a legal case are grouped into briefs or parts (called [*peças*]{} in Portuguese) which are categorized into a set of classes. In order to speed up the analysis of cases, the first step needed is to automate the classification of these documents.
This paper reports results of a preliminary evaluation on a dataset containing 6,814 documents ([*peças*]{}) from the STF. We propose a Bidirectional Long Short-Term Memory network architecture for this task and show that it obtains 84% F$_1$ score on this dataset with no pre-processing.
Proposed method \[sec:methodology\]
===================================
Text extraction
---------------
The first step is to extract text from the PDF file. If the content of a document page is a raster image, we apply the Tesseract OCR system [@smith_Tesseract_icdar2007] and store the text. If the page embeds its text as metadata, its quality is verified using regular expressions. If the quality level is acceptable the text is stored, otherwise the OCR is applied as if the page was in raster image format and the result is stored.
Bidirectional Long Short-Term Memory Model {#sec:bilstm}
------------------------------------------
The proposed model is based on recurrent models to deal with text as sequential information. Long Short-Term Memory (LSTM) models use the information from the previous status of neurons [@hochreiter_schmidhuber_origin_lstm_nc_1997]. More contextual information can be extracted using a bidirectional LSTM (Bi-LSTM) [@graves_framewise_origin_bilstm_nn_2005]. The data processing flows forward and backward at same time [@schuster_etal_bidirectional_TSP_1997] and the output of each LSTM are merged using their sum [@graves_framewise_origin_bilstm_nn_2005].
Bi-LSTM models have shown to be effective in problems of speech recognition [@Ghaeini_etal_lstm_arXiv_2018; @Kiperwasser_etal_lstm_arXiv_2016; @Tai_etal_lstm_arXiv_2015; @rao_spasojevic_lstm_arxiv; @huang_etal_lstm_arXiv_2015], being a state-of-the-art model to classify sequential data into multiple classes. Documents are sequences of words and our problem is also multi-class, therefore Bi-LSTM is a suitable model.
\[sec:architecture\]
Our architecture (shown in Figure \[fig:diagram\]) is composed of three main layers with 1000 tokens of input. We followed [@rao_spasojevic_lstm_arxiv; @huang_etal_lstm_arXiv_2015] and used word embedding as an input layer of the Bi-LSTM. This layer transforms each token in a distributed array of 100 dimensions. The recurrent layer has two hidden LSTM, a forward and backward layer model, each with 200 memory blocks and one-cell. The output of this layer uses a ReLu activation. The two hidden LSTMs are combined by adding their outputs. The last layer is dense, with 6 output neurons and a Softmax activation function.
![Bi-LSTM Architecture diagram[]{data-label="fig:diagram"}](./media/model.pdf){width=".7\columnwidth"}
VICTOR project’s dataset of brief parts ([*peças*]{}) {#sec:dataset}
=====================================================
We used the dataset described in [@daSilva_Braz_CNN_ICoFS_2018]. Our work focuses on classifying five main types of legal documents handled by the Brazilian supreme court (STF). Documents that do not belong to these classes are grouped in a class called [*Others*]{}. Table \[tab:dataset\] presents the classes used and the number of samples in each of them.
Label (in Portuguese) Description Samples
---------------------------------------------- ---------------------------------- ---------
[*Acórdão*]{} Appelate Decision 82
[*Recurso Extraordinário (RE)*]{} Extraordinary Appeal 63
[*Agravo de Recurso Extraordinário (ARE)*]{} Extraordinary Appeal Bill/Review 92
[*Despacho*]{} Administrative Orders 55
[*Sentença*]{} Judgement 110
[*Outros*]{} Other Documents Types 280
: Set of classes in VICTOR project’s dataset of document types ([*peças*]{}). []{data-label="tab:dataset"}
A total of 6,814 text documents were manually labeled by a team of four specialist lawyers. This dataset was split into three parts: 70% of the samples for training, 20% for validation and 10% for test.
This dataset is quite challenging as it has a high level of within class diversity. The documents do not follow a standard, not only in terms of layout but also in terms of scanned image quality and whether or not they embed digital text or have only raster images. Furthermore, a significant part of these documents often contained handwritten annotations, stamps, stains etc. As discussed earlier, the PDF files are often not indexed and some of them include several documents.
Experiments and results
=======================
Although legal documents can be quite long, the first page is usually the most informative. Furthermore, the files in VICTOR dataset often include multiple documents, so later pages in a PDF file can be unreliable. For these reasons, our Bi-LSTM model was designed to take inputs of 1000 tokens, which usually covers most of the contents of one page. Our evaluations on the validation set show that this was discriminative enough. One major advantage of this w.r.t. using text from the whole document is that the main bottleneck of our system is the OCR system, which takes 1s per processing core to run on each page. By using a method that efficiently exploits the first 1000 tokens, we only need to run the OCR on up to two pages per document.
Since most of the text was obtained by running an OCR raster scanned noisy documents (rather than extracting text from PDFs metadata), many out-of-vocabulary ‘words’ appeared in the text. One can deal with this problem by using a number of regular expressions and stemming, as done in [@daSilva_Braz_CNN_ICoFS_2018] and/or by running a named entity recognition system such as that of [@luz_etal_propor2018]. In this paper, we simply limited the tokenizer’s vocabulary to the 100,000 most frequent distinct words. This was enough to include all relevant words as well as specific symbols of the judiciary system, such as law numbers and Latin words.
Our model was trained for 20 epochs with a batch size of 64 samples and a learning rate of 0.001. The total training time was of 120.02s on a NVidia Titan XP, which has 12GB of RAM. In this setup, prediction on is done in 1.47ms per document. In contrast, the CNN model of [@daSilva_Braz_CNN_ICoFS_2018] takes 5.87ms per document, excluding the time to run the OCR (1s per page per CPU core).
Figures \[tab:results\] and \[fig:matrix\] detail our results. Our BI-LSTM model archives a mean precision of 85% and f$_1$-Score 84% without requiring any preprocessing heuristics and regular expressions. This contrasts with the CNN model of [@daSilva_Braz_CNN_ICoFS_2018], which uses a set of hand crafted rules in the tokenization process as well as regular expressions to remove noisy text.
prec. rec. F$_1$
--------- ------- ------ -------
ARE 0.82 0.84 0.83
Acordão 0.71 0.89 0.79
Desp. 0.74 0.82 0.78
Outro 0.91 0.82 0.87
RE 0.77 0.70 0.73
Sent. 0.92 0.95 0.93
average 0.85 0.84 0.84
![Confusion matrix.\[fig:matrix\]](./media/confusion_matrix.pdf){width="\textwidth"}
Conclusion {#sec:conclusion}
==========
We proposed a tool to significantly speed up the first steps of the analysis of legal documents that reach the [*Supremo Tribunal Federal*]{} (STF, from Brazil), which is the world’s most clogged up supreme court. The task consists in classifying legal briefs ([*peças*]{}) into a set of 6 classes. For that we introduced a Bidirectional Long Short-Term Memory model which processes the first 1000 tokens of the documents, i.e., usually just the first page. This model is strong enough to classify these documents with an F$_1$ score of 84%, dismissing the need to run an OCR on the remaining pages of the document.
The next step of this project consists in designing a tool that combines information from all documents that compose a legal case in order to aid the decision making in the judgment process.
### Acknowledgments {#acknowledgments .unnumbered}
We acknowledge the support of “Projeto de Pesquisa & Desenvolvimento de aprendizado de máquina (machine learning) sobre dados judiciais das repercussões gerais do Supremo Tribunal Federal - STF”.
[^1]: The authors belong to three diferent insitutions of UnB: Faculdade de Engenharias do Gama, Departamento de Ciência da Computação and Faculdade de Direito. Further information at https://cic.unb.br/\~teodecampos/ViP
|
---
abstract: 'Polluted white dwarfs are [generally accreting terrestrial-like material that may originate from a debris belt like the asteroid belt in the solar system]{}. [The fraction of white dwarfs that are polluted drops off significantly for white dwarfs with masses $M_{\rm WD}\gtrsim 0.8\,\rm M_\odot$]{}. This implies that asteroid belts and planetary systems around main-sequence stars with mass $M_{\rm MS}\gtrsim 3\,\rm M_\odot$ may not form because of the intense radiation from the star. This is in agreement with current debris disc and exoplanet observations. [ The fraction of white dwarfs that show pollution also drops off significantly for low mass white dwarfs $(M_{\rm WD}\lesssim 0.55\,\rm M_\odot)$.]{} However, the low-mass white dwarfs that do show pollution are not currently accreting but have accreted in the past. We suggest that asteroid belts around main sequence stars with masses $M_{\rm MS}\lesssim 2\,\rm M_\odot$ [ are not likely to]{} survive the stellar evolution process. The destruction likely occurs during the AGB phase and could be the result of interactions of the asteroids with the stellar wind, the high radiation or, for the lowest mass stars that have an unusually close-in asteroid belt, scattering during the tidal orbital decay of the inner planetary system.'
author:
- |
Rebecca G. Martin[^1], Mario Livio, Jeremy L. Smallwood and Cheng Chen\
Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA
bibliography:
- 'ms.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Asteroid belt survival through stellar evolution: dependence on the stellar mass'
---
\[firstpage\]
minor planets, asteroids: general – planets and satellites: dynamical evolution and stability – white dwarfs – stars: AGB and post-AGB
Introduction {#intro}
============
White dwarfs are about $10^5$ times more dense than the Earth and so gravitational settling of heavy elements in the atmosphere is fast, less than around a few tens of Myr [e.g. @Paquette1986; @Wyatt2014]. If the cooling age is older than this, but less than about $500\,\rm Myr$, the atmosphere consists of hydrogen and/or helium only. Thus, the detection of metals in the atmosphere suggests accretion of material on to the white dwarf [e.g. @Veras2016]. Observations show that at least 27% of white dwarfs with cooling ages of $20-200\,\rm Myr$ are currently accreting debris and an additional 29% have accreted material in the past [@Koester2014].
The composition of the white dwarf polluting material is similar to that of the bulk Earth and solar system meteorites [e.g. @Gansicke2012; @Jura2014; @Xu2014; @Farihi2016; @Harrison2018; @Hollands2018; @Swan2019; @Doyle2019; @Bonsor2020]. [Therefore it must have formed inside of the snow line radius, the radius outside of which water is found in the form of ice that occurs at temperatures $\sim 170\,\rm K$ in the protoplanetary disc [e.g. @Podolak2004; @Lecaretal2006; @Kennedy2008; @Min2011; @MartinandLivio2012; @MartinandLivio2013snow].]{} [The material may be delivered to the white dwarf from a planetesimal belt similar to the asteroid belt in the solar system rather than a Kuiper belt equivalent. ]{} Only in rare cases is volatile rich material accreted [e.g., @Xu2017]. [Other suggested sources include delivery by moons [@Payne2016; @Payne2017] or fragments of broken up terrestrial planets (or moons) [@Malamud2020; @Malamud2020b]. ]{} Given the observed white dwarf accretion rates it is expected that most polluted white dwarfs have a reservoir of mass at least comparable to the mass in the asteroid belt in the solar system [@Zuckerman2010].
Asteroidal material is delivered to the white dwarf through a debris disc close to the white dwarf that forms through tidal disruptions [e.g. @Jura2003; @Debes2012; @Veras2014; @Veras2015; @Xu2018; @Malamud2020; @Malamud2020b]. Asteroids may be perturbed into highly eccentric orbits through interactions with undetected planets [e.g. @Debes2012; @Frewen2014; @Bonsor2015; @Smallwood2018]. We suggest that for a white dwarf to be polluted over long timescales there are two requirements. First, an asteroid belt must form around the main-sequence star. Secondly, the asteroid belt must survive the stellar evolution process. In Section \[planetarysystems\] we examine the properties of main-sequence stars that host debris discs and planetary systems. We further examine observational evidence for planetary systems and debris discs around evolved stars including polluted white dwarfs. In Section \[connect\] we propose that asteroid belts around low mass main-sequence stars [(those with mass less than about $2\,\rm M_\odot$)]{} are destroyed during stellar evolution leading to a lack of polluting material around low-mass white dwarfs [(those with mass less than about $0.55\,\rm M_\odot$)]{}. We draw our conclusions in Section \[concs\].
Observations of debris discs and planetary systems {#planetarysystems}
==================================================
In this Section we first examine observational evidence for the formation of asteroid belts around main-sequence stars. The detection of asteroid belts themselves is difficult, but stars that host planetary systems may have undetected asteroid belts. We consider here the range of masses of main-sequence stars that host planets and debris discs. We then explore the evidence for planetary systems around giant stars. Finally, we investigate observational evidence that suggests that asteroid belts that form around [stars with mass greater than about $2\,\rm M_\odot$]{} can survive through the stellar evolution process.
Main-sequence star systems {#mainsequence}
--------------------------
Most stars in the Milky Way host planetary systems [@Cassan2012]. Nearly all observed exoplanets have been found around stars with masses $M_{\rm MS}\lesssim 3\,\rm M_\odot$. There are only a few exceptions that have higher stellar mass. The highest mass star with a well determined mass that hosts a planet is UMa that has mass $3.09 \pm 0.07\,\rm M_\odot$ [@Sato2012]. [This upper mass limit is not sharp transition but a tail where the number of planets discovered decreases with host star mass [e.g. @Reffert2015; @Ghezzi2018]. Observing planets around O-type and B-type stars is difficult and so the limit is a combination of detection limitations and where planets can form and survive around more massive stars [@Kennedy2008; @Veras2020b].]{}
Debris discs are detected around about 25% of main-sequence stars [@Hughes2018]. Discs are observed around stars with masses $\lesssim 2.4\,\rm M_\odot$ [@Koenig2011]. Discs around more massive stars may be photoevaporated on timescales which are too short to be observed due to intense radiation from the host star.
[Debris discs are observed through the thermal emission of the dust and may be characterised by the infrared excess observed in their SED. The excess may generally be modelled with one or two blackbody components [e.g. @Su2009; @Su2013]. The cold components have temperatures $<130\,\rm K$ while the warm components have temperatures $\sim 190\,\rm K$ [@Morales2011; @Ballering2013; @Chen2014]. The warm and cold components come from different radial locations with different temperatures [@Kennedy2014].]{} Debris disc observations show that two-component structures, like the asteroid belt and the Kuiper belt in the solar system, are common [e.g. @Kennedy2014; @Rebollido2018]. The cause for the gap between the belts is likely the formation of planets in the gap that remove all the planetesimals from the region. [@Geiler2017] found that that $98\%$ of observed debris disc systems can be explained with a two-component structure rather than a one-component structure. The few sources for which warm dust in the systems cannot be explained by this structure must be a result of cometary sources or a recent major collision or planetary system instability.
Giant planets are thought to form outside of the snow line radius, since there is a higher density of solid material there [e.g. @Pollack1996]. Thus, asteroid belts may coincide with the location of the snow line radius [@MartinandLivio2013asteroids]. [@Ballering2017] found that the warm dust components in single-component systems (those without a cold component) are aligned with the primordial snow line, meaning the snow line in the protoplanetary disc. However, in two-component systems, the location is more diverse. The belts, at least in the one-component system, may be formed of terrestrial material. The location of the warm dust belts in one-component models have a best fit $R_{\rm dust}/{\rm au}=3.68(M/{\rm M_\odot})^{1.08}$ [@Ballering2017]. The shaded region in Fig. \[snowline\] shows the $1\sigma$ scatter around the best fitting line to the radius of warm dust belts [@Ballering2017]. In the two-component models the warm dust components show little correlation with stellar mass and are scattered in the approximate range $0.5-30\,\rm au$. We discuss this figure in more detail in Section \[connect\].
![The shaded region shows the observed best fitting region for the location of warm dust belts (those with temperature $\sim 190\,\rm K$.) around MS stars [data from @Ballering2017]. The dotted blue line shows the critical initial semi-major axis above which a jovian planet survives the AGB phase. The dashed red line shows the critical initial semi-major axis above which a terrestrial planet survives the AGB phase. These theoretical lines are approximated from the stellar evolution models of [@Mustill2012]. The dot-dashed lines show theoretical survival radii for $100\,\rm m$ (upper), $1\,\rm km$ (middle) and $10\,\rm km$ sized asteroids approximated by equation (\[asteroid\]) [@Dong2010]. []{data-label="snowline"}](data4.eps){width="\columnwidth"}
Giant star systems
------------------
[To date, 112 substellar companions[^2] around 102 G and K giant stars have been found [e.g. @Reichert2019]. [@Grunblatt2019] investigated 2476 low luminosity red giant branch stars observed by the K2 mission [@Howell2014]. They found a higher occurence rate of planets with size greater than Jupiter in orbital periods less than 10 days compared to around dwarf Sun-like stars. This suggests that the effects of stellar evolution on the occurence of close in planets that are larger than Jupiter are not significant until the star moves significantly up the red giant branch.]{} Debris discs have also been observed around giants suggesting that they can also survive the stellar evolution [e.g. @Bonsor2013; @Bonsor2014]. Debris discs around giant stars are more difficult to detect than around main sequence stars because radiation pressure removes small-particle dust around higher luminosity stars [@Bonsor2010].
White dwarf systems
-------------------
The fraction of white dwarfs that host debris discs is somewhere between a few percent up to 100%, but most discs are too faint to detect [e.g. @Barber2012; @Veras2016; @Bonsor2017; @Swan2019]. The current observational limit is reached for discs around white dwarfs with cooling ages $t_{\rm cool}>0.5\,\rm Myr$ [@Bergfors2014]. Compact debris discs are thought to be formed by the tidal disruption of small bodies around the white dwarf [@Jura2003; @Farihi2009; @Veras2014]. Atomic emission lines suggest the existence of gaseous discs co-located with the compact circumstellar dust [@Gansicke2006; @Guo2015]. [There is one exception to the compactness of gaseous discs, the disc around WD J0914+1914 is thought to be formed from an evaporating giant planet on a close-in orbit around the white dwarf [@Gansicke2019].]{}
Since white dwarfs are intrinsically faint, transit searches for debris and planets are difficult. However, the light curve of WD 1145+017 shows transit features thought to be produced by dust clouds released by planetesimals that orbit the white dwarf with an orbital period of about $4.5\,\rm hr$ [@Vanderburg2015; @Gansicke2016]. There is also evidence for solid objects orbiting around white dwarfs SDSS J1228+1040 [@Manser2019] [and ZTF J0139+5245 [@Vanderbosch2019]. The discoveries made so far have arisen from ZTF, GTC and SDSS.]{} [@vansluijs2018] examined a sample of 1148 white dwarfs observed by K2 and did not identify any substellar body transits with orbital separation $<0.5\,\rm au$.
Fig. \[wddata\] shows the fraction of observed white dwarfs that are either currently accreting or show evidence for past accretion using the data from [@Koester2014]. [ The highest mass white dwarf with evidence for pollution has mass $M_{\rm WD}=0.91\,\rm M_\odot$ [@Fusillo2019]. This corresponds to a progenitor main-sequence star mass of about $4\,\rm M_\odot$ [@Koester2014].]{} [However, there is a transition where the fraction of white dwarfs that are polluted falls off significantly at a mass of]{} around $M_{\rm WD}=0.8\,\rm M_\odot$. This corresponds to a main-sequence star of around $M_{\rm MS}=3\,\rm M_\odot$. This suggests that asteroid belt formation or survival around high mass stars (those with mass $\gtrsim 3\,\rm M_\odot$) is difficult. [Stars with mass greater than about $3\,\rm M_\odot$]{} are too hot for the formation of a long-lived dusty disc. This is consistent with the observations of debris discs and planetary systems around main-sequence stars discussed in Section \[mainsequence\]. [Recently, [@Veras2020b] explored the limits on the locations of planets that would be able to survive to the white dwarf phase around stars with masses in the range $6-8\,\rm M_\odot$. They found that a major planet must be located at orbital distance greater than about $3-6\,\rm au$ at the end of the main-sequence lifetime in order to survive stellar evolution. The orbital radius outside of which minor planets survive is in the range $10-1000\,\rm au$ depending on planet size. Thus, if white dwarf pollution is to be observed around higher mass white dwarfs in the future it would come from already fragmented debris since the minor planets would likely not be still intact. ]{}
While [the number of white dwarfs included in the data drops off at low masses]{}, there does also appear to be a transition at small masses for which the [fraction]{} of white dwarfs that are polluted drops, at around $M_{\rm WD}=0.55\,\rm M_\odot$. The white dwarfs with these low masses tend to be younger and no longer accreting. We therefore suggest that asteroid belts around low mass main-sequence stars [with mass less than $2\,\rm M_\odot$]{} may form, but they do not survive the stellar evolution process to the formation of the white dwarf. We discuss possible theoretical explanations for this scenario in the next Section.
![The fraction of the observed white dwarfs that show evidence for accretion (dashed line) and the fraction of currently accreting white dwarfs (solid line) as a function of the white dwarf mass. The data are taken from [@Koester2014]. []{data-label="wddata"}](data.eps){width="\columnwidth"}
Asteroid belt destruction around low mass stars {#connect}
===============================================
In this Section we examine theoretical models for the evolution of asteroid orbits through stellar evolution. Our goal is to explain why the asteroid belts around stars [with mass less than about $2\,\rm M_\odot$]{} may not survive the process, while those around more massive stars (those with mass $2-3\,\rm M_\odot$) do.
Planet survival
---------------
Planets and debris that are close to the star during the main-sequence (MS) will not survive stellar evolution to the white dwarf phase as they may be engulfed or evaporated by a giant star [e.g. @Villaver2007; @Villaver2009; @Kunitomo2011]. Bodies that become engulfed by the star are expected to be destroyed unless their mass is a Jupiter mass or more [e.g. @Livio1984; @Mustill2018]. The star is largest during the AGB phase and at that time its size in AU is about equal to its initial main-sequence masses in $\rm M_\odot$ for mass in the range $1-5\,\rm M_\odot$ [e.g. @Mustill2018]. For higher stellar mass, there is more mass loss that occurs during the AGB phase. The mass loss leads to the expansion of the orbits of substellar bodies and therefore allows them to survive even if they begin at radii such that the stellar radius subsequently expands beyond [e.g. @Livio1984; @Mustill2012].
There are two competing effects that determine where the critical survival orbital radius is for a planet mass body. The tidal force pulls the object towards the expanded envelope while the effects of stellar mass loss push the planet away [e.g. @Mustill2012]. Tidal forces are stronger for more massive planets and so the survival radius increases with planet mass. The more massive the main-sequence star, the farther out planets must be to survive engulfment. Fig. \[snowline\] shows approximate survival radii for initially circular orbit terrestrial planets (dashed red line) and Jupiter mass planets (dotted blue line) [@Mustill2012]. The survival radii are larger for eccentric planets [@Villaver2014]. The corresponding lines for the RGB would be much closer in [e.g. @Kunitomo2011; @Villaver2014].
Terrestrial planets form inside of the snow line, and hence inside of the warm dust belt location [e.g. @Raymond2009]. [The critical orbital radius for which terrestrial planets survive stellar evolution (the red dashed line in Fig. \[snowline\]) is close to the location of the warm dust belts (the shaded region in Fig. \[snowline\]), ]{} Consequently, terrestrial planets around stars [with mass less than about $1\,\rm M_\odot$]{} may not survive until the white dwarf phase. However, around higher mass stars, there is a range of orbital radii between the location of the warm dust belt and the critical survival radius where terrestrial planets may survive.
Close-in giant planets also do not survive. If there is a giant planet that is engulfed, as it spirals in it may disrupt an interior asteroid belt. Thus, it would seem that for asteroid belt survival we require the giant planets to survive the engulfment process. However, as shown in Fig. \[snowline\], the observed asteroid belts are at radii larger than the critical survival radius for a Jupiter mass planet. Thus, scattering of asteroids from a planet undergoing tidal decay is unlikely, except perhaps for the lowest mass stars that have close in asteroid belts.
Planetesimal survival
---------------------
For a planetesimal to survive stellar evolution to the white dwarf phase, it must not be engulfed by the star itself. Considering only the effects of stellar mass loss and tidal forces, this is a less stringent constraint than that which applies to the survival of giant planets, since the planetesimal exerts only a weak tidal torque. The orbital locations of the warm dust belts are much larger than the maximum size of an AGB star (see Fig. \[snowline\]) and so engulfment is not likely unless there is an unusually strong gas drag in the stellar wind.
The adiabatic approximation for the expansion of the orbits of planet and asteroid objects may be employed within about $100\,\rm au$ [@Veras2011; @Veras2016b]. Orbital eccentricity is conserved and the relative semi-major axis increase scales with the relative stellar mass loss. Additionally, asteroids may interact strongly with the radiation from the AGB star. The interaction is complex since it depends upon the shape, orientation and albedo of each asteroid. Asteroids can be radiatively pushed by the Yarkovsky effect [@Bottke2001; @Bottke2006; @Veras2019]. The Yarkovsky drift may be several orders of magnitude larger than that from Poynting-Robertson and radiation pressure [@Veras2015c]. Asymmetric asteroids can be spun up through the YORP effect [e.g. @Rubincam2000; @Vokroulicky2002]. The YORP effect may destroy asteroids with sizes $100{\,\rm m}-10\,\rm km$ at orbital radii $\lesssim7\,\rm au$ [@Veras2014b; @Veras2020]. For such conditions, the YORP affect alone may be responsible for the destruction of asteroid belts around low-mass main-sequence stars [(those with mass less than about $2\,\rm M_\odot$) ]{}, while those around more massive stars survive because they are at larger orbital radii.
For asteroids with sizes small enough that the Yarkovsky effect is not important, the drag force becomes dominant [@Veras2015c]. The survival of planetesimals during the AGB phase may be determined by balancing the expansion of the orbit due to the stellar mass loss and the gas resistance. The critical radius for survival is $$R_{\rm crit}=2.57\,\left(\frac{M_{\rm MS}}{\rm M_\odot}\right)^{3/5}\left(\frac{s}{0.1\,\rm km}\right)^{-2/5}\,\rm au
\label{asteroid}$$ [@Dong2010], where $s$ is the asteroid size and we assume a wind speed $v_{\rm wind}=10\,\rm km\,s^{-1}$ and an asteroid density of $\rho=3\,\rm g\, cm^{-3}$. In Fig. 1 we show the critical survival radius for asteroids of size $100\,\rm m$ (upper dot-dashed line), $1\,\rm km$ (middle dot-dashed line) and $10\,\rm km$ (lower dot-dashed line). The smaller asteroids in most belts may not survive the wind loss, while larger asteroids can. Asteroid belts around low-mass stars [(with mass less than about $1\,\rm M_\odot$) ]{} may be removed for sizes $\lesssim 1\,\rm km$. Thus, asteroid belts around low mass stars may be severely depleted in mass through the interaction with the stellar wind.
Conclusions {#concs}
===========
There is strong observational and theoretical evidence that white dwarf pollution occurs from asteroid-belt-like material. [There are significant drop offs in the fraction of white dwarfs that are polluted at masses higher than about $0.8\,\rm M_\odot$ and lower than about $0.55\,\rm M_\odot$]{}. We have therefore proposed that (i) asteroid belts (and planetary systems) do not form around stars more massive than about $3\,\rm M_\odot$ and (ii) asteroid belts around stars less massive than about $2\,\rm M_\odot$ do not survive stellar evolution to the white dwarf stage. There are several mechanisms that can contribute to asteroid belt destruction during the AGB phase. These include the interaction of asteroids with the stellar wind through gas drag and the YORP effect, both of which affect the close-in asteroid belts around lower-mass main sequence stars. The orbital decay of a giant planet due to tides may scatter an inner asteroid belt for the very lowest mass stars.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank an anonymous referee for providing a useful and thorough review. RGM acknowledges support from NASA through grant NNX17AB96G. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: https://www.lsw.uni-heidelberg.de/users/sreffert/giantplanets/giantplanets.php
|
---
author:
- |
Valerio Lucarini$^*$$^\dagger$\
`[email protected]`,\
Tobias Kuna$^\dagger$\
`[email protected]`,\
Jeroen Wouters$^*$\
`[email protected]`,\
Davide Faranda$^*$\
`[email protected]`\
bibliography:
- 'response.bib'
title: 'Relevance of sampling schemes in light of Ruelle’s linear response theory'
---
*$^*$Klimacampus, Universität Hamburg, Hamburg, Germany\
$^\dagger$Department of Mathematics and Statistics, University of Reading, Reading, UK*
Introduction
============
The study of how the properties of general non-equilibrium statistical mechanical systems change when considering a generic perturbation, usually related to variations either in the value of some internal parameters or in the external forcing, is of great relevance, both in purely mathematical terms and with regards to applications to the natural and social sciences. Whereas in quasi-equilibrium statistical mechanics it is possible to link the response of a system to perturbations to its unforced fluctuations thanks to the fluctuation-dissipation theorem [@kubo_66; @zubarev], in the general non-equilibrium case it is not possible to frame rigorously an equivalence between internal fluctuations and forcings. At a fundamental level, this is closely related to the fact that forced and dissipative systems feature a singular invariant measure. Whereas natural fluctuations of the system are restricted to the unstable manifold, because, by definition, asymptotically there is no dynamics along the stable manifold, perturbations will induce motions - of exponentially decaying amplitude - out of the attractor with probability one, as discussed in, e.g., [@ruelle_review_2009; @ruelle_98; @luc08]. It is worth noting that Lorenz anticipated some of these ideas when studying the difference between free and forced variability of the climate system [@lorenz79]. This crucial difficulty inherent to out-of-equilibrium systems is lifted if the external perturbation is, rather artificially, everywhere tangent to the unstable manifold, or if the system includes some stochastic forcing, which smooths out the resulting the invariant measure [@laco07].
Recently, Ruelle [@ruelle_differentiation_1997; @ruelle_98; @ruelle_review_2009] paved the way to the study of the response of general non-equilibrium systems to perturbations by presenting rigorous results leading to the formulation of a response theory for Axiom A dynamical systems [@ruelle89], which possess a Sinai-Ruelle-Bowen invariant measure [@young_what_2002]. Given a measurable observable of the system, the change in its expectation value due to an $\epsilon$-perturbation in the flow (or in the map, in the case of discrete dynamics) can be written as a perturbative series of terms proportional to $\epsilon^n$, where each term of the series can be written as the expectation value of some well-defined observable over the unperturbed state. Ruelle’s formula is identical to Kubo’s classical formula [@kubo_57] when a Hamiltonian system is considered [@luc08].
Whereas Axiom A systems are mathematically non-generic, the applicability of the Ruelle theory to a variety of actual models is supported by the so-called *chaotic hypothesis* [@galla], which states that systems with many degrees of freedom behave as if they were Axiom A systems when macroscopic statistical properties are considered. The chaotic hypothesis has been interpreted as the natural extension of the classic ergodic hypothesis to non-Hamiltonian systems [@galla06].
In the last decade great efforts have been directed at extending and clarifying the degree of applicability of the response theory for non-equilibrium systems along five main lines:
- extension of the theory for more general classes of dynamical systems [@dolgopyat; @baladi];
- introduction of effective algorithms for computing the response in dynamical systems with many degrees of freedom [@abra07], in order to support the numerical analyses pioneered by [@reick_linear_2002; @cessac];
- investigation of the frequency-dependent response - the susceptibility - for the linear and nonlinear cases, with the ensuing introduction of a new theory of Kramers-Kronig relations and sum rules for non-equilibrium systems [@luc08; @shimizu10] supported by numerical experiments [@luc09];
- study of the response to external perturbations of non-equilibrium systems undergoing stochastic dynamics [@majda_stochastic_2010; @yuge10];
- use of Ruelle’s response theory to study the impact of adding stochastic forcing to otherwise deterministic systems [@luc11b].
In particular, the response theory seems especially promising for tackling notoriously complex problems such as those related to studying the response of geophysical systems to perturbations, which include the investigation of climate change; see discussions in [@abra07; @luc09; @luc11]. In particular, in [@luc11], it is discussed that by deriving from the linear susceptibility the time-dependent Green function, it is possible to devise a strategy to compute climate change for a general observable and for a general time-dependent pattern of forcing. Recently, response theory is becoming of great interest also in social sciences such as economics [@hawkins_2009].
When developing a response theory, there are two possible ways to frame the temporal impact of the additional perturbation to the dynamics. Either one considers the impact at a given time $t$ of a perturbation affecting the system since a very distant past, or one considers the impact in the distant future of perturbations starting at the present time. When deriving the response formula, [Ruelle]{} takes the first approach and delivers the *correct* formula [@ruelle_differentiation_1997]. Taking a different point of view and considering the specific case of periodic perturbations – which, anyway, tell us the whole story about the response by linearity –, Reick [@reick_linear_2002] derives a formula that is well suited for analyzing the output of numerical experiments [@luc09; @luc11].
Given the great relevance and increasing popularity in applications of the response theory introduced by Ruelle, in this paper, we reconsider the theory of the linear response of non-equilibrium steady states to perturbations and try to bridge the theoretical derivations and the strategies for designing numerical experiments and analyzing efficiently their outputs.
In Section 2, we study the relevance of the choice of the time horizon for evaluating the impact of the perturbation and we demonstrate by direct calculation that the Ruelle approach is the correct one. We clarify some of the assumptions implicitly considered in his derivations. We then discuss the special case of periodic perturbations, showing that using them as basis for a response theory greatly simplifies the formulas and the conditions under which the formulas are derived. In Section 3, we discuss the implications of our results in terms of strategies for improving the quality of numerical simulations and of the analysis of their output signals and reconsider Reick’s formula [@reick_linear_2002]. In Section 4 we present our conclusions and perspectives for future work.
Linear Response Theory, revised {#sec:linear_response_theory}
===============================
Separable perturbations
-----------------------
We study the linear response of a discrete dynamical system to general time-dependent perturbations. All calculations are formal, in the sense that we neglect all higher orders in the perturbation without deriving an estimate for these terms and we assume that all sums converge in all senses necessary.
The unperturbed dynamical system is given by $$x_{t+1} = f(x_t) \,,$$ with $t \in \mathbb{Z}$, $x_t \in M$, $M$ being a smooth manifold and $f: M \rightarrow M$ a differentiable map. For simplicity we consider a time-independent unperturbed dynamics, although the following can be extended to a time-dependent case in a straightforward manner. Moreover, the analysis of the case of a continuous time flow $\dot{x} = f(x)$ is perfectly analogous to what is presented in the following and the main corresponding results will be mentioned in Appendix \[app:continuous\].
The dynamical system is perturbed by a time-dependent forcing $X(t,x)$ as follows: $$\begin{aligned}
\tilde{x}_{t+1} = \tilde{f}_{t+1}(\tilde{x}_t) := f(\tilde{x}_t) + X(t+1,f(\tilde{x}_t)) \,. \label{eq:perturbation}\end{aligned}$$
The effect of the perturbation on individual trajectories is in general difficult to describe. More can however be said about the statistical properties of the system. One can look at the expectation values of observables under invariant states of the dynamical system: $$\begin{aligned}
\rho(A) := \int \rho(dx) A(x) \,,\end{aligned}$$ where $\rho(dx)$ is an invariant measure of the unperturbed dynamics i.e. $$\begin{aligned}
\rho(A \circ f) = \rho (A) \,.\end{aligned}$$ for any observable A. In general, a dynamical system can possess many invariant measures. The physically relevant measure for dynamical systems is the SRB measure [@young_what_2002]. This measure is physical in the sense that for a set of initial conditions of full Lebesgue measure the time averages $\lim_{t \rightarrow \infty}
\frac{1}{t} \sum_{k=1}^t A(f^k(x))$ converge to the expectation value under $\rho$. In other words, for any measure $l(dx)$ that is absolutely continuous w.r.t. Lebesgue, we have that $$\begin{aligned}
\rho(A) = \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{k=1}^t \int l(dx) A( f^k (x)) \,. \label{eq:SRB}\end{aligned}$$
We want to determine the linear response of expectation values under the SRB measure to perturbations of the dynamical system as in Eq. \[eq:perturbation\]. We denote by $\delta_T \rho$ the difference in the expectation value between the perturbed and unperturbed system at time $T$. In [@ruelle_review_2009], Ruelle presents a formula for the linear response due to perturbations that are separable in time and space: $$\begin{aligned}
X(t,x) = \phi(t) \chi(x) \,.\end{aligned}$$ The leading order term of the expansion of $\delta_T \rho(A)$ in $X$ is given by $$\begin{aligned}
\delta_T \rho(A) \approx \sum_{j\in \mathbb{Z}} G_A(j) \phi(T-j) \,, \label{eq:ruelle_time_sep}\end{aligned}$$ with $$\begin{aligned}
G_A(j)= \theta(j) \int \rho(dx) \chi(x) D(A \circ f^j)(x) \,, \label{eq:response_func}\end{aligned}$$ where $\theta$ is the Heaviside function. Since $\delta_T \rho(A)$ is expressed as a convolution product of $G_A$ and $\phi$, the Fourier transform of the response $\delta_\omega \rho (A) =
\sum_{T \in \mathbb{Z}} e^{i T \omega} \delta_T \rho (A)$ is given by a product of the Fourier transform $\hat{\phi}(\omega)$ of the time factor $\phi(t)$ and a susceptibility function $\hat{\kappa}_A (\omega)$: $$\begin{aligned}
\delta_\omega \rho (A) &\approx \hat{\kappa}_A (\omega) \hat{\phi} (\omega) \,. \label{eq:response}\end{aligned}$$ where $$\begin{aligned}
\hat{\phi}(\omega)&=\sum_{j \in \mathbb{Z}} e^{i j \omega} \phi(j) \,, \nonumber \\
\hat{\kappa}_A(\omega) &= \sum_{j \in \mathbb{Z}} e^{i j \omega} G_A(j) \nonumber \\
&= \sum_{j \geq 0} e^{i j \omega} \int \rho (dx) \chi(x) D(A \circ f^j)(x) \,.
\label{eq:suscep}\end{aligned}$$ Due to the causality of the response function $G_A(j)$ (i.e. $G_A(j)=0$, $j< 0)$), the susceptibility $\hat{\kappa}_A(\omega)
$ is analytic in the upper complex plane and satisfies Kramers-Kronig relations [@ruelle_98; @luc08].
General perturbations
---------------------
If the perturbation is of a more general nature (i.e. not separable), we can deduce a linear response formula from Eq. \[eq:response\], solely based on linearity in the following way. Let $\phi_r(t)$ be a Schauder basis [@lindenstrauss_classical_1996] of time-dependent functions and $\psi_s(x)$ a Schauder basis of space-dependent functions. One can take for example the Fourier basis in time and a wavelet basis in space, or whatever basis may be suitable for the system at hand. The product functions $\phi_r(t)
\psi_s(x)$ then form a basis of the time and space dependent functions as a tensor product [@ryan_2002]. More concretely, we may for an appropriate sense of convergence assume that any function $X(t,x)$ can be decomposed in the product basis $\phi_r(t)
\psi_s(x)$ with coefficients $a_{r,s}$: $$\begin{aligned}
X(t,x) = \sum_{r,s \geq 0} a_{r,s} \phi_r(t) \psi_s(x) \,.\end{aligned}$$ Since each of the factors in this sum is separable, we can use Eq. \[eq:response\] and the linearity of the response to get that the response is given by $$\begin{aligned}
\delta_\omega \rho (A) \approx \sum_{r,s \geq 0} a_{r,s}
\hat{\phi}_r(\omega) \hat{\kappa}_{s,A}(\omega) \,, \label{eq:suscav}\end{aligned}$$ where $\hat{\kappa}_{s,a}$ is the susceptibility function of observable $A$, corresponding to the forcing pattern given by $\psi_s(x)$. Since the vectors $\psi_s(x)$ constitute a basis, the functions $\hat{\kappa}_{s,a}$ are *elementary* linear susceptibilities that allow to construct the response of the system to any pattern of forcing.
By inserting the expression of $\hat{\kappa}_{s,A}(\omega)$ from Eq. \[eq:suscep\] into Eq. \[eq:suscav\], it is possible to deduce the frequency-dependent response of the system. It is expressed as an ensemble average of a dot product of Fourier transforms, namely the transforms of the perturbation term and of the linear tangent of the observable, $G_A(\omega,x)$: $$\begin{aligned}
\delta_\omega \rho (A) &\approx \sum_{j \geq 0} e^{i j \omega} \int \rho (dx) \hat{X}(\omega,x)
D(A \circ f^j)(x) \nonumber
\\
&= \int \rho(dx) \hat{X}(\omega,x) G_{A}(\omega,x) \,, \label{eq:suscav2}\end{aligned}$$ with $$\begin{aligned}
G_{A}(\omega,x) &= \sum_{j \geq 0} e^{i j \omega} D(A \circ f^j)(x) \nonumber \\
\hat{X}(\omega,x) &= \sum_{T \in \mathbb{Z}} X(T,x) e^{i \omega T} \nonumber \\
&= \sum_{r,s \geq 0} a_{r,s} \hat{\phi}_r (\omega) \psi_s (x). \label{eq:suscav3}\end{aligned}$$ Instead from Eqs. \[eq:ruelle\_time\_sep\]-\[eq:response\_func\] in the time domain $$\begin{aligned}
\delta_T \rho(A) \approx \int \rho(dx) \sum_{j\geq 0} X(T-j,x) D(A \circ f^j)(x) \,. \label{eq:ruelle_time}\end{aligned}$$ For the case of a periodic perturbation $X(t+\tau,x)=X(t,x)$, where $\tau\in\mathbb{N}$, we get as linear response $$\begin{aligned}
\delta_T \rho (A) &\approx \int \rho(dx) \sum_{n=1}^\tau \sum_{m=0}^\infty X(T-n-m \tau,x) D(A \circ f^{n+m \tau})(x) \nonumber \\
%&= \int \rho(dx) \sum_{n=1}^\tau X(T-n,x) \sum_{m=0}^ \infty D(A \circ f^ {n + m \tau}) (x) \nonumber \\
&= \int \rho(dx) \sum_{n=1}^\tau X(T-n,x) G_{A,n} (x) \,, \label{eq:ruelle_periodic}\end{aligned}$$ with $$\begin{aligned}
G_{A,n} (x) = \sum_{m=0}^ \infty D(A \circ f^ {n + m \tau}) (x) \,.\end{aligned}$$
In order to elucidate some crucial aspects of the Ruelle’s response theory, we now propose a direct derivation of the linear response to the perturbation $X(t,x)$ by considering the history of the perturbed and unperturbed trajectory of the system and verify under which conditions we find agreement with Eqs. \[eq:suscav2\]-\[eq:ruelle\_time\]. Our goal is to derive the leading order term of the expansion of $\delta_T \rho(A)$ with respect to $X$ from first principle, i.e. without resorting to the Schauder decomposition as above. Such a derivation should of course arrive at the same results as those in Eqs. \[eq:suscav2\]-\[eq:ruelle\_time\].
### Response at a moving time horizon {#sec:forward}
We describe the perturbed measure $\tilde{\rho}_T(A)$ such that the system is initialized at time $T$ in an initial condition according to the measure $l$. We move the time horizon at which we observe forward and average the time-evolved measurements. The system is prepared and then observed while it is evolving over a sufficiently long time. The measure $\tilde{\rho}_T$ is time-dependent as the dynamics $\tilde{f}$ is also time-dependent. Formally we take $\tilde{\rho}_T$ to be the ergodic mean of the expectation values of $A$, starting at time $T$: $$\begin{aligned}
\tilde{\rho}_T (A) &= \lim_{t \rightarrow \infty}
\frac{1}{t}\sum_{k=1}^t \int l(dx) A(\tilde{f}_T^{k}(x)) \,. \label{eq:time_SRB}
% \tilde{\rho}_T(dx) &= \lim_{t \rightarrow \infty}
% \frac{1}{t}\sum_{k=1}^t (\tilde{f}_T^{k})^* l(dx)\end{aligned}$$ Here $l(dx)$ is an initial measure that is absolutely continuous with respect to Lebesgue and $\tilde{f}^k_T$ represents $k$ iterations of the perturbed dynamics from time $T$ to $T+k$: $$\begin{aligned}
\tilde{f}^k_T (x) = \tilde{f}_{T+k} \circ \ldots \circ \tilde{f}_{T+1} (x) \label{eq:forward} \,.\end{aligned}$$ The difference in expectation values $\delta_T \rho$ is the given by $$\begin{aligned}
\delta_T \rho(A) = \tilde{\rho}_T(A) - \rho(A). \label{eq:deltarho}\end{aligned}$$
Following the computation presented in [@ruelle_review_2009] for the separable case, we can expand the perturbed dynamics $\tilde{f}$ around the unperturbed dynamics $f$. We then try to rewrite the response of the perturbed system in terms of the SRB measure of the unperturbed system by finding an expression for $A(\tilde{f}_T^k(x))$ in terms of $A(f^k(x))$.
We can approximate up to first order in $X$ the two time step future evolution by expanding around the unperturbed dynamics $f^2(x)$: $$\begin{aligned}
\tilde{x}_{T+2} &= \tilde{f}_{T+2} \circ \tilde{f}_{T+1} (\tilde{x}_T) \\
& \approx f^2(\tilde{x}_T) + X(T+1,f(\tilde{x}_T)).Df(f(\tilde{x}_T)) + X(T+2,f^2(\tilde{x}_T)) \,.\end{aligned}$$ For $k$ time steps we similarly get: $$\begin{aligned}
\tilde{x}_{T+k} &= \tilde{f}_{T+k} \circ \ldots \circ \tilde{f}_{T+1} (\tilde{x}_T) \\
&\approx f^k (\tilde{x}_T) + \sum_{j=1}^k X(T+j,f^{j}(\tilde{x}_T)).(Df^{k-j})(f^{j} (\tilde{x}_T) )\,.\end{aligned}$$ Thus, we can approximate $A(\tilde{f}_{T+k} \circ \ldots \circ \tilde{f}_{T+1} (x))$ to first order in $X$ as follows: $$\begin{aligned}
A(\tilde{f}_{T+k} \circ \ldots \circ \tilde{f}_{T+1} (x)) \approx & A(f^k(x)) \nonumber \\
&+ A^\prime (f^k(x)) \left( \sum_{j=1}^k X(T+j,f^{j}(x)).(Df^{k-j})(f^{j} (x) ) \right) \nonumber \\
&= A(f^k(x)) \nonumber \\
&+ \sum_{j=1}^{k} X(T+j,f^{j}(x)) D(A \circ f^{k-j}) (f^{j}(x)) \,. \label{eq:A_approx}\end{aligned}$$
The linear response of $A$ is obtained by substituting Eq. \[eq:A\_approx\] into Eq. \[eq:deltarho\], through Eq. \[eq:SRB\] and Eq. \[eq:time\_SRB\]: $$\begin{aligned}
\delta_T \rho(A) &\approx \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{k=1}^t \int l(dx) \sum_{j=1}^{k} X(T+j,f^{j}(x)) D(A \circ f^{k-j}) (f^{j}(x)) \nonumber \\
%& = \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j = 1}^{t} \sum_{i=0}^{t-j} \int l(dx) X(T+j,f^j(x)) D(A \circ f^{i}) (f^j(x)) \nonumber \\
& = \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{i = 0}^{t-1} \sum_{j=1}^{t-i} \int l(dx) X(T+j,f^j(x)) D(A \circ f^{i}) (f^j(x)) \,. \nonumber\end{aligned}$$ Using that for $ i \geq t $ the expression is zero, we have $$\begin{aligned}
\delta_T \rho(A) & \approx \sum_{i \geq 0} \int \left( \lim_{t \rightarrow \infty} \frac{1}{t}
\sum_{j=1}^{t-i} (f^*)^{j} l(dx) X(T+j, x) \right) D(A \circ f^{i}) (x) \,. \label{eq:future}\end{aligned}$$ Note that it is not possible to rewrite the sum in $j$ as the ergodic time mean of $l$ due to the time dependence of the perturbation $X(T+j,x)$. Therefore, surprisingly, Eq. \[eq:future\] does not in general agree with Eq. \[eq:ruelle\_time\]. In particular, by taking the limit on the right hand side, we obtain that the $T$-dependence disappears. Say we shift $T$ to $T-T^\prime$ in the limit appearing in the above equation: $$\begin{aligned}
&\lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j=1}^{t-i} (f^*)^{j} l(dx) X(T-T^\prime + j, x) \\
&= \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j^\prime = 1-T^\prime}^{t-i-T^\prime} (f^*)^{j^\prime + T^\prime} l(dx) X(T + j^\prime, x) \\
&= \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j^\prime = 1}^{t-i} (f^*)^{j^\prime} \left((f^*)^{T^\prime} l(dx) \right) X(T + j^\prime, x) \,.\end{aligned}$$ Taking the reasonable assumption that the result in Eq. \[eq:future\] does not depend on the initial measure $l(dx)$ (this cannot be obtained from the uniqueness of the SRB measure), the obtained response of the system is time-independent even if the forcing is time-dependent.
Let us compare the result contained in Eq. \[eq:future\] with Eq. \[eq:ruelle\_time\_sep\] in the special case of a time-independent perturbation $X(t,x)=\chi(x)$. Now Eq. \[eq:future\] and Eq. \[eq:ruelle\_time\_sep\] agree since Eq. \[eq:future\] simplifies to: $$\begin{aligned}
\delta_T \rho (A) \approx \sum_{i \geq 0} \int \rho(dx) \chi(x) D(A\circ
f^i)(x) \,,\end{aligned}$$ because $\lim_{t \rightarrow \infty} 1/t \sum_{j=1}^{t-i}
(f^*)^{j} l(dx))=\rho(dx)$, by the definition of the SRB measure. The formula given by Ruelle [@ruelle_98] is recovered, as can be seen by substituting $\phi(t)=1$ into Eq. \[eq:ruelle\_time\_sep\].
However, already in the case of a time-periodic perturbation $$X(t,x)=X(t+\tau,x)$$ there is no agreement between Eq. \[eq:future\] and Eq. \[eq:ruelle\_time\]. In this case the sum over $j$ appearing in Eq. \[eq:future\] can be written as a double sum, one over $k$ periods, indexed by $m$, and one over the $\tau$ phases in each period, indexed by $n$: $$\begin{aligned}
& \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j=1}^t (f^*)^j l(dx) X(T+j,x) \nonumber \\
& = \lim_{k \rightarrow \infty} \frac{1}{k \tau} \sum_{m=1}^k \sum_{n=1}^\tau (f^*)^{m \tau} (f^*)^{n} l(dx) X(T+n,x) \nonumber \\
& = \frac{1}{\tau}\sum_{n=1}^\tau \rho_n(dx) X(t+n,x) \,, \label{eq:moving_periodic}\end{aligned}$$ with $$\rho_n(dx) = \lim_{k \rightarrow \infty} \frac{1}{k } \sum_{m=1}^k (f^*)^{m \tau} (f^*)^{n} l(dx) \,.$$ Under the assumption that $\rho_n = \rho$ for all $n \in \lbrace 1,\ldots,\tau \rbrace $, i.e. sub-sampling does not impact the unperturbed invariant measure, the response gives a similar result as the Ruelle formula, but with an averaged perturbation. Substituting Eq. \[eq:moving\_periodic\] into Eq. \[eq:future\], we obtain a formula of the form of Eq. \[eq:ruelle\_time\], with the difference that instead of the true forcing $X(t,x)$ the averaged forcing $$\frac{1}{\tau} \sum_{n=1}^\tau X(t+n,x)$$ appears. The disagreement is apparent, e.g. when one considers a perturbation of the form $X(t,x)=\sin(\frac{2\pi l}{\tau}t) \chi(x)$, which obviously results in a zero response. This effect has a clear intuitive interpretation. The response at a given time depends mostly on the immediate past, hence if one does not keep fixed the horizon, one risks to average out the variability. The previous formula reflects this intuition.
One way to obtain agreement with Formula \[eq:ruelle\_periodic\] is to choose a specific sampling procedure. We sample with the same periodicity $\tau$ of the forcing, thus altering the definition of the response. We define the measures for the perturbed and unperturbed system as $$\begin{aligned}
\tilde{\rho}^\prime_{T,p} (dx) &:= \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=0}^N (\tilde{f}^{k \tau + p}_T)^* l(dx) \nonumber \\
\rho^\prime (dx) &:= \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=0}^N (f^{k \tau + p})^* l(dx) \,. \label{eq:periodic_sampling}\end{aligned}$$ With this definition we obtain using Eq. \[eq:A\_approx\]: $$\begin{aligned}
\delta \rho^\prime_{T,p} (A) &= \tilde{\rho}^\prime_{T,p} (A) - \rho^\prime (A) \nonumber\\
& \approx \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=0}^N \left( \sum_{m=-1}^{N-1} \sum_{i=1}^{N-m} \theta(m \tau + n + p) \right) \int \left( f^{m \tau + n + p} \right)^* l (dx) \nonumber \\
& \hspace{3cm} X(T+ m \tau + n + p, x ) D(A \circ f^{k \tau - m \tau - n})(x) \,. \nonumber\end{aligned}$$ Using the periodicity of $X$ one can obtain $$\begin{aligned}
%&= \sum_{n=1}^\tau \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{m=0}^{N-1} \sum_{k=m+1}^N \int (f^{n+m \tau})^* l(dx) X(T+n, x) D(A \circ f^{(k-m) \tau -n})(x) \nonumber \\
%\delta \rho^\prime_T &\approx \sum_{n=1}^\tau \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{i \geq 1} \sum_{m=0}^{N-i} \int (f^{n+m \tau})^* l(dx) X(T+n, x) D(A \circ f^{i \tau -n})(x) \nonumber \\
\delta \rho^\prime_{T,p} (A) &\approx \sum_{n=1}^\tau \sum_{i\geq 1} \int \left( \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{m=-1}^{N-i} (f^{n+m \tau+p})^* l(dx) \theta(m \tau + p +n) \right) \\
& \qquad \quad X(T+n+p, x) \left( D(A \circ f^{i \tau -n})(x) \right) \nonumber \\
&= \sum_{n=0}^{\tau-1} \int \rho(dx) X(T+p-n,x) G_{A,n} (x) = \delta_{T+p} \rho(A) \,. \nonumber\end{aligned}$$ Hence by choosing the initial phase $p$ at which we start sampling, we can obtain the response at this phase. This means that we only need to start one long simulation of $f$ and $\tilde{f}$ and do summations of the differences $(A\circ \tilde{f} - A\circ f)(x)$ according to Eq. \[eq:periodic\_sampling\] at all phases $p$ in one period to obtain the entire response to the periodic forcing. By applying a forcing that contains several frequencies, such as a block wave, we can extract the susceptibility at all present frequencies in one run by taking the Fourier transform of the response.
Note that if we sample the signal with a periodicity $\eta$ which is prime with respect to the period $\tau$ of the forcing, we will obtain no $p$-dependence (with $p$, in this case, ranging from 0 to $\eta-1$) in the response. For all values of $p$ we will obtain as a result the response to the time-averaged forcing. Therefore, the case of sampling at all time steps discussed above is just the special case given by $\eta=1$, where we are basically considering the case of the Nyquist frequency. Instead, if $\tau$ and $\eta$ are not prime with respect to each other, the sampling procedure will be able to ascertain the $p$-dependence of the response of the system at the periodicity given by the common harmonic terms.
If the periodicity of the forcing is not known, the above discussion tells us that by doing a sampling at larger and larger periods $\eta$ and checking for each of those the phase-dependence of the response, it is possible to deduce the fundamental period of the forcing. If the procedure does not converge, we are facing a quasi-periodic or continuous-spectrum forcing for which this approach fails.
Therefore, this situation is unsatisfactory. Why do we only get the correct result for periodic perturbations and fine-tuning the sampling or by taking constant perturbations?
### Response at a fixed time horizon {#sec:backward}
This paradox can be resolved by defining the time-dependent SRB measure in Eq. \[eq:time\_SRB\] using a different method of sampling. We now consider the time evolution $\tilde{f}^k_T$ in this definition to go from time $T-k$ in the past up to the fixed time horizon $T$, so instead of Eq. \[eq:forward\], we have: $$\begin{aligned}
\tilde{f}^k_T = \tilde{f}_{T} \circ \ldots \circ \tilde{f}_{T-k} \label{eq:backward} \,.\end{aligned}$$ Note that this approach does not use the reversed time dynamics but rather a different time perspective in which the final time is fixed as the current time and the perturbation starts in the remote past.
The expansion to first order in $X$ around the dynamics of $f$ now becomes: $$\begin{aligned}
\tilde{x}_T &= \tilde{f}_T \circ \ldots \circ \tilde{f}_{T-k+1} (\tilde{x}_{T-k}) \nonumber \\
&\approx f^k (\tilde{x}_{T-k}) + \sum_{j=0}^{k-1} X(T-j,f^{k-j}(\tilde{x}_{T-k})).(Df^{j})(f^{k-j} (x_{T-k})) \,. \label{eq:fixed_exp}\end{aligned}$$ Hence, the linear response of $\rho(A)$ at time $T$ is given by $$\begin{aligned}
\delta_T \rho(A) &\approx \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{k=1}^t \int l(dx) \sum_{j=0}^{k-1} X(T-j,f^{k-j}(x)) D(A \circ f^{j}) (f^{k-j}(x)) \nonumber \\
%& = \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j = 0}^{t-1} \sum_{i=1}^{t-j} \int l(dx) X(T-j,f^i(x)) D(A \circ f^{j}) (f^i(x)) \nonumber \\
& = \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{j \geq 0} \int \sum_{i=1}^{t-j} (f^*)^{i} l(dx) X(T-j, x) D(A \circ f^{j}) (x) \,. \nonumber \\\end{aligned}$$ Note that in contrast to Eq. \[eq:future\] the indices are such that the time average of the measure and the perturbation are decoupled. This crucially depends on the choice of the sampling. This allows us to use the definition of the SRB measure in Eq. \[eq:SRB\] and replace the time average in the limit by $\rho$: $$\begin{aligned}
\delta_T \rho(A) & \approx \sum_{j \geq 0} \int \rho(dx) X(T-j,x) D(A \circ f^{j}) (x) \,. \label{eq:lin_response}\end{aligned}$$ This agrees with Eq. \[eq:ruelle\_time\]. Here we do get the anticipated result. Note that this expression gives also a non-zero response for a perturbation which is non-zero only for a finite time, as opposed to Eq. \[eq:future\].
This sampling is the natural one fore deducing the general linear response theory. Doing the calculation for constant forcing does not elucidate the relevance of the choice of sampling. This sampling corresponds to a Gedankenexperiment where the system is prepared in the distant past and we observe the difference of the perturbed and unperturbed evolution up to a given instant $T$.
Numerics {#sec:numerics}
========
Reick’s formula
---------------
For perturbations that are separable ($X(t,x)=\phi(t)\chi(x)$) and have a single driving frequency $\Omega$ ($\phi(t)=\epsilon
cos(\Omega t)$), the following sampling scheme for computing the susceptibility for a given observable $A$ has been proposed by Reick [@reick_linear_2002]: $$\begin{aligned}
\hat{\kappa}_A(\Omega) = \lim_{\epsilon \rightarrow 0} \lim_{N \rightarrow
\infty} \frac{1}{N \epsilon} \sum_{t=1}^{N } e^{i \Omega
t} \int \rho(dx) \left( A( \tilde{f}_0^t (x)) - A(f^t (x))
\right). \label{eq:reick}\end{aligned}$$ This formula has been later adopted to analyze the output of a simple climate model [@luc11] and a generalization has been proposed to study the nonlinear susceptibilities describing harmonic generation [@luc09]. Applicability of this formula depends on performing numerical experiments where the initial samples approximate the unperturbed SRB measure $\rho$.
Using our previous calculations, we want to circumstantiate the validity of the formula. We apply Ruelle’s response theory to obtain a perturbative expression of Eq. \[eq:reick\] in terms of quantities of the unperturbed dynamics. $$\begin{aligned}
& \lim_{N\rightarrow \infty} \frac{1}{N } \sum_{t=1}^{N } e^{i \Omega t} \int l(dx) \left( A(\tilde{f}^t_0(x)) - A(f^t(x)) \right) \nonumber \\
&\approx \lim_{N\rightarrow \infty} \frac{1}{N } \sum_{t=1}^{N } e^{i \Omega t} \int \sum_{j=1}^t (f^j)^* l(dx) X(j,f^j(x)) D( A \circ f^{t-j}) (x) \,. \nonumber \\\end{aligned}$$ Here we encounter the same problem as in Eq. \[eq:future\], namely the coupling of the averages of the measure and the perturbation. Indeed, using Reick’s formula sampling from an initial measure different from the unperturbed SRB measure, one does not get a reasonable response, as reported in [@luc11].
By sampling according to the unperturbed SRB measure $\rho$ instead of $l$, the above equation becomes $$\begin{aligned}
&\lim_{N\rightarrow \infty} \frac{1}{N } \sum_{j=1}^{N } \sum_{k=0}^{N - j} e^{i \Omega k} e^{i \Omega j} \int \rho(dx) X(j,x) D( A \circ f^{k}) (x) \nonumber \\
% &= \lim_{N\rightarrow \infty} \frac{1}{N } \int \rho(dx) \sum_{k \geq 0} e^{i \Omega k} D(A \circ f^k)(x) \sum_{j=1}^{N -k} e^{i \Omega j} X(j,x) \nonumber \label{eq:step1}
&= \lim_{N\rightarrow \infty} \frac{1}{N } \int \rho(dx) G_A(\omega,x) \sum_{j=1}^{N} e^{i \Omega j} X(j,x) \,. \label{eq:step1}\end{aligned}$$ We insert the inverse discrete time Fourier transform $$\begin{aligned}
X(j,x) = \frac{1}{2 \pi} \int_{-\pi}^\pi \hat{X}(\omega,x) e^{-i \omega j} d\omega\end{aligned}$$ into Eq. \[eq:step1\]: $$\begin{aligned}
&\int \rho(dx) G_A(\omega,x) \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{j=1}^{N } e^{i \Omega j} X(j,x) \nonumber \\
&= \int \rho(dx) G_A(\omega,x) \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{j=1}^{N } e^{i \Omega j} \frac{1}{2 \pi} \int_{-\pi}^\pi \hat{X}(\omega,x) e^{-i \omega j} d\omega \nonumber \\
&= \int \rho(dx) G_A(\omega,x) \lim_{N \rightarrow \infty} \frac{1}{2\pi} \int_{-\pi}^\pi \hat{X}(\omega,x) u_N(\Omega - \omega) d\omega \nonumber \\
&= \lim_{N \rightarrow \infty} \frac{1}{2 \pi} \int_{-\pi}^\pi d\omega u_N(\Omega-\omega) \hat{\kappa}_A (\omega)
\,. \label{eq:lim1}\end{aligned}$$ where $$u_N(\Omega-\omega)=\frac{1}{N } \sum_{j=1}^{N} e^{i(\Omega - \omega)j}$$ This can be rewritten by making use of $$\begin{aligned}
x + \ldots + x^N = x \frac{1-x^N}{1-x} \,. \end{aligned}$$ as $$\begin{aligned}
u_N(\Omega-\omega)=\frac{1}{N} e^{i(\Omega-\omega)} \frac{1-e^{i N (\Omega-\omega)}}{1-e^{i (\Omega-\omega)}} \,,\end{aligned}$$ which converges to $0$ as $N$ goes to infinity, except for $\Omega=\omega$. At $\Omega=\omega$ the sum over $j$ gives $N$. Hence, if $\hat{X}(\omega,x)$ is integrable, we can take the limit in Eq. \[eq:lim1\] inside the integral $$\begin{aligned}
\frac{1}{2\pi} \int_{-\pi}^\pi \hat{X}(\omega,x) \mathbf{1}_{\{ \Omega \} } (\omega) d\omega = 0 \,,\end{aligned}$$ where $\mathbf{1}_{\{ \Omega \} }$ is the indicator function on $\{ \Omega \}$. We deduce that in the case of a general perturbation with a continuous Fourier spectrum, Reick’s numerical approach cannot be applied. Note also that for finite time steps $N$ (as is always the case for numerical experiments), there is an additional broadening of the signal of order $1/N$, as is apparent from Eq. \[eq:lim1\].
If on the other hand the Fourier transform is singular, for example $$\begin{aligned}
X(\omega,x)&=\delta(\Omega-\omega) \chi(x) \,,\end{aligned}$$ which corresponds to the monochromatic signal $$\begin{aligned}
X(j,x) &= \frac{1}{2 \pi} e^{-i \Omega j} \chi(x) \,,\end{aligned}$$ we have that $$\begin{aligned}
\lim_{N\rightarrow \infty} \frac{1}{N} \sum_{j=1}^{N} e^ {i \Omega j} X(j,x) &= \frac{\chi(x)}{2 \pi} \,.\end{aligned}$$
Therefore, Eq. \[eq:step1\] becomes $$\begin{aligned}
\int \rho(dx) G_A(\omega,x) \chi(x) = \hat{\kappa}_A(\Omega) \,,\end{aligned}$$ as predicted by Reick.
The above calculation shows how the explicit expansion of Reick’s response formula allows us to interpret its finite time behaviour. Eq. \[eq:lim1\] shows how this sampling scheme amounts to filtering the susceptibility $\hat{\kappa}_A$ with the function $u_N$.
Note that in the case of a several frequencies contributing to the forcing, we are again in the case of general periodic forcing. The susceptibility can in this case be computed in two ways. Either one uses Reick’s formula at every frequency present in the signal, which amounts to doing spectroscopy. On the other hand, one can also compute the full response $\delta_T \rho$ at all phases over on period and apply a Fourier transform to this time-dependent function. The response at any one specific phase can be efficiently computed with the periodic sampling strategy proposed in Eq. \[eq:periodic\_sampling\]. In this approach each value for the difference of $A$ between perturbed and unperturbed is processed only once, compared to the summation being done for every frequency with Reick’s formula.
Sampling continuous spectra
---------------------------
The discussion in the previous subsection demonstrates how sampling according to Formula \[eq:reick\] can only give a correct result in cases where the Fourier spectrum of the perturbation is discrete. In this subsection we explore how the discussion on the expansion at a fixed time horizon can help us find a sampling for the case of a continuous Fourier spectrum.
One possibility is to sample the response directly from the full formula of the perturbation of the SRB measure: $$\begin{aligned}
\delta_T \rho (A) = \lim_{t \rightarrow \infty} \frac{1}{t} \sum_{k=1}^t \int l(dx) A(\tilde{f}_{T} \circ \ldots \circ \tilde{f}_{T-k} (x)) - A(f^k (x))\end{aligned}$$ This sampling however entails some practical difficulties. As can be seen from the formula, an ergodic average is taken over the length of the numerical run $k$. Increasing $k$ to $k+1$ is equivalent to altering the initial conditions from $x$ to $\tilde{f}_{T-k-1}(x)$. This trajectory cannot be recovered from the previously calculated trajectories of length $k$. Hence one needs to redo the calculations of the trajectories for every value of $k$. As we will see, less costly sampling methods can be devised.
To study the behaviour of different sampling methods, let us define the following quantity: $$\begin{aligned}
\delta^{(k,n)} \rho_T (A) = \int l(dx) A( \tilde{f}_{T} \circ \ldots \circ \tilde{f}_{T-k} \circ f^n (x) ) - A(f^{k+n}(x))\end{aligned}$$ By changing $k$, we control the length of time over which we observe the difference between the perturbed and unperturbed dynamics. The initial measure $l$ is furthermore transformed by $n$ applications of the unperturbed dynamics. By increasing $n$, the initial measure $l$ converges to the unperturbed SRB measure $\rho$.
Making use of Equation \[eq:fixed\_exp\], we can expand $\delta^{(k,n)} \rho_T$ to get a better idea of the behaviour of this quantity under different limits and ergodic averages: $$\begin{aligned}
\delta^{(k,n)} \rho_T (A) = \sum_{j=0}^{k-1} \int {f^{(k-j+n)}}^*l(dx) X(T-j,x) D(A\circ f^j)(x) \label{eq:expanded_diff}\end{aligned}$$ The aim when constructing a sampling scheme is to take limits and ergodic averages over $k$ and $n$ in such a way that the response given by Equation \[eq:ruelle\_time\] is obtained. As we have seen with Reick’s formula, this convergence can depend on the perturbation $X$. Furthermore, as exemplified by the discussion in this section, numerical cost should be considered. The following ergodic mean: $$\begin{aligned}
\lim_{t \rightarrow \infty} \frac{1}{t} \sum_{k=1}^t \delta^{(k,n)} \rho_T (A) \label{eq:k_ergodic}\end{aligned}$$ converges to the response $\delta_T \rho(A)$ given by Eq. \[eq:expanded\_diff\] for any value of $n$. This can be shown following the discussion provided in Section \[sec:backward\], where we discuss the case $n=0$. Another possibility to obtain the SRB measure $\rho$ in Eq. \[eq:expanded\_diff\] is to take the limit of $n$ going to infinity for a fixed $k$. We obtain: $$\begin{aligned}
\lim_{n \rightarrow \infty}\delta^{(k,n)} \rho_T(A) &= \sum_{j=0}^{k-1} \int \rho(dx) X(T-j,x) D(A\circ f^j)(x) \label{eq:expanded_diff2}.\end{aligned}$$ where we have assumed that $\lim_{m\rightarrow \infty} {f^m}^* l = \rho$. This expression tends to $\delta_T \rho(A)$ in the limit of $k\rightarrow\infty$. From a theoretical point of view, an increase in the value of $n$ simply translates into a change in the initial measure $l$. Numerically, though, doing a long initial unperturbed run will evolve the initial measure towards the invariant measure $\rho$, hence improving convergence when Eqs. - are considered. In fact, there are a number of different options when attempting to reach a good numerical convergence. These include
- increasing the length of unperturbed and perturbed trajectories ($n$ and $k$)
- enlarging the number of initial conditions (chosen according to $l$)
- deciding whether or not ergodic averaging is performed over $n$ and $k$.
In the limit of infinitely long perturbed runs, these approaches give the same result. However, for finite time they will perform differently.
Conclusions
===========
In this paper we have reconsidered Ruelle’s linear response theory by analyzing the impact of choosing different methods of sampling in relation to different classes of forcings. Explicitly doing an expansion of the perturbed dynamics around the unperturbed measures allows us to explore which sampling methods converge and under which conditions.
The general response formula is obtained by choosing a specific sampling where the system is prepared in the distant past and we observe the difference of the perturbed and the unperturbed dynamics up to a given time $T$. By proposing a general decomposition of space-time dependent forcings using a Schauder decomposition, we have elucidated that it is possible to define *elementary* linear susceptibilities that allow to construct the response of the system to any pattern of forcing.
The other possible sampling strategy, where the time horizon is not fixed, does not give rise to a natural response theory except for constant perturbations. In the case of periodic forcings one can obtain a meaningful formula by redefining appropriately the response, finely tuned to the forcing under investigation. One needs to subsample the signal with the same period of the forcing and explore all the initial phases. By taking this approach, it is in principle possible to discover the fundamental period of the external perturbations by varying the sampling period. Thanks to our approach we get a deeper understanding of the range of applicability of Reick’s formula, which has been used as a signal processing tool to study the linear response of numerical models.
Nonetheless, this approach fails if the forcing is not periodic, in which case we must resort to the fixed-time horizon framework to get a meaningful answer. In fact, our findings explain why considering the fixed-time horizon it is possible to analyze a response to forcings that have a continuous Fourier spectrum. The clarifications presented in this paper may be of relevance for devising the data processing for actual laboratory experiments on nonlinear systems.
We also clarify that it is crucial in practical terms to use an ensemble approach where the initial conditions sample approximately the unperturbed SRB measure. Our calculation is explicitly performed for discrete time, but the analogous results for continuous time are presented in Appendix \[app:continuous\]. Moreover our considerations seem to be appropriate also for the case of nonlinear response [@ruelle_d._nonequilibrium_1998; @luc09].
To summarize, we have shown the following
- Sampling a general response from an initial time up to a moving time horizon does not lead to a well-defined sampling method.
- Starting the simulation at times in the distant past and averaging the response at a fixed time horizon always results in the full response of the system at the fixed point in time. This approach can be computationally inefficient.
- Sampling a periodic response with a moving time horizon results in a response of the system as if it were forced with an averaged forcing.
- In the periodic case, the full response can be computed by sampling with a horizon moving forward in time with steps of one period. This response depends on the initial phase. The susceptibility can be computed through a Fourier transform of the response.
- A constant forcing can be considered as a periodic forcing with period $1$ and can thus be sampled with a horizon moving with time steps of $1$.
- For periodic forcings, Reick’s spectroscopic formula also allows to discern the response at different frequencies, i.e. the susceptibility. It gives a zero susceptibility for forcings with a continuous spectrum.
We believe that the results presented in this article can be of interest to researchers interested in studying the response of complex systems to modulations of their internal parameters or to external perturbations. For various reasons, climate science is an especially promising field of application. First of all, we clarify crucial differences between sampling periodic and aperiodic forcings. This is a crucial issue if one wants to apply linear response theory to study different scenarios such as the response of the system to monotonically increasing CO$_2$ levels (see a forthcoming paper by the authors) versus its response to periodic forcings such as those due to astronomical and astrophysical phenomena. In particular, we have proposed a parsimonious but effective way for analysing periodic - but non-monochromatic - forcings. Again, this setting is applicable to climate science due to the presence of cycles with different time scale, such as the daily, yearly and solar cycles.
Future work will address the investigation of the response of a non-equilibrium system to a general random field. Moreover, we will analyze the impact of the various sampling schemes described in this paper when studying the output of numerical models.
{#section .unnumbered}
VL, JW, DF acknowledge the financial support of the EU-ERC project NAMASTE-Thermodynamics of the climate system. TK acknowledges F. Bonetto for fruitful discussions.
Continuous time response formulas {#app:continuous}
=================================
Here we give the formulas for continuous time systems corresponding to the ones presented in the main text. The time evolution is in this setting given by a differential equation $$\begin{aligned}
\frac{dx}{dt} = F(x) \,,\end{aligned}$$ resulting in a flow $ x(t+s) = f^s (x(t)) $. The SRB measure is given by $$\begin{aligned}
\rho(A) = \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t ds \int l(dx) A(f^s(x)) \,.\end{aligned}$$ For a separable perturbation $$\begin{aligned}
\frac{dx}{dt} = F(x) + \chi(x) \phi(t)\end{aligned}$$ the susceptibility is given by $$\begin{aligned}
\hat{\kappa}_A(\omega) = \int_0^\infty dt e^{i \omega t} \int \rho(dx) \chi(x) D(A \circ f^t)(x) \,.\end{aligned}$$ In case of a general perturbation $F(x) \rightarrow F(x) + X(t,x)$ the linear response becomes: $$\begin{aligned}
\delta_T \rho(A) \approx \int_0^\infty d\tau \int \rho(dx) X(T-\tau,x) D(A \circ f^\tau)(x)\end{aligned}$$
The SRB measure with a moving time horizon is: $$\begin{aligned}
\tilde{\rho}_T = \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t ds \int l(dx) A( \tilde{f}^{T+t}_T (x))\end{aligned}$$ and the SRB measure with a fixed time horizon: $$\begin{aligned}
\tilde{\rho}_T = \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t ds \int l(dx) A( \tilde{f}^{T}_{T-t} (x))\end{aligned}$$ where $\tilde{f}_{t_1}^{t_2}(x)$ is a trajectory of the perturbed system, starting at time $t_1$ in $x$ and evolving up to time $t_2$.
Reick’s formula now becomes: $$\begin{aligned}
\hat{\kappa}_A(\omega) = \lim_{\epsilon \rightarrow 0} \lim_{\nu \rightarrow \infty} \frac{1}{\nu \epsilon} \int_0^\nu dt e^{i \Omega t} \int \rho(dx) \left( A(\tilde{f}_0^t(x)) - A(f_0^t(x)) \right)\end{aligned}$$
![The convergence of $u_N$ to the indicator function $\mathbf{1}_{ \{ 0 \} }$. The dashed blue line shows $u_{10}$, the full red line $u_{50}$.[]{data-label="fig:u_N"}](u_N)
|
---
abstract: 'Heisenberg’s original uncertainty relation is related to measurement effect, which is different from the preparation uncertainty relation. However, it has been shown that Heisenberg’s error-disturbance uncertainty relation can be violated in some cases. We experimentally test the error-tradeoff uncertainty relation by using a continuous-variable Einstein-Podolsky-Rosen (EPR) entangled state. Based on the quantum correlation between the two entangled optical beams, the errors on amplitude and phase quadratures of one EPR optical beam coming from joint measurement are estimated respectively, which are used to verify the error-tradeoff relation. Especially, the error-tradeoff relation for error-free measurement of one observable is verified in our experiment. We also verify the error-tradeoff relations for nonzero errors and mixed state by introducing loss on one EPR beam. Our experimental results demonstrate that Heisenberg’s error-tradeoff uncertainty relation is violated in some cases for a continuous-variable system, while the Ozawa’s and Brainciard’s relations are valid.'
author:
- 'Yang Liu$^{1,2}$'
- 'Zhihao Ma$^{3}$'
- 'Haijun Kang$^{1,2}$'
- 'Dongmei Han$^{1,2}$'
- 'Meihong Wang$^{1,2}$'
- 'Zhongzhong Qin$^{1,2}$'
- 'Xiaolong Su$^{1,2}$'
- 'Kunchi Peng$^{1,2}$'
title: 'Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state'
---
Introduction
=============
As one of the cornerstones of quantum mechanics, uncertainty relation describes the measurement limitation on two incompatible observables [Heisenberg]{}. It should be emphasized that the uncertainty relation actually states an intrinsic property of a quantum system, rather than a statement about the observational success of current technology. Uncertainty relation has deep connection with many special characters in quantum mechanics, such as Bell non-locality and entanglement [@Bell2; @EPR2], which cannot occur in classical world. With rapid progress in quantum technology, such as quantum communication and quantum computation [@COMMUN; @FURU], in recent years, it is important for us to know the fundamental limitations in the achievable accuracy of quantum measurement.
Note that there are two different types of uncertainty relations, one is the preparation uncertainty relation, which studies the minimal dispersion of two quantum observables before measurement [@Kennard; @Rob29]. The Robertson uncertainty relation [@Rob29], reads as $\sigma (x)\sigma
(p)\geq \hbar /2$, is a typical example in this sense, where $\sigma (x)$ and $\sigma (p)$ are the standard deviations of position and momentum of a particle. For such uncertainty relation, the measurements of $x$ and $p$ are performed on an ensemble of identically prepared quantum systems. While in the original spirit of Heisenberg’s idea [@Heisenberg], the Heisenberg’s uncertainty principle should be based on the observer’s effect, which means that measurement of a certain system cannot be made without affecting the system. So this leads to the second type of uncertainty relation: measurement uncertainty relation, which studies to what extent the accuracy of position measurement of a particle is related to the disturbance of the particle’s momentum, so called the error-disturbance uncertainty relation [@Ozawa03]. It is also called the error-tradeoff relation in the approximate joint measurements of two incompatible observables [Ozawa04,Branciard]{}.
Heisenberg’s error-tradeoff uncertainty relation for joint measurement is generally expressed as$$\varepsilon (A)\varepsilon (B)\geq C_{AB} \label{H}$$where $C_{AB}=\left\vert \left\langle [A,B]\right\rangle \right\vert /2$, $[A,B]=AB-BA$. However, it has been shown that this relation is not valid in some cases [@Balllentine]. For this reason, Ozawa and Hall proposed new measurement uncertainty relations which have been theoretically proven to be universally valid for any incompatible observables, respectively [Ozawa03,Ozawa04,Hall04]{}. After that, Branciard proposed a new uncertainty relation, which is universally valid and tighter than the Ozawa’s relation [@Branciard]. There are also other types of measurement uncertainty relations generalizing Heisenberg’s original idea, which can be found in Refs. [Werner2,PhysRevA022106,PhysRevA032,PhysRevLett050401,lu,Barchielli2017]{}. Experimental tests of the measurement uncertainty relations have been demonstrated in photonic [EXPphotons1,EXPphotons2,EXPphotons3,EXPphotons4,EXPphotons5,EXPphotons6]{}, spin [@EXPpolarizedneutrons1; @EXPphotons7; @EXPphotons8; @EXPW1], and ion trap systems [@EXPW2]. All of these experiments are limited in discrete-variable systems. Up to now, experimental test of the measurement uncertainty relation based on continuous-variable system has not been reported.
{width="80mm"}
In this paper, we present the first experimental test of the error-tradeoff relation for two incompatible variables, amplitude and phase quadratures of an optical mode, using a continuous-variable EPR entangled state. Based on quantum correlations of the EPR entangled beams, the error-tradeoff relation with zero error (error-free) of one observable is verified directly by performing joint measurement on two EPR beams. In this case, Heisenberg’s error-tradeoff uncertainty relation is violated, while Ozawa’s and Branciard’s relations are valid. We also test the error-tradeoff relations for nonzero errors and mixed state by introducing loss on signal mode. Our experimental test of the continuous-variable error-tradeoff relations makes the test of the measurement uncertainty relation more complete.
Theoretical framework
=====================
One mode of EPR entangled state is used as signal state $\rho $ and two incompatible observables are taken as $A=\hat{x}_{1}$ and $B=\hat{p}_{1}$, respectively \[Fig. 1(a)\], where $\hat{x}_{1}=(\hat{a}+\hat{a}^{\dag })/2$ and $\hat{p}_{1}=(\hat{a}-\hat{a}^{\dag })/2i$ denote the amplitude and phase quadratures of $\rho $, respectively. Another mode of EPR entangled state is used as the meter state $\rho _{M}$. Two compatible observables $C$ and $D$ are measured simultaneously to approximate $A$ and $B.$ The quality of the approximations are characterized by defining the root-mean-square errors $\varepsilon (A)=\langle (C-A)^{2}\rangle ^{1/2}$ and $\varepsilon
(B)=\langle (D-B)^{2}\rangle ^{1/2}$. Ozawa’s error-tradeoff relation is expressed by [@Ozawa03; @Ozawa04] $$\varepsilon (A)\varepsilon (B)+\varepsilon (A)\sigma (B)+\sigma
(A)\varepsilon (B)\geqslant C_{AB} \label{O}$$where $\sigma (A)$ is the standard deviation of observable $A$. The Branciard’s error-tradeoff relation is given by [@Branciard]$$\begin{aligned}
&&\big[\varepsilon ^{2}(A)\sigma ^{2}(B)+\sigma ^{2}(A)\varepsilon ^{2}(B)
\notag \label{B} \\
&&+2\varepsilon (A)\varepsilon (B)\sqrt{\sigma ^{2}(A)\sigma
^{2}(B)-C_{AB}^{2}}\big]^{1/2}\geqslant C_{AB}\end{aligned}$$where the parameter $C_{AB}=1/4$ denote that $A$ and $B$ cannot be jointly measured on $\rho $ simultaneously. The variances of the amplitude and phase quadratures of two EPR beams are expressed as $\sigma ^{2}(\hat{x}_{1})=\sigma ^{2}(\hat{p}_{1})=\sigma ^{2}(\hat{x}_{2})=\sigma ^{2}(\hat{p}_{2})=(e^{2r}+e^{-2r})/8$, where $r$ is the squeezing parameter [@FURU]. In the experiment, we test Heisenberg’s, Ozawa’s and Branciard’s error-tradeoff uncertainty relations in three cases, i.e., error-free measurement of one observable, nonzero error and mixed state cases.
Experimental implementation and results
=======================================
In the experiment, an EPR entangled state with $-$2.9 dB squeezing and 3.9 dB antisqueezing is prepared by a nondegenerate optical parametric amplifier (NOPA), as shown in Fig. 1(b), which consists of an $a$-cut type-II KTP crystal and a concave mirror [@EPRSU]. The front face of the KTP crystal is used as the input coupler, and the concave mirror with 50 mm curvature serves as the output coupler. The front face of the KTP crystal is coated with the transmission of 42% at 540 nm and high reflectivity at 1080 nm. The end face of the KTP crystal is antireflection coated for both 540 nm and 1080 nm. In the measurement, a sample size of 5$\times
10^{5\text{ }}$data points is used for all quadrature measurements with sampling rate of 500 K/s. The interference efficiency between signal and local oscillatior is 99% and the quantum efficiency of photodiodes are 99.6%.
At first, we consider a situation that the observable $A$ is measured accurately (error-free measurement of observable $A$), i.e., the optimal estimation $C=A$. The measured phase quadrature $D=\hat{p}_{2}$ is used to approximate the observable $B$. Because the amplitude quadrature $\hat{x}_{1}
$ of $\rho $ and the phase quadrature $\hat{p}_{2}$ of $\rho _{M}$ are compatible, they can be measured simultaneously. The errors for approximating $A$ and $B$ are expressed as $\varepsilon (A)=\sqrt{\langle
(C-A)^{2}\rangle }=0$, and $\varepsilon (B)=\sqrt{\langle (D-B)^{2}\rangle }=\sqrt{\sigma ^{2}(\hat{p}_{2}-\hat{p}_{1})}=e^{-r}/\sqrt{2}$, respectively. Since $\varepsilon (A)=0$ and $\varepsilon (B)<\infty $, we have $$\varepsilon (A)\varepsilon (B)=0.$$It is obvious that Heisenberg’s error-tradeoff uncertainty relation is violated. The Ozawa’s and Branciard’s relations are the same for $\varepsilon (A)=0$, which are $$\sigma (A)\varepsilon (B)=\sqrt{1+e^{-4r}}/4\geqslant 1/4.
\label{error-free}$$
The amplitude quadrature $\hat{x}_{1}$ of the signal state is measured by a homodyne detector HD1 in the time domain, as shown in Fig. 1(b). To evaluate the error $\varepsilon (B)$, we experimentally measure the observables $B$ and $D$, i.e. the phase quadratures $\hat{p}_{1}$ and $\hat{p}_{2}$, by two homodyne detectors (HD1 and HD2) simultaneously.
{width="80mm"}
{width="80mm"}
In our experiment, the achievable lower bound is limited by the quantum correlation of the EPR entangled state \[Eq. (\[error-free\])\]. In order to demonstrate this property, we change the quantum correlation of signal state and meter state by changing the relative phase $\theta $ between the two mode of EPR entangled state. Thus, the error $\varepsilon (B)=\sqrt{\sigma
^{2}(e^{i\theta }\hat{p}_{2}-\hat{p}_{1})}$ is measured in experiment. When the relative phase $\theta =0^{\circ }$ and $\theta =360^{\circ }$, the minimum error is obtained \[Fig. 2(a)\] and the left-hand-side (LHS) of the relation reaches its minimum value \[Fig. 2(b)\], which is determined by the present squeezing level. When$\ \theta =180^{\circ }$, the maximum error is obtained, which corresponds to the measurement of anti-correlated noise $\sqrt{\sigma ^{2}(\hat{p}_{2}+\hat{p}_{1})}$. The results confirm that the Ozawa’s and Branciard’s relations are the same and valid for the error-free measurement of observable $A$.
{width="80mm"}
Then, we test the error-tradeoff relation with nonzero errors. When both errors are not equal to zero, Ozawa’s and Branciard’s relations are different. In the experiment, we apply a linear operation on the signal mode, which is done by transmitting the signal mode through a lossy channel, as shown in the inset of Fig. 1(b). In this case, the amplitude and phase quadratures of the signal mode are changed to $\hat{x}_{1}^{^{\prime }}=\sqrt{T}\hat{x}_{1}+\sqrt{1-T}\hat{x}_{v}$ and $\hat{p}_{1}^{^{\prime }}=\sqrt{T}\hat{p}_{1}+\sqrt{1-T}\hat{p}_{v}$, respectively, after transmitted over the lossy channel, where $\hat{x}_{v}$ and $\hat{p}_{v}$ represent the amplitude and phase quadratures of vacuum. By choosing $C=\hat{x}_{1}^{^{\prime }}$ and $D=\hat{p}_{2}$, which are compatible, the errors for the two incompatible observables $A=\hat{x}_{1}$ and $B=\hat{p}_{1}$ are $\varepsilon (A)=\sqrt{\sigma ^{2}(\hat{x}_{1}^{^{\prime }}-\hat{x}_{1})}$ and $\varepsilon (B)=\sqrt{\sigma ^{2}(\hat{p}_{2}-\hat{p}_{1})}$, respectively.
In this case, the error $\varepsilon (A)$ increases with the decreasing of channel efficiency, while the error $\varepsilon (B)$ is not affected by the channel efficiency \[Fig. 3(a)\]. Heisenberg’s error-tradeoff unceratinty relation is violated when the transmission efficiency is higher than 0.3. While the Ozawa’s and Branciard’s relations are always valid \[Fig. 3(b)\]. By comparing the LHS of Ozawa’s and Branciard’s relation, we confirm that Branciard’s relation is tighter than Ozawa’s relation.
{width="80mm"}
Finally, we demonstrate the error-tradeoff relation for mixed state, i.e., the state $\rho $ transmitted over a lossy channel. Here, observables $C=A=\hat{x}_{1}^{^{\prime }}$, $B=\hat{p}_{1}^{^{\prime }}$, and $D=\hat{p}_{2}$ are chosen, and thus errors for the mixed state are $\varepsilon (A)=0$ and $\varepsilon (B)=\sqrt{\sigma ^{2}(\hat{p}_{2}-\hat{p}_{1}^{^{\prime }})}$, respectively. In this case, Ozawa’s and Branciard’s relations are the same. The error $\varepsilon (B)$ and the LHS of the relation increase along with the decreasing of transmission efficiency as shown in Fig. 4(a) and 4(b), respectively. The error and LHS of the relation get the minimum value when the transmission efficiency is unit.
The predicted lower bounds for Heisenberg’s \[Eq. (\[H\])\], Ozawa’s \[ Eq. (\[O\])\] and Brinciard’s \[Eq. (\[B\])\] error-tradeoff relations are compared in the plane ($\varepsilon (A),\varepsilon (B)$), as shown in Fig. 5. For the Heisenberg’s error-tradeoff uncertainty relation (bounded by the blue dashed curve), one of the error must be infinite when the other goes to zero. While in our experiment, for the case of error $\varepsilon (A)=0$, the finite error $\varepsilon (B)$ is observed (red circles), which violates the Heisenberg’s error-tradeoff uncertainty relation, yet satisfies the Ozawa’s and Branciard’s relation. For the case of nonzero errors, only one of the observed values satisfies the Heisenberg’s error-tradeoff uncertainty relation (the data with 0.2 transmission efficiency). Our experimental data do not reach the lower bound of the relations for the limitation of the experiment condition, for example the limited squeezing parameter.
Conclusion
==========
We experimentally test the Heisenberg’s, Ozawa’s and Branciard’s error-tradeoff relations for continuous-variable observables, i.e., amplitude and phase quadratures of an optical mode. Especially, we investigate the error-tradeoff relation in case of zero error by using Gaussian EPR entangled state. Three different measurement apparatus are applied in our experiment, which are used to test the error-tradeoff relation for three different cases. The results demonstrate that the Heisenberg’s error-tradeoff uncertainty relation is violated in some cases while the Ozawa’s and the Brinciard’s relations are valid. Our work is useful not only in understanding fundamentals of physical measurement but also in developing of continuous variable quantum information technology.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This research was supported by the NSFC (Grant Nos. 11834010,and 61601270), the program of Youth Sanjin Scholar, the Applied Basic Research Program of Shanxi Province (Grant No. 201601D202006), National Basic Research Program of China (Grant No. 2016YFA0301402), and the Fund for Shanxi “1331 Project" Key Subjects Construction.
Y.L. and Z.M. contributed equally to this work.
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|
---
abstract: 'We discuss some of the recent developments of ${\cal N}=1$ super Yang-Mills theories in the context of the gauge-string correspondence.'
address: 'Nordita Institute, Copenhagen, Denmark'
author:
- Paolo Merlatti
title: 'Non-perturbative physics of ${\cal N}=1$ super Yang-Mills theories'
---
Introduction
============
In recent years ${\cal N}=1$ super Yang-Mills theories have been much investigated and a number of non trivial results about their non-perturbative dynamics have been reached.
The motivation of this intensive investigation is string theory. Indeed it has emerged that the embedding of gauge theory in string theory (via D-branes) is a powerful tool to explore the fascinating infrared dynamics of the field theory under consideration. Within this framework, various approaches and several line of investigations have been developed, all going under the rather generic name of gauge-string correspondence. One of the most interesting result that has emerged in this contest is the relation of ${\cal N}=1$ super Yang-Mills to matrix models. This relation has been found following a long path, going through topological string theory, superstring theory and D-branes.
However, after the matrix model structure of gauge theories has been conjectured in this set-up [@dv], it has also been possible to recover the same results in a purely field theoretical approach [@cdsw; @csw; @dglvz] (for a nice and detailed review see [@fer]).
In this talk I will mainly describe the field theoretical approach based on the chiral ring structure characterizing ${\cal N}=1$ supersymmetric theories and on a generalization of the Konishi anomaly [@cdsw]. Following [@cdsw; @csw] we will just sketch how things work in the general case, recovering the main results about the solution of the theory. By means of solution here we mean the determination of all the chiral quantities in the supersymmetric vacua. However the solution will be found in a rather implicit form (as it is given in [@csw]) and a lot of effort has still to be done to work out physical quantities case by case. We will specialize thus to the case of supersymmetric QCD (SQCD, namely ${\cal N}=1$ super Yang-Mills theories with $N_f$ flavors in the fundamental and anti-fundamental representation of the gauge group), with $SU(N)$ gauge group and we will always be considering the case $N_f<N$.
For this theory we will determine the structure of the supersymmetric vacua and, using the general solution we were referring to in the last paragraph, we will compute the low-energy chiral quantity called gaugino condensate. This can sound trivial as this quantity has been already determined using different techniques and it has been argued in [@cdsw; @mat] that it follows from a more general operator statement holding in any supersymmetric vacuum of the theory. However all the previous computations of this quantity have been done using non-perturbative techniques (such as instantons, monopoles or referring to the non-perturbative information encoded in the parent ${\cal N}=2$ Seiberg Witten curve)[^1]. Also in [@cdsw], the operator statement from which the gaugino condensate follows has been found doing a non-perturbative generalization of a classical result, this generalization coming from a well known instanton computation but being not related to the solution of the theory that has been found in that paper. Thus, it’s necessary (or at least reassuring) to see that things work properly for the solution given in [@csw]. And moreover it is rather meaningfull that, differently from the other approaches, we will determine the gaugino condensate using only perturbative properties of the theory.
We will turn then to consider again the gauge-string correspondence, that, as I already said, was the original motivation for this field theoretical investigation. We will see thus that, using the knowledge that the gaugino does condensate in a supersymmetric vacuum, the gauge-string correspondence seems to imply some non-perturbative modification of the standard field theoretical description of ${\cal N}=1$ pure Yang-Mills theories. In particular we will be lead to argue that there is some non-perturbative modification of the standard $U(1)_R$ anomaly of these theories.
The general case
================
Let us consider ${\cal N}=1$ supersymmetric $U(N)$ gauge theory with a chiral superfield $\Phi$ in the adjoint representation of the gauge group, $N_f$ matter fields $Q_f$ in the fundamental representation and $N_f$ ($\tilde{Q}_{\tilde{f}}$) in the anti-fundamental one and a generic tree-level superpotential: $$\label{tree}
W_{tree} = Tr~W(\Phi)+\tilde{Q}_{\tilde{f}}~m_f^{\tilde{f}}(\Phi)Q^f.$$ where $W(z)$ is a degree $n+1$ polynomial $$W(z)=\sum_{k=0}^n \frac{1}{k+1}g_kz^{k+1}\ \ \mbox{and}\ \ W'(z)=g_n\prod_{i=1}^n(z-a_i)$$ It’s easy to see that in a classical vacuum the gauge group breaks to $\prod_{i=1}^l U(N_i)$ with $l\leq n$.
This theory has a chiral structure that allows us to define the chiral ring. Chiral operators are simply operators that are annihilated by the supersymmetries $\bar{Q}_{\dot{\alpha}}$ of one chirality. The product of two chiral operators is also chiral. Chiral operators are usually considered modulo operators of the form $\{\bar{Q}_{\dot{\alpha}},\ldots\}$. The expectation value of a chiral operator in a supersymmetric vacuum depends only on its equivalence class because the vacuum is annihilated by the supersymmetry generators $\bar{Q}_{\dot{\alpha}}$. The equivalence classes can be multiplied, and form a ring called the chiral ring. A superfield whose lowest component is a chiral operator is called a chiral superfield.
A crucial property of the chiral operators is that the expectation value of a product of chiral operators is independent of each of their positions. Then, using also cluster decomposition, we have $$\left\langle\prod_IO^I(x_I)\right\rangle~=~\left\langle\prod_IO^I\right\rangle~=~\prod_I\left\langle O^I\right\rangle$$
Using some equalities holding in the chiral ring [@cdsw], it is possible to give a complete list of independent single-trace chiral operators. They are $Tr~\Phi^k,\ Tr~\Phi^kW_{\alpha},\ Tr~\Phi^kW_{\alpha}W^{\alpha}$, and $\tilde{Q}_{\tilde{f}}\Phi^kQ^f$ for $f,\ \tilde{f}=1\ldots N_f$. It is possible to write in a compact way all these bosonic operators generating the chiral ring in terms of the gauge invariant quantities: $$\begin{aligned}
T(z)~=~Tr\frac{1}{z-\Phi}\\ R(z)~=~-\frac{1}{32\pi^2} Tr \frac{W_{\alpha}^{\alpha}}{z-\Phi}\\ M(z)^f_{\tilde{f}}~=~\tilde{Q}_{\tilde{f}}\frac{1}{z-\Phi}Q^f\end{aligned}$$ Our interest is in the relations that the chiral operators satisfy. These relations are operator statements that hold in any supersymmetric vacuum. Then, if we have enough such relations, we can solve the theory determining all the chiral observables. At a classical level, such relations are simply the classical equations of motion. A quantum generalization of them is given by the perturbative Ward identities that come from the one-loop Konishi anomaly, that is an anomaly in the variation of the superfield $\Phi$, $\delta\Phi=\epsilon\Phi$. But, as in [@cdsw], we can consider the more general variation $\delta \Phi = f(\Phi,{\cal W}_\alpha)$. It turns then out [@cdsw] that such variation leads to an anomaly for the current $J_f=Tr~ \bar \Phi e^{ad V} f(\Phi,{\cal W}_\alpha)$, from which the Ward identity follows: $$\left< \Tr f(\Phi,{\cal W}_\alpha) \frac{\partial
W}{\partial \Phi} \right> = -\frac{1}{32 \pi^2} \left<
\sum_{i,j}\left[{\cal W}_\alpha , \left[ {\cal W}^\alpha ,
\frac{\partial f}{\partial \Phi_{ij}} \right] \right]_{ji} \right>$$ Now, by choosing appropriate variations of the matter field, for instance $\delta \Phi = f(\Phi,{\cal W}_{\alpha }) = R(z)$, we obtain equations for the generating functions $R(z)$ and ${\cal T}(z)$. For example, it is easy to work out the equation for $R(z)$: $$R^2(z) = W'(z) R(z) + {1\over 4}f(z)$$ with $f(z)$ a polynomial of degree $n-1$. The solution is $$2~ R(z)~=~W'(z)-\sqrt{W'(z)^2+f(z)}$$
The appearance of the square-root and the natural requirement that physical quantities (such as $R(z)$) be single-valued imply now that $z$ takes value on the double-sheeted complex plane, a Riemann surface of genus $n-1$. If we define it as $y(z)~=~W'(z)-2~R(z)$, its equation is simply $$y(z)^2~=~W'(z)^2+f(z).$$ The appearance of this Riemann surface is the result of the quantum dynamics of the theory and to find the exact solution corresponds now to solve for the polynomial $f(z)$.
The solution
------------
Let’s see now how to find constraints on the generating function ${\cal T}(z)$. We begin considering that some semiclassical gauge theory data allows to determine its analytic structure [@csw]. In particular it turns out that ${\cal T}(z)$ has only simple poles and the residue at them is integer. Thanks to complex geometry, this is enough to fix it uniquely, apart from a constant. Indeed, considering the one form $T(z)~dz$, it is possible to find [@csw] $$T(z)~dz ~=~d\ln\psi(z)$$where$$\psi(z)~=~P(z)+\sqrt{P(z)^2-\alpha B(z)}$$ with $$\frac{\alpha}{4}~=~c~\Lambda^{2N-N_f},\ \ c\neq 0,\ \ B(z)~=~Det~ m(z)$$ $m$ being the matrix in flavor space appearing in the superpotential and $c$ is an arbitrary dimensionless constant. Moreover, the polynomial $P(z)$ has to satisfy the following equations: $$\begin{aligned}
\label{complete}
P(z)^2-\alpha B(z)&=&y(z)^2 H(z)^2\\ W'(z)^2+f(z)&=&y(z)^2\end{aligned}$$ where the degree of $P(z),\ F(z),\ W(z)$ and $H(z)$ are, respectively, $N,\ 2n,\ n+1$ and $N-n$.
This is the solution of the problem. Indeed by means of these equations now we can fix also $f(z)$, and then all the observable of the theory are determined. Even if this form of the solution is rather implicit and it has to be worked out case by case, this is rather surprising. Indeed we only used the property of the chiral ring and the (generalized) Konishi anomaly. But now all the observable are determined in the full quantum theory, where also non-perturbative effects are taken into account.
It’s true that we have still one undetermined constant appearing in front of $\Lambda$, the dynamically generated scale. To fix it simply corresponds to choose a renormalization scheme (it can safely be absorbed in the definition of $\Lambda$).
As a final comment, let me just emphasize that all this perturbative field theoretical analysis has been motivated by string theory results. This is indeed one of the example of the very beautiful and rich interplay that there is between gauge and string theories. In the following, after a simple example, we will go back to string theory and we will see if we can get further insights on the field theory just discussed.
Example: ${\cal N}=1$ super-QCD
===============================
Let’s now use [@pm] the generic solution of the last subsection to recover information on ${\cal N}=1$ super Yang-Mills theory (with gauge group $SU(N)$) coupled to $N_f$ massive flavors (always $N_f<N$). We choose a superpotential with a large mass term for the adjoint chiral superfield, so that it is reasonable to integrate it out and to be left with only the fundamental flavors and the glue fields. We then choose the tree-level superpotential to be: $$W_{\mbox{tree}}~=~\frac{1}{2}M~ Tr\Phi^2~+~\tilde{Q}_{\tilde{f}}~m^{\tilde{f}}_f Q^f$$ with $m^{\tilde{f}}_f~=~m_f\delta^{\tilde{f}}_f$ and $M>>m_i$. At energy $E$ such that $m_i<E<<M$, we are then left with ${\cal N}=1$ super-QCD with $N_f$ massive flavors and physical scale $$\label{scale}\Lambda_I^{3N-N_f}~=~\Lambda^{2N-N_f}~M^N.$$
The problem (\[complete\]) in the case at hand reduces to $$\begin{aligned}
P_N(z)^2~-~4~\Lambda^{2N-N_f}\prod_{i=1}^{N_f}~m_i~=~F_2(z)~H_{N-1}^2(z)
\label{probl}\\
F_2(z)~=~z^2~+~\frac{f_0}{M^2}\end{aligned}$$ where $f_0$ is a constant, related to the gaugino condensate $S$ by [@Cachazo:2001jy]: $$S~=~-\frac{1}{4M}~f
\label{condf}$$
This problem can be easily solved [@pm] if we consider Chebyshev polynomials (${\cal T}_l(x)$ and ${\cal U}_m(x)$, respectively of first and second kind). Indeed they satisfy the following identity $${\cal T}_l(x)^2-1~=~(x^2-1)~{\cal U}_{l-1}(x)^2$$
The solution to (\[probl\]) is thus: $$\begin{aligned}
P_N(z)~=~2\tilde{\Lambda}^N\eta^N{\cal T}_N\left(\frac{z}{2\eta\tilde{\Lambda}}\right),\hspace{0.5cm}F_2(z)~=~z^2-4\eta^2\tilde{\Lambda}^2,\label{solar}\\ H_{N-1}(z)~=~\eta^{N-1}\tilde{\Lambda}^{N-1}{\cal U}_{N-1}\left(\frac{z}{2\eta\tilde{\Lambda}}\right)\end{aligned}$$ where $$\tilde{\Lambda}^{2N}~=~\Lambda^{2N-N_f}\prod_{i=1}^{N_f}m_i\hspace{2cm}\mbox{and}\hspace{2cm}\eta^{2N}=1;$$ What is most important for us is that the solution (\[solar\]) also implies: $$f~=~-4\eta^2 M^2 (\Lambda^{2N-N_f}\prod_{i=1}^{N_f}m_i)^{1/N},
\label{f}$$ Moreover, we know that the equation (\[probl\]) has been proved to have a unique solution in a very analogous case [@Cachazo:2002pr] and that proof is also valid for the case we are considering here. From (\[condf\]) and (\[f\]) it is now easy to see that the gaugino condensate is non vanishing and it is equal to $$S~=~\eta^2 M \left(\Lambda^{2N-N_f}\prod_{i=1}^{N_f}m_i\right)^{1/N}
\label{condtot}$$
If we now integrate out the adjoint field $\Phi$, using (\[scale\]) we easily find $$S~=~\eta^2 \left(\Lambda_I^{3N-N_f}\prod_{i=1}^{N_f}m_i\right)^{1/N}
\label{condflav}$$ This is the expected result in the presence of fundamental flavors [@Intriligator:1995au].
Let’s also notice that for the specific case $N_f=0$, (pure ${\cal N}=1$ gluodynamics) we get: $$S~=~\eta^2\Lambda_I^3
\label{gv}$$ and this is in perfect agreement with the expectations of the so called ‘weak coupling computation’ for pure ${\cal N}=1$ gluodynamics (see [@Konishi:2003ts] for a related way of getting the same result; see instead [@Dorey:2002ik] for a general discussion).
Using standard techniques to integrate in and out various fields, it is possible to relate these results to known superpotentials[^2] [@Intriligator:1995au]. For example, considering the case $N_f=0$, from (\[gv\]) and the knowledge of the one-loop $\beta$-function $\beta(g_{YM})=-\frac{3N}{16\pi^2}~g_{YM}^3$, it is easy to recover the Veneziano-Yankielowicz superpotential $$W_{V.-Y.}~=~S\left(\ln\frac{\Lambda^{3N}}{S^N}+N\right).$$ This is the low-energy superpotential for pure ${\cal N}=1$ Yang-Mills theory. It is written in terms of the composite field $S$ and of the dynamically generated scale $\Lambda$. From it we cannot learn nothing about the running of the Yang-Mills coupling constant, whose knowledge we had to assume a priori.
In the framework of the gauge-string correspondence that motivated all this field theory analysis, it is instead possible to write directly the superpotential in terms of $g_{YM}$ and an ultraviolet cut-off $\Lambda_0$. Then, as we are going to see, it is possible to determine directly the $\beta$-function.
Back to the gauge-geometry correspondence
=========================================
The central idea of the gauge-geometry correspondence is [@Gopakumar:1998ki; @Vafa:2000wi; @Cachazo:2001jy] to engineer geometrically the field theory via D-branes wrapped over certain cycles of a non-trivial Calabi-Yau geometry. Thus, the low energy dual arises from a geometric transition of the Calabi-Yau, where the branes have disappeared and have been replaced by suitable fluxes.
The simplest example is pure ${\cal N}=1$ super Yang-Mills theory, engineered on the conifold. Before the transition we have $N$ D5 branes wrapped on the blown-up $S^2$ of a resolved conifold. After the transition we are instead left with $N$ units of Ramon-Ramon flux through the $S^3$ of a deformed conifold.
From the knowledge of the geometry after the transition and the map between the original microscopic field theoretical degrees of freedom and the new geometrical data, it is possible to determine the effective gauge theory superpotential [@Cachazo:2001jy]. It turns out to be: $$\begin{aligned}
\nonumber W_{eff}(S)~=~\frac{8\pi^2}{g_{YM}^2}S+\frac{1}{2}N\Lambda_0^3\sqrt{1-\frac{4S}{\Lambda_0^3}}+\\ \label{fullsupot} +NS\left[\ln\frac{S}{\Lambda_0^3}-2\ln\frac{1}{2}\left(1+\sqrt{1-\frac{4S}{\Lambda_0^3}}\right)\right]\end{aligned}$$ We can now expand it in powers of $\Lambda_0$ (the ultra-violet cut-off at which $g_{YM}$ is evaluated) and then get: $$\frac{1}{2}N\Lambda_0^3+W_{V.Y.}(S,\Lambda_0,g^2_{YM})+O(\frac{1}{\Lambda_0})$$ where $$W_{V.Y.}(S,\Lambda_0,g^2_{YM})=NS\left(\ln\frac{S}{\Lambda_0^3}+\frac{8\pi^2}{N g_{YM}^2}-1\right)
\label{vy}$$
From this superpotential and the knowledge of gaugino condensation that we have from the last paragraph, it is possible now to determine the $\beta$-function of the theory.
Indeed, we can minimize (\[vy\]) with respect to the gaugino bilinear superfield $S$ and impose that the result is the gaugino condensation: $$S=\Lambda_0^3~ {\mbox e}^{-\frac{8\pi^2}{N g_{YM}^2}}~~=~\star\Lambda^3$$ Differentiating this relation we can now get the $\beta$-function $$\beta (g_{YM})=\frac{\partial g_{YM}}{\partial\ln\frac{\Lambda_0}{\Lambda}}=~-\frac{3N}{16\pi^2}~g_{YM}^3.$$ This is the right $\beta$ function in the Wilsonian scheme, where it receives contributions only at one loop.
We can now apply the same procedure also to the full superpotential (\[fullsupot\]): $$S=\frac{\Lambda_0^3}{4\cosh^2\frac{4\pi^2}{N g_{YM}^2}}~~=~\star\Lambda^3$$ from which we can read the $\beta$-function $$\beta (g_{YM})=\frac{\partial g_{YM}}{\partial\ln\frac{\Lambda_0}{\Lambda}}=~-\frac{3N}{16\pi^2}g_{YM}^3~\frac{1+ {\mbox e}^{-\frac{8\pi^2}{N g_{YM}^2}}}{1- {\mbox e}^{-\frac{8\pi^2}{N g_{YM}^2}}}$$ This result seems to imply that there are non-perturbative correction to the perturbative running of $g_{YM}$. In a supersymmetric theory this means that there are non-perturbative modification also to the $U(1)_R$ anomaly.
This is not the only case in which the gauge-gravity correspondence suggests that such non-perturbative effects are present. Indeed also in other cases, where the supergravity solution is explicitely known (the Klebanov Strassler solution [@ks] and the Maldacena Nu$\tilde{n}$ez one [@mn]), it has been emphasized [@dlm; @bm; @i] that the gravitational description seems to include non-perturbative configurations (the so-called fractional instantons) that affect the running of $g_{YM}$.
This analysis would then imply that there are non-perturbative contributions to the anomaly. If this would be the case, the Konishi anomaly itself (the starting point of the field theory analysis we briefly discussed here) should probably be modified. Then also the solution of the gauge theory discussed in this talk should be modified to take into account the non-perturbative corrections.
This is just one example of how the interplay between gauge and string theories can be very profitable.
References {#references .unnumbered}
==========
[99]{}
R. Dijkgraaf and C. Vafa, “A perturbative window into non-perturbative physics”, arXiv:hep-th/0208048.
F. Cachazo, M. R. Douglas, N. Seiberg and E. Witten, “Chiral rings and anomalies in supersymmetric gauge theory,” JHEP [**0212**]{} (2002) 071 \[arXiv:hep-th/0211170\].
F. Cachazo, N. Seiberg and E. Witten, “Chiral Rings and Phases of Supersymmetric Gauge Theories,” JHEP [**0304**]{} (2003) 018 \[arXiv:hep-th/0303207\]. R. Dijkgraaf, M. T. Grisaru, C. S. Lam, C. Vafa and D. Zanon, “Perturbative computation of glueball superpotentials,” Phys. Lett. B [**573**]{} (2003) 138 \[arXiv:hep-th/0211017\].
R. Argurio, G. Ferretti and R. Heise, “An introduction to supersymmetric gauge theories and matrix models,” arXiv:hep-th/0311066.
M. Matone and L. Mazzucato, “On the chiral ring of N = 1 supersymmetric gauge theories,” JHEP [**0310**]{} (2003) 011 \[arXiv:hep-th/0307130\]. F. Cachazo, K. A. Intriligator and C. Vafa, “A large N duality via a geometric transition,” Nucl. Phys. B [**603**]{} (2001) 3 \[arXiv:hep-th/0103067\]. P. Merlatti, “Gaugino condensate and phases of N = 1 super Yang-Mills theories,” arXiv:hep-th/0307115. F. Cachazo and C. Vafa, “N = 1 and N = 2 geometry from fluxes,” arXiv:hep-th/0206017. K. A. Intriligator and N. Seiberg, “Lectures on supersymmetric gauge theories and electric-magnetic duality,” Nucl. Phys. Proc. Suppl. [**45BC**]{} (1996) 1 \[arXiv:hep-th/9509066\]. K. Konishi and A. Ricco, “Calculating gluino condensates in N = 1 SYM from Seiberg-Witten curves,” arXiv:hep-th/0306128.
N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “The calculus of many instantons,” Phys. Rept. [**371**]{} (2002) 231 \[arXiv:hep-th/0206063\]. R. Argurio, V. L. Campos, G. Ferretti and R. Heise, Phys. Rev. D [**67**]{} (2003) 065005 \[arXiv:hep-th/0210291\].
Y. Demasure and R. A. Janik, Phys. Lett. B [**553**]{} (2003) 105 \[arXiv:hep-th/0211082\]. Y. Demasure, arXiv:hep-th/0307082. R. Gopakumar and C. Vafa, Adv. Theor. Math. Phys. [**3**]{} (1999) 1415 \[arXiv:hep-th/9811131\]. C. Vafa, J. Math. Phys. [**42**]{} (2001) 2798 \[arXiv:hep-th/0008142\].
I. R. Klebanov and M. J. Strassler, JHEP [**0008**]{} (2000) 052 \[arXiv:hep-th/0007191\]. J. M. Maldacena and C. Nunez, Phys. Rev. Lett. [**86**]{} (2001) 588 \[arXiv:hep-th/0008001\].
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M. Bertolini and P. Merlatti, Phys. Lett. B [**556**]{} (2003) 80 \[arXiv:hep-th/0211142\]. E. Imeroni, Phys. Lett. B [**541**]{} (2002) 189 \[arXiv:hep-th/0205216\].
[^1]: Not all of these techniques still apply in the presence of matter, the case discussed here
[^2]: Direct relations between effective superpotentials and matrix models were investigated in [@Argurio:2002xv; @Demasure:2002sc; @Demasure:2003sk]
|
---
author:
- 'Roland K. W. Roeder [^1] [^2], John H. Hubbard, and William D. Dunbar'
bibliography:
- 'andreev.bib'
title: 'Andreev’s Theorem on hyperbolic polyhedra'
---
**Abstract**
In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles [@AND; @AND2]. Given a combinatorial description of a polyhedron, $C$, Andreev’s Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry.
Andreev’s Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston’s Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev’s Theorem resembles (in a simpler way) the proof of Thurston.
We correct a fundamental error in Andreev’s proof of existence and also provide a readable new proof of the other parts of the proof of Andreev’s Theorem, because Andreev’s paper has the reputation of being “unreadable”.
**Résumé**
E. M. Andreev a publié en 1970 une classification des polyèdres hyperboliques compacts de dimension trois (autrement que les tétraèdres) dont les angles dièdres sont non-obtus [@AND; @AND2]. Etant donné une description combinatoire d’un polyèdre $C$, le Théorème d’Andreev dit que les angles dièdres possibles sont exactement décrits par cinq classes d’inégalités linéaires. Le Théorème d’Andreev démontre également que le polyèdre résultant est alors unique à isométrie hyperbolique près.
D’une part, le Théorème de Andreev est évidemment un énoncé intéressant de la géométrie de l’espace hyperbolique en dimension 3; d’autre part c’est un outil essentiel dans la preuve du Théorème d’Hyperbolization de Thurston pour les variétés Haken de dimension 3. Il est d’ailleurs remarquable à quel point la démonstration d’Andreev rappelle (en plus simple) la démonstration de Thurston.
La démonstration d’Andreev contient une erreur importante. Nous corrigeons ici cette erreur et nous fournissons aussi une nouvelle preuve lisible des autres parties de la preuve, car le papier d’Andreev a la réputation d’être “illisible”.
Statement of Andreev’s Theorem
==============================
Andreev’s Theorem provides a complete characterization of compact hyperbolic polyhedra having non-obtuse dihedral angles. This classification is essential for proving Thurston’s Hyperbolization theorem for Haken 3-manifolds and is also a particularly beautiful and interesting result in its own right. Complete and detailed proofs of Thurston’s Hyperbolization for Haken 3-manifolds are available written in English by Jean-Pierre Otal [@OT] and in French by Michel Boileau [@BOI].
In this paper, we prove Andreev’s Theorem based on the main ideas from his original proof [@AND]. However, there is an error in Andreev’s proof of existence. We explain this error in Section 6 and provide a correction. Although the other parts of the proof are proven in much the same way as Andreev proved them, we have re-proven them and re-written them to verify them as well as to make the overall proof of Andreev’s Theorem clearer. This paper is based on the doctoral thesis of the first author [@ROE], although certain proofs have been streamlined, especially in Sections 4 and 5.
The reader may also wish to consider the similar results of Rivin and Hodgson [@RH; @H], Thurston [@T_NOTES Chapter 13], Marden and Rodin [@MR], Bowers and Stephenson [@STEVE], Rivin [@RIV_IDEAL2; @RIV_IDEAL1; @RIV_IDEAL3], and Bao and Bonahon [@BAO]. In [@RH], the authors prove a more general statement than Andreev’s Theorem and in [@H] Hodgson deduces Andreev’s Theorem as a consequence of their previous work. The proof in [@RH] is similar to the one presented here, except that the conditions classifying the polyhedra are written in terms of measurements in the De Sitter space, the space dual to the hyperboloid model of hyperbolic space. Although a beautiful result, the main drawback of this proof is that the last sections of the paper, which are necessary for their proof that such polyhedra exist, are particularly hard to follow.
Marden and Rodin [@MR] and Thurston [@T_NOTES Chapter 13] consider configurations of circles with assigned overlap angles on the Riemann Sphere and on surfaces of genus $g$ with $g > 0$. Such a configuration on the Riemann Sphere corresponds to a configuration of hyperbolic planes in the Poincaré ball model of hyperbolic space. Thus, there is a direct connection between circle patterns and hyperbolic polyhedra. The proof of Thurston provides a classification of configurations of circles on surfaces of genus $g > 0$. The proof of Marden and Rodin [@MR] is an adaptation of Thurston’s circle packing theorem to the Riemann Sphere, resulting in a theorem similar to Andreev’s Theorem. Although Thurston’s statement has analogous conditions to Andreev’s classical conditions, Marden and Rodin require that the sum of angles be less than $\pi$ for every triple of circles for which each pair intersects. This prevents the patterns of overlapping circles considered in their theorem from corresponding to compact hyperbolic polyhedra.
Bowers and Stephenson [@STEVE] prove a “branched version” of Andreev’s Theorem, also in terms of circle patterns on the Riemann Sphere. Instead of the continuity method used by Thurston and Marden-Rodin, Bowers and Stephenson use ideas intrinsic to the famous Uniformization Theorem from complex analysis. The unbranched version of their theorem provides a complete proof of Andreev’s Theorem, which provides an alternative to the proof presented here.
Rivin has proven beautiful results on ideal hyperbolic polyhedra having arbitrary dihedral angles [@RIV_IDEAL2; @RIV_IDEAL1] (see also Gu[é]{}ritaud [@GUE] for an alternative viewpoint, with exceptionally clear exposition.) Similar nice results are proven for hyperideal polyhedra by Bao and Bonahon [@BAO]. Finally, the papers of Vinberg on discrete groups of reflections in hyperbolic space [@AVS; @VIN; @VINREFL; @VINVOL; @VS] are also closely related, as well as the work of Bennett and Luo [@LUO] and Schlenker [@SCH2; @SCH1; @SCH3].
Let $E^{3,1}$ be $\mathbb{R}^4$ with the indefinite metric $\Vert {\bf x}
\Vert^2 = -x_0^2+x_1^2+x_2^2+x_3^2$. The space of ${\bf x}$ for which this indefinite metric vanishes is typically referred to as the lightcone, which we denote by $C$.
In this paper, we work in the hyperbolic space $\mathbb{H}^3$ given by the component of the subset of $E^{3,1}$ given by $$\Vert {\bf x} \Vert^2 = -x_0^2+x_1^2+x_2^2+x_3^2 = -1$$
having $x_0 > 0$, with the Riemannian metric induced by the indefinite metric $$-dx_0^2+dx_1^2+dx_2^2+dx_3^2.$$
Hyperbolic space $\mathbb{H}^3$ can be compactified by adding the set of rays in $\{{\bf x} \in C \mbox{ : } x_0 \ge 0 \},$ which clearly form a topological space $\partial \mathbb{H}^3$ homeomorphic to the sphere $\mathbb{S}^{2}$. We will refer to points in $\partial \mathbb{H}^3$ as [*points at infinity*]{} and refer to the compactification as $\overline{\mathbb{H}^3}$. For more details, see [@THURSTON_BOOK p. 66].
The hyperplane orthogonal to a vector ${\bf v} \in
E^{3,1}$ intersects $\mathbb{H}^3$ if and only if $\langle{\bf v},{\bf
v}\rangle> 0$. Let ${\bf v} \in E^{3,1}$ be a vector with $\langle{\bf
v},{\bf v}\rangle > 0$, and define $$\begin{aligned}
P_{\bf v} = \{{\bf w} \in \mathbb{H}^3 | \langle{\bf w},{\bf v}\rangle
= 0\} \mbox{ and }
H_{\bf v} = \{{\bf w} \in \mathbb{H}^3 | \langle{\bf w},{\bf v}\rangle \leq
0 \}\end{aligned}$$
to be the hyperbolic plane orthogonal to ${\bf v}$ and the corresponding closed half space, oriented so that ${\bf v}$ is the outward pointing normal.
If one normalizes $\langle{\bf v},{\bf v}\rangle = 1$ and $\langle{\bf w},{\bf
w}\rangle = 1$ the planes $P_{\bf v}$ and $P_{\bf w}$ in $\mathbb{H}^3$ intersect in a line if and only if $\langle{\bf v},{\bf w}\rangle^2 <
1$, in which case their dihedral angle is $\arccos(-\langle{\bf v},{\bf
w}\rangle)$. They intersect in a single point at infinity if and only if $\langle{\bf v},{\bf w}\rangle^2 = 1$; in this case their dihedral angle is $0$.
A [*hyperbolic polyhedron*]{} is an intersection
$$P = \bigcap_{i=0}^N H_{\bf v_i}$$
having non-empty interior. Throughout this paper we will make the assumption that ${\bf v}_1,\cdots,{\bf v}_N$ form a minimal set of vectors specifying $P$. That is, we assume that none of the half-spaces $H_{\bf v_i}$ contains the intersection of all the others.
It is not hard to verify that if $H_{\bf v_i}, H_{\bf v_j},H_{\bf v_k}$ are three distinct halfspaces appearing in the definition of the polyhedron $P$, then the three vectors ${\bf v_i}$, ${\bf v_j}$, and ${\bf v_k}$ will always be linearly independent. For example, if ${\bf v_i} = {\bf v_j} + {\bf v_k}$, then $H_{\bf v_i}$ would be a subset of $H_{\bf v_j} \cap H_{\bf v_k}$, contradicting minimality.
We will often use the Poincaré ball model of hyperbolic space, given by the open unit ball in $\mathbb{R}^3$ with the metric
$$4\frac{dx_1^2+dx_2^2+dx_3^2}{(1 -\Vert {\bf x}\Vert^2)^2}$$
and the upper half-space model of hyperbolic space, given by the subset of $\mathbb{R}^3$ with $x_3 > 0$ equipped with the metric $$\frac{dx_1^2+dx_2^2+dx_3^2}{x_3^2}.$$
Both of these models are isomorphic to $\mathbb{H}^3$.
Hyperbolic planes in these models correspond to Euclidean hemispheres and Euclidean planes that intersect the boundary perpendicularly. Furthermore, these models are conformally correct, that is, the hyperbolic angle between a pair of such intersecting hyperbolic planes is exactly the Euclidean angle between the corresponding spheres or planes.
Below is an image of a hyperbolic polyhedron depicted in the Poincaré ball model with the sphere at infinity shown for reference. It was displayed in the excellent computer program Geomview [@GEO].
[*Abstract polyhedra and Andreev’s Theorem*]{}
Some elementary combinatorial facts about hyperbolic polyhedra are essential before we can state Andreev’s Theorem. Notice that a compact hyperbolic polyhedron $P$ is topologically a 3-dimensional ball, and its boundary a 2-sphere $\mathbb{S}^2$. The face structure of $P$ gives $\mathbb{S}^2$ the structure of a cell complex $C$ whose faces correspond to the faces of $P$, and so forth.
Considering only hyperbolic polyhedra with non-obtuse dihedral angles simplifies the combinatorics of any such $C$:
\[TRIVALENT\] (a) A vertex of a non-obtuse hyperbolic polyhedron $P$ is the intersection of exactly 3 faces. (b) For such a $P$, we can compute the angles of the faces in terms of the dihedral angles; these angles are also $\leq \pi/2$.
[**Proof:**]{} Let $v$ be a finite vertex where $n$ faces of $P$ meet. After an appropriate isometry, we can assume that $v$ is the origin in the Poincaré ball model, so that the faces at $v$ are subsets of Euclidean planes through the origin. A small sphere centered at the origin will intersect $P$ in a spherical $n$-gon $Q$ whose angles are the dihedral angles between faces. Call these angles $\alpha_1,...,\alpha_n$. Re-scale $Q$ so that it lies on the sphere of unit radius, then the Gauss-Bonnet formula gives $\alpha_1+\cdots+\alpha_n = \pi(n-2) +
{{\rm Area}}(Q)$. The restriction to $\alpha_i \leq \pi/2$ for all $i$ gives $n\pi/2
\geq \pi(n-2) + {{\rm Area}}(Q)$. Hence, $n\pi/2 < 2\pi$. We conclude that $n =
3$.
The edge lengths of $Q$ are precisely the angles in the faces at the origin. Supposing that $Q$ has angles $(\alpha_i,\alpha_j,\alpha_k)$ and edge lengths $(\beta_i,\beta_j,\beta_k)$ with the edge $\beta_l$ opposite of angle $\alpha_l$ for each $l$, The law of cosines in spherical geometry states that:
$$\begin{aligned}
\label{TLC}
\cos(\beta_i) = \frac{\cos(\alpha_i)
+\cos(\alpha_j)\cos(\alpha_k)} {\sin(\alpha_j)\sin(\alpha_k)}.\end{aligned}$$
Hence, the face angles are calculable from the dihedral angles. They are non-obtuse, since the right-hand side of the equation is positive for $\alpha_i,\alpha_j,\alpha_k$ non-obtuse. (Equation (\[TLC\]) will be used frequently throughout this paper.) [$\Box$ ]{}
The fundamental axioms of incidence place the following, obvious, further restrictions on the complex $C$:
- Every edge of $C$ belongs to exactly two faces.
- A non-empty intersection of two faces is either an edge or a vertex.
- Every face contains not fewer than three edges.
We will call any trivalent cell complex $C$ on $\mathbb{S}^2$ that satisfies the three conditions above an [*abstract polyhedron*]{}. Notice that since $C$ must be a trivalent cell complex on $\mathbb{S}^2$, its dual, $C^*$, has only triangular faces. The three other conditions above ensure that the dual complex $C^*$ is a simplicial complex, which we embed in the same $\mathbb{S}^2$ so that the vertex corresponding to any face of $C$ is an element of the face, etc. (Andreev refers to this dual complex as the [*scheme of the polyhedron*]{}.) The figure below shows an abstract polyhedron $C$ drawn in the plane (i.e. with one of the faces corresponding to the region outside of the figure.) The dual complex is also shown, in dashed lines.
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\#1\#2\#3\#4\#5[ @font ]{}
(4015,2919)(139,-2143)
We call a simple closed curve $\Gamma$ formed of $k$ edges of $C^*$ a [*k-circuit*]{} and if all of the endpoints of the edges of $C$ intersected by $\Gamma$ are distinct, we call such a circuit a [*prismatic k-circuit*]{}. The figure below shows the same abstract polyhedron as above, except this time the prismatic 3-circuits are dashed, the prismatic 4-circuits are dotted, and the dual complex is not shown.
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\#1\#2\#3\#4\#5[ @font ]{}
(4152,2571)(80,-1853)
Before stating Andreev’s Theorem, we prove two basic lemmas about abstract polyhedra:
\[3CIRC\] If $\gamma$ is a 3-circuit that is not prismatic in an abstract polyhedron $C$ intersecting edges $e_1,e_2$, and $e_3$, then edges $e_1,e_2$, and $e_3$ meet at a vertex.
[**Proof:**]{} Since $\gamma$ is 3-circuit that is not prismatic, a pair of the edges meet at a vertex. We suppose that $e_1$ and $e_2$ meet at this vertex, which we label $v_1$. Since the vertices of $C$ are trivalent, there is some edge $e'$ meeting $e_1$ and $e_2$ at $v_1$. We suppose that $e'$ is not the edge $e_3$ to obtain a contradiction. Moving $\gamma$ past the vertex $v_1$, we can obtain a new circuit $\gamma'$ intersecting only the two edges $e_3$ and $e'$.
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\#1\#2\#3\#4\#5[ @font ]{}
(4957,962)(649,-722) (826,112)[(0,0)\[lb\]]{} (856,-691)[(0,0)\[lb\]]{} (895,-286)[(0,0)\[lb\]]{} (1334,-198)[(0,0)\[lb\]]{} (2057, 88)[(0,0)\[lb\]]{} (2313, 7)[(0,0)\[lb\]]{} (2547,-299)[(0,0)\[lb\]]{} (4133,138)[(0,0)\[lb\]]{} (4163,-665)[(0,0)\[lb\]]{} (4202,-260)[(0,0)\[lb\]]{} (5377,134)[(0,0)\[lb\]]{} (1865,-190)[(0,0)\[lb\]]{} (4872,-173)[(0,0)\[lb\]]{} (5035,-422)[(0,0)\[lb\]]{} (1735,-463)[(0,0)\[lb\]]{}
The curve $\gamma'$ intersects only two edges, hence it only crosses two faces of $C$. However, this implies that these two faces of $C$ intersect along the two distinct edges $e'$ and $e_3$, contrary to fact that two faces of an abstract polyhedron which intersect do so along a single edge. [$\Box$ ]{}
\[4CIRC\] Let $C$ be an abstract polyhedron having no prismatic 3-circuits. If $\gamma$ is a 4-circuit which is not prismatic, then $\gamma$ separates exactly two vertices of $C$ from the remaining vertices of $C$.
[**Proof:**]{} Suppose that $\gamma$ crosses edges $e_1,e_2,e_3$, and $e_4$ of $C$. Because $\gamma$ is not a prismatic 4-circuit, a pair of these edges meet at a vertex. Without loss of generality, we suppose that edges $e_1$ and $e_2$ meet at this vertex, which we denote $v_1$. Since $C$ is trivalent, there is some edge $e'$ meeting $e_1$ and $e_2$ at $v_1$. Let $\gamma '$ be the 3-circuit intersecting edges $e_3, e_4$ and $e'$, obtained by sliding $\gamma$ past the vertex $v_1$. Since $C$ has no prismatic 3-circuits, $\gamma'$ is not prismatic, so by Lemma \[3CIRC\], edges $e_3,e_4$, and $e'$ meet at another vertex $v_2$. The entire configuration is shown in the diagram below.
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\#1\#2\#3\#4\#5[ @font ]{}
(4647,1139)(649,-722) (4434,-143)[(0,0)\[lb\]]{} (3946,165)[(0,0)\[lb\]]{} (3976,-638)[(0,0)\[lb\]]{} (5116,-562)[(0,0)\[lb\]]{} (5161,248)[(0,0)\[lb\]]{} (5296,-202)[(0,0)\[lb\]]{} (5034,-374)[(0,0)\[lb\]]{} (4830,-233)[(0,0)\[lb\]]{} (2581,-306)[(0,0)\[lb\]]{} (2610, 21)[(0,0)\[lb\]]{} (4023,-233)[(0,0)\[lb\]]{} (826,112)[(0,0)\[lb\]]{} (856,-691)[(0,0)\[lb\]]{} (1996,-615)[(0,0)\[lb\]]{} (2041,195)[(0,0)\[lb\]]{} (2176,-255)[(0,0)\[lb\]]{} (1914,-427)[(0,0)\[lb\]]{} (895,-286)[(0,0)\[lb\]]{} (1246,-203)[(0,0)\[lb\]]{}
Therefore, the 4-circuit $\gamma$ separates the two vertices $v_1$ and $v_2$ from the remaining vertices of $C$. [$\Box$ ]{}
[**Andreev’s Theorem**]{}
Let $C$ be an abstract polyhedron with more than 4 faces and suppose that non-obtuse angles $\alpha_i$ are given corresponding to each edge $e_i$ of $C$. There is a compact hyperbolic polyhedron $P$ whose faces realize $C$ with dihedral angle $\alpha_i$ at each edge $e_i$ if and only if the following five conditions all hold:
1. For each edge $e_i$, $\alpha_i > 0$.
2. Whenever 3 distinct edges $e_i,e_j,e_k$ meet at a vertex, $\alpha_i+\alpha_j+\alpha_k > \pi$.
3. Whenever $\Gamma$ is a prismatic 3-circuit intersecting edges $e_i,e_j,e_k$, $\alpha_i+\alpha_j+\alpha_k < \pi$.
4. Whenever $\Gamma$ is a prismatic 4-circuit intersecting edges $e_i,e_j,e_k,e_l$, then $\alpha_i+\alpha_j+\alpha_k+\alpha_l < 2\pi$.
5. Whenever there is a four sided face bounded by edges $e_1,$ $e_2,$ $e_3,$ and $e_4$, enumerated successively, with edges $e_{12}, e_{23}, e_{34}, e_{41}$ entering the four vertices (edge $e_{ij}$ connects to the ends of $e_i$ and $e_j$), then: $$\alpha_1 + \alpha_3 + \alpha_{12} + \alpha_{23} + \alpha_{34} +
\alpha_{41} < 3\pi, \hspace{.2in} {\rm and}$$ $$\alpha_2 + \alpha_4 + \alpha_{12} + \alpha_{23} + \alpha_{34} +
\alpha_{41} < 3\pi.$$
Furthermore, this polyhedron is unique up to isometries of $\mathbb{H}^3$.
In addition to the role that Andreev’s theorem plays, as a bootstrap in the proof of Thurston’s hyperbolization theorem, it is worth noting that, in the context of orbifolds, the former can be thought of as a very special case of the latter (extended to Haken orbifolds as in [@BP Chapter 8] or [@CHK]). Consider closed $3$-orbifolds with underlying topological space a $3$-ball, and with singular set equal to the boundary sphere. That singular set will consist of a trivalent graph, together with “mirrors” on the complementary regions. Each edge of the graph is labeled with an integer $k>1$, corresponding to a dihedral angle of $\pi/k$. The definition of a $3$-orbifold implies that the angle sum at each vertex will satisfy condition (2) in Andreev’s theorem [@KAP sections 6.1 and 6.3].
Restrict the combinatorics of the singular set, slightly more than in the statement of Andreev’s theorem: $C$ must be an abstract polyhedron with more than 5 faces. Such an orbifold is Haken if and only if it is irreducible [@T_NOTES Proposition 13.5.2]. Condition (3) in Andreev’s theorem guarantees irreducibility, and also, together with condition (4), guarantees that every Euclidean 2-suborbifold is compressible. Therefore, for Haken orbifolds of this topological type, Andreev’s theorem says precisely that having no incompressible Euclidean 2-suborbifolds is equivalent to the existence of a hyperbolic structure [@T_NOTES Section 13.6]; see also [@KAP Section 6.4].
For a given $C$ let $E$ be the number of edges of $C$. The subset of $(0,\pi /2]^E$ satisfying these linear inequalities will be called the [*Andreev Polytope*]{}, $A_C$. Since $A_C$ is determined by linear inequalities, it is convex.
Andreev’s restriction to non-obtuse dihedral angles is emphatically necessary to ensure that $A_C$ be convex. Without this restriction, the corresponding space of dihedral angles, $\Delta_C$, of compact (or finite volume) hyperbolic polyhedra realizing a given $C$ is not convex [@DIAZ]. In fact, the recent work by Díaz [@DIAZ_ANDREEV] provides a detailed analysis of this space of dihedral angles $\Delta_C$ for the class of abstract polyhedra $C$ obtained from the tetrahedron by successively truncating vertices. Her work nicely illustrate the types of non-linear conditions that are necessary in a complete analysis of the larger space of dihedral angles $\Delta_C$.
The work of Rivin [@RIV_IDEAL2; @RIV_IDEAL1] shows that the space of dihedral angles for ideal polyhedra forms a convex polytope, [*without the restriction to non-obtuse angles*]{}. (See also [@GUE].)
Notice also that the hypothesis that the number of faces is greater than four is also necessary because the space of non-obtuse dihedral angles for compact tetrahedra is not convex [@ROE_TET]. Conditions (1-5) remain necessary conditions for compact tetrahedra, but they are no longer sufficient.
\[FIVE\] If $C$ is not the triangular prism, condition (5) of Andreev’s Theorem is a consequence of conditions (3) and (4).
[**Proof:**]{} Given a quadrilateral face, if the four edges leading from it form a prismatic 4-circuit, $\Gamma_1$, as depicted on the left hand side of the figure below, clearly condition (5) is a result of condition (4). Otherwise, at least one pair of the edges leading from it meet at a vertex. If only one pair meets at a point, we have the diagram below in the middle. In this case, the curve $\Gamma_2$ can easily be shown to be a prismatic 3-circuit, so that $\alpha_{34} + \alpha_{41} + \beta < \pi$, so that condition (5) is satisfied because $\alpha_{34}$ and $\alpha_{41}$ cannot both be $\pi/2$.
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\#1\#2\#3\#4\#5[ @font ]{}
(3800,1237)(552,-681) (2381,445)[(0,0)\[lb\]]{} (3967,-174)[(0,0)\[lb\]]{} (2683,-182)[(0,0)\[lb\]]{} (2119,-632)[(0,0)\[lb\]]{} (2645,-632)[(0,0)\[lb\]]{} (1240,-179)[(0,0)\[lb\]]{} (1165,-629)[(0,0)\[lb\]]{} (1165,234)[(0,0)\[lb\]]{} (639,-629)[(0,0)\[lb\]]{} (634,276)[(0,0)\[lb\]]{}
Otherwise, if two pairs of the edges leaving the quadrilateral face meet at vertices, we have the diagram on the right-hand side. The only way to complete this diagram is with the edge labeled $e_0$, resulting in the triangular prism. [$\Box$ ]{}
Hence, we need only check condition (5) for the triangular prism, which corresponds to the only five-faced $C$.
Given some $C$, it may be a difficult problem to determine whether $A_C =
\emptyset$ and correspondingly, whether there are any hyperbolic polyhedra realizing $C$ with non-obtuse dihedral angles. In fact, for the abstract polyhedron in the following figure, conditions (2) and (3) imply respectively that $\alpha_1+\cdots+\alpha_{12} > 4\pi$ and $\alpha_1+\cdots+\alpha_{12} < 4\pi$. So, for this $C$, we have $A_C =
\emptyset$. However, for more complicated $C$, it can be significantly harder to determine whether $A_C = \emptyset$.
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(1862,1979)(455,-1172) (1471,370)[(0,0)\[lb\]]{} (1846,464)[(0,0)\[lb\]]{} (1306,-164)[(0,0)\[lb\]]{} (455,-304)[(0,0)\[lb\]]{} (2317,-861)[(0,0)\[lb\]]{} (1428,651)[(0,0)\[lb\]]{} (1377,106)[(0,0)\[lb\]]{} (994, -6)[(0,0)\[lb\]]{} (777,-349)[(0,0)\[lb\]]{} (543,-590)[(0,0)\[lb\]]{} (1890,-863)[(0,0)\[lb\]]{} (1964,-1114)[(0,0)\[lb\]]{}
Luckily, there are special cases, including:
\[EX3APR\] If there are no prismatic 3-circuits in $C$, there exists a unique hyperbolic polyhedron realizing C with dihedral angles $2\pi/5$.
[**Proof:**]{} Since there are no prismatic 3-circuits in $C$, condition (3) of the theorem is vacuous and clearly $\alpha_i = 2\pi/5$ satisfy conditions (1), (2), (4), and (5). [$\Box$ ]{}
Setup of the Proof {#SETUP}
==================
Let $C$ be a trivalent abstract polyhedron with $N$ faces. We say that a hyperbolic polyhedron $P \subset \mathbb{H}^3$ [*realizes $C$*]{} if there is a cellular homeomorphism from $C$ to $\partial P$ (i.e., a homeomorphism mapping faces of $C$ to faces of $P$, edges of $C$ to edges of $P$, and vertices of $C$ to vertices of $P$). We will call each isotopy class of cellular homeomorphisms $\phi : C \rightarrow \partial P$ a [*marking*]{} on $P$.
We will define ${\cal P}_C$ to be the set of pairs $(P,\phi)$ so that $\phi$ is a marking with the equivalence relation that $(P,\phi) \sim
(P',\phi ')$ if there exists an isomorphism $\alpha : \mathbb{H}^3
\rightarrow \mathbb{H}^3$ such that $\alpha(P) = P'$ and both $\phi '$ and $\alpha \circ \phi$ represent the same marking on $P'$.
The space ${\cal P}_C$ is a manifold of dimension $3N-6$ (perhaps empty).
[**Proof:**]{} Let ${\cal H}$ be the space of closed half-spaces of $\mathbb{H}^3$; clearly ${\cal H}$ is a 3-dimensional manifold. Let ${\cal O}_C$ be the set of marked hyperbolic polyhedra realizing $C$. Using the marking to number the faces from $1$ to $N$, an element of ${\cal O}_C$ is an $N$-tuple of half-spaces that intersect in a polyhedron realizing $C$. This induces a mapping from ${\cal
O}_C$ to ${\cal H}^N$ whose image is an open set. We give ${\cal O}_C$ the topology that makes this mapping from ${\cal O}_C$ into ${\cal H}^N$ a local homeomorphism. Since ${\cal H}^N$ is a $3N$-dimensional manifold, ${\cal O}_C$ must be a $3N$-dimensional manifold as well.
If $\alpha(P,\phi) = (P,\phi)$, we have that $\alpha \circ \phi$ is isotopic to $\phi$ through cellular homeomorphisms. Hence, the isomorphism $\alpha$ must fix all vertices of $P$, and consequently restricts to the identity on all edges and faces. However, an isomorphism of $\mathbb{H}^3$ which fixes four non-coplanar points must be the identity. Therefore $Isom(\mathbb{H}^3)$ acts freely on ${\cal O}_C$. This quotient space of this action is ${\cal P}_C$, hence ${\cal P}_C$ is a manifold with dimension equal to $dim({\cal O}_C) -
dim(Isom(\mathbb{H}^3)) = 3N-6.$ [$\Box$ ]{}
An $m$-sided polygon $Q \subset \mathbb{H}^2$ with sides $s_i$ supported by lines $l_i$ for $i=1,\dots,m$ is called a [*parallelogram*]{} if, after adjoining any ideal endpoints in $\partial \mathbb{H}^2$ to these sides and lines, $s_i \cap s_j
= \emptyset$ implies $l_i \cap l_j = \emptyset$. In other words, if two sides of $Q$ don’t meet in $\overline{\mathbb{H}^2}$, then their supporting lines have a common perpendicular. We then define ${\cal P}^1_C$ to be the subset of ${\cal
P}_C$ consisting of those polyhedra all of whose faces are parallelograms.
Let’s check that ${\cal P}_C^1$ is an open subset of ${\cal P}_C$. If $P_{\bf
v_i},$ $i=1,2,3$ are three planes carrying faces of $P \in {\cal P}_C$, then $\{{\bf v}_i \mbox{, } i=1,2,3 \}$ will be a linearly independent set spanning a subspace $V$. Such a triple of planes has no common intersection point in $\overline{\mathbb{H}^3}$ if and only if the metric is indefinite when restricted to $V$, or equivalently, if and only if every vector orthogonal to $V$ has positive inner product with itself. This is an open condition on triples of half-spaces; hence, it is an open condition on ${\cal P}_C$ to require that the planes supporting three fixed faces of $P$ have no intersection in $\overline{\mathbb{H}^3}$.
Requiring that a single face of $P$ be a parallelogram is a finite intersection of such open conditions, for triples formed of that face and two faces whose intersections with that face form non-adjacent edges. For $P$ to lie in ${\cal
P}^1_C$ is a further finite intersection over its faces, so ${\cal P}^1_C$ is an open subset of ${\cal P}_C$, and hence ${\cal P}^1_C$ is a manifold of dimension $3N-6$, as well.
In fact, we be most interested in the subset ${\cal P}_C^0$ of polyhedra with non-obtuse dihedral angles. Notice that ${\cal P}_C^0$ is not, [*a priori*]{}, a manifold or even a manifold with boundary. However, as a consequence of Proposition \[TRIVALENT\] (b) and the fact that polygons with non-obtuse interior angles are parallelograms, we have the inclusion ${\cal P}_C^0 \subset {\cal
P}_C^1$.
Using the fact that the edge graph of $C$ is trivalent, one can check that $E$, the number of edges of $C$, is the same as the dimension of ${\cal
P}_C^1$. Since exactly three edges enter each vertex and each edge enters exactly two vertices, $3V = 2E$. The Euler characteristic gives $2=N - E +
V = N - E + 2E/3$ implying $E = 3(N-2)$, the dimension of ${\cal P}_C$ and ${\cal P}_C^1$.
Given any $P \in {\cal P}_C$, let $\alpha(P) =
(\alpha_1,\alpha_2,\alpha_3,...)$ be the $E$-tuple consisting of the dihedral angles of $P$ at each edge (according to some fixed numbering of the edges of $C$). This map $\alpha$ is obviously continuous with respect to the topology on ${\cal P}_C$, which it inherits from its manifold structure.
Our goal is to prove the following theorem, of which Andreev’s Theorem is a consequence:
\[HOMEO\] For every abstract polyhedron $C$ having more than four faces, the mapping $\alpha: {\cal P}_C^0 \rightarrow A_C$ is a homeomorphism.
We will say that [*Andreev’s Theorem holds for $C$*]{} if $\alpha: {\cal
P}_C^0 \rightarrow A_C$ is a homeomorphism for a specific abstract polyhedron $C$.
We begin the proof of Theorem \[HOMEO\] by checking that $\alpha({\cal
P}_C^0) \subset A_C$ in Section \[SEC\_INEQSAT\]. In Section \[SEC\_INJ\], we prove that $\alpha$ restricted to ${\cal P}_C^1$ is injective, and in Section \[SEC\_PROP\], we prove that $\alpha$ restricted to ${\cal P}_C^0$ is proper. In the beginning of Section \[SEC\_NONEMPT\] we combine these results to show that $\alpha: {\cal P}_C^0 \rightarrow A_C$ is a homeomorphism onto its image and that this image is a component of $A_C$. The remaining, and most substantial part of Section \[SEC\_NONEMPT\], is to show that $A_C \neq
\emptyset$ implies ${\cal P}_C^0 \neq \emptyset$.
The inequalities are satisfied. {#SEC_INEQSAT}
===============================
\[INEQSAT\] Given $P \in {\cal P}_C^0$, the dihedral angles $\alpha(P)$ satisfy conditions (1-5).
We will need the following two lemmas about the basic properties of hyperbolic geometry.
\[INTERSECT\] Suppose that three planes $P_{\bf v_1},P_{\bf v_2},P_{\bf v_3}$ intersect pairwise in $\mathbb{H}^3$ with non-obtuse dihedral angles $\alpha, \beta$, and $\gamma$. Then, $P_{\bf v_1},P_{\bf v_2},P_{\bf v_3}$ intersect at a vertex in $\overline{\mathbb{H}^3}$ if and only if $\alpha+\beta+\gamma \geq \pi.$ The planes intersect in $\mathbb{H}^3$ if and only if the inequality is strict.
[**Proof:**]{} The planes intersect in a point of $\overline{\mathbb{H}^3}$ if and only if the inner product is either positive definite or semi-definite on the subspace $V$ spanned by $\{ {\bf v}_i \mbox{, } i=1,2,3\}$. In the former case the intersection point is in $\mathbb{H}^3$, and in the latter case it is in $\partial \mathbb{H}^3$; in both cases the point is determined by the orthogonal complement of $V$. The matrix describing the inner product on $V$ is
$$\begin{aligned}
\left[
\begin{array}{ccc}
1 & \langle{\bf v_1},{\bf v_2}\rangle & \langle{\bf v_1},{\bf v_3}\rangle \\
\langle{\bf v_1},{\bf v_2}\rangle & 1 & \langle{\bf v_2},{\bf v_3}\rangle \\
\langle{\bf v_1},{\bf v_3}\rangle & \langle{\bf v_2},{\bf v_3}\rangle & 1\\
\end{array}
\right]
=
\left[
\begin{array}{ccc}
1 & -\cos\alpha & -\cos\beta \\
-\cos\alpha & 1 & -\cos\gamma \\
-\cos\beta & -\cos\gamma & 1 \\
\end{array}
\right]\end{aligned}$$
where $\alpha,\beta,$ and $\gamma$ are the dihedral angles between the pairs of faces $(P_{\bf v_1},P_{\bf v_2})$, $(P_{\bf v_1},P_{\bf v_3}),$ and $(P_{\bf
v_2},P_{\bf v_3})$, respectively.
Since the principal minor is positive definite for $0 < \alpha \leq \pi/2$, it is enough to find out when the determinant $$1 -2\cos\alpha\cos\beta\cos\gamma -\cos^2\alpha-\cos^2\beta-\cos^2\gamma$$ is non-negative.
A bit of trigonometric trickery (we used complex exponentials) shows that the expression above is equal to $$\begin{aligned}
\label{COSEQN}
-4\cos\left(\frac{\alpha+\beta+\gamma}{2}\right)\cos\left(\frac{\alpha-\beta+\gamma}{2}\right)\cos\left(\frac{\alpha+\beta-\gamma}{2}\right)\cos\left(\frac{-\alpha+\beta+\gamma}{2}\right)\end{aligned}$$
Let $\delta = \alpha+\beta+\gamma$. When $\delta < \pi$, (\[COSEQN\]) is strictly negative; when $\delta = \pi$, (\[COSEQN\]) is clearly zero; and when $\delta > \pi$, (\[COSEQN\]) is strictly positive. Hence the inner product on the space spanned by ${\bf v_1},{\bf
v_2},{\bf v_3 }$ is positive semidefinite if and only if $\delta \geq \pi$. It is positive definite if and only if $\delta > \pi$.
Then it is easy to see that the three planes $P_{\bf v_1},P_{\bf v_2},P_{\bf v_3}
\subset \mathbb{H}^3$ intersect at a point in $\overline{\mathbb{H}^3}$ if and only if they intersect pairwise in $\mathbb{H}^3$ and the sum of the dihedral angles $\delta \geq \pi$. It is also clear that they intersect at a finite point if and only if the inequality is strict. [$\Box$ ]{}
\[INTERSECT2\] Let $P_1,P_2,P_3 \subset \mathbb{H}^3$ be planes carrying faces of a polyhedron $P$ that has all dihedral angles $\leq \pi/2$. (a) If $P_1,P_2,P_3$ intersect at a point in $\mathbb{H}^3$, then the point $p
= P_1 \cap P_2 \cap P_3$ is a vertex of $P$.
\(b) If $P_1,P_2,P_3$ intersect at a point in $\partial \mathbb{H}^3$, then $P$ is not compact, and the point of intersection is in the closure of $P$.
[**Proof:**]{} (a) Consider what we see in the plane $P_1$. Let $H_i$ be the half-space bounded by $P_i$ which contains the interior of $P$, and let $Q = P_1 \cap H_2 \cap H_3$. If $p \notin P$, then let $U$ be the component of $Q-P$ that contains $p$ in its closure. This is a non-convex polygon; let $p,p_1,...,p_k$ be its vertices. The exterior angles of $U$ at $p_1,...,p_k$ are the angles of the face of $P$ carried by $P_1$, hence $\leq \pi/2$ by part (b) of Proposition \[TRIVALENT\]. See the following figure:
(0,0)
\#1\#2\#3\#4\#5[ @font ]{}
(1945,2165)(983,-1633) (984,-1183)[(0,0)\[lb\]]{} (1800,-1606)[(0,0)\[lb\]]{} (1776,-680)[(0,0)\[lb\]]{} (1120,-1268)[(0,0)\[lb\]]{} (1695,248)[(0,0)\[lb\]]{} (2373,-1331)[(0,0)\[lb\]]{} (2698,-1114)[(0,0)\[lb\]]{}
Suppose that $\alpha_1,...\alpha_k$ are the angles of $P$ at $p_1,...,p_k$, and let $\alpha$ be the angle at $p$. Then the Gauss-Bonnet formula tells us that: $$(\pi-\alpha)+\alpha_1-((\pi-\alpha_2)+\cdots+(\pi-\alpha_{k-1})) +\alpha_k
-{{\rm Area}}(U) = 2\pi,$$ which can be rearranged to read $$(\alpha_1+\alpha_k-\pi) -\alpha -\sum_{j=2}^{k-1} (\pi-\alpha_j) = {{\rm Area}}(U).$$ This is clearly a contradiction. All of the terms on the left are non-positive, and ${{\rm Area}}(U) > 0$. If $p$ is at infinity (i.e., $\alpha = 0$), this expression is still a contradiction, proving part (b). [$\Box$ ]{}
[**Proof of Proposition \[INEQSAT\]:**]{} For condition (1), notice that if two adjacent faces intersect at dihedral angle $0$, they intersect at a point at infinity. If this were the case, $P$ would be non-compact.
For condition (2), let $x$ be a vertex of $P$. Since $P$ is compact, $x \in
\mathbb{H}^3$ and by Lemma \[INTERSECT\], the sum of the dihedral angles between the three planes intersecting at $x$ must be $> \pi$.
For condition (3), note first that by Lemma \[INTERSECT\] if three faces forming a 3-circuit have dihedral angles summing to a number $\geq \pi$, then they meet in $\overline{\mathbb{H}^3}$. If they meet at a point in $\mathbb{H}^3$, by Lemma \[INTERSECT2\](a) this point is a vertex of $P$, so these three faces do not form a prismatic 3-circuit. Alternatively, if the three planes meet in $\partial \mathbb{H}^3$ by Lemma \[INTERSECT2\](b), then $P$ is non-compact, contrary to assumption. Hence, any three faces forming a prismatic 3-circuit in $P$ must have dihedral angles summing to $< \pi$.
For condition (4), let $H_{\bf v_1},H_{\bf v_2},H_{\bf v_3},H_{\bf v_4}$ be half spaces corresponding to the faces which form a prismatic 4-circuit; obviously condition (4) is satisfied unless all of the dihedral angles are $\pi/2$, so we suppose that they are. We will assume the normalization $\langle{\bf v_i},{\bf v_i}\rangle = 1$ for each ${\bf i}$. The Gram matrix $Q
= \left[\langle{\bf v_i},{\bf v_j}\rangle \right]_{i,j} =$
[$$\begin{aligned}
\left[
\begin{array}{cccc}
1 & 0 & \langle{\bf v_1},{\bf v_3}\rangle & 0 \\
0 & 1 & 0 & \langle{\bf v_2},{\bf v_4}\rangle \\
\langle{\bf v_3},{\bf v_1}\rangle & 0 & 1 & 0 \\
0 & \langle{\bf v_4},{\bf v_2}\rangle & 0 & 1 \\
\end{array}
\right]\end{aligned}$$ ]{}
has determinant $0$ if the ${\bf v}$’s are linearly dependent, and otherwise represents the inner product of $E^{3,1}$ and hence has negative determinant. In both cases we have $$\det Q = (1 - \langle{\bf v_1},{\bf v_3}\rangle^2)(1 - \langle{\bf v_2},{\bf
v_4}\rangle^2) \leq 0.$$ So $\langle{\bf v_1},{\bf v_3}\rangle^2 \leq 1$ and $\langle{\bf v_2},{\bf
v_4}\rangle^2 \geq 1$ or vice versa (perhaps one or both are equalities). This means that one of the opposite pairs of faces of the 4-circuit intersect, perhaps at a point at infinity. We can suppose that this pair is $H_{\bf v_1}$ and $H_{\bf
v_3}$.
If $H_{\bf v_1}$ and $H_{\bf v_3}$ intersect in $\mathbb{H}^3$, they do so with positive dihedral angle. Since $H_{\bf v_2}$ intersects each $H_{\bf v_1}$ and $H_{\bf v_3}$ orthogonally, the three faces pairwise intersect and have dihedral angle sum $> \pi$. By Lemmas \[INTERSECT\] and \[INTERSECT2\] these three faces intersect at a point in $\mathbb{H}^3$ which is a vertex of $P$. In this case, the 4-circuit $H_{\bf v_1},H_{\bf v_2},H_{\bf v_3},H_{\bf v_4}$ is not prismatic.
Otherwise, $H_{\bf v_1}$ and $H_{\bf v_3}$ intersect at a point at infinity. In this case, since $H_{\bf v_2}$ intersects each $H_{\bf v_1}$ and $H_{\bf v_3}$ with dihedral angle $\pi/2$ the three faces intersect at this point at infinity by Lemma \[INTERSECT\] and then by Lemma \[INTERSECT2\] $P$ is not compact, contrary to assumption.
Hence, if $H_{\bf v_1},H_{\bf v_2},H_{\bf v_3},H_{\bf v_4}$ forms a prismatic 4-circuit, the sum of the dihedral angles cannot be $2\pi$.
For condition (5), suppose that the quadrilateral is formed by edges $e_1,
e_2, e_3, e_4$. Violation of one of the inequalities would give that the dihedral angles at each of the edges $e_{ij}$ leading to the quadrilateral is $\pi/2$ and that the dihedral angles at two of the opposite edges of the quadrilateral are $\pi/2$. See the diagram below:
(0,0)
\#1\#2\#3\#4\#5[ @font ]{}
(1332,1312)(496,-986) (752,-612)[(0,0)\[lb\]]{} (843,-142)[(0,0)\[lb\]]{} (1399,-629)[(0,0)\[lb\]]{} (1191,-941)[(0,0)\[lb\]]{} (1191,205)[(0,0)\[lb\]]{} (496,-281)[(0,0)\[lb\]]{} (1608,-281)[(0,0)\[lb\]]{} (1364,-136)[(0,0)\[lb\]]{}
Each vertex of this quadrilateral had three incident edges labeled $e_i$, $e_j$, and $e_{ij}$. Violation of the inequality gives that $\alpha_{ij} = \pi/2$ and either $\alpha_i = \pi/2$ or $\alpha_j = \pi/2$. Using Equation (\[TLC\]) from section 1, we see that each face angle in the quadrilateral must be $\pi/2$. So, we have that each of the face angles of the quadrilateral is $\pi/2$, which is a contradiction to the Gauss-Bonnet Theorem. Hence both of the inequalities in condition (5) must be satisfied.
This was the last step in proving Proposition \[INEQSAT\]. [$\Box$ ]{}
The mapping $\alpha: {\cal P}_C^1 \rightarrow \mathbb{R}^E$ is injective. {#SEC_INJ}
=========================================================================
Recall from section \[SETUP\] an $m$-sided polygon $Q \subset \mathbb{H}^2$ with sides $s_i$ supported by lines $l_i$ for $i=1,\dots,m$ is called a [*parallelogram*]{} if, after adjoining any ideal endpoints in $\partial \mathbb{H}^2$ to these sides and lines, $s_i \cap s_j
= \emptyset$ implies $l_i \cap l_j = \emptyset$.
In section \[SETUP\] that we defined ${\cal P}^1_C$ to be the space of polyhedra realizing $C$ whose faces are parallelograms. We then checked that ${\cal P}^1_C$ is an open subset of ${\cal P}_C$ and that ${\cal P}^0_C \subset {\cal P}^1_C$. The goal of this section is to prove:
\[INJECTIVE\] The mapping $\alpha : {\cal P}^1_C \rightarrow \mathbb{R}^E$ is injective.
[**Proof:**]{} Suppose that $P, P' \in {\cal P}^1_C$ are two polyhedra such that $\alpha(P) = \alpha(P')$. We can label each edge $e$ of $C$ by $-,0,$ or $+$ depending on whether the length of $e$ in $P'$ is less than, equal to, or greater than the length of $e$ in $P$.
We will prove that if $\alpha(P') = \alpha(P')$ then each pair of corresponding edges has the same length. This gives that the faces of $P$ and $P'$ are congruent since the face angles are determined by the dihedral angles (Proposition \[TRIVALENT\]). Then, since $P$ and $P'$ have congruent faces and the same dihedral angles, they are themselves congruent.
Each edge of $C$ corresponds to a unique edge of the dual complex, $C^*$, which we label with $-,0$, or $+$, accordingly. Consider the graph $\cal{G}$ consisting of the edges of $C^*$ labeled either $+$ or $-$, but not $0$, together with the vertices incident to these edges. Since $C^*$ is a simplicial complex on $\mathbb{S}^2$, $\cal{G}$ is a simple planar graph. (Here, simple means that there is at most one edge between any distinct pair of vertices and no edges from a vertex to itself.) We assume that $\cal{G}$ is non-empty, in order to find a contradiction.
\[CAUCHY\] Let $\cal{G}$ be a simple planar graph whose edges are labeled with $+$ and $-$. There is a vertex of $\cal{G}$ with at most two sign changes when following the cyclic order of the edges meeting at that vertex.
Proposition \[CAUCHY\] provides the global statement necessary for Cauchy’s rigidity theorem on Euclidean polyhedra, see [@PFB Chapter 12], and also the global statement necessary here. The proof, see [@PFB page 68], is a clever, yet elementary counting argument, combined with Euler’s Formula.
Therefore, at some vertex of $\cal{G}$, there are either zero or two changes of sign when following the cyclic order of edges meeting at that vertex. In the case that there are zero changes in sign, we may assume, without loss in generality that all of the signs at this vertex are $+$’s, by switching the roles of $P$ and $P'$, if necessary. Thus, in either case, there is a face $F$ of $C$ not marked entirely with $0$’s so that, once the edges labeled $-$ are removed from $\partial F$, the edges labeled $+$ all lie in the same component of what remains.
Let $Q$ and $Q'$ be the faces in $P$ and $P'$ corresponding to $F$. By the assumption that $P, P' \in {\cal P}^1_C$, $Q$ and $Q'$ are parallelograms and because $P$ and $P'$ have the same dihedral angles, $Q$ and $Q'$ must have the same face angles. We will now show that $Q$ and $Q'$ cannot have side lengths differing according to the distribution of $+$’s and $-$’s on $\partial F$ that was deduced above.
The following lemma, from [@AND page 422] but with a new proof, shows that (roughly speaking) stretching edges in a piece of the boundary will pull apart the two edges at the ends of that piece. It is important to keep in mind that the parallelograms $R$ and $R'$ need not be compact and need not have finite volume, since there are no restrictions on whether the first and $m$-th sides intersect.
[**(Andreev’s Auxiliary Lemma)**]{} \[AUX\] Let $R$ and $R'$ be $m$-sided parallelograms further assume that $R$ and $R'$ have finite vertices $A_i = s_i \cap s_{i+1}$ and $A_i' = s_i' \cap s_{i+1}'$ for $i=1,\cdots m-1$. If
- The interior angle at vertex $A_i$ and vertex $A_i'$ are equal for $i=2,\cdots m-1$, and
- $|s_j| \leq |s_j'|$ for $j=2,\cdots,m-1$,
with at least one of the inequalities strict. Then $\langle {\bf v}_1, {\bf v}_m \rangle > \langle {\bf v}_1', {\bf v}_m' \rangle$, where ${\bf v}_i$ and ${\bf v}_m'$ are the outward pointing normal to the edge $s_i$ of $R$ and $R'$, respectively.
[**Proof:**]{} We will prove the lemma first in the case where the side lengths differ only at one side $|s_j| < |s_j'|$ and then observe that the resulting polygon again satisfies the hypotheses of the lemma so that one can repeat as necessary for each pair of sides that differ in length.
We can situate side $s_j$ on the line $x_2=0$ centered at $(1,0,0)$ within the upper sheet of the hyperboloid $-x_0^2 +x_1^2+x_2^2 = -1$ and assume that $R$ is entirely “above” this line, that is at points with $x_2 \geq 0$. We also assume that the sides of $R$ are labeled counterclockwise, i.e., $s_{i+1}$ is counterclockwise from $s_i$ for each $i$.
Applying the isometry: $$\begin{aligned}
I(t) = \left[\begin{array}{ccc} \cosh(t) & \sinh(t) & 0 \\ \sinh(t) & \cosh(t) & 0 \\ 0 & 0 & 1 \end{array} \right]\end{aligned}$$
to the sides $s_i$ with index $i > j$ for $t > 0$ performs the desired deformation of $R$.
One can check that $$\begin{aligned}
\frac{d}{dt} \langle {\bf v}_1, I(t) {\bf v}_m \rangle = \sinh(t) (v_{m1}v_{11}-v_{m0} v_{10})-\cosh(t)(v_{m1} v_{10}-v_{m0} v_{11}),\end{aligned}$$ if we write ${\bf v_1} = (v_{10},v_{11},v_{12})$ and ${\bf v_m} = (v_{m0},v_{m1},v_{m2})$. Since $(1,0,0)$ is in the interior of $s_j$ we must have that $\langle (1,0,0),
{\bf v}_1 \rangle < 0$, which is equivalent to $v_{10} > 0$. For the same reason we also have $v_{m0} > 0$.
We will first check that this derivative $\frac{d}{dt} \langle {\bf v}_1, I(t) {\bf v}_m \rangle$ is negative at $t=0$, or equivalently that: $$\begin{aligned}
0 < \det\left[\begin{array}{ccc}
v_{10} & 0 & v_{m0} \\
v_{11} & 0 & v_{m1} \\
v_{12} & -1 & v_{m2} \end{array} \right]
= v_{m1}v_{10} - v_{m0}v_{11}.\end{aligned}$$
Imagine the three column vectors, in order, in a right-handed coordinate system with the $x_0$ axis pointing up, the $x_1$ axis pointing forward and the $x_2$ axis pointing to the right. Because of the choice of orientation made above, the duals to the geodesics carrying $s_1,s_i,s_m$, when viewed from “above” and in that order, will also turn counter-clockwise. Since these duals all have non-negative $x_0$ coordinates, and two of these are positive, they form a right-handed frame, and the determinant is therefore positive.
We now check that $\frac{d}{dt} \langle {\bf v}_1, I(t) {\bf v}_m \rangle < 0$ for an arbitrary $t > 0$. Because $v_{m1} v_{10} - v_{m0} v_{11} > 0$ this is equivalent to: $$\begin{aligned}
\label{WANT}
\frac{v_{m1}v_{11}-v_{m0} v_{10}}{v_{m1} v_{10} - v_{m0} v_{11}} < \frac{\cosh(t)}{\sinh(t)}.\end{aligned}$$ Furthermore, since $\frac{\cosh(t)}{\sinh(t)} > 1$ it is sufficient to show that $\frac{v_{m1}v_{11}-v_{m0} v_{10}}{v_{m1} v_{10} - v_{m0} v_{11}} \leq 1$.
Since $R$ is a parallelogram oriented counter-clockwise, $l_1$ and $l_j$ cannot intersect in $\mathbb{H}^2$ to the right of $O$, since only $l_{j+1}$ can intersect $l_j$ there; nor can $l_1$ and $l_j$ be asymptotic at the right ideal endpoint of $l_j$ (represented by the vector $(1,1,0) \in E^{2,1}$). This means that we never intersect the boundary of the half-space in $E^{2,1}$ corresponding to ${\bf v}_1$ when we move in a straight line from $(1,0,0)$ to $(1,1,0)$, forcing the latter point to lie in the interior of that half-space. In other words, $0 > \langle (1,1,0),(v_{10},v_{11},v_{12})\rangle
= -v_{10} + v_{11}.$
The following diagrams (radially projecting $E^{2,1}$ to $\{x_0 = 1\}$) illustrate some of the ways $l_1$, $l_j$, and $l_m$ can be arranged, but are not intended to be a comprehensive list. Configuration 1 is allowed (but will only occur if $2 < j < m-1$). Configuration 2 is only allowed when $m=j+1$. Configuration 3 is ruled out by the intersection of $l_1$ with $l_j$ to the right of $O$, violating the parallelogram condition, and Configuration 4 is forbidden by the orientation condition.
(0,0)
\#1\#2\#3\#4\#5[ @font ]{}
(4988,4421)(211,-3699) (921,-1889)[(0,0)\[lb\]]{} (650,-1403)[(0,0)\[lb\]]{} (723,-3661)[(0,0)\[lb\]]{} (1066,471)[(0,0)\[lb\]]{} (814,-348)[(0,0)\[lb\]]{} (3491,-1373)[(0,0)\[lb\]]{} (4177,455)[(0,0)\[lb\]]{} (3699,-317)[(0,0)\[lb\]]{} (3395,-3668)[(0,0)\[lb\]]{} (3880,-2871)[(0,0)\[lb\]]{} (3901,-1833)[(0,0)\[lb\]]{} (3717,-2625)[(0,0)\[lb\]]{} (946,-2651)[(0,0)\[lb\]]{} (3631,483)[(0,0)\[lb\]]{} (1116,-2887)[(0,0)\[lb\]]{} (1049,-587)[(0,0)\[lb\]]{} (3916,-567)[(0,0)\[lb\]]{} (405,-2066)[(0,0)\[lb\]]{} (3413,-1905)[(0,0)\[lb\]]{} (596,362)[(0,0)\[lb\]]{}
An analogous argument shows $l_j$ and $l_m$ cannot intersect in $H^2$ to the left of $O$ and that they cannot be asymptotic at $(1,-1,0)$, so $(1,-1,0)$ is contained in the interior of the half-space dual to $(v_{m0},v_{m1},v_{m2})$, or in other words, $-v_{m0} - v_{m1} < 0$.
Combining these two observations yields $0 > (v_{m1} + v_{m0})(v_{11} - v_{10})
= v_{m1}v_{11} +v_{m0} v_{11} - v_{m1}v_{10} - v_{m0}v_{10} $ which is equivalent to $v_{m1}v_{11} - v_{m0}v_{10} < v_{m1}v_{10} - v_{m0}v_{11}$. In combination with the fact that the right hand side of this inequality is positive, this shows that Equation (\[WANT\]) holds.
For $i=1,\cdots m-1$ the adjacent sides $s_i$ and $s_{i+1}$ of $R$ continue to intersect at finite vertices with the same interior angles as before this deformation. Applying what we have just proved to an appropriate sub-polygon of $R$ we can see that $\langle {\bf v}_k, {\bf v}_l \rangle$ is non-increasing for pairs of sides of $s_k$ and $s_l$ that did not intersect before this deformation. Because these sides satisfied $\langle {\bf v}_k,
{\bf v}_l \rangle < 0$ before the deformation, they continue to do so, and the resulting polygon satisfies the hypotheses of Lemma \[AUX\]. Hence, one can increase the lengths of sides $s_j$ sequentially, in order to prove Lemma \[AUX\] in full generality.
[$\Box$ ]{}
We continue the proof of Proposition \[INJECTIVE\]. Suppose that $F$ has $n$ sides. We can renumber the sides of $F$ so that the second through $(m-1)$-st sides of $F$ are labeled with $+$’s and $0$’s and at least one of them is labeled with a $+$, so that the first and $m$-th sides are arbitrarily labeled, and, if $m < n$, so that the remaining sides are all labeled with $-$’s and $0$’s. There is usually more than one way to do this (often with differing values of $m$), any of which will suffice. See the diagrams below for examples. In the former, there are many alternate choices; in the latter, there is essentially one choice, up to combinatorial symmetry.
(0,0)
\#1\#2\#3\#4\#5[ @font ]{}
(5032,2069)(376,-4371) (4273,-4310)[(0,0)\[lb\]]{} (4230,-2508)[(0,0)\[lb\]]{} (5171,-3644)[(0,0)\[lb\]]{} (3671,-3494)[(0,0)\[lb\]]{} (3596,-2819)[(0,0)\[lb\]]{} (5021,-2819)[(0,0)\[lb\]]{} (4271,-2669)[(0,0)\[lb\]]{} (3821,-2894)[(0,0)\[lb\]]{} (3896,-3419)[(0,0)\[lb\]]{} (4346,-3719)[(0,0)\[lb\]]{} (4391,-3940)[(0,0)\[lb\]]{} (4807,-2967)[(0,0)\[lb\]]{} (4470,-3451)[(0,0)\[lb\]]{} (1501,-2386)[(0,0)\[lb\]]{} (751,-2536)[(0,0)\[lb\]]{} (376,-2836)[(0,0)\[lb\]]{} (376,-3361)[(0,0)\[lb\]]{} (676,-3886)[(0,0)\[lb\]]{} (601,-2911)[(0,0)\[lb\]]{} (826,-3661)[(0,0)\[lb\]]{} (901,-2686)[(0,0)\[lb\]]{} (1555,-2579)[(0,0)\[lb\]]{} (2331,-2753)[(0,0)\[lb\]]{} (1347,-4023)[(0,0)\[lb\]]{} (1268,-3791)[(0,0)\[lb\]]{} (1874,-2836)[(0,0)\[lb\]]{} (630,-3341)[(0,0)\[lb\]]{} (1881,-3632)[(0,0)\[lb\]]{} (2098,-3359)[(0,0)\[lb\]]{} (2340,-3438)[(0,0)\[lb\]]{} (2033,-3791)[(0,0)\[lb\]]{} (1142,-4340)[(0,0)\[lb\]]{}
Because $Q$ and $Q'$ are parallelograms, the (possibly non-compact) polygons bounded by the union of sides $s_1,\cdots,s_m$ from $Q$ and the union of the sides $s_1',\cdots,s_m'$ from $Q'$ satisfy the hypotheses of Lemma \[AUX\]. Hence, if we denote the outward pointing normals to $s_1$ and $s_m$ in $Q$ by ${\bf v}_1$ and ${\bf v}_m$ and in $Q'$ by ${\bf v}_1'$ and ${\bf v}_m '$, the lemma guarantees that $\langle {\bf v}_1', {\bf v}_m' \rangle > \langle {\bf
v}_1, {\bf v}_m \rangle$.
However, either $n=m$ so that the first and $m$-th sides of $F$ meet at a vertex giving $\langle {\bf v}_1', {\bf v}_m' \rangle = \langle {\bf v}_1, {\bf
v}_m \rangle$, or, if $n > m$ all of the sides in $F$ with index greater than $m$ are labeled $0$ or $-$. Applying Lemma \[AUX\] to the polygons bounded by the sides $s_{m+1},\cdots,s_n$ from $Q$ and $s_{m+1}',\cdots,s_n'$ from $Q'$ we find that $\langle {\bf v}_1', {\bf v}_m' \rangle \leq \langle {\bf v}_1, {\bf v}_m
\rangle$. In both cases we obtain a contradiction.
We have not used any restriction on the dihedral angles, only the restriction that $P_1$ and $P_2$ are have parallelogram faces, so we shown that $\alpha :
{\cal P}^1_C \rightarrow \mathbb{R}^E$ is injective. [$\Box$ ]{}Proposition \[INJECTIVE\].
Because ${\cal P}^0_C \subset {\cal P}^1_C$, it follows immediately that:
$\alpha : {\cal P}^0_C \rightarrow A_C$ is injective.
This gives the uniqueness part of Andreev’s Theorem.
The mapping $\alpha: {\cal P}_C^0 \rightarrow A_C$ is proper. {#SEC_PROP}
=============================================================
In this section, we prove that the mapping $\alpha: {\cal P}_C^0
\rightarrow A_C$ is a proper map. In fact, we will prove a more general statement (Proposition \[GENERAL\]) which will be useful later in the paper.
\[FACEANG\] Let $\cal{F}$ be a face of a hyperbolic polyhedron $P$ with non-obtuse dihedral angles. If a face angle of $\cal{F}$ equals $\pi/2$ at the vertex $v$, then the dihedral angle of the edge opposite the face angle (the edge that enters $v$ and is not in $F$) is $\pi/2$ and the dihedral angle of one of the two edges in $\cal{F}$ that enters $v$ is $\pi/2$.
[**Proof:**]{} This will follow from Equation (\[TLC\]) in Proposition \[TRIVALENT\], which one can use to calculate face angles from the dihedral angles at a vertex. In Equation (\[TLC\]), if $\beta_i = \pi/2$ we have:
$$0 = \frac{\cos(\alpha_i)
+\cos(\alpha_j)\cos(\alpha_k)} {\sin(\alpha_j)\sin(\alpha_k)},$$
where $\alpha_i$ is the dihedral angle opposite the face angle $\beta_i$ and $\alpha_j,\alpha_k$ are the dihedral angles of the other two edges entering $v$. Both $\cos(\alpha_i) \geq 0$ and $\cos(\alpha_j)\cos(\alpha_k) \geq 0$ for non-obtuse $\alpha_i,\alpha_j,$ and $\alpha_k$, so that $\cos(\alpha_i) = 0$ and $\cos(\alpha_j)\cos(\alpha_k)=0$. Hence $\alpha_i = \pi/2$ and either $\alpha_j = \pi/2$ or $\alpha_k = \pi/2$. [$\Box$ ]{}
\[NORMALIZE\] Given three points $v_1,v_2,v_3$ that form a non-obtuse, non-degenerate triangle in the Poincaré model of $\mathbb{H}^3$, there is a unique isometry taking $v_1$ to a positive point on the $x$-axis, $v_2$ to a positive point on the $y$-axis, and $v_3$ to a positive point on the $z$-axis.
[**Proof:**]{} The points $v_1,v_2,$ and $v_3$ form a triangle $T$ in a plane $P_T$. It is sufficient to show that there is a plane $Q_T$ in the Poincaré ball model that intersects the positive octant in a triangle isomorphic to $T$. The isomorphism taking $v_1,v_2,$ and $v_3$ to the $x,y$, and $z$-axes will then be the one that takes the plane $P_T$ to the plane $Q_T$ and the triangle $T$ to the intersection of $Q_T$ with the positive octant.
Let $s_1,s_2,$ and $s_3$ be the side lengths of $T$. The plane $Q_T$ must intersect the $x,y,$ and $z$-axes distances $a_1,a_2,$ and $a_3$ satisfying the hyperbolic Pythagorean theorem:
$$\cosh(s_1) = \cosh(a_2) \cosh(a_3)$$ $$\cosh(s_2) = \cosh(a_3) \cosh(a_1)$$ $$\cosh(s_3) = \cosh(a_1) \cosh(a_2)$$
These equations can be solved for $\left(\cosh^2(a_1),\cosh^2(a_2),\cosh^2(a_3)\right)$, obtaining $$\left( \frac{\cosh(s_2) \cosh(s_3)}{\cosh(s_1)},
\frac{\cosh(s_3) \cosh(s_1)}{\cosh(s_2)}, \frac{\cosh(s_1)
\cosh(s_2)}{\cosh(s_3)} \right),$$ The only concern in solving for $a_i$ is that each of these expressions is $\geq 1$. However, this follows from the triangle $T$ being non-obtuse, using the hyperbolic law of cosines. [$\Box$ ]{}
All of the results in this chapter are corollaries to the following proposition:
\[GENERAL\] Given a sequence of compact polyhedra $P_i$ realizing $C$ with $\alpha(P_i) = {\bf a}_i \in A_C$. If ${\bf a}_i$ converges to ${\bf
a} \in \overline{A_C}$, satisfying conditions (1,3-5), then there exists a polyhedron $P_0$ realizing $C$ with dihedral angles ${\bf a}$.
[**Proof:** ]{} Throughout this proof, we will denote the vertices of $P_i$ by $v_1^i,\cdots,v_n^i$. Let $v_a^i,v_b^i$, and $v_c^i$ be three vertices on the same face of $P_i$, which will form a triangle with non-obtuse angles for all $i$. According to Lemma \[NORMALIZE\], for each $i$ we can normalize $P_i$ in the Poincaré ball model so that $v_a^i$ is on the $x$-axis, $v_b^i$ is on the $y$-axis, and $v_c^i$ is on the $z$-axis.
For each $i$, the vertices $v^i_1,\cdots,v^i_n$ are in $\overline{
\mathbb{H}^3}$, which is a compact space under the Euclidean metric. Therefore, by taking a subsequence, if necessary, we can assume that each vertex of $P_i$ converges to some point in $\overline{\mathbb{H}^3}$. We denote the collection of all of these limit points of $v^i_1,\cdots,v^i_n$ by ${\cal A}_1,\cdots,{\cal A}_q
\in \overline{\mathbb{H}^3}$. Let $P_0$ be the convex hull of ${\cal
A}_1,\cdots,{\cal A}_q$. Since each $P_i$ realizes $C$, if we can show that each ${\cal A}_m$ is the limit of a single vertex of $P_i$, then $P_0$ will realize $C$ and have dihedral angles ${\bf a}$.
In summary, we must show that no more than one vertex converges to each ${\cal A}_m$, using that ${\bf a}$ satisfies conditions (1), (3), (4), and (5).
[*We first check that if ${\cal A}_m \in \partial
\mathbb{H}^3$, then there is a single vertex of $P_i$ converging to ${\cal A}_m$:* ]{}
Therefore, suppose that there are $k>1$ vertices of $P_i$ converging to ${\cal A}_m$ to find a contradiction to the fact that ${\bf a}$ satisfies conditions (1), and (3-5). Without loss of generality, we assume that $v_1^i,\cdots,v_k^i$ converge to ${\cal A}_m$ and $v_{k+1}^i,\cdots,v_n^i$ converge to other points ${\cal A}_j$ for $j
\neq m$.
Since ${\cal A}_m$ is at infinity, our normalization (restricting $v_a^i,v_b^i,$ and $v_c^i$ to the $x,y,$ and $z$-axes respectively) ensures that at least two of these vertices ($v_a^i,v_b^i,$ and $v_c^i$) do not converge to ${\cal A}_m$. This fact will be essential throughout this part of the proof.
Doing a Euclidean rotation of the entire Poincaré ball, we can assume that ${\cal A}_m$ is at the north pole of the sphere, without changing the fact that there are at least two vertices of $P_i$ that do not converge to ${\cal A}_m$.
We will do a sequence of normalizations of the position of $P_i$ in the Poincaré ball model to study the geometry of $P_i$ near ${\cal A}_m$ as $i$ increases.
For all sufficiently large $i$, there is some hyperbolic plane $Q$, which is both perpendicular to the $z$-axis and has ${\cal A}_m$ and $v_1^i,\cdots,v_k^i$ on one side of $Q$ and the remaining ${\cal A}_j$ ($j \neq m$) and all of the vertices $v_{k+1}^i,\cdots,v_n^i$ that do not converge to ${\cal A}_m$ on the other side of $Q$. This is possible because ${\cal A}_1,\cdots,{\cal A}_q$ are distinct points in $\overline{\mathbb{H}^3}$, and because ${\cal A}_m \in \partial
\mathbb{H}^3$.
For each $i$, let $R_i$ be the hyperbolic plane which intersects the $z$-axis perpendicularly, and at the point farthest from the origin such that the closed half-space toward ${\cal A}_m$ contains all vertices which will converge to ${\cal A}_m$ (see figure below). Let $D_i$ denote the distance from $R_i$ to $Q$ along the $z$-axis; as $i
\rightarrow \infty$ and the vertices $v^i_1,\cdots,v^i_k$ tend to ${\cal A}_m \in \partial \mathbb{H}^3$, $D_i \rightarrow \infty$. Let $S_i$ be the hyperbolic plane intersecting the $z$-axis perpendicularly, halfway between $R_i$ and $Q$.
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For each $i$, we normalize the polyhedra $P_i$ by translating the plane $S_i$ along the $z$-axis to the equatorial plane, $H = \{z=0\}$. We consider this a change of viewpoint, so the translated points and planes retain their former names. Hence, under this normalization, we have planes $R_i$ and $Q$ both perpendicular to the $z$-axis, and hyperbolic distance $D_i/2$ from the origin. Each vertex $v_1^i,\cdots,v_k^i$ is bounded above the plane $R_i$, and all of the vertices $v_{k+1}^i,\cdots,v_n^i$ of $P_i$ that do not converge to ${\cal A}_m$ are bounded below the plane $Q$.
As $i$ tends to infinity, $R_i$ and $Q$ intersect the $z$-axis at arbitrarily large hyperbolic distances from the origin. Denote by ${\cal N}$ and by ${\cal S}$ the half-spaces that these planes bound away from the origin. Given arbitrarily small (Euclidean) neighborhoods of the pole, for all sufficiently large $i$ the half-spaces will be contained in these neighborhoods. Hence, any edge running from ${\cal
N}$ to ${\cal S}$ will intersect $H$ almost orthogonally, and close to the origin, as illustrated in the figure below. Let $e^i_1,\cdots,e^i_l$ denote the collection of such edges.
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By assumption, there are $k \geq 2$ vertices in ${\cal N}$ and due to our normalization, there are two or more vertices in ${\cal S}$.
Thus the intersection $P_i \cap H$ will be almost a Euclidean polygon and its angles will be almost the dihedral angles $\alpha(e_1^i),...\alpha(e_l^i)$; in particular, for $i$ sufficiently large they will be at most only slightly larger than $\pi/2$. This implies that $l_i$, the number of such edges, is 3 or 4 for $i$ sufficiently large, as a Euclidean polygon with at least 5 faces has at least one angle $\geq 3\pi/5$.
If $l=3$, the edges $e_1^i,e_2^i,e_3^i$ intersecting $H$ are a prismatic circuit because there are two or more vertices in both ${\cal
N}$ and in ${\cal S}$. The sum $\alpha(e_1^i)+\alpha(e_2^i)+\alpha(e_3^i)$ tends to $\pi$ as $i$ tends to infinity, hence ${\bf a}$ cannot satisfy condition (3), contrary to assumption.
Similarly, if $l_i = 4$, and if $e_1^i,e_2^i,e_3^i,e_4^i$ to form a prismatic 4-circuit, then the corresponding sum of dihedral angles tends to $2\pi$ violating condition (4).
So we are left with the possibility that $l = 4$, and that $e_1^i,e_2^i,e_3^i,e_4^i$ do not form a prismatic 4-circuit. In this case, a pair of these edges meet at a vertex which may be in either ${\cal N}$, as shown in the diagram below, or in ${\cal S}$. Without loss of generality, we assume that $e_1^i$ and $e_2^i$ meet at this vertex, which we call $x^i$ and we assume that $x^i \in {\cal N}$. Because we assume that there are two or more vertices converging to ${\cal A}_m$, there most be some edge $e_j^i$ ($j \neq 1,2,3,4$) meeting $e_1^i$ and $e_2^i$ at $x^i$. We denote by $f^i$ the face of $P_i$ containing $e_1^i$, $e_2^i$ and $x^i$. An example of this situation is drawn in the diagram below, although the general situation can be more complicated.
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Since the sum of the dihedral angles along this 4-circuit limits to $2\pi$, and each dihedral angle is non-obtuse, each of the dihedral angles $\alpha(e_1^i), \cdots,\alpha(e_4^i)$ limits to $\pi/2$. One can use Equation (\[TLC\]) to check that the dihedral angle $\alpha(e_j^i)$ will converge to the face angle $\beta^i$ in the face $f^i$ at vertex $v^i$. This is because the right-hand side of this equation limits to $\cos(\alpha(e_j^i))$, while the right hand side of the equation equals $\cos(\beta_i)$. Then, as $i$ goes to infinity, the neighborhood ${\cal
N}$ (containing $x^i_1$) converges to the north pole while the neighborhood ${\cal S}$ (containing the other two vertices in $f^i$ which form the face angle $\beta_i$ at $x^i_1$) converges to the south pole. This forces the face angle $\beta^i$ to tend to zero, and hence the dihedral angle $\alpha(e_j^i)$ must tend to zero as well, contradicting the fact that all coordinates of the limit point ${\bf
a}$ are positive, by condition (1).
Therefore, we can conclude that any ${\cal A}_m \in \partial
\mathbb{H}^3$ is the limit of a single vertex of the $P_i$. Hence, the vertices of $P_i$ that converge to points at infinity (in the original normalization) converge to distinct points at infinity.
[*It remains to show that if ${\cal A}_m \in \mathbb{H}^3$, then there is a single vertex of $P_i$ converging to ${\cal A}_m$:* ]{}
First, we check that none of the faces of the $P_i$ can degenerate to either a point, a line segment, or a ray.
Because we have already proven that each of the vertices of $P_i$ that converge to points in $\partial \mathbb{H}^3$ converge to distinct points, any face ${\cal F}_i$ of $P_i$ that degenerates to a point, a line segment, or a ray, must have area that limits to zero. Hence, by the Gauss-Bonnet formula, the sum of the face angles would have to converge to $\pi(n-2)$, if the face has $n$ sides. The fact that the face angles are non-obtuse, allows us to see that $\pi(n-2)
\leq n\pi/2$ implying that $n \leq 4$. This restricts such a degenerating face ${\cal F}_i$ to either a triangle or a quadrilateral, the only Euclidean polygons having non-obtuse angles.
If ${\cal F}_i$ is a triangle, then the three edges leading to ${\cal F}_i$ form a prismatic 3-circuit, because our hypothesis $N>4$ implies that $C$ is not the simplex. If ${\cal F}_i$ degenerates to a point, the three faces adjacent to ${\cal F}_i$ would meet at a vertex, in the limit. Therefore, by Lemma 3.2, the sum of the dihedral angles at the edges leading to ${\cal F}_i$ would limit to a value $\geq \pi$, contrary to condition (3). Otherwise, if ${\cal F}_i$ is a triangle which degenerates to a line segment or ray in the limit, then two of its face angles will tend to $\pi/2$. Then, by Lemma \[FACEANG\], the dihedral angles at the edges opposite of these face angles become $\pi/2$. However, these edges are part of the prismatic 3-circuit of edges leading to ${\cal F}_i$, resulting in an angle sum $\geq \pi$, contrary to condition (3).
If, on the other hand, ${\cal F}_i$ is a quadrilateral, each of the face angles would have to limit to $\pi/2$. By Lemma \[FACEANG\], the dihedral angles at each of the edges leading from ${\cal F}_i$ to the rest of $P_i$ would limit to $\pi/2$, as well as at least one edge of ${\cal F}_i$ leading to each vertex of ${\cal F}_i$. Therefore, the dihedral angles at each of the edges leading from ${\cal F}_i$ to the rest of $P_i$ and at least one opposite pair of edges of ${\cal F}_i$ limit to $\pi/2$, in violation of condition (5).
Since none of the faces of the $P_i$ can degenerate to a point a line segment, or a ray, neither can the $P_i$. Suppose that the $P_i$ degenerate to a polygon, $\cal{G}$. Because the dihedral angles are non-obtuse, only two of the faces of the $P_i$ can limit to the polygon $\cal{G}$. Therefore the rest of the faces of the $P_i$ must limit to points, line segments, or rays, contrary to our reasoning above.
We are now ready to show that any ${\cal A}_m$ that is a point in $\mathbb{H}^3$ is the limit of a single vertex of the $P_i$. Let $v_1,\cdots,v_k$ be the distinct vertices that converge to the same point ${\cal A}_m$. Then, since the $P_i$ do not shrink to a point, a line segment, a ray, or a polygon, there are at least three vertices $v_p^i,
v_q^i$ and $v_r^i$ that don’t converge to ${\cal A}_m$ and that don’t converge to each other. Perform the appropriate isometry taking ${\cal
A}_m$ to the origin in the ball model. Place sphere $S$ centered at the origin, so small that $v_p^i, v_q^i$ and $v_r^i$ never enter $S$.
For all sufficiently large values of $i$, the intersection $P_i \cap S$ approximates a spherical polygon whose angles approximate the dihedral angles between the faces of $P_i$ that enter $S$. These spherical polygons cannot degenerate to a point or a line segment because the polyhedra $P_i$ do not degenerate to a line segment, a ray, or a polygon. By reasoning similar to that of Proposition 1.1, one can check that this polygon must have only three sides and angle sum $> \pi$. If $k>1$, the edges of this triangle form a prismatic 3-circuit in $C^*$, since for each $i$, $P_i$ has more than one vertex inside the sphere ($k >
1$) and at least the three vertices corresponding to the points represented by $v_p^i, v_q^i$ and $v_r^i$. So, the $P_i$ would have a prismatic 3-circuit whose angle sum limits to a value $> \pi$. However, this contradicts our assumption that ${\bf a}$ satisfies condition (3). Therefore, we must have $k=1$.
Therefore, we conclude that each ${\cal A}_m$ is the limit of a single vertex of $P_i$. [$\Box$ ]{}
\[PROPER\] The mapping $\alpha : {\cal P}_C^0 \rightarrow A_C$ is proper.
[**Proof:**]{} Suppose that $P_i$ is a sequence of polyhedra in ${\cal P}_C^0$ with $\alpha(P_i) = {\bf a_i} \in A_C$. If the sequence ${\bf a_i}$ converges to a point in ${\bf a} \in A_C$, we must show that there is a subsequence of the $P_i$ that converges in ${\cal P}_C^0$.
Since ${\bf a}$ satisfies conditions (1) and (3-5), by Proposition \[GENERAL\], there is a subsequence of the $P_i$ converging to a polyhedron $P_0$ that realizes $C$ with dihedral angles ${\bf a}$. The sum of the dihedral angles at each vertex of $P_0$ is $> \pi$ since ${\bf a}$ satisfies condition (2) as well. Therefore, by Lemma 3.2, each vertex of $P_0$ is at a finite point in $\mathbb{H}^3$, giving that $P_0$ is compact. Hence $P_0$ realizes $C$, is compact, and has non-obtuse dihedral angles. Therefore $P_0 \in {\cal P}_C^0$. [$\Box$ ]{}
\[INFVERTEX\] Suppose that Andreev’s Theorem holds for $C$ and suppose that there is a sequence ${\bf a}_i \in A_C$ converging to ${\bf a} \in \partial
A_C$. If ${\bf a}$ satisfies conditions (1,3-5) and if condition (2) is satisfied for vertices $v_1,\cdots,v_k$ of $C$, but not for $v_{k+1},\cdots,v_n$ for which the dihedral angle sum is exactly $\pi$, then there exists a non-compact polyhedron $P_0$ realizing $C$ with dihedral angles ${\bf a}$. $P_0$ has vertices $v_1,\cdots,v_k$ at distinct finite points and the $v_{k+1},\cdots,v_n$ at distinct points at infinity.
[**Proof:** ]{} Because we assume that Andreev’s Theorem holds for $C$, there exists a sequence of polyhedra $P_i$ with $\alpha(P_i) = {\bf a}_i$. The proof continues in the same way as that of Corollary \[PROPER\], except that in this case, it then follows from Lemma 3.2 that $v_1,\cdots,v_k$ lie in $\mathbb{H}^3$, while $v_{k+1},\cdots,v_n$ lie in $\partial
\mathbb{H}^3.$ [$\Box$ ]{}
Notice that if an abstract polyhedron has no prismatic 3-circuits, it has no triangular faces, so collapsing a single edge to a point results in a cell complex satisfying all of the conditions of an abstract polyhedron, except that one of the vertices is now 4-valent.
\[PREWHITEHEAD\] Let $C$ be an abstract polyhedron having no prismatic 3-circuits for which Andreev’s Theorem is satisfied. For any edge $e_0$ of $C$, let $C_0$ be the complex obtained by contracting $e_0$ to a point. Then, there exists a non-compact polyhedron $P_0$ realizing $C_0$ with the edge $e_0$ contracted to a single vertex at infinity and the rest of the vertices at finite points in $\mathbb{H}^3$.
[**Proof of Proposition \[PREWHITEHEAD\]:**]{} Let $v_1$ and $v_2$ be the vertices at the ends of $e_0$, let $e_1,e_2,e_3,e_4$ the edges emanating from the ends of $e_0$, and $f_1,f_2,f_3,f_4$ be the four faces containing either $v_1$ or $v_2$, as illustrated below.
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For $\epsilon \in (0,\pi/2]$ the angles: $\alpha(e_0) = \epsilon,$ $\alpha(e_1) = \alpha(e_2) =
\alpha(e_3) = \alpha(e_4) = \pi/2,$ and $\alpha(e) = 2\pi/5$, for all other edges $e$, are in $A_C$ since $C$ has no prismatic 3-circuits. Therefore, because we assume that Andreev’s Theorem holds for $C$, there is a polyhedron $P_{\epsilon} \in {\cal P}_C^0$ realizing these angles. Choose a sequence $\epsilon_i > 0$ converging to $0$.
As in the proof of Proposition \[GENERAL\], we choose three vertices of $P_i$ that are on the same face (but not on $f_2$ or $f_4$) and normalize the polyhedra so that each $P_i$ has these three vertices on the $x$-axis, the $y$-axis, and the $z$-axis, respectively. The vertices $v_1^i,\cdots,v_n^i$ of each $P_i$ are in the compact space $\overline{\mathbb{H}^3}$ so, as before, we can take a subsequence so that each vertex converges to some point in $\overline{\mathbb{H}^3}$.
The strategy of the proof is to consider the limit points ${\cal
A}_1,\cdots,{\cal A}_q$ of these vertices and to show that $v_1^i$ and $v_2^i$ converge to the same limit point, say ${\cal A}_1$ and that exactly one of each of the other vertices $v_j^i$ (for $j>2$) converges to each of the other ${\cal A}_m$, with $m>1$.
We first check that that $v_1^i$ and $v_2^i$ converge to the same point at infinity. Since the dihedral angle at edge $e_0$ converges to $0$, the two planes carrying faces $f_2$ and $f_4$ will intersect at dihedral angles converging to $0$. In the limit, these planes will intersect with dihedral angle $0$, therefore, at a single point in $\partial {\mathbb{H}^3}$. The edge $e_0$ is contained within the intersection of these two planes, hence the entire edge $e_0$ converges to a single point in $\partial \mathbb{H}^3$, and hence $v_1^i$ and $v_2^i$ converge to this point. We label this point by ${\cal A}_1$.
We must now show $v_1^i$ and $v_2^i$ are the only vertices that converge to ${\cal A}_1$. As in the proof of Proposition \[GENERAL\], one can do further normalizations so that all of the vertices that converge to ${\cal A}_1$ are in an arbitrarily small neighborhood ${\cal N}$ of the north pole and all of the other vertices are in an arbitrarily small neighborhood ${\cal S}$ of the south pole. Since ${\bf a}$ satisfies conditions (3) and (4) we deduce that there there are exactly four edges $e_1^i,e_2^i,e_3^i$, and $e_4^i$ that connect ${\cal N}$ to ${\cal
S}$ and that do not form a prismatic 4-circuit. Then, by Lemma \[4CIRC\], this 4-circuit separates exactly two vertices from the remaining vertices of $P_i$, hence only $v_1^i$ and $v_2^i$ converge to ${\cal A}_1$.
The proof that a single vertex of $P_i$ converges to each ${\cal A}_m$, for $m>1$ is almost the same as the proof of Proposition \[GENERAL\], because the dihedral angles ${\bf a}_i$ are non-zero for all edges other than $e_0$ and because conditions (3-5) are satisfied.
The only difference is that one must directly check that faces $f_2$ and $f_4$ do not collapse to line segments or rays, even though on each of these faces, two vertices converge to the same point at infinity. A vertex of $f_2$ or $f_4$ that is not $v_1$ or $v_2$ must have face angle that is strictly acute. Otherwise, Lemma \[FACEANG\] would give the dihedral angles at two of the edges meeting at this vertex is $\pi/2$, contrary to the fact that at least two of these dihedral angles are assigned to be $2\pi/5$. From here, the Gauss-Bonnet Theorem can be used to check that neither $f_2$ nor $f_4$ collapses to a line segment or a ray.
Therefore, we conclude that the vertices $v_j^i$ ($i>3$) converge to distinct points in $\overline{\mathbb{H}^3}$ away from the limit of $v_1^i$ and $v_2^i$, which converge to the same point in $\partial
\mathbb{H}^3$. Since condition (2) is satisfied at each vertex $v_j^i$ ($i>3$), Lemma 3.2 guarantees that each of these ${\cal A}_m$ is at a finite point. Therefore, $v_1^i$ and $v_2^i$ converge to the same point at infinity, and each of the other vertices converges to a distinct finite point. [$\Box$ ]{}
$A_C \neq \emptyset$ implies ${\cal P}_C^0 \neq \emptyset$ {#SEC_NONEMPT}
==========================================================
At this point, we know the following result:
\[NONEMPTYIMPLIESAND\] If ${\cal P}^0_C \neq \emptyset$, then $\alpha : {\cal P}^0_C \rightarrow
A_C$ is a homeomorphism.
[**Proof:**]{} We have shown in preceding sections that $\alpha: {\cal P}_C^1 \rightarrow
\mathbb{R}^E$ is a continuous and injective map whose domain and range are manifolds (without boundary) of the same dimension, so it follows from invariance of domain that $\alpha$ is a local homeomorphism. Local homeomorphisms restrict nicely to subspaces, giving that $\alpha: {\cal P}_C^0 \rightarrow \mathbb{R}^E$ is a local homeomorphism, as well. In fact, because $\alpha: {\cal P}_C^0
\rightarrow \mathbb{R}^E$ is also injective it is a homeomorphism onto its image, which we showed in Section \[SEC\_INEQSAT\] is a subset of $A_C$.
Since $A_C$ is convex and therefore connected, it suffices to show that $\alpha({\cal P}_C^0)$ is both open and closed in $A_C$, for it will then follow (since the image is non-empty by hypothesis) that the image is the entire set $A_C$. But local homeomorphisms are open maps, so $\alpha({\cal
P}_C^0)$ is open in $A_C$; and since Corollary \[PROPER\] shows that $\alpha: {\cal P}_C^0 \rightarrow A_C$ is proper, it immediately follows that the limit of any sequence in the image of $\alpha$ which converges in $A_C$ must lie in the image of $\alpha$, so $\alpha({\cal P}_C^0)$ is closed in $A_C$. [$\Box$ ]{}
Indeed, any local homeomorphism between metric spaces which is also proper will be a finite-sheeted covering map [@DOUADY p. 23] and [@LIMA p. 127]. This gives an alternative route to proving the above result.
But what is left is absolutely not obvious, and is the hardest part of the whole proof: proving that if $A_C \neq \emptyset$, then ${\cal P}^0_C \neq
\emptyset$. We have no tools with which to approach it and must use bare hands. We follow the proof of Andreev, although the proof of his key lemma contains a significant error. We provide our own correction.
First recall that in Corollary \[EX3APR\], we saw that if $C$ has no prismatic 3-circuits, $A_C \neq \emptyset$. We will call polyhedra that have no prismatic 3-circuits [*simple polyhedra*]{}. We will also say that the dual graph $C^*$ is simple if it satisfies the dual condition, that every $3$-cycle is the boundary of a single face. (This usage of “simple” follows Andreev, but differs from that used by others, including Vinberg [@VINREFL p. 47], for polyhedra in all dimensions greater than two.)
We first prove that ${\cal P}^0_C \neq \emptyset$ for simple polyhedra, and hence by Proposition \[NONEMPTYIMPLIESAND\] that Andreev’s Theorem holds for simple polyhedra. We then show that for any $C$ having prismatic 3-circuits, if $A_C \neq \emptyset$, then ${\cal P}^0_C \neq \emptyset$ by making a polyhedron realizing $C$ from (possibly many) simple polyhedra. By Proposition \[NONEMPTYIMPLIESAND\], this final step will complete the proof of Andreev’s theorem.
[*[Proof of Andreev’s Theorem for Simple Polyhedra]{}*]{}
\[APRISNONEMPTY\] If $C$ is simple and has $N > 5$ faces, ${\cal P}^0_C \neq \emptyset$. In words: every simple polyhedron is realizable.
[**Proof:**]{} The proof comprises three lemmas. We will first state the lemmas and prove this proposition using them. Then we will prove the lemmas.
\[EXPRISM\] Let $Pr_N$ and $D_N$ be the abstract polyhedra corresponding to the $N$-faced prism and the $N$-faced “split prism”, as illustrated below. If $N > 4$, ${\cal P}^0_{Pr_N}$ is nonempty and if $N > 7$, ${\cal P}^0_{D_N}$ is nonempty.
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A Whitehead move on an edge $e$ of an abstract polyhedron is given by the local change $Wh(e)$ described by the following diagram. The Whitehead move in the dual complex is dashed. (Sometimes we will find it convenient to describe the Whitehead move entirely in terms of the dual complex, in which case we write $Wh(f)$).
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\[WHITEHEAD\] Let the abstract polyhedron $C'$ be obtained from the simple abstract polyhedron $C$ by a Whitehead move $Wh(e)$. Then if ${\cal P}^0_C$ is non-empty, so is ${\cal P}^0_{C'}.$
[**(Whitehead Sequence)**]{} \[ROEDER\] Let $C$ be a simple abstract polyhedron on $\mathbb{S}^2$ which is not a prism. If $C$ has $N > 7$ faces, one can simplify $C$ by a finite sequence of Whitehead moves to $D_N$ such that [*all of the intermediate abstract polyhedra are simple.*]{}
[**Proof of Proposition \[APRISNONEMPTY\], assuming these three lemmas:**]{} Given simple $C$ with $N > 5$ faces; if $C$ is a prism, the statement is proven by Lemma \[EXPRISM\]. One can check that if $C$ has $7$ or fewer faces (and is not the tetrahedron) it is a prism. So, if $C$ is not a prism, we have $N > 7$. Then, according to Lemma \[ROEDER\], one finds a reduction by (say $m$) Whitehead moves to $D_N$, with each intermediate abstract polyhedron simple. Applying Lemma \[WHITEHEAD\] $m$ times, one sees that ${\cal P}^0_C$ is non-empty if and only if ${\cal P}^0_{D_N}$ is non-empty. However, ${\cal P}^0_{D_N}$ is non-empty by Lemma \[EXPRISM\]. [$\Box$ ]{}
Theorem 6 from Andreev’s original paper corresponds to our Proposition \[APRISNONEMPTY\]. The hard technical part of this is the proof of Lemma \[ROEDER\]. Andreev’s original proof of Theorem 6 in [@AND; @AND2] provides an algorithm giving the Whitehead moves needed for this lemma but the algorithm [*just doesn’t work*]{}. It was implemented as a computer program by the first author and failed on the first test case, $C$ being the dodecahedron. On one of the final steps, it produced an abstract polyhedron which had a prismatic 3-circuit. This error was then traced back to a false statement in Andreev’s proof of the lemma. We will explain the details of this error in the proof of Lemma \[ROEDER\].
[**Proof of Lemma \[EXPRISM\]:**]{} We construct the $N$-faced prism explicitly. First, construct a regular polygon with $N-2$ sides centered at the origin in the disc model for $\mathbb{H}^2$. ($N-2 \geq 3$, since $N \geq 5$.) We can do this with the angles arbitrarily small. Now view $\mathbb{H}^2$ as the equatorial plane the ball model of $\mathbb{H}^3$; and consider the hyperbolic planes which are perpendicular to the equatorial plane and contain one side of the polygon. In Euclidean geometry these are hemispheres with centers on the boundary of the equatorial disc. The dihedral angles between intersecting pairs of these planes are the angles of the polygon.
Now consider two hyperbolic planes close to the equatorial plane, one slightly above and one slightly beneath, both perpendicular to the $z$-axis. These will intersect the previous planes at angles slightly smaller than $\pi/2$. The region defined by these $N$ planes makes a hyperbolic polyhedron realizing the cell structure of the prism, so ${\cal P}_{Pr_N} \neq \emptyset$. In particular, using Proposition \[NONEMPTYIMPLIESAND\], Andreev’s Theorem holds for $C = Pr_N$, $N \geq 5$.
When $N>7$, the split prism $D_N$ can be constructed by gluing together a prism and its mirror image, each having $N-1$ faces. The dihedral angles are given in the figure below.
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These angles satisfy Andreev’s conditions (1–5), and Andreev’s Theorem holds for $Pr_{N-1}$ since $N-1>6>5$, so there exists such a hyperbolic prism. When this prism is glued to its mirror image, along the $(N-3)$-gon given by the outermost edges in the figure, the corresponding dihedral angles all double. So the edges on the outside which were labeled $\pi/2$ “disappear” into the interior of a “merged” face, and the edge which was labeled $\pi/4$ now corresponds to a dihedral angle of $\pi/2$. Hence, ${\cal P}^0_{D_N} \neq
\emptyset$, when $N>7$. Notice that when $N = 7$, the construction yields $Pr_7$ (which is combinatorially equivalent to $D_7$). [$\Box$ ]{}
[**Proof of Lemma \[WHITEHEAD\]:**]{} We are given $C$ and $C'$ simple with $C'$ obtained by a Whitehead move on the edge $e_0$ and we are given that $P_C \neq \emptyset$. By Proposition \[NONEMPTYIMPLIESAND\], since $P_C \neq \emptyset$, we conclude that Andreev’s Theorem holds for $C$. Let $C_0$ be the complex obtained from $C$ by shrinking the edge $e_0$ down to a point. By Proposition \[PREWHITEHEAD\], there exists a non-compact polyhedron $P_0$ realizing $C_0$ since Andreev’s Theorem holds for $C$.
We use the upper half-space model of $\mathbb{H}^3$, and normalize so that $e_0$ has collapsed to the origin of $\mathbb{C} \subset \partial
\mathbb{H}^3$. The faces incident to $e_0$ are carried by 4 planes $H_1,...,H_4$ each intersecting the adjacent ones at right angles, and all meeting at the origin. Their configuration will look like the center of the following figure. (Recall that planes in the upper half-space model of $\mathbb{H}^3$ are hemispheres which intersect $\partial \mathbb{H}^3$ in their boundary circles. The dihedral angle between a pair of planes is the angle between the corresponding pair of circles in $\partial
\mathbb{H}^3$.)
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The pattern of circles in the center of the figure can by modified forming the figures on the left and the right with each of the four circles intersecting the adjacent two circles orthogonally. If we leave the other faces of $P_0$ fixed we can make a small enough modification that the edges $e_1,e_2,e_3,e_4$ still have finite non-zero lengths. Since each of the dihedral angles corresponding to edges other than $e_0,e_1,e_2,e_3,$ and $e_4$ were chosen to be $2\pi/5$, this small modification will not increase any of these angles past $\pi/2$. One of these modified patterns of intersecting circles will correspond to a polyhedron in ${\cal P}^0_C$ and the other to a polyhedron in ${\cal P}^0_{C'}$. [$\Box$ ]{}
[**Proof of Lemma \[ROEDER\]:**]{} We assume that $C \neq Pr_N$ is a simple abstract polyhedron with $N>7$ faces. We will construct a sequence of Whitehead moves that change $C$ to $D_N$, so that no intermediate complex has a prismatic 3-circuit.
Find a vertex $v_\infty$ of $C^*$ which is connected to the greatest number of other vertices. We will call the link of $v_\infty$, a cycle of $k$ vertices and $k$ edges, the [*outer polygon*]{}. Notice that $k \leq N-2$, with equality precisely when $C = Pr_N$. Therefore, since $C \neq Pr_N$ by hypothesis, our first goal is to find Whitehead moves which increase $k$ to $N-3$ without introducing any prismatic $3$-circuits along the way. Once this is completed, an easy sequence of Whitehead moves changes the resulting complex to $D^*_N$.
Let us set up some notation. Draw the dual complex $C^*$ in the plane with the vertex $v_\infty$ at infinity, so that the outer polygon $P$ surrounding the remaining vertices and triangles. We call the vertices inside of $P$ [*interior vertices*]{}. All of the edges inside of $P$ which do not have an endpoint on $P$ are called [*interior edges*]{}.
Note that the graph of interior vertices and edges is connected, since $C^*$ is simple. An interior vertex which is connected to only one other interior vertex will be called an [*endpoint*]{}.
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Throughout this proof we will draw $P$ in black and we draw interior edges and vertices of $C^*$ in black, as well. The connections between $P$ and the interior vertices will be in grey. Connections between $P$ and $v_{\infty}$ will be black, if shown at all.
The link of an interior vertex $v$ will intersect $P$ in a number of components $F_v^1,\cdots,F_v^n$. (Possibly $n = 0$.) See the above figure. We will say that $v$ is [*connected to $P$ in these components.*]{} Notice that since $C^*$ is simple, an endpoint is always connected to $P$ in exactly one such component.
\[CHECK\] If a Whitehead move on the dual $C^*$ of an abstract polyhedron yields $C'^*$ (replacing $f$ by $f'$), and if $\delta$ is a simple closed path in $C^*$, which separates one endpoint of $f'$ from the other, then any newly-created 3-circuit will contain some vertex of $\delta$ which shares edges with both endpoints of $f'$.
[**Proof:**]{} A newly created 3-circuit $\gamma$ must contain the new edge $f'$ as well as two additional edges $e_1$ and $e_2$ connecting from a single vertex $V$ to the two endpoints of $f'$. By the Jordan Curve Theorem, since $\delta$ separates the endpoints of $f'$, the path $e_1e_2$ intersects $\delta$. The path $e_1e_2$ is entirely in $C^*$ since $f'$ is the only new edge in $C'^*$, so the vertex $V$ is in $\delta$. [$\Box$ ]{}
We now prove three additional sub-lemmas that specify certain Whitehead moves that, when performed on a simple abstract polyhedron $C$ (which is not a prism and has more than 7 faces), do not introduce any prismatic 3-circuits. Hence the resulting abstract polyhedron $C'^*$ is simple. More specifically, we will use Sub-lemma \[CHECK\] to see that each Whitehead move introduces exactly two newly-created 3-circuits in $C'^*$, the two triangles containing the new edge $f'$.
\[SUBLEMMA1\] Suppose that there is an interior vertex $A$ of $C^*$ which is connected to $P$ in exactly one component consisting of exactly two consecutive vertices $Q$ and $R$. The Whitehead move $Wh(QR)$ on $C^*$ increases the length of the outer polygon by one, and introduces no prismatic $3$-circuit.
[**Proof:**]{}
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Clearly this Whitehead move increases the length of $P$ by one. We apply Sub-lemma \[CHECK\] to see that this move introduces no prismatic 3-circuits. We let $\delta = P$, the outer polygon, which clearly separates the interior vertex $A$ from $v_\infty$ in $C^*$. Any new 3-circuit would consist of a point on $P$ connected to both $A$ and $v_\infty$. By hypothesis, there were only the two points $Q$ and $R$ on $P$ connected to $A$. These two points result in the new triangles $QAv_\infty$ and $RAv_\infty$ in ${C'}^*$. Therefore $Wh(QR)$ result in no prismatic 3-circuits. [$\Box$ ]{}
In the above sub-lemma, the condition that $A$ is connected to exactly two consecutive vertices of $P$ prevents $A$ from being an endpoint. For if $A$ is an endpoint, let $B$ denote the unique interior vertex connected to $A$. Then $BQR$ would be a $3$-circuit in $C^*$ separating $A$ from the other vertices and hence would contradict the hypothesis that $C$ is simple. Therefore any endpoint must be connected to $P$ in a single component having three or more vertices.
\[SUBLEMMA2\] Suppose that there is an interior vertex $A$ that is connected to $P$ in a component consisting of $M$ consecutive vertices $Q_1,\cdots,Q_M$ of $P$ (and possibly other components).
\(a) If $A$ is not an endpoint and $M > 2$, the sequence of Whitehead moves $Wh(AQ_M),\ldots,Wh(AQ_3)$ results in a complex in which $A$ is connected to the same component of $P$ in only $Q_1$ and $Q_2$. These moves leave $P$ unchanged, and introduce no prismatic 3-circuit.
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\(b) If $A$ is an endpoint and $M > 3$, the sequence of Whitehead moves $Wh(AQ_M),\ldots,Wh(AQ_4)$ results in a complex in which $A$ is connected to the same component of $P$ in only $Q_1,Q_2$, and $Q_3$. These moves leave $P$ unchanged and introduce no prismatic 3-circuits.
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[**Proof:**]{} Part (a) $A$ is not an endpoint. Clearly the move $Wh(AQ_M)$ decreases $M$ by one. We check that if $M>2$, this move introduces no prismatic 3-circuits. We let $\delta$ be the path $v_\infty Q_{M-2}AQ_M$ which separates $Q_{M-1}$ from $E$ in $C^*$. By Sub-lemma \[CHECK\], any new 3-circuit contains a vertex on $\delta$ connected to both $E$ and $Q_{M-1}$. Clearly $v_\infty$ is not connected to the interior vertex $E$. If $M > 3$, $Q_{M-2}$ is connected only to $Q_{M-1}$, $A$, $Q_{M-3}$, and $v_\infty$. Otherwise, when $M=3$, a connection of $Q_1$ to $E$ would mean that $C^*$ had a $3$-cycle $EQ_1A$ which would have to separate $D$ (in the figure above corresponding to part (a)) from $Q_2$, contrary to the hypothesis that $C^*$ is simple.
Hence, the only two vertices on $\delta$ that are connected to both $E$ and $Q_{M-1}$ are $A$ and $Q_M$, forming the two triangles $AQ_{M-1}E$ and $Q_MQ_{M-1}E$ in $C'^*$. Hence there are no new prismatic 3-circuits, so we can reduce $M$ by one, when $M > 2$.
Part (b) $A$ is an endpoint. We again use $\delta = v_\infty Q_{M-2}AQ_M$ to check that the move $Wh(AQ_M)$ introduces no prismatic $3$-circuits. The proof is identical to Part (a), except that $M > 3$ is needed to conclude that $Q_{M-2}$ is not connected to $E$ since $A$ is now an endpoint.
Therefore, as long as $M
> 3$ we can reduce $M$ by one without introducing prismatic 3-circuits. Recall that an endpoint of a simple complex cannot be connected to fewer than three points of $P$, so this is optimal. [$\Box$ ]{}
Note: In both parts (a) and (b), each of the Whitehead moves $Wh(AQ_M)$ transfers the connection between $A$ and $Q_M$ to a connection between the neighboring interior vertex $E$ and $Q_{M-1}$. In fact $Q_{M-1}$ gets added to the component containing $Q_M$ in which $E$ is connected to $P$. This is helpful later on, in Case 2 of Lemma \[ROEDER\].
\[SUBLEMMA3\] Suppose that there is an interior vertex $A$ whose link contains two distinct vertices $X$ and $Y$ of $P$. Then there are Whitehead moves which eliminate any component in which $A$ is connected to $P$, if that component does not contain $X$ or $Y$. $P$ is unchanged, and no prismatic $3$-circuits will be introduced.
[**Example:**]{}
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Here $A$ is connected to $P$ in four components containing six vertices. We can eliminate connections of $A$ to all of the components except for the single-point components $X$ and $Y$.
[**Proof:**]{} Let $O$ be a component not containing $X$ or $Y$. If $O$ contains more than two vertices, we can reduce it to two vertices by Sub-lemma \[SUBLEMMA2\](a).
Suppose that $O$ contains exactly two vertices, $V$ and $W$. We check that the move $Wh(AW)$ eliminates the connection from $A$ to $W$ without introducing prismatic $3$-circuits. Let $D$ be the unique interior vertex forming triangle $ADW$, as in the figure below. The move $Wh(AW)$ creates the new edge $DV$. Let $\delta$ be the loop $v_\infty Y A W$ which separates $D$ from $V$ in $C^*$. See the dashed curve in the figure below. By Sub-lemma \[CHECK\], any new 3-circuit contains a point on $\delta$ that is connected to both $D$ and $V$. Clearly $v_\infty$ is not connected to the interior vertex $D$. Since $Y$ and $V$ are in different components of connection between $A$ and $P$, $Y$ is not connected to $V$. Therefore, only $A$ and $W$ are connected to both $D$ and $V$, forming the triangles $ADV$ and $WVD$ in $C'^*$. Therefore, $Wh(AW)$ results in no prismatic 3-circuits.
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Thus, we can suppose that $O$ contains a single vertex $V$. We check that the move $Wh(AV)$, which eliminates this connection, does not introduce any prismatic $3$-circuits. Let $D$ and $E$ be the unique interior vertices forming the triangles $ADV$ and $AEV$. Let $\delta_1$ be the curve $v_\infty
YAV$ and $\delta_2$ be the curve $v_\infty XAV$ in $C^*$. See the two dashed curves in the figure below. Both of these curves separate $D$ and $E$ in $C^*$. Applying Sub-lemma \[CHECK\] twice, we conclude that any newly created 3-circuit contains a point that is [*both on $\delta_1$ and on $\delta_2$*]{} and that connects to both $D$ and $E$. The only points on both $\delta_1$ and $\delta_2$ are $v_\infty, A$, and $V$. Since $D$ and $E$ are interior, $v_\infty$ cannot connect to either of them. The connections from $A$ and from $V$ to $D$ and $E$ result in the triangles $ADE$ and $VDE$ in $C'^*$. Therefore, $Wh(AV)$ results in no prismatic 3-circuits.
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[$\Box$ ]{}
The proof that this move does not introduce any prismatic 3-circuit depends essentially on the fact that $A$ is connected to $P$ in at least two other vertices $X$ and $Y$. Andreev describes a nearly identical process to Sublemma \[SUBLEMMA3\] in his paper [@AND] on pages 333-334. However, he merely assumes that $A$ is connected to $P$ in at least one component in addition to the components being eliminated. He does not require that $A$ is connected to $P$ in at least [*two vertices*]{} outside of the components being eliminated. Andreev then asserts: “It is readily seen that all of the polyhedra obtained in this way are simple...” In fact, the Whitehead move demonstrated below clearly creates a prismatic 3-circuit. (Here, $M$ and $N$ lie in $P$.)
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Having assumed this stronger (and incorrect) version of Sub-lemma \[SUBLEMMA3\], the remainder of Andreev’s proof is relatively easy. Unfortunately, the situation pictured above is not uncommon (as we will see in Case 3 below!) Restricted to the weaker hypotheses of Sub-lemma \[SUBLEMMA3\] we will have to work a little bit harder.
We will now use these three sub-lemmas to show that if the length of $P$ is less than $N-3$ (so that there are at least 3 interior vertices), then we can do Whitehead moves to increase the length of $P$ by one, without introducing any prismatic $3$-circuits.
[**Case 1:**]{} An interior point which isn’t an endpoint connects to $P$ in a component with two or more vertices, and possibly in other components.
Apply Sub-lemma \[SUBLEMMA2\](a) decreasing this component to two vertices. We can then apply Sub-lemma \[SUBLEMMA3\], eliminating any other components since this component contains two vertices. Finally, apply Sub-lemma \[SUBLEMMA1\] to increase the length of the outer polygon by 1.
[**Case 2:**]{} An interior vertex that is an endpoint is connected to more than three vertices of $P$.
We assume that each of the interior points that are not endpoints are connected to $P$ in components consisting of single points, otherwise we are in Case 1. Let $A$ be the endpoint which is connected to more than three vertices of $P$. By Sub-lemma \[SUBLEMMA2\](b), there is a Whitehead move that transfers one of these connections to the interior vertex $E$ that is next to $A$. The point $E$ is not an endpoint since the interior graph is connected and the assumption $N-k > 3$ implies that there are at least three interior vertices. Now, one of the components in which $E$ is connected to $P$ has exactly two vertices, so we can then apply Case 1 for vertex $E$.
[**Case 3:**]{} Each interior vertex which is an endpoint is connected to exactly $3$ points of $P$ and every other interior vertex is connected to $P$ in components each consisting of a single vertex.
First, notice that if the interior vertices and edges form a line segment, this restriction on how interior points are connected to $P$ results in the prism, contrary to hypothesis of this lemma. However, there are many complexes satisfying the hypotheses of this case which have interior vertices and edges forming a graph more complicated than a line segment:
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For such complexes we need a very special sequence of Whitehead moves to increase the length of $P$.
Pick an interior vertex which is an endpoint and label it $I_1$. Denote by $P_1$, $P_2$, and $P_3$ the three vertices of $P$ to which $I_1$ connects. $I_1$ will be connected to a sequence of interior vertices $I_2, I_3, \cdots
I_m, m \ge 2$, with $I_m$ the first interior vertex in the sequence that is connected to more than two other interior vertices. Vertex $I_m$ must exist by the assumption that the interior vertices don’t form a line segment, the configuration that we ruled out above. By hypothesis, $I_2,\cdots,I_m$ can only connect to $P$ in components which each consist of a vertex, hence each must be connected to $P_1$ and to $P_3$. Similarly, there is an interior vertex (call it $X$) which connects both to $I_m$ and to $P_1$ and another vertex $Y$ which connects to $I_m$ and $P_3$. Vertex $I_m$ may connect to other vertices of $P$ and other interior vertices, as shown on the left side of the following diagram.
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(4185,1639)(16,-1844) (1861,-915)[(0,0)\[lb\]]{} (2851,-361)[(0,0)\[lb\]]{} (4201,-961)[(0,0)\[lb\]]{} (2701,-1786)[(0,0)\[lb\]]{} (1051,-361)[(0,0)\[lb\]]{} (1051,-1711)[(0,0)\[lb\]]{} (136,-961)[(0,0)\[lb\]]{} (1201,-833)[(0,0)\[lb\]]{} (3601,-916)[(0,0)\[lb\]]{} (2784,-909)[(0,0)\[lb\]]{} (3084,-908)[(0,0)\[lb\]]{} (2194,-914)[(0,0)\[lb\]]{}
Now we describe a sequence of Whitehead moves that can be used to connect $I_m$ to $P$ in only $P_1$ and $P_2$. This will allow us to use Sub-lemma \[SUBLEMMA1\] to increase the length of $P$ by one.
First, using Sub-lemma \[SUBLEMMA3\], one can eliminate all possible connections of $I_m$ to $P$ in places other than $P_1$ and $P_3$. Next, we do the move $Wh(I_mP_3)$ so that $I_m$ connects to $P$ only in $P_1$. We check that this Whitehead move does not create any prismatic 3-circuits. Let $\delta$ be the curve $v_\infty P_1 I_m P_3$ separating $I_{m-1}$ from $Y$. By Sub-lemma \[CHECK\], any newly created prismatic 3-circuit would contain a point on $\delta$ connected to both $I_{m-1}$ and $Y$. Since $Y$ and $I_{m-1}$ are interior, $v_\infty$ does not connect to them. Also, $P_1$ is not connected to $Y$ as this would correspond to a pre-existing prismatic 3-circuit $P_1I_{m}Y$, contrary to assumption. So, the only vertices of $\delta$ that are connected to both $I_{m-1}$ and $Y$ are $I_m$ and $P_3$, which result in the triangles $I_mI_{m-1}Y$ and $P_3I_{m-1}Y$, hence do not correspond to newly created prismatic 3-circuits. Therefore $Wh(I_mP_3)$ introduces no prismatic 3-circuits.
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(4185,1639)(16,-1844) (1605,-1179)[(0,0)\[lb\]]{} (2851,-361)[(0,0)\[lb\]]{} (4201,-961)[(0,0)\[lb\]]{} (2701,-1786)[(0,0)\[lb\]]{} (3601,-886)[(0,0)\[lb\]]{} (3076,-886)[(0,0)\[lb\]]{} (2776,-886)[(0,0)\[lb\]]{} (1201,-886)[(0,0)\[lb\]]{} (1051,-361)[(0,0)\[lb\]]{} (1051,-1711)[(0,0)\[lb\]]{} (2228,-901)[(0,0)\[lb\]]{} (151,-961)[(0,0)\[lb\]]{}
Next, one must do the moves $Wh(I_{m-1}P_1)$,...,$Wh(I_1P_1)$, in that order. (See the figure below). We check that each of these moves creates no prismatic 3-circuits. Fix $1 \le k \le m-1$, and let $\delta$ be the loop $v_\infty P_1
I_k P_3$. $Wh(I_k P_1)$ creates a new edge $I_{k-1}I_m$ if $k>1$, or $P_2 I_m$ if $k=1$, the vertices of which are separated by $\delta$. Since $I_m$ is interior, $v_\infty$ does not connect to $I_m$. Also, $I_m$ is no longer connected to $P_3$. Therefore the only points of $\delta$ that are both connected to $I_m$ and $I_{k-1}$ are $I_k$ and $P_1$. Those connections form the new triangles $P_1I_mI_{k-1}$ and $I_kI_{k-1}I_m$ (when $k=1$, replace $I_{k-1}$ with $P_2$). Hence no prismatic $3$-circuits were created, as claimed.
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(4215,3607)(-14,-3608) (2851,-2161)[(0,0)\[lb\]]{} (2723,-2919)[(0,0)\[lb\]]{} (1576,-2986)[(0,0)\[lb\]]{} (151,-2761)[(0,0)\[lb\]]{} (1051,-3511)[(0,0)\[lb\]]{} (1051,-2161)[(0,0)\[lb\]]{} (1201,-2611)[(0,0)\[lb\]]{} (4201,-2761)[(0,0)\[lb\]]{} (2191,-1816)[(0,0)\[lb\]]{} (2851,-61)[(0,0)\[lb\]]{} (2723,-819)[(0,0)\[lb\]]{} (1576,-886)[(0,0)\[lb\]]{} (151,-661)[(0,0)\[lb\]]{} (1051,-1411)[(0,0)\[lb\]]{} (1051,-61)[(0,0)\[lb\]]{} (1201,-511)[(0,0)\[lb\]]{} (4201,-661)[(0,0)\[lb\]]{} (2701,-1486)[(0,0)\[lb\]]{} (3076,-586)[(0,0)\[lb\]]{} (3076,-2686)[(0,0)\[lb\]]{} (2414,-819)[(0,0)\[lb\]]{} (2701,-3586)[(0,0)\[lb\]]{} (2414,-2919)[(0,0)\[lb\]]{} (3587,-798)[(0,0)\[lb\]]{} (3592,-2893)[(0,0)\[lb\]]{}
After this sequence of Whitehead moves, we obtain the diagram below, with $I_m$ connected to $P$ exactly at $P_1$ and $P_2$. We can then apply Sub-lemma \[SUBLEMMA1\] to increase the length of $P$ by the move $Wh(P_1P_2)$, as shown below.
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(4185,1639)(16,-1844) (3603,-1092)[(0,0)\[lb\]]{} (2851,-361)[(0,0)\[lb\]]{} (4201,-961)[(0,0)\[lb\]]{} (2701,-1786)[(0,0)\[lb\]]{} (1051,-361)[(0,0)\[lb\]]{} (1051,-1711)[(0,0)\[lb\]]{} (151,-961)[(0,0)\[lb\]]{} (3076,-1118)[(0,0)\[lb\]]{} (2723,-1119)[(0,0)\[lb\]]{} (2415,-1126)[(0,0)\[lb\]]{} (1201,-811)[(0,0)\[lb\]]{} (1576,-1186)[(0,0)\[lb\]]{}
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(4185,1639)(16,-1844) (3596,-1094)[(0,0)\[lb\]]{} (2851,-361)[(0,0)\[lb\]]{} (2701,-1786)[(0,0)\[lb\]]{} (1051,-361)[(0,0)\[lb\]]{} (1051,-1711)[(0,0)\[lb\]]{} (151,-961)[(0,0)\[lb\]]{} (1576,-1186)[(0,0)\[lb\]]{} (3076,-1118)[(0,0)\[lb\]]{} (2723,-1119)[(0,0)\[lb\]]{} (2415,-1126)[(0,0)\[lb\]]{} (1201,-811)[(0,0)\[lb\]]{} (4201,-961)[(0,0)\[lb\]]{}
This concludes Case 3.
Since $C^*$ must belong to one of these cases, we have seen that if the length of $P$ is less than $N-3$, we can do Whitehead moves to increase it to $N-3$ without creating prismatic $3$-circuits. Hence we can reduce to the case of two interior vertices, both of which must be endpoints. Then we can apply Sublemma \[SUBLEMMA2\](b) to decrease the number of connections between one of these two interior vertices and $P$ to exactly $3$. The result is the complex $D_N$, as shown to the right below.
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(4074,1599)(439,-1798) (1189,-641)[(0,0)\[lb\]]{} (3751,-1186)[(0,0)\[lb\]]{} (1240,-1148)[(0,0)\[lb\]]{} (3684,-661)[(0,0)\[lb\]]{}
[$\Box$ ]{}Lemma \[ROEDER\].
[*[Proof of Andreev’s Theorem for general polyhedra]{}*]{}
We have seen that Andreev’s Theorem holds for every simple abstract polyhedron $C$. Now we consider the case of $C$ having prismatic 3-circuits. So far, the only example we have seen has been the triangular prism. Recall that there are some such $C$ for which $A_C = \emptyset$, so we can only hope to prove that ${\cal P}^0_C \neq \emptyset$ when $A_C \neq \emptyset$.
\[PI3\] If $A_C \neq \emptyset$, then there are points in $A_C$ arbitrarily close to $(\pi/3,\pi/3,\cdots,\pi/3)$.
[**Proof:**]{} Let ${\bf a} \in A_C$ and let ${\bf a_t} = {\bf a}(1-t)
+ (\pi/3,\pi/3,\cdots,\pi/3)t$. To see that for each $t \in [0,1)$, ${\bf a_t}
\in A_C$, we check conditions (1-5): each coordinate is clearly $>0$ so condition (1) is satisfied. Given edges $e_i, e_j, e_k$ meeting at a vertex we have $({\bf a_i + a_j + a_k})(1-t) + \pi t > \pi(1-t) +
\pi t = \pi$ for $t < 1$, since $({\bf a_i + a_j + a_k}) > \pi$. So, condition (2) is satisfied. Similarly, given a prismatic 3-circuit intersecting edges $e_i, e_j, e_k$ we have $({\bf a_i + a_j + a_k})(1-t) + \pi t < \pi(1-t) + \pi
t = \pi$ for $t < 1$, so condition (3) is satisfied. Conditions (4) and (5) are satisfied since each component of ${\bf a_t}$ is $< \pi/2$ for $t > 0$ and since ${\bf a}$ satisfies these conditions for $t = 0$. [$\Box$ ]{}
The polyhedra corresponding to these points in $A_C$, if they exist, will have “spiky” vertices and think “necks”, wherever there is a prismatic 3-circuit.
We will distinguish two types of prismatic 3-circuits. If a prismatic 3-circuit in $C^*$ separates one point from the rest of $C^*$, we will call it a [*truncated triangle*]{}, otherwise we will call it an [*essential 3-circuit*]{}. The name truncated triangle comes from the fact that such a 3-circuit in $C^*$ corresponds geometrically to the truncation of a vertex, forming a triangular face. We will first prove the following sub-case:
\[TRUNCEXIST\] Let $C$ be an abstract polyhedron (with $N > 4$ faces) in which every prismatic 3-circuit in $C^*$ is a truncated triangle. If $A_C$ is non-empty, then ${\cal P}_C^0$ is non-empty.
We will need the following three elementary lemmas in the proof:
\[PERPPLANE\] Given three planes in $\mathbb{H}^3$ that intersect pairwise, but which do not intersect at a point in $\overline{\mathbb{H}^3}$, there is a fourth plane that intersects each of these planes at right angles.
[**Proof:**]{} Suppose that the three planes are given by $P_{\bf v_1},P_{\bf v_2},$ and $P_{\bf v_3}$. Since there is no common point of intersection in $\overline{\mathbb{H}^3}$, the line $P_{\bf v_1} \cap P_{\bf v_2} \cap P_{\bf v_3}$ in $E^{3,1}$ is outside of the light-cone, so the hyperplane $(P_{\bf v_1} \cap P_{\bf v_2} \cap P_{\bf v_3})^\perp$ intersects $\mathbb{H}^3$ and hence defines a plane orthogonal to $P_{\bf v_1},P_{\bf v_2},$ and $P_{\bf v_3}$. [$\Box$ ]{}
\[DECREASE\_ANGLE\] Given two halfspaces $H_1$ and $H_2$ intersecting with dihedral angle $a \in
(0,\pi/2]$ and a point $p$ in the interior of $H_1 \cap H_2$. Let $l_1$ be the ray from $p$ perpendicular to $\partial H_1$ and let $\widetilde{H_1}$ be the half-space obtained by translating $\partial H_1$ a distance $\delta$ further from $p$ perpendicularly along $l_1$. For $\delta > 0$ and sufficiently small $\widetilde{H_1}$ and $H_2$ intersect with dihedral angle $\alpha(\delta)$ where $\alpha$ is a decreasing continuous function of $\delta$.
[**Proof:**]{} In this and the following lemma, we use the fact that in a polyhedron $P$ with non-obtuse dihedral angles (here $P = H_1 \cap H_2$), the foot of the perpendicular from an arbitrary interior point of $P$ to a plane containing a face of $P$ will lie in that face (and indeed will be an interior point of that face). See, for example, [@VINREFL p. 48].
We assume that $\delta$ is sufficiently small so that $\partial
\widetilde{H_1}$ and $\partial H_2$ intersect. Let $l_2$ be the ray from $p$ perpendicular to $\partial H_2$ and let $Q$ be the plane containing $l_1$ and $l_2$. By construction, $Q$ intersects $\partial H_1$, $\partial H_2$ and $\partial \widetilde{H_1}$ each perpendicularly so that $Q \cap H_1$ and $Q
\cap H_2$ are half-planes in $Q$ intersecting with angle $a$ and $Q \cap
\widetilde{H_1}$ and $Q \cap H_2$ are half-planes in $Q$ intersecting with angle $\alpha(\delta)$. See the figure below:
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(3114,2129)(1170,-6787) (2020,-5829)[(0,0)\[lb\]]{} (2138,-6145)[(0,0)\[lb\]]{} (1247,-5295)[(0,0)\[lb\]]{} (1924,-5395)[(0,0)\[lb\]]{} (1360,-5072)[(0,0)\[lb\]]{} (2720,-5631)[(0,0)\[lb\]]{} (1312,-6022)[(0,0)\[lb\]]{} (1958,-4799)[(0,0)\[lb\]]{} (2289,-6582)[(0,0)\[lb\]]{} (2494,-5854)[(0,0)\[lb\]]{}
Let $R$ be the compact polygon in $Q$ bounded by $\partial \widetilde{H_1}$, $\partial H_2$, $l_1$ and $l_2$. (In the figure above, $R$ is the shaded region.) Because $R$ has non-obtuse interior angles it is a parallelogram. If we denote the outward pointing normal vectors to $\widetilde{H_1}$ and $H_2$, by ${\bf v}_1(\delta)$ and ${\bf v}_2$, then Lemma \[AUX\] gives that $\langle {\bf v}_1(\delta), {\bf v}_2 \rangle$ is a decreasing continuous function of $\delta$. Since $\alpha(\delta) = -\cos \left( \langle {\bf
v}_1(\delta), {\bf v}_2 \rangle \right)$, we see that $\alpha$ is a decreasing continuous function of $\delta$, as well. [$\Box$ ]{}
\[TRUNCATION\] Given a finite volume hyperbolic polyhedron $P$ with dihedral angles in $(0,\pi/2]$ and with trivalent ideal vertices. Suppose that the vertices $v_1,...,v_n$ are at distinct points at infinity and the remaining are at finite points in $\mathbb{H}^3$. Then there exists a polyhedron $P'$ which is combinatorially equivalent to the result of truncating $P$ at its ideal vertices, such that the new triangular faces of $P'$ are orthogonal to each adjacent face, and each of the remaining dihedral angles of $P'$ lies in $(0,\pi/2]$.
[**Proof:**]{} Let $p$ be an arbitrary point in the interior of $P = \bigcap_{i=0}^N H_{i}$ and let $l_i$ be the ray from $p$ to $\partial H_i$ that is perpendicular to $\partial H_i$.
By Lemma \[DECREASE\_ANGLE\], we can decrease the dihedral angles between the face carried by $\partial H_i$ and all adjacent faces an arbitrarily small non-zero amount by translating $\partial H_i$ a sufficiently small distance further from $p$. Appropriately repeating for each $i=1,\ldots,N$, we can shift each of the half-spaces an appropriate small distance further away from $p$ in order to decrease all of the dihedral angles by some non-zero amount bounded above by any given $\epsilon > 0$.
We choose $\epsilon$ sufficiently small that the sum dihedral angles at finite vertices of $P$ remains $> \pi$ and dihedral angles between each pair of faces remains $> 0$. At each infinite vertex $v_i$ of $P$ the sum of dihedral angles becomes $< \pi$ because each dihedral angle is decreased by a non-zero amount. Consequently, Lemma \[PERPPLANE\] gives a fourth plane perpendicular to each of the three planes previously meeting at $v_i$. This resulting polyhedron $P'$ has the same combinatorics as $P$, except that each of the infinite vertices of $P$ is replaced by a triangular face perpendicular to its three adjacent faces. By construction the dihedral angles of $P'$ are in $(0,\pi/2]$. [$\Box$ ]{}
[**Proof of Proposition \[TRUNCEXIST\]:**]{} By the hypothesis that $N>4$ from Andreev’s Theorem, $C$ is not the tetrahedron and since we have already seen that Andreev’s Theorem holds for the triangular prism, we assume that $C$ has more than 5 faces. In this case, one can replace all (or all but one) of the truncated triangles by single vertices to reduce $C$ to a simple abstract polyhedron (or to a triangular prism). The latter case is necessary when replacing all of the truncated triangles would lead to a tetrahedron, ($Pr_5$ is a truncated tetrahedron.) In either case, we call the resulting abstract polyhedron $C^0$.
The idea of the proof quite simple. Since Andreev’s Theorem holds for $C^0$, we construct a polyhedron $P^0$ realizing $C^0$ with appropriately chosen dihedral angles. We then decrease the dihedral angles of $P^0$, using Lemmas \[TRUNCATION\] and \[PERPPLANE\] to truncate vertices of $P^0$ as they “go past $\infty$,” eventually obtaining a compact polyhedron realizing $C$ with non-obtuse dihedral angles.
Using that $A_C \neq \emptyset$ and Lemma \[PI3\], choose a point $\beta \in
A_C$ so that each coordinate of $\beta$ is within $\delta$ of $\pi/3$, with $\delta < \pi/18$. It will be convenient to number the edges of $C$ and $C^0$ in the following way: If there is a prismatic 3-circuit in $C^0$ (i.e., $C^0=Pr_5$), we number these edges $1,2,$ and $3$ in $C$ and $C^0$.
Otherwise, we can use the facts that $C^0$ is trivalent and is not a tetrahedron, to find three edges whose endpoints are six distinct vertices. Next, we number the remaining edges common to $C$ and $C^0$ by $4,5,\cdots,k$. Finally, we number the edges of $C$ that were removed to form $C^0$ by $k+1,\cdots,n$ so that the edges surrounded by prismatic 3-circuits of $C$ with smaller angle sum (given by $\beta$) come before those surrounded by prismatic 3-circuits with larger angle sum.
To see that the point $\gamma =
(\beta_1,\beta_2,\beta_3,\beta_4+2\delta,\beta_5+2\delta,...,\beta_k+2\delta)$ is an element of $A_{C^0}$, we check conditions (1-5). Each of the dihedral angles specified by $\gamma$ is in $(0,\pi/2)$ because $0 <
\beta_i + 2\delta < \pi/3 + 3\delta < \pi/3 + \pi/6 = \pi/2$. Therefore, condition (1) is satisfied, as well as conditions (4) and (5) because the angles are acute. Two of the edges labeled $4$ and higher will enter any vertex of $C^0$ so the sum of the three dihedral angles at each vertex is at least $4\delta$ greater than the sum of the three dihedral angles given by $\beta$, which is $> \pi -3\delta$. Therefore condition (2) is satisfied. If there is a prismatic 3-circuit in $C^0$, it crosses the first three edges of $C^0$ and is also a prismatic 3-circuit in $C$. Since $\beta \in A_C$, $\beta_1+\beta_2+\beta_3 < \pi$, and it follows that condition (3) is satisfied by $\gamma$.
Now define $\alpha(t) = (1-t){\gamma} + t(\beta_1,...\beta_k)$. Let $T_0,...,T_{l-1} \in
(0,1)$ be the values of $t$ at which there is at least one vertex of $C^0$ for which $\alpha(t)$ gives an angle sum of $\pi$. We also define $T_{-1} = 0, T_l = 1$. We will label the vertices that have angle sum $\pi$ at $T_i$ by $v^i_1,\cdots,v_{n(i)}^i$. Let $C^{i+1}$ be $C^{i}$ with $v^i_1,\cdots,v_{n(i)}^i$ truncated for $i=0,\cdots,l-1$. Hence $C^l$ is combinatorially equivalent to $C$.
Since the number of edges increases as we move from $C^0$ toward $C$, we will redefine $\alpha(t)$, appending $\sum_{j=1}^i n(j)$ coordinates, all constant and equal to $\pi/2$, for values of $t$ between $T_{i-1}$ and $T_i$.
We know that Andreev’s Theorem holds for $C^0$ because $C^0$ is either simple, or the triangular prism. So, it is sufficient to show that if Andreev’s Theorem is satisfied for $C^i$ then it is satisfied for $C^{i+1}$, for each $i=0,\cdots,l-1$. To do this, we must generate a polyhedron realizing $C^{i}$ with the vertices $v^i_1,...,v_{n(i)}^i$ at infinity and the other vertices at finite points in $\mathbb{H}^3$. This will be easy with our definition of $\alpha(t)$ and Corollary \[INFVERTEX\]. We will then use Lemma \[TRUNCATION\] to truncate these vertices. Details follow.
To use Corollary \[INFVERTEX\], we must check that $\alpha(t) \in A_{C^i}$ when $t \in (T_{i-1},T_i)$. This follows directly from the definition of $\alpha(t)$. To check condition (1), notice that both $\beta_j$ and $\gamma_j$ are non-zero and non-obtuse, for each $j (1 \le j \le k)$, so $\alpha_j(t)$ must be as well. To check condition (2), note first that if a vertex of $C^i$ belongs also to both $C$ and $C^0$, then it will have dihedral angle sum greater than $\pi$ because both the corresponding sums in both $\beta$ and $\gamma$ have that property. If the vertex corresponds to a truncated triangle of $C$, but not of $C^i$, then by the definition of $T_i$, the angle sum is greater than $\pi$. If the vertex lies on one of the truncated triangles of $C^i$, then at least two of the incident edges will have angles equal to $\pi/2$. In each case, condition (2) is satisfied.
Any prismatic 3-circuit in $C^i$ is is either a prismatic 3-circuit in both $C^0$ and $C$ (the special case where $C^0$ is the triangular prism) or is a prismatic circuit of $C^i$ which wasn’t a prismatic circuit of $C^0$. In the first case, the dihedral angle sum is $<\pi$ because condition (3) is satisfied by both $\beta$ and $\gamma$ and in the second case, the angle sum is $< \pi$ by definition of the $T_i$.
For each $j=1,\cdots,k$ we have $\beta_j, \gamma_j \in (0,\pi/2)$, so $\alpha_j(t) \in (0,\pi/2)$. However, $\alpha_j(t) = \pi/2$ for $j > k$, corresponding to the edges of the added triangular faces. Fortunately, a prismatic 4-circuit cannot cross edges of these triangular faces, since it would have to cross two edges from the same triangle, which meet at a vertex, which is contrary to the definition of a prismatic circuit. Thus, a prismatic $4$-circuit can only cross edges numbered less than or equal to $k$, each of which is assigned an acute dihedral angle, giving that condition (4) is satisfied.
Lemma \[FIVE\] gives that condition (5) is a consequence of conditions (3) and (4) for $C^1,\cdots,C^l$, because they cannot be triangular prisms. If $C^0$ happens to be the triangular prism, condition (5) holds, since each of its edges $e_1,\cdots,e_k$ is assigned an acute dihedral angle.
Consider the sequence of dihedral angles $\alpha_{m,i} = \alpha(T_{i-1} +
(1-1/m)(T_i-T_{i-1}))$. By our above analysis, $\alpha_{m,i} \in A_{C^i}$ for each $m,i$. In fact, by definition $\alpha_{m,i}$ limits to the point $\alpha(T_{i}) \in \partial A_{C^i}$ (as $m \rightarrow \infty$), which satisfies conditions (1-5) to be in $A_{C^i}$, except that the sum of the dihedral angles at each vertex $v^i_1,\cdots,v_{n(i)}^i$ is exactly $\pi$. We assume that Andreev’s Theorem holds for $C^i$, so by Corollary \[INFVERTEX\], there exists a non-compact polyhedron $P^i$ realizing $C^i$ with each of the vertices $v^i_1,\cdots,v_{n(i)}^i$ at infinity and the rest of the vertices at finite points.
By Lemma \[TRUNCATION\], the existence of $P^{i}$ implies that there is a polyhedron realizing $C^{i+1}$ and therefore, by Proposition \[NONEMPTYIMPLIESAND\], that Andreev’s Theorem holds for the abstract polyhedron $C^{i+1}$. Repeating this process until $i+1 = l$, we see that Andreev’s Theorem holds for $C^l$, which is our original abstract polyhedron $C$. [$\Box$ ]{}
\[FINAL\] If $A_C \neq \emptyset$, then ${\cal P}_C^0 \neq
\emptyset$.
Combined with Proposition \[NONEMPTYIMPLIESAND\], this proposition concludes the proof of Andreev’s Theorem.
[**Proof:**]{} By Proposition \[APRISNONEMPTY\] and Proposition \[TRUNCEXIST\] we we are left with the case that there are $k>0$ essential 3-circuits. We will show that a polyhedron realizing $C$ with dihedral angles ${\bf a} \in A_C$ can be formed by gluing together $k+1$ appropriate sub-polyhedra, each of which has only truncated triangles.
We will work entirely within the dual complex $C^*$. Label the essential 3-circuits $\gamma_1,...,\gamma_k$. The idea will be to replace $C^*$ with $k+1$ separate abstract polyhedra $C_1^*,...,C_{k+1}^*$ each of which has no essential 3-circuits. The $\gamma_i$ separate the sphere into exactly $k+1$ components. Let $C_i^*$ be the $i$-th of these components. To make $C_i^*$ a simplicial complex on the sphere we must fill in the holes, each of which is bounded by 3 edges (some $\gamma_l$). We glue in the following figure (the dark outer edge is $\gamma_l$).
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(945,676)(866,-522) (1611,-113)[(0,0)\[lb\]]{} (1369,-255)[(0,0)\[lb\]]{}
so that where there was an essential prismatic 3-circuit in $C^*$ there are now two truncated triangles in distinct $C_i^*$. Notice that none of the $C_i$ is a triangular prism, since we have divided up $C$ along essential prismatic 3-circuits.
In $C_i$, we will call each vertex, edge, or face obtained by such filling in a [*new vertex, new edge,*]{} or [*new face*]{} respectively. We will call all of the other edges [*old edges*]{}. We label each such new vertex with the number $l$ corresponding to the 3-circuit $\gamma_l$ that was filled in. For each $l$, there will be exactly 2 new vertices labeled $l$ which are in two different $C_i^*$, $C_j^*$. See the following diagram for an example.
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(3621,3518)(574,-2892) (3923,-1854)[(0,0)\[lb\]]{} (1872,-12)[(0,0)\[lb\]]{} (1628,-1819)[(0,0)\[lb\]]{} (1733,-950)[(0,0)\[lb\]]{} (1594,-2861)[(0,0)\[lb\]]{} (3471,-2687)[(0,0)\[lb\]]{} (3367,-1124)[(0,0)\[lb\]]{} (1802,-1714)[(0,0)\[lb\]]{}
The choice of angles ${\bf a} \in A_C$ gives dihedral angles assigned to each old edge in each $C_i^*$. Assign to each of the new edges $\pi/2$. This gives a choice of angles ${\bf a_i}$ for for which it is easy to check that ${\bf a_i} \in A_{C_i}$ for each $i$:
Clearly condition (1) is satisfied since these angles are non-zero and none of them obtuse. The angles along each triangle of old edges in $C_i^*$ already satisfy condition (2) since ${\bf a} \in A_C$. For each of the new triangles added, two of the edges are assigned $\pi/2$ and the third was already assigned a non-zero angle, according to ${\bf a}$, so condition (2) is satisfied for these triangles, too. None of the new edges in $C_i^*$ can be in a prismatic 3-circuit or a prismatic 4-circuit since such a circuit would have to involve two such new edges, which form two sides of a triangle. Therefore, each prismatic 3- or 4-circuit in $C_i^*$ has come from such a circuit in $C^*$, so the choice of angles ${\bf a_i}$ will satisfy (3) and (4). Since none of the $C_i^*$ is a triangular prism, condition (5) is a consequence of condition (4), and hence is satisfied.
Therefore by Proposition \[TRUNCEXIST\] there exist polyhedra $P_i$ realizing the data $(C_i,{\bf a_i})$, for $1 \le i \le k+1$. For each pair of new vertices labeled $l$, the two faces dual to them are isomorphic, since by Proposition \[TRIVALENT\], the face angles are the same (giving congruent triangular faces). So one can glue all of the $P_i$ together according to the labeling by $l$. Since the edges of these triangles were assigned dihedral angles of $\pi/2$, the faces coming together from opposite sides of such a glued pair fit together smoothly. The result is a polyhedron $P$ realizing $C$ and angles ${\bf a}$. See the following diagram.
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[$\Box$ ]{}
That concludes the proof that $\alpha: {\cal P}_C^0 \rightarrow A_C$ is a homeomorphism for every abstract polyhedron $C$ having more than four faces and hence concludes the proof of Andreev’s Theorem.
Example of the combinatorial algorithm from Lemma \[ROEDER\]
============================================================
We include an example of the combinatorial algorithm described in Lemma \[ROEDER\], which gives a sequence of Whitehead moves to reduce the dual complex of a simple abstract polyhedron, $C^*$, to $D_N^*$. We then explain how the sequence of Whitehead moves described in Andreev’s paper [@AND] would result in prismatic 3-circuits for this $C$ and for many others.
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It is interesting to note that Andreev’s version of our Lemma \[ROEDER\] (his Theorem 6 in [@AND]) would fail for this abstract polyhedron $C^*$. The major difficulty is to achieve the first increase in the length of the outer polygon $P$. We carefully chose the vertex $v$ where the graph of interior vertices and edges branches and then did Whitehead moves to reduce the number of components where this vertex is connected to $P$. This is done in sub-figures (1)-(4). If we had started with any other interior vertex and tried to decrease the number of components where it is connected to $P$, prismatic 3-circuits would develop as shown below (the dashed curve) in sub-figures (1) and (2).
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Andreev states that one must do Whitehead moves until each interior vertex is connected to $P$ in a single component (creating what he calls the inner polygon), but he [*does not*]{} indicate that one must do this for certain interior vertices before others. Using a very specific order of Whitehead moves to reduce the number of components of connection from each interior vertex to $P$ to be either one or zero would work, but Andreev does not prove this. Instead of doing this, we find that it is simpler just to do Whitehead moves so that $v$ is connected to $P$ in a single component consisting of two vertices and an edge, instead of creating the whole “inner polygon” as Andreev would. Once this is done, doing the Whitehead move on this edge of $P$ increases the length of $P$ by one.
There are cases where the graph of interior vertices and edges has no branching points and Andreev’s proof could not work, even having chosen to do Whitehead moves to decrease the components of connection between each interior vertex and $P$ to one in some specific order. This is true for the following abstract polyhedron $C^*$, shown in sub-figure (1) below:
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Each interior vertex of $C^*$ is either an endpoint, or connected to $P$ in exactly two components, each of which is a single point. Doing a Whitehead move to eliminate any of these connections would result in a prismatic 3-circuit like the dashed one in sub-figure (2) above.
Instead, Case 2 of our Lemma \[ROEDER\] “borrows” a point of connection from one of the endpoints (sub-figures (1) and (2) below), making one of the interior vertices connected to $P$ in two components, one consisting of two vertices and an edge, and the other consisting of one vertex. One then can eliminate the single point of connection as in the Whitehead move (2) to (3). Then, it is simple to increase the length of $P$ by doing the Whitehead move on the single edge in the single component of connection between this interior vertex and $P$, as shown in the change from sub-figure (3) to (4).
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[**Acknowledgments**]{}\
This paper is a abridged version of the thesis written by the first author for the Université de Provence [@ROE]. He would like to thank first of all John Hamal Hubbard, his thesis director, for getting him interested and involved in the beautiful subject of hyperbolic geometry.
In 2002, J. H. Hubbard suggested that the first author write a computer program that implements Andreev’s Theorem using the ideas from Andreev’s proof. He was given a version of this proof written by J. H. Hubbard and William D. Dunbar as a starting point. From the ideas in that manuscript, he wrote a program which, while faithfully implementing Andreev’s algorithm, was unsuccessful in computing the polyhedra with given dihedral angles and combinatorial structure. From there, he found the error in Andreev’s proof, which led him to first to a way to correct the program so that the polyhedra for which it was tested were actually computed correctly, and later to the arguments needed close the gap in Andreev’s proof.
All of the work that the first author has done has developed out of ideas that he learned from the manuscript written by J. H. Hubbard and W. D. Dunbar. Therefore, he thanks both J. H. Hubbard and Bill Dunbar for their extensive work on that preliminary manuscript, and for their significant subsequent help and interest in his corrections of the proof by including them as co-authors.
The authors would also like to thank Adrien Douady, Michel Boileau, Jean-Pierre Otal, Hamish Short, and Jérôme Los for their helpful comments and their interest in our work.
[^1]: [email protected]
[^2]: Supported by a U.S. National Defense Science and Engineering Fellowship and later by a NSF Integrative Graduate Research and Training (IGERT) Fellowship.
|
---
abstract: 'We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman-Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.'
address:
- 'Applied Mathematics Program, Yale University, New Haven, CT 06511'
- 'Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925'
author:
- 'M. Rachh'
- 'T. Askham'
bibliography:
- 'dirichlet-biharm-2017.bib'
nocite: '[@*]'
title: Integral equation formulation of the biharmonic Dirichlet problem
---
integral equations ,biharmonic ,Dirichlet ,multiply connected
Introduction and problem formulation
====================================
A variety of problems of mathematics and physics require the computation of a biharmonic potential subject to Dirichlet boundary conditions. The pure bending problem for an isotropic and homogeneous thin clamped plate is a classical application. Another application is the computation of a $C^1$ extension of a given function from its domain of definition to a larger, enclosing domain (we discuss these applications further in \[sec:apps\]).
The Dirichlet problem is given as follows. For a domain $D$ with boundary $\Gamma$, find a function $w$ such that
$$\begin{aligned}
\Delta^2 w &= 0 \mbox{ in } D \label{eq:biharmD1} \; ,\\
w &= f \mbox{ on } \Gamma \label{eq:biharmD2} \; ,\\
\frac{\partial w}{\partial n} &= g \mbox{ on } \partial
D, \label{eq:biharmD3}\end{aligned}$$
where $f$ and $g$ are continuous functions defined on $\Gamma$.
The use of standard finite difference methods for the solution of \[eq:biharmD1,eq:biharmD2,eq:biharmD3\] is complicated greatly by the fact that the differential equation is fourth order. For instance, the resulting linear system for a discretization with $N$ nodes in each dimension would have a condition number proportional to $N^4$, which poses several concerns for obtaining high accuracy solutions for large problems.
Integral equation methods, on the other hand, have many advantages for such problems. Because \[eq:biharmD1,eq:biharmD2,eq:biharmD3\] is homogeneous, the resulting integral equation is defined on the boundary alone and there is a reduction in the dimension of the problem. Complex geometries are handled more easily by an integral equation and, with appropriate choice of representation, the discrete problem tends to be as well conditioned as the underlying physical problem, independent of the system size [@kress1999linear]. One challenge for integral equation methods is that the resulting linear systems are dense. However, there are many well developed fast algorithms for the solution of these systems, most descending from the fast multipole method (FMM) [@greengard1987].
Integral representations for the solution of \[eq:biharmD1,eq:biharmD2,eq:biharmD3\] have been developed previously. In particular, the problem is addressed in Peter Farkas’ thesis [@Farkas89] and the method presented there has been extended to three dimensions in [@Jiang11]. The integral kernels derived in [@Farkas89] are taken to be linear combinations of derivatives of the fundamental solution of the biharmonic problem. Assuming the boundary is a smooth curve, the combinations are chosen to maximize the smoothness of the integral kernel as a function on the boundary (for smooth domains). However, the integral kernels derived for \[eq:biharmD1,eq:biharmD2,eq:biharmD3\] have a leading order singularity of $r^{-2}$ on a domain with a corner. Because of this singularity, designing quadrature rules for discretizing the integral equation is difficult for domains with corners. Furthermore, the resulting discretized system has large condition numbers for domains whose boundaries have high curvature.
For the related problem of two dimensional steady Stokes flow, the stream function formulation results in a biharmonic equation with the gradient of the biharmonic potential specified on the boundary. Let $w$ be the stream function for Stokes flow with no slip boundary conditions, then $$\begin{aligned}
\Delta^2 w &= 0 \mbox{ in } D \; , \label{eq:biharmV1}\\
\frac{\partial w}{\partial \tau} &= f \mbox{ on }
\Gamma \label{eq:biharmV2} \; , \\
\frac{\partial w}{\partial n} &= g \mbox{ on }
\Gamma \, , \label{eq:biharmV3}\end{aligned}$$ for appropriately chosen functions $f$ and $g$. Over the past century, much work has been done to develop integral representations for the biharmonic problem in this setting, as well as the similar setting of the Airy stress function formulation of the plane theory of elasticity [@greengard1996integral; @power1993completed; @power1987second; @michlin1957integral]. The representations given in the above references typically have more benign singularities than the representation presented in [@Farkas89]. In particular, the representation used in this paper, taken from [@greengard1996integral; @power1993completed], has a leading order singularity of $r^{-1}$ on domains with corners. Moreover, this representation (and others from the above references) can be expressed in terms of Goursat functions, allowing for a convenient representation of the stream function. Because of these advantages, we choose to adapt the representation of [@greengard1996integral; @power1993completed] to solve \[eq:biharmD1,eq:biharmD2,eq:biharmD3\].
This adaptation is not immediate. First, in two dimensional Stokes flow, the physical quantities of interest are derivatives of the biharmonic potential $w$ and not $w$ itself; the representation of $w$ from [@greengard1996integral; @power1993completed] is not necessarily single-valued. Second, in converting the boundary conditions \[eq:biharmD2,eq:biharmD3\] into the boundary conditions \[eq:biharmV2,eq:biharmV3\], the data is differentiated along the curve so that the original boundary condition is only met up to a constant. These issues are addressed here, with particular attention paid to the case of multiply connected domains. More precisely, we will show that the desired (and uniquely defined) potential can be expressed in terms of (possibly) multi-valued Goursat functions.
The rest of the paper is organized as follows. In \[sec:prelim\], we present some mathematical preliminaries, including the notation used throughout the paper, a review of the Farkas integral representation, and a review of the completed layer potential representation for solving \[eq:biharmV1,eq:biharmV2,eq:biharmV3\] in terms of the Goursat functions. In \[sec:anapp\], we explain how to adapt the Stokes layer potentials for the Dirichlet problem, present an integral representation for solving \[eq:biharmD1,eq:biharmD2,eq:biharmD3\], and prove the invertibility of the resulting integral equation. We outline the numerical tools we used to solve this integral equation and present some numerical results in \[sec:results\]. In \[sec:conclusion\], we provide some concluding remarks and ideas for future research.
Preliminaries {#sec:prelim}
=============
The notation for the following concepts can be cumbersome and an attempt has been made to stay consistent. Vector-valued quantities are denoted by bold, lower-case letters (e.g. ${{\boldsymbol h}}$), while tensor-valued quantities are bold and upper-case (e.g. $\mathbf{T}$). Subscript indices of the non-bold character (e.g. $h_i$ or $T_{ijk}$) are used to denote the entries within a vector or tensor. We use the standard Einstein summation convention, i.e., there is an implied sum taken over the repeated indices of any term. The vectors ${{\boldsymbol x}}$ and ${{\boldsymbol y}}$ are reserved for spatial variables in ${{\mathbb R}}^2$, while $z$ and $\xi$ are reserved for spatial variables in ${{\mathbb C}}$. We make the standard identification between points in ${{\mathbb R}}^2$ and points in ${{\mathbb C}}$, i.e. the point ${{\boldsymbol x}}= (x_1,x_2)^\intercal \in {{\mathbb R}}^2$ is equivalent to the point $z = x_1+i x_2$, and we switch between the two notions implicitly in much of what follows. For integration, the symbol $dS$ is used to denote an integral with respect to arc length and the symbol $d\xi$ is used to denote a complex contour integral. Script letters $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ are reserved for Banach spaces. $I_{\mathcal{X}}: \mathcal{X}\to\mathcal{X}$ denotes the identity operator on $\mathcal{X}$.
Let $D$ will denote a bounded, possibly multiply-connected, domain in ${{\mathbb R}}^2$ with a smooth boundary $\Gamma$ (unless otherwise noted). For a domain with $N$ holes, we will denote the outer boundary by $\Gamma_0$ and the boundary of each hole by $\Gamma_1, \ldots, \Gamma_N$, so that $\Gamma = \cup_{i=0}^N \Gamma_i$. Let ${{\boldsymbol n}}({{\boldsymbol x}})$ denote the outward unit normal and $\boldsymbol{\tau}({{\boldsymbol x}})$ the positively-oriented unit tangent for ${{\boldsymbol x}}\in \Gamma$. If we need to distinguish between the exterior and interior of $\Gamma$, we will let $D^- = D$ denote the interior and $D^+ = {{\mathbb R}}^2 \setminus (D \cup \Gamma)$ denote the exterior.
Applications of the biharmonic Dirichlet problem {#sec:apps}
------------------------------------------------
Consider the pure bending of an isotropic and homogeneous thin clamped plate. In the Kirchoff-Love theory, the vertical displacement of the plate, $w$, satisfies the equations $$\begin{aligned}
-\Delta^2 w = q &\quad {{\boldsymbol x}}\in D \\
w = 0 &\quad {{\boldsymbol x}}\in \Gamma \\
\frac{\partial w}{\partial n} = 0 &\quad {{\boldsymbol x}}\in \Gamma, \end{aligned}$$ where $D \subset \mathbb{R}^2$ represents the midline of the thin plate, $\Gamma$ is its boundary, and $q$ is the transverse load applied to the plate. Using standard techniques, the above problem can be reduced to a homogeneous biharmonic problem of the form \[eq:biharmD1,eq:biharmD2,eq:biharmD3\].
In a recent paper, [@Askham2017], it was shown that the solution of polyharmonic Dirichlet problems can be used as part of the solution of *inhomogeneous* PDEs on complex geometries. We will breifly review this procedure here.
Consider the Poisson equation
$$\begin{aligned}
\Delta u &= f &{{\boldsymbol x}}\in D \; \label{eq:poiss1}, \\
u &= g &{{\boldsymbol x}}\in \Gamma \; .\end{aligned}$$
A particular solution, $v$, which satisfies \[eq:poiss1\] can be obtained from the formula
$$v({{\boldsymbol x}}) = -\dfrac1{2\pi} \int_\Omega \log |{{\boldsymbol x}}-{{\boldsymbol y}}|
\tilde{f}({{\boldsymbol y}}) \, dy \; , \label{eq:volint}$$
where $\Omega$ is some domain such that $D\subset \Omega$ and $\tilde{f}$ is a function defined on $\Omega$ which satisfies $\tilde{f}|_D = f$.
There are rapid methods for evaluating the integral \[eq:volint\] in the case that $\Omega$ is a box. However, it is unclear how best to define (and compute) the values of $\tilde{f}$ on $\Omega \setminus D$, particularly such that $\tilde{f}$ is smooth across the boundary of $D$. One approach is to compute the extension as the solution of a homogeneous PDE on the exterior.
Suppose that $w$ solves
$$\begin{aligned}
\Delta^2 w = 0 &\quad {{\boldsymbol x}}\in \Omega \setminus D
\nonumber\\
w = f &\quad {{\boldsymbol x}}\in \Gamma \nonumber \\
\frac{\partial w}{\partial n} =\frac{\partial f}{\partial n}
&\quad {{\boldsymbol x}}\in \Gamma \label{eq:biharm2}\\
w = 0 &\quad {{\boldsymbol x}}\in \partial \Omega \nonumber\\
\frac{\partial w}{\partial n} = 0 &\quad {{\boldsymbol x}}\in \partial
\Omega \nonumber \; ,\end{aligned}$$
which is a problem of the form \[eq:biharmD1,eq:biharmD2,eq:biharmD3\]. Then, setting $\tilde{f}|_D = f$ and $\tilde{f}|_{\Omega \setminus D} = w$ makes $\tilde{f}$ a $C^1$ function across $\Gamma$.
In [@Askham2017], a $C^0$ extension was computed as the solution of a Laplace problem on $\Omega \setminus D$. This was found to accelerate the convergence of the Poisson solver over discontinuous extension (i.e. $\tilde{f}$ to be zero outside of $D$). By computing a smoother extension, as in the solution of the problem above, the efficiency and robustness of the Poisson solver could be further improved. For a PDE-based version of this approach, see [@stein2016].
The Farkas integral representation
----------------------------------
As mentioned in the introduction, there are existing integral representations for the solution of \[eq:biharmD1,eq:biharmD2,eq:biharmD3\]. In [@Farkas89], the solution is given as the sum of two layer potentials, i.e.
$$w({{\boldsymbol x}}) = \int_\Gamma K_1^F ({{\boldsymbol x}},{{\boldsymbol y}}) \sigma_1({{\boldsymbol y}}) \, dS({{\boldsymbol y}})
+ \int_\Gamma K_2^F({{\boldsymbol x}},{{\boldsymbol y}}) \sigma_2({{\boldsymbol y}}) \, dS({{\boldsymbol y}}) \; ,
\label{eq:farkRep}$$
where $\sigma_1$ and $\sigma_2$ are unknown densities.
The integral kernels, $K_1^F$ and $K_2^F$ are based on derivatives of the Green’s function for the biharmonic equation. For two points on the plane, ${{\boldsymbol x}}$ and ${{\boldsymbol y}}$, the Green’s function is given by
$$G^B({{\boldsymbol x}},{{\boldsymbol y}}) = \dfrac1{8\pi} |{{\boldsymbol x}}-{{\boldsymbol y}}|^2 \log |{{\boldsymbol x}}-{{\boldsymbol y}}| \; .$$
Let ${{\boldsymbol r}}= {{\boldsymbol y}}- {{\boldsymbol x}}$ and $r = |{{\boldsymbol y}}-{{\boldsymbol x}}|$. Then,
$$\begin{aligned}
K_1^F({{\boldsymbol x}},{{\boldsymbol y}}) &= G^B_{n_yn_yn_y}({{\boldsymbol x}},{{\boldsymbol y}})
+ 3 G^B_{n_y\tau_y\tau_y}({{\boldsymbol x}},{{\boldsymbol y}}) \; , \label{eq:kf1} \\
K_2^F({{\boldsymbol x}},{{\boldsymbol y}}) &= -G^B_{n_yn_y}({{\boldsymbol x}},{{\boldsymbol y}})
+ G^B_{\tau_y\tau_y}({{\boldsymbol x}},{{\boldsymbol y}}) \label{eq:kf2} \; .\end{aligned}$$
More explicitly, we have
$$\begin{aligned}
K_1^F({{\boldsymbol x}},{{\boldsymbol y}}) &= \dfrac1{\pi}
\dfrac{({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol y}}))^3}{({{\boldsymbol r}}\cdot {{\boldsymbol r}})^2} \; ,\\
K_2^F({{\boldsymbol x}},{{\boldsymbol y}}) &= \dfrac1{2\pi} \left ( \dfrac12 -
\dfrac{({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol y}}))^2}{{{\boldsymbol r}}\cdot {{\boldsymbol r}}} \right ) \; .\end{aligned}$$
On enforcing the Dirichlet boundary conditions for $w$, we obtain the integral equation
$$\begin{split}
\begin{pmatrix}
f({{\boldsymbol x}}) \\
g({{\boldsymbol x}})
\end{pmatrix} &=
\int_\Gamma
\begin{pmatrix}
K_{11}^F({{\boldsymbol x}},{{\boldsymbol y}}) & K_{12}^F({{\boldsymbol x}},{{\boldsymbol y}}) \\
K_{21}^F({{\boldsymbol x}},{{\boldsymbol y}}) & K_{22}^F({{\boldsymbol x}},{{\boldsymbol y}})
\end{pmatrix}
\begin{pmatrix}
\sigma_1({{\boldsymbol y}})\\
\sigma_2({{\boldsymbol y}})
\end{pmatrix}
\, dS({{\boldsymbol y}}) \\
&\quad + \begin{pmatrix}
1/2 & 0 \\
-\kappa({{\boldsymbol x}}) & 1/2
\end{pmatrix}
\begin{pmatrix}
\sigma_1({{\boldsymbol x}})\\
\sigma_2({{\boldsymbol x}})
\end{pmatrix} \; , \label{eq:farkIntEq}
\end{split}$$
where $\kappa$ denotes the signed curvature as a function on $\Gamma$ and ${{\boldsymbol x}}$ is a point on $\Gamma$. The kernels are given by $K_{11}^F = K_1^F$, $K_{12}^F = K_2^F$, $$\begin{aligned}
K_{21}^F({{\boldsymbol x}},{{\boldsymbol y}}) &= \left(K_1^F({{\boldsymbol x}},{{\boldsymbol y}})\right )_{n_x} \nonumber \\
&= \dfrac1{\pi} \left ( -3 \dfrac{ ({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol y}}))^2
({{\boldsymbol n}}({{\boldsymbol x}}) \cdot {{\boldsymbol n}}({{\boldsymbol y}}))}{({{\boldsymbol r}}\cdot {{\boldsymbol r}})^2}
+4 \dfrac{ ({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol y}}))^3 ({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol x}}))}
{ ({{\boldsymbol r}}\cdot {{\boldsymbol r}})^3} \right ) \; \label{eq:farkK21},\\
K_{22}^F({{\boldsymbol x}},{{\boldsymbol y}}) &= \left(K_2^F({{\boldsymbol x}},{{\boldsymbol y}})\right )_{n_x} \nonumber \\
&= \dfrac1{\pi} \left ( \dfrac{ ({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol y}})) ({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol x}}))}
{{{\boldsymbol r}}\cdot{{\boldsymbol r}}} - \dfrac{({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol y}}))^2 ({{\boldsymbol r}}\cdot {{\boldsymbol n}}({{\boldsymbol x}}))}
{({{\boldsymbol r}}\cdot{{\boldsymbol r}})^2} \right ) \; .\end{aligned}$$ For a sufficiently smooth and simply connected domain, the integral equation \[eq:farkIntEq\] is invertible. The case of a multiply connected domain is not treated fully in [@Farkas89], but some of the issues are considered.
As mentioned above, the kernels $K^F_1$ and $K^F_2$ are constructed with the goal that the $K^F_{ij}$ are as smooth as possible. Suppose that the boundary $\Gamma$ is of class $C^k$. Then, the kernels, $K^F_{ij}({{\boldsymbol x}},{{\boldsymbol y}})$, are $C^{k-2}$ functions on the boundary for each ${{\boldsymbol y}}\in \Gamma$ [@Farkas89]. Therefore, on a smooth boundary, these kernels are smooth. However, on a domain with a corner, it is clear from the formula \[eq:farkK21\] that the kernel $K_{21}^F$ has a singularity with strength $r^{-2}$. This singularity, in addition to the term in \[eq:farkIntEq\] which explicitly involves the curvature, makes the representation \[eq:farkRep\] unstable for domains with high curvature (or corners).
Stokes flow in the plane {#subsec:stokesflow}
------------------------
The equations of incompressible Stokes flow with no-slip boundary conditions on a domain $D$ with boundary $\Gamma$ are $$\begin{aligned}
-\Delta{{\boldsymbol u}}+\nabla p & =0\quad\mbox{in D},
\label{eq:StokesFlowEq} \\
\nabla\cdot{{\boldsymbol u}}& =0\quad\mbox{in D},
\label{eq:MassConservation}\\
{{\boldsymbol u}}&= {{\boldsymbol h}}\quad\text{on }\Gamma,
\label{eq:bcStokesFlow}\end{aligned}$$ where ${{\boldsymbol u}}$ is the velocity of the fluid and $p$ is the pressure. Following standard practice, the velocity ${{\boldsymbol u}}$ can be represented by a stream function $w$. Let $${{\boldsymbol u}}= \nabla^{\perp} w = \left(
\begin{array}{r} \frac{\partial w}{\partial x_2} \\
-\frac{\partial w}{\partial x_1} \end{array} \right) \label{eq:StreamFunc},$$ so that the divergence-free condition, \[eq:MassConservation\], is satisfied automatically. Taking the dot product of $\nabla^\perp$ with \[eq:StokesFlowEq\] results in a biharmonic equation for $w$. In particular, $w$ is a biharmonic function which satisfies \[eq:biharmV1,eq:biharmV2,eq:biharmV3\] with $f = -h_i n_i$ and $g = h_i \tau_i$.
Goursat functions
-----------------
Goursat showed that any biharmonic function $w$ can be represented by two analytic functions $\phi$ and $\psi$ (called Goursat functions) as $$w \left(x_1,x_2\right) = \text{Re} \left(\bar{z} \phi \left(z \right)
+ \chi \left(z\right) \right) , \label{eq:wGoursat}$$ where $\chi^{'} = \psi$ and $z=x_1+i x_2$ [@goursat1898equation]. In solving equation \[eq:biharmV1,eq:biharmV2,eq:biharmV3\], we are interested in $\frac{\partial w}{\partial x_1}$ and $\frac{\partial w}{\partial x_2}$. Muskhelishvili’s formula [@muskhelishvili1977some] gives an expression for these quantites in terms of the Goursat functions as $$\frac{\partial w}{\partial x_1}+ i \frac{\partial w}{\partial x_2} =
\phi\left(z\right) + z\overline{\phi^{'} \left(z\right)} +
\overline{\psi \left(z\right)} \, . \label{eq:MushkiFormula}$$ We say that a pair of Goursat functions $\phi$ and $\psi$ is *equivalent* to a Stokes velocity field ${{\boldsymbol u}}$ if the biharmonic function $w$ defined by \[eq:wGoursat\] is such that ${{\boldsymbol u}}=
\nabla^\perp w$.
The references [@greengard1996integral; @power1993completed; @power1987second; @michlin1957integral] give many options for the representation of $\phi$ and $\psi$ as layer potentials of a complex density given on the boundary of the domain. Of computational interest, representations for $\phi$ and $\psi$ exist such that enforcing the boundary conditions of \[eq:biharmV1,eq:biharmV2,eq:biharmV3\] results in an invertible second-kind integral equation (SKIE) for the density.
Integral representations for Stokes flow in the plane
-----------------------------------------------------
We will first present the single layer and double layer potentials of Stokes flow in the Stokeslet/stresslet formulation, which may be more familiar. For details concerning these ideas, see, inter alia, [@pozrikidis1992boundary]. We will then present their equivalent potentials in the classical Goursat function formulation. The reason for doing so is two-fold: first, the Goursat formulation makes it more natural to evaluate the stream function $w$; second, the complex variables-based Goursat formulation is readily adaptable for efficient fast multipole methods.
### Stokes layer potentials {#sec:stokeslayer}
The Green’s function $\mathbf{G}$ for the incompressible Stokes equations in free space, or *Stokeslet*, is given by $$G_{ij}\left({{\boldsymbol x}},{{\boldsymbol y}}\right)=\frac{1}{4\pi}\left[
-\log\left|{{\boldsymbol x}}-{{\boldsymbol y}}\right| \delta_{ij} +
\frac{\left( x_{i} - y_{i} \right) \left( x_{j} - y_{j}
\right)} {\left| {{\boldsymbol x}}- {{\boldsymbol y}}\right|^2}\right] \quad
i,j \in \{1,2\} \, .$$ The vector field $u_i = G_{ij}\left({{\boldsymbol x}},{{\boldsymbol y}}\right) f_j$ represents a Stokes velocity field at ${{\boldsymbol x}}$ due to a point force $\mathbf{f}$ applied at ${{\boldsymbol y}}$. For a continuous distribution of surface forces $\boldsymbol\mu$ on a curve $\Gamma$, the induced Stokes field, called a single layer potential, is given by $$\label{stokesslpdef}
\left [ {S}_{\Gamma} \mu \right ]_i
\left({{\boldsymbol x}}\right)=
\int_{\Gamma} G_{ij} \left({{\boldsymbol x}},{{\boldsymbol y}}\right)
\mu_{j}\left({{\boldsymbol y}}\right) \, dS({{\boldsymbol y}}) \quad i=1,2 \, .$$
The following lemma describes the behavior of the Stokes single layer potential as a function on ${{\mathbb R}}^2$, see [@pozrikidis1992boundary] for details.
\[lem:slppropspart1\] Let $\mathbf{S}_{\Gamma}\mathbf{{{\boldsymbol \mu}}}({{\boldsymbol x}})$ denote a single layer Stokes potential of the form \[stokesslpdef\]. Then, $\mathbf{S}_{\Gamma}{{\boldsymbol \mu}}\left({{\boldsymbol x}}\right)$ satisfies the Stokes equations in ${{\mathbb R}}^{2}\backslash\Gamma$ and $\mathbf{S}_{\Gamma}{{\boldsymbol \mu}}({{\boldsymbol x}})$ is continuous in ${{\mathbb R}}^{2}$.
The Stokes single layer potential has equivalent Goursat functions, $\phi_{S}$ and $\psi_{S}$, which can be expressed in terms of complex layer potentials: $$\begin{aligned}
\phi_{S}(z) & =-\frac{1}{8\pi}\int_{\Gamma}\rho\left(\xi\right)
\log\left(\xi-z\right)\, dS(\xi) +
\frac{1}{8\pi} \int_{\Gamma} \rho\left(\xi\right)
\, dS(\xi) \, , \label{eq:SLGoursatPhi}\\
\chi_{S}(z) & =\frac{1}{8\pi}\int_{\Gamma}
\overline{\rho\left(\xi\right)}\left(\xi-z\right)
\left[\log\left(\xi-z\right)-1\right] \, dS(\xi) \nonumber \\
&\quad +\frac{1}{8\pi}\int_{\Gamma}\overline{\xi}\rho(\xi)\log
\left(\xi-z\right) \, dS(\xi) \, ,\label{eq:SLGoursatChi}\\
\psi_{S}(z) & =-\frac{1}{8\pi}\int_{\Gamma}
\overline{\rho\left(\xi\right)}\log\left(\xi-z\right) \, dS(\xi)-
\frac{1}{8\pi}\int_{\Gamma}\frac{\overline{\xi}\rho\left(\xi\right)}
{\xi-z}\, dS(\xi) \, , \label{eq:SLGoursatPsi}\end{aligned}$$ where $z=x_1 + ix_2$, $\xi = y_1 + iy_2$, and $\rho = \mu_2 - i \mu_1$. The stream function $w_S$ corresponding to this Goursat function pair is then $$\begin{split}
w_{S}(z) &=\mbox{Re}\left[\frac{1}{4\pi}\int_{\Gamma}\mbox{Re}
\left[\overline{\left(\xi-z\right)}\rho\left(\xi\right)\right]
\log\left(\xi-z\right) \, dS(\xi) \right. \\
&\quad \left. -\frac{1}{8\pi}\int_{\Gamma}\overline{\rho
\left(\xi\right)}\left(\xi-z\right)\, dS(\xi) + \overline{z}\frac{1}{8\pi}
\int_{\Gamma}
\rho\left(\xi\right) \, dS(\xi) \right] \\
& =: S^w_\Gamma \rho \, . \label{eq:SLStreamFunc}
\end{split}$$ Note that, the velocity field associated with the stream function $w_{S}$ is given by $$\nabla^{\perp} w_{S} \left(z\right)=
\nabla^{\perp} S^w_\Gamma \rho \left(z\right)
= \mathbf{S}_{\Gamma} \boldsymbol{\mu} \left({{\boldsymbol x}}\right) \, .
\label{eq:SLStreamFuncEquiv}$$
Another quantity of physical interest in Stokes flow is the stress tensor $\boldsymbol\sigma$; for a Stokes velocity field ${{\boldsymbol u}}$ and pressure $p$, it is given by $$\sigma_{ij} =-p\delta_{ij}+\left(\frac{\partial u_{i}}{\partial x_{j}}+
\frac{\partial u_{j}}{\partial x_{i}}\right) \, .$$ The stress tensor $\mathbf{T}$, or *stresslet*, associated with the Green’s function $\mathbf{G}$ is given by $$T_{ijk} \left({{\boldsymbol x}},{{\boldsymbol y}}\right) = -\frac{1}{\pi}
\frac{\left( x_{i} - y_{i} \right) \left( x_{j} - y_{j} \right)
\left( x_{k} - y_{k} \right)}{\left| {{\boldsymbol x}}- {{\boldsymbol y}}\right|^{4}}
\, .$$ The vector field $u_i = T_{ijk}({{\boldsymbol y}},{{\boldsymbol x}}) n_k f_j$ represents the velocity field resulting from a stresslet with strength $\bf$ oriented in the direction ${{\boldsymbol n}}$ at ${{\boldsymbol y}}$. For a distribution of stresslets $\boldsymbol\mu$ on a curve $\Gamma$, the induced Stokes field, called a double layer potential, is given by $$\label{stokesdlpdef}
\left [ {D}_{\Gamma} {\mu} \right ]_i
\left({{\boldsymbol x}}\right) =
\int_{\Gamma}
T_{ijk}\left({{\boldsymbol y}},{{\boldsymbol x}}\right) n_k
\left({{\boldsymbol y}}\right)
\mathbf{\mu}_j \left({{\boldsymbol y}}\right) \, dS({{\boldsymbol y}})
\, .$$
The following lemma describes the behavior of the Stokes double layer potential as a function on ${{\mathbb R}}^2$, see [@pozrikidis1992boundary] for details.
\[lem:dlppropspart1\] Let $\Gamma$ be a curve, $D^+$ denote the exterior of the curve, $D^-$ denote the interior, $\mathbf{D}_{\Gamma}{{\boldsymbol \mu}}\left({{\boldsymbol x}}\right)$ denote a double layer Stokes potential of the form \[stokesdlpdef\], and ${{\boldsymbol x}}_0 \in \Gamma$. Then, $\mathbf{D}_{\Gamma}{{{\boldsymbol \mu}}}\left({{\boldsymbol x}}\right)$ satisfies the Stokes equations in ${{\mathbb R}}^{2}\backslash\Gamma$ and the jump relations: $$\begin{aligned}
\lim_{\substack{{{\boldsymbol x}}\to{{\boldsymbol x}}_{0}\\
{{\boldsymbol x}}\in D^{\pm}}}
\left [ {D}_{\Gamma} {\mu}\right ]_{i} ({{\boldsymbol x}}) &=
\pm\frac{1}{2}\mathbf{\mu}_{i}
\left({{\boldsymbol x}}_{0}\right)+\oint_{\Gamma}{T}_{j,i,k} \left({{\boldsymbol y}},
{{\boldsymbol x}}_{0}\right) n_{k}\left({{\boldsymbol y}}\right) {\mu_j}
\left({{\boldsymbol y}}\right) \, dS({{\boldsymbol y}}) \\
&=: \pm\frac{1}{2}\mathbf{\mu}_{i}({{\boldsymbol x}}_0)
+ \left [ {D}_{\Gamma}^{PV}
{\mu} \right ]_{i} ({{\boldsymbol x}}_0) \, .\end{aligned}$$ In the above, $\oint$ denotes a Cauchy principal value integral and $\mathbf{D}_\Gamma^{PV} {{\boldsymbol \mu}}$ denotes the double layer potential with the integral taken in the principal value sense.
The double layer potential above has equivalent Goursat functions, $\phi_D$ and $\psi_D$, which can be expressed in terms of complex layer potentials: $$\begin{aligned}
\phi_D(z) & =-\frac{1}{4\pi i}\int_{\Gamma}\frac{\rho
\left(\xi\right)}{\xi-z}d\xi \label{eq:DLGoursatPhi} \, ,\\
\chi_D(z) & =\frac{1}{4\pi i}\int_{\Gamma}\left(\overline{\rho
\left(\xi\right)}d\xi+\rho\left(\xi\right)\overline{d\xi}\right)\log
\left(\xi-z\right)
+\frac{1}{4\pi i}\int_{\Gamma}\frac{\overline{\xi}\rho
\left(\xi\right) \, d\xi}{\xi-z} \, , \label{eq:DLGoursatChi} \\
\psi_D(z) & =-\frac{1}{4\pi i}\int_{\Gamma}\frac{\overline{\rho
\left(\xi\right)}d\xi+\rho\left(\xi\right)\overline{d\xi}}{\xi-z}
+\frac{1}{4\pi i}\int_{\Gamma}\frac{\overline{\xi}\rho\left(\xi\right) \, d\xi}
{\left(\xi-z\right)^{2}} \, , \label{eq:DLGoursatPsi}\end{aligned}$$ where $z, \xi$, and $\rho$ are as above. The stream function $w_{D}$ corresponding to this Goursat function pair is then $$\begin{split}
w_D(z) &=\mbox{Re}\left[\frac{1}{4\pi i}\int_{\Gamma}
\frac{\overline{\xi-z}}{\xi-z}\rho\left(\xi\right)d\xi
+\frac{1}{4\pi i}\int_{\Gamma}\left(\overline{\rho\left(\xi\right)}d\xi
+\rho\left(\xi\right)\overline{d\xi}\right)\log\left(\xi-z\right)\right]
\\
&=: D^w_\Gamma \rho\label{eq:wDL} \, .
\end{split}$$ As before, the velocity field associated with the stream function $w_D$ is given by $$\nabla^{\perp} w_D\left(z\right) =
\nabla^{\perp} D^w_\Gamma \rho \left(z\right)=
\mathbf{D}_{\Gamma} \boldsymbol{\mu} \left({{\boldsymbol x}}\right) \, .
\label{eq:DLStreamFuncEquiv}$$
### The completed double layer representation for Stokes flow
Using the layer potentials described above, we can represent the solution of $\cref{eq:biharmV1,eq:biharmV2,eq:biharmV3}$, or equivalently the system \[eq:StokesFlowEq\], \[eq:MassConservation\], and \[eq:bcStokesFlow\], in terms of a density $\boldsymbol\mu$ given on the boundary of the domain. The completed double layer representation [@power1993completed] for the velocity is $${{\boldsymbol u}}({{\boldsymbol x}}) = \mathbf{S}_{\Gamma} {\boldsymbol \mu} ({{\boldsymbol x}})
+ \mathbf{D}_{\Gamma} {\boldsymbol \mu} ({{\boldsymbol x}}) +
\mathbf{W}_\Gamma {\boldsymbol \mu} \label{eq:IntRepStokes} \; ,$$ where $\mathbf{W}_\Gamma \boldsymbol{\mu} =
\int_\Gamma \boldsymbol{\mu} \, dS$, and the representation of an equivalent pair of Goursat functions, giving a stream function $w$, can be inferred from the formulas of the previous subsection. When the no-slip boundary conditions are enforced for this representation, the result is an invertible SKIE for the density $\boldsymbol\mu$. The reader can refer to [@power1993completed] for a detailed discussion of the Fredholm alternative for this representation. We summarize it as
\[lem:stokesrep\] Let ${{\boldsymbol u}}$ be defined as in \[eq:IntRepStokes\] and ${{\boldsymbol x}}_0 \in \Gamma$. Then
$$\begin{split}
\lim_{\substack{{{\boldsymbol x}}\to{{\boldsymbol x}}_{0}\\
{{\boldsymbol x}}\in D^{\pm}}} {{\boldsymbol u}}({{\boldsymbol x}}) &=
\pm\frac{1}{2}\boldsymbol{\mu} ({{\boldsymbol x}}_0)
+ \mathbf{S}_\Gamma \boldsymbol{\mu} ({{\boldsymbol x}}_0)
+ \mathbf{D}^{PV}_\Gamma \boldsymbol{\mu} ({{\boldsymbol x}}_0)
+ \mathbf{W}_\Gamma \boldsymbol{\mu} \\
& =: \pm\frac{1}{2}\boldsymbol{\mu}({{\boldsymbol x}}_0)
+ \mathbf{K}_\Gamma \boldsymbol{\mu} ({{\boldsymbol x}}_0) \, .
\end{split}$$
For a sufficiently smooth curve $\Gamma$, the operator $\mathbf{K}_\Gamma$ is a compact operator on $\mathcal{X} \times \mathcal{X}$, where $\mathcal{X}$ is $L^2(\Gamma)$ or $C^{0,\alpha}(\Gamma)$ for $\alpha \in (0,1)$. Further, the integral equation
$$\left (- \dfrac{1}{2} \mathbf{I}_{\mathcal{X}\times\mathcal{X}}
+ \mathbf{K}_\Gamma \right ) \boldsymbol{\mu} = {{\boldsymbol h}}\;$$
is invertible, even for multiply connected domains.
For the above integral equation, the singularities of the integral kernels which define $\mathbf{K}_\Gamma$ are at worst order $r^{-1}$, even for a boundary with a corner.
Integral equation derivation {#sec:anapp}
============================
We would like to adapt the completed double layer representation for solutions of Stokes flow \[eq:biharmV1,eq:biharmV2,eq:biharmV3\] to solve the clamped plate problem \[eq:biharmD1,eq:biharmD2,eq:biharmD3\]. Let $f$ and $g$ be the boundary data as in \[eq:biharmD1,eq:biharmD2,eq:biharmD3\]. By computing tangential derivatives of $f$ on each boundary component, we get the following related Stokes problem: $$\begin{aligned}
\Delta^2 \tilde{w} = 0 &\quad {{\boldsymbol x}}\in D \, ,\nonumber\\
\frac{\partial \tilde{w}}{\partial \tau} =
\frac{\partial f}{\partial \tau}
&\quad {{\boldsymbol x}}\in \Gamma \label{eq:biharm5} \, ,\\
\frac{\partial \tilde{w}}{\partial n} = g &\quad {{\boldsymbol x}}\in \Gamma
\, .
\nonumber\end{aligned}$$ There are two main issues to be addressed in using the completed double layer representation in this context. First, the representation is designed for Stokes flow, in which the quantities of interest are derivatives of the potential $\tilde{w}$ and not $\tilde{w}$ itself; the representation for $\tilde{w}$ may not be single-valued. We will establish that, in the context of \[eq:biharm5\], the stream function is necessarily single-valued. We also discuss some numerical issues related to evaluating the stream function. The second issue to address is that the solution $\tilde{w}$ only satisfies the original boundary condition for the value of $\tilde{w}$ up to a constant on each boundary component. In fact, for multiply connected domains, the completed double layer representation is incomplete for the Dirichlet problem \[eq:biharmD1,eq:biharmD2,eq:biharmD3\]. We present a remedy for this issue and provide some physical intuition.
Single-valued stream functions
------------------------------
To solve the Dirichlet problem \[eq:biharmD1,eq:biharmD2,eq:biharmD3\], it is necessary to compute a single-valued biharmonic potential. In the case of a multiply connected domain, there is no guarantee that a single-valued stream function exists for a given velocity field.
Consider the following example. Let $(r,\theta)$ denote standard polar coordinates. It is easy to verify that the velocity field ${{\boldsymbol u}}=\frac{1}{r}\hat{r}$ solves the equations of Stokes flow in an annulus centered at the origin. A stream function for this flow is $w=\theta$, which is not single-valued; indeed, there are no single-valued stream functions which generate this flow.
Let $D$ be a multiply connected domain with boundary $\Gamma = \cup_{i=0}^N \Gamma_i$, as in the previous section. We note that the gradient of a stream function is determined by the velocity field, i.e.
$$\nabla w = -{{\boldsymbol u}}^\perp := \begin{pmatrix}
- u_2 \\ u_1
\end{pmatrix} \; .$$
Therefore, a velocity field has single-valued stream functions if and only if ${{\boldsymbol u}}^\perp$ is conservative. Using standard results from multivariable calculus, we can characterize such flows.
Suppose that ${{\boldsymbol u}}$ is a divergence-free velocity field which is $C^1$ on $D$ and continuous on $D\cup \Gamma$. The field ${{\boldsymbol u}}^\perp$ is conservative if and only if
$$\int_{\Gamma_i} {{\boldsymbol u}}\cdot {{\boldsymbol n}}\, dS
= 0 \quad i=0,1,\ldots N \, .
\label{eq:NecSuffStreamFuncExistence}$$
The equalities \[eq:NecSuffStreamFuncExistence\] constitute $N$ linearly independent constraints on the boundary data because the divergence-free condition \[eq:MassConservation\] implies that $\int_{\Gamma} {{\boldsymbol u}}\cdot {{\boldsymbol n}}\, dS = 0$. It turns out that these conditions are satisfied when the Dirichlet problem is recast as a Stokes flow \[eq:biharm5\], as it is easily verified that $$\int_{\Gamma_i} {{\boldsymbol u}}\cdot {{\boldsymbol n}}\, dS
= \int_{\Gamma_i} \frac{\partial f}{\partial \tau} \, dS =0 \, .$$ Thus, any stream function $\tilde{w}$ obtained for the Stokes flow \[eq:biharm5\] is necessarily single-valued.
Evaluating the stream function {#subsec:stream}
------------------------------
Given compatible boundary data for the velocity field ${{\boldsymbol u}}$, the completed double layer representation for Stokes flow \[eq:IntRepStokes\] guarantees the existence of a solution density $\boldsymbol\mu$ and a corresponding stream function $\tilde{w}$. The Goursat function formula for $\tilde{w}$, see \[sec:stokeslayer\], is necessarily single-valued, as explained above, but it is not immediately obvious from the formula that this should be true.
The difficulty in the representation of $\tilde{w}$ comes from the part of the stream function corresponding to the double layer potential \[eq:wDL\]. The second term in the expression for the double layer potential is $$v_1(z) = \mbox{Re}\left[\frac{1}{4\pi i}\int_{\Gamma}\left(
\overline{\rho\left(\xi\right)}d\xi+\rho\left(\xi\right)
\overline{d\xi}\right)\log\left(\xi-z\right)\right] \, .$$ To compute this term, in a naïve numerical implementation, the question of which is the appropriate branch of the logarithm to use would arise at many steps. To avoid this complication, it is possible instead to compute $v_1$, up to a constant, as the harmonic conjugate of the function $$v_2 = \frac{1}{4\pi}\int_{\Gamma}\left(\overline{\rho\left(\xi\right)}
d\xi+\rho\left(\xi\right)\overline{d\xi}\right)\log
\left(\left|\xi-z\right|\right) \, . \label{eq:harmconjg}$$
We will use this approach to evalute $v_1$ numerically. As a result of the Cauchy-Riemann equations, the harmonic conjugate of $v_2$, satisfies the following Neumann problem for the Laplace equation: $$\begin{aligned}
\Delta v_1 &= 0 &\quad x\in D \, ,\\
\frac{\partial v_1}{\partial n} &= -\frac{\partial v_2}{\partial \tau}
&\quad x\in\Gamma \, .\end{aligned}$$ It is possible then to use standard integral equation methods to compute $v_1$.
Let $v_1 = S^L_\Gamma \sigma$, where $S^L_\Gamma \sigma$ is the single layer potential for Laplace’s equation, given by $$S^L_\Gamma\sigma ({{\boldsymbol x}}) = -\frac{1}{2\pi} \int_{\Gamma} \log
\left|{{\boldsymbol x}}- {{\boldsymbol y}}\right|\sigma({{\boldsymbol y}})\, dS \left({{\boldsymbol y}}\right) \, ,$$ where $\sigma\in \mathcal{X}= C^{0,\alpha}\left(\Gamma\right)$, for some $\alpha \in (0,1)$, is an unknown density (see [@kress1999linear; @guenther1988partial]). Imposing the Neumann boundary conditions results in the following boundary integral equation for $\sigma$: $$\begin{aligned}
-\frac{\partial v_2}{\partial \tau} ({{\boldsymbol x}}) &= \frac{1}{2} \sigma \left({{\boldsymbol x}}\right) - \frac{1}{2\pi}\oint_{\Gamma}
\frac{\partial}{\partial n_x}
\log \left| {{\boldsymbol x}}- {{\boldsymbol y}}\right |\sigma({{\boldsymbol y}})\, dS \left({{\boldsymbol y}}\right)
\, , \\
-\frac{\partial v_2}{\partial \tau} &= \left( \frac{1}{2}I_{\mathcal{X}} + K^L_\Gamma \right) \sigma
\, ,
\label{eq:BlockSystemRow2P1tmp}\end{aligned}$$ where the operator $K^L_\Gamma$ is compact, so that the integral equation is second kind. For a derivation of this result, see [@kress1999linear].
It is well known that the operator $\frac{1}{2}I_{\mathcal{X}} + K^L_\Gamma$ has a one dimensional null space. Thus, we choose to solve the above integral equation subject to the constraint $\int_{\Gamma} \sigma \, dS = 0$. Furthermore, it is known that solving the Neumann problem subject to the above constraint is equivalent to solving $$\begin{aligned}
\left( \frac{1}{2}I_{\mathcal{X}} + K^L_\Gamma + W_\Gamma \right) \sigma
= -\frac{\partial v_2}{\partial \tau}
\label{eq:BlockSystemRow2P1} \end{aligned}$$ where $W_\Gamma\sigma = \int_{\Gamma} \sigma \, dS$.
Making the representation complete
----------------------------------
As mentioned above, the solution $\tilde{w}$ of the auxiliary Stokes problem \[eq:biharm5\] only satisfies the boundary conditions of the original Dirichlet problem \[eq:biharmD1,eq:biharmD2,eq:biharmD3\] up to a constant on each boundary component. For a simply connected domain, this constant can be recovered from the fact that adding an arbitrary constant to a stream function does not change the velocity field. Thus, in simply connected domains, there is an equivalence in the solutions of \[eq:biharm5\] and \[eq:biharmD1,eq:biharmD2,eq:biharmD3\].
To analyze the case of a multiply connected domain, we first consider radially symmetric solutions on an annulus centered at the origin. Let $w\left(r\right)$ be a radially symmetric biharmonic potential. Then $w(r)$ solves the ordinary differential equation (ODE) $$\frac{d}{dr}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}r
\frac{dw}{dr} = 0 \, .$$ Four linearly independent solutions of this ODE are $1$, $r^2$, $\log \left(r\right)$, and $r^2 \log\left(r\right)$. For each solution, we can compute the associated velocity field ${{\boldsymbol u}}= \nabla^{\perp} w$. By construction, ${{\boldsymbol u}}$ satisfies the continuity condition \[eq:MassConservation\]. For the momentum equation \[eq:StokesFlowEq\] to be satisfied, we need that $\Delta{{\boldsymbol u}}$ is a conservative vector field, which is equivalent to the condition that $\int_{\gamma} \Delta{{\boldsymbol u}}\cdot d{{\boldsymbol \ell}}= 0$ for any closed loop $\gamma$ in the annulus. For the first three linearly independent solutions, $\Delta {{\boldsymbol u}}=0$ so that $\Delta {{\boldsymbol u}}$ is trivially a conservative vector field. The fourth solution, on the other hand, has $\Delta {{\boldsymbol u}}=\frac{4}{r} \hat{\theta}$. By considering a curve $\gamma$ encircling the origin, we see that $\Delta {{\boldsymbol u}}$ is not a conservative vector field and that any pressure for the velocity field associated with $r^2 \log\left(r\right)$ is not single-valued.
The function $\frac{1}{8\pi} r^2 \log\left(r\right)$ is the Green’s function for the biharmonic equation and is the equivalent of a *charge* for such problems. The above analysis can be extended to show that any solution of the biharmonic equation with net charge cannot be represented as a Stokes velocity field. In simply connected domains, since $\Delta^2 w=0$, there can be no net biharmonic charge in the domain. For multiply connected domains with genus $N$, the set of stream functions for Stokes velocity fields misses an $N$ dimensional space of solutions, corresponding to biharmonic charges located in the holes of the domain. Following this reasoning, we obtain a complete representation for biharmonic potentials on multiply connected domains by adding $N$ charges, one per each hole of the domain, to the representation for $w$. The details of this approach, and the proof that it is sufficient, is in the next section.
The integral representation
---------------------------
Following the discussion in the previous two sections, it is now possible to present an integral representation for the Dirichlet problem of the biharmonic equation based on the completed double layer representation for the Stokes problem. We first fix some notation. Let $D$ be a multiply connected domain, with boundary $\Gamma = \cup_{k=0}^N \Gamma_k$, as in the previous sections. For each boundary component $\Gamma_k$, let $D_k$ be its interior and $z_k$ be a point in $D_k$. Then, let the solution $w$ be represented in terms of layer potentials and biharmonic charges as $$w\left(z\right) = S^w_\Gamma \rho \left(z\right)
+ D^w_\Gamma \rho \left(z\right)
+ \text{Re}
\left[\overline{z} \int_{\Gamma} \rho \left(\xi \right) \, dS \right]
+ c_0 + \sum_{k=1}^{N} c_k r_{k}^2
\log \left(r_k\right)
\, ,
\label{eq:IntRepW}$$ where $\rho$ is an unknown density, the $c_k$ are unknown constants, the distance from $z$ to $z_k$ is $r_k = |z-z_k|$, and the operators $S^w_\Gamma$ and $D^w_\Gamma$ map complex densities to the stream functions corresponding to single and double layer potentials, as defined in \[sec:prelim\].
As discussed in \[subsec:stream\], we will only evaluate the operator $D^w_\Gamma$ up to a constant in our numerical implementation. However, this does not affect the analysis of this section because of the freedom in choosing $c_0$.
As before, we can identify a real, vector-valued density $\boldsymbol{\mu} = (\mu_1,\mu_2)^\intercal$ with $\rho$ by setting $\mu_2({{\boldsymbol x}}) -
i \mu_1({{\boldsymbol x}}) = \rho(z)$. Let ${{\boldsymbol u}}= \nabla^\perp w$ be the velocity field corresponding to the stream function $w$. Then, in terms of $\boldsymbol{\mu}$, we have
$${{\boldsymbol u}}\left({{\boldsymbol x}}\right)=
\mathbf{S}_{\Gamma} \boldsymbol{\mu} \left({{\boldsymbol x}}\right) +
\mathbf{D}_{\Gamma} \boldsymbol{\mu} \left({{\boldsymbol x}}\right) +
\mathbf{W}_\Gamma \boldsymbol{\mu} +
\nabla^{\perp} \sum_{k=1}^{N} c_k r_{k}^2 \log (r_k)
\, ,
\label{eq:IntRepU}$$
where $\mathbf{S}_{\Gamma} \boldsymbol{\mu}$ and $\mathbf{D}_{\Gamma} \boldsymbol{\mu}$ are the single and double layer potentials for the density $\boldsymbol{\mu}$, as defined in section 2.
Let $\alpha \in \left(0,1 \right)$ and $\mathcal{X} = C^{0,\alpha} \left(\Gamma \right)$. Assume that the boundary data for the Dirichlet problem \[eq:biharmD1,eq:biharmD2,eq:biharmD3\] satisfies $f \in C^{1,\alpha} \left(\Gamma \right)$ and $g \in \mathcal{X}$, a slightly stronger assumption on the regularity of $f$ than given in the original problem statement. Denote the integrals of $f$ around each boundary component by $b_k = \int_{\Gamma_k} f \, dS$. To solve equation \[eq:biharmD1,eq:biharmD2,eq:biharmD3\], we impose the boundary conditions on the gradient of $w$ as in \[eq:biharm5\] on the above representation for $w$, or, equivalently, the no-slip boundary conditions \[eq:bcStokesFlow\] on the above representation for ${{\boldsymbol u}}$ with $${{\boldsymbol h}}= \left(-\left(\frac{\partial f}{\partial \tau}\tau_2
+ g n_2\right),
\frac{\partial f}{\partial \tau}\tau_1 + g n_1 \right)^\intercal \, .$$ Under the assumptions on $f$ and $g$, the boundary data ${{\boldsymbol h}}\in \mathcal{X} \times \mathcal{X}$.
Let ${{\mathbf B}}{{\boldsymbol c}}({{\boldsymbol x}})$ denote the part of the velocity field due to the charges, i.e.
$${{\mathbf B}}{{\boldsymbol c}}({{\boldsymbol x}}) =
\nabla^{\perp} \sum_{k=1}^{N} c_k r_{k}^2 \log ( r_k) \; .$$
Then, due to \[lem:slppropspart1,lem:dlppropspart1\], enforcing the boundary condition ${{\boldsymbol u}}({{\boldsymbol x}}) = {{\boldsymbol h}}({{\boldsymbol x}})$ for each ${{\boldsymbol x}}\in \Gamma$ results in the following boundary integral equation $$\begin{aligned}
{{\boldsymbol h}}\left({{\boldsymbol x}}\right) &= -\frac{1}{2} \boldsymbol{\mu}
\left({{\boldsymbol x}}\right) +
\mathbf{S}_{\Gamma} \boldsymbol{\mu} \left({{\boldsymbol x}}\right) +
\mathbf{D}_{\Gamma}^{PV} \boldsymbol{\mu} \left({{\boldsymbol x}}\right) +
\mathbf{W}_\Gamma \boldsymbol{\mu}({{\boldsymbol x}})
\nonumber \\
&\quad + \nabla^{\perp} \sum_{k=1}^{N} c_k r_{k}^2
\log (r_k)
\\
{{\boldsymbol h}}({{\boldsymbol x}}) &= \left(-\frac{1}{2} \mathbf{I}_{\mathcal{X}\times\mathcal{X}}
+ \mathbf{K}_\Gamma\right) \boldsymbol{\mu}({{\boldsymbol x}}) +
{{\mathbf B}}{{\boldsymbol c}}({{\boldsymbol x}})
\label{eq:BlockSystemRow1new}
\, .\end{aligned}$$
To ensure that the values of $w$ are correct on the boundary, further constraints are needed. We impose $N+1$ additional conditions on the value of $w$ $$\int_{\Gamma_k} w \, dS = b_k \quad k=0,1,2,\ldots , N \label{eq:IntConstNew}
\, ,$$ where the constants $b_k$ are as defined above. The integral of $w$ about each component can be written in terms of the unknowns as
$$\begin{aligned}
\int_{\Gamma_k} w\left({{\boldsymbol x}}\right) \, dS({{\boldsymbol x}}) &=
\int_{\Gamma_k} \left[
S^w_\Gamma \left[-\mu_2 + i\mu_1 \right] \left( \xi \right )
+ D^w_\Gamma \left[-\mu_2 + i\mu_1 \right] \left( \xi \right )
\right ] \, dS_{\xi} \nonumber \\
&\quad + \int_{\Gamma_k} \left [ \alpha \int_{\Gamma}
\boldsymbol{\mu} \left({{\boldsymbol y}}\right)
\cdot {{\boldsymbol x}}\, dS_{{{\boldsymbol y}}} \right ] \, dS({{\boldsymbol x}})
\nonumber \\
&\quad + \int_{\Gamma_k} \left[ c_0 +
\sum_{l=1}^{N} c_l r_{l}^2 \log r_l \right] \, dS({{\boldsymbol x}}) \\
& =: D_{k} \boldsymbol{\mu} + F_{k}{{\boldsymbol c}}\label{eq:BlockSystemRow2new}\end{aligned}$$
Combining equations \[eq:BlockSystemRow1new\], \[eq:IntConstNew\], and \[eq:BlockSystemRow2new\], we get the following linear system for the unknowns $\boldsymbol{\mu}$ and ${{\boldsymbol c}}$ $$\begin{bmatrix}
-\frac{1}{2}\mathbf{I}_{\mathcal{X}\times\mathcal{X}}
+\mathbf{K}_\Gamma & {{\mathbf B}}\\
\mathbf{D} & \mathbf{F}
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{\mu} \\
{{\boldsymbol c}}\end{bmatrix} =
\begin{bmatrix}
{{\boldsymbol h}}\\
{{\boldsymbol b}}\end{bmatrix}
\label{eq:IntEqBiharm1new} \; ,$$ where $\mathbf{D} = \left(D_0, \ldots D_N\right)^\intercal$, $\mathbf{F} = \left(F_0, \ldots F_N\right)^\intercal$, and ${{\mathbf B}}= \left(b_0, \ldots b_N\right)^\intercal$.
The block system \[eq:IntEqBiharm1new\] is an invertible Fredholm operator.
It is simple to show that the linear system \[eq:IntEqBiharm1new\] is Fredholm. The block which contains $-1/2 \mathbf{I}_{\mathcal{X}\times\mathcal{X}}
+ \mathbf{K}_\Gamma$ is Fredholm due to \[lem:stokesrep\]. The off-diagonal blocks, denoted by ${{\mathbf B}}$ and $\mathbf{D}$, are trivially compact because either the domain or range of the operator is finite dimensional. Finally, $\mathbf{F}$ is Fredholm because it is a finite-dimensional linear operator. Therefore, the full system is Fredholm.
Due to the Fredholm alternative, it is only necessary to establish the injectivity of the system \[eq:IntEqBiharm1new\] to prove that it is invertible. It is clear that if $\boldsymbol{\mu}$ and ${{\boldsymbol c}}$ solve equation \[eq:IntEqBiharm1new\], then the resulting solution, $w$, given by \[eq:IntRepW\], solves the original Dirichlet problem \[eq:biharmD1,eq:biharmD2,eq:biharmD3\]. By construction, $w$ is biharmonic in $D$. Moreover, $w$ satisfies $\frac{\partial w}{\partial \tau} = \frac{\partial f}{\partial \tau}$ and $\frac{\partial w}{\partial n} = g$ on the whole boundary $\Gamma$ and $\int_{\Gamma_{k}} w = \int_{\Gamma_k} f$ for each boundary component $\Gamma_k$, so that the boundary conditions are satisfied.
In the case that ${{\boldsymbol h}}\equiv {{\boldsymbol 0}}$ and ${{\boldsymbol b}}= {{\boldsymbol 0}}$, we have that $f = g \equiv 0$ for the Dirichlet problem. By the uniqueness of solutions to \[eq:biharmD1,eq:biharmD2,eq:biharmD3\], this implies that $w \equiv 0$ in $D$. It is, however, less immediate that $w \equiv 0$ implies that $\boldsymbol{\mu} \equiv
{{\boldsymbol 0}}$ and ${{\boldsymbol c}}= {{\boldsymbol 0}}$.
For each $k = 1,\ldots,N$, let $\tilde{\Gamma}_{k} \subset D$ be a curve which satisfies $n\left(z_{j}, \tilde{\Gamma}_{k}\right) = \delta_{jk}$, where $n\left(z,\gamma\right)$ represents the winding number of the curve $\gamma$ about z. Because ${{\boldsymbol u}}= \nabla^{\perp} w$ and $w\equiv 0$ in $D$, we have $$\int_{\tilde{\Gamma}_k} \Delta {{\boldsymbol u}}\cdot \boldsymbol{\tau} \, dS = 0 .$$
Let ${{\boldsymbol u}}^\mu = \mathbf{S}_\Gamma \boldsymbol{\mu}
+ \mathbf{D}_\Gamma \boldsymbol{\mu} = {{\boldsymbol u}}- {{\mathbf B}}{{\boldsymbol c}}- \mathbf{W}_\Gamma \boldsymbol{\mu}$. We observe that ${{\boldsymbol u}}^\mu$ corresponds to a Stokes velocity field in $D$ for any $\boldsymbol{\mu}$. Let $p$ be its associated pressure. Then $$\int_{\tilde{\Gamma}_k} \Delta {{\boldsymbol u}}^\mu
\cdot \boldsymbol{\tau}
\, dS = \int_{\tilde{\Gamma}_k} \nabla p \cdot \boldsymbol{\tau} \, dS = 0 \, .$$ Further, a simple calculation shows that $$\int_{\tilde{\Gamma}_k} \Delta \nabla^{\perp} c_j
r_j^2 \log (r_j) \cdot \boldsymbol{\tau} \, dS =
8 \pi c_{j} \delta_{jk} \, ,$$ for $j = 1, \ldots, N$. Combining these equations, we conclude that $$0 = \int_{\tilde{\Gamma}_k} \Delta {{\boldsymbol u}}\cdot
\boldsymbol{\tau} \, dS =
\int_{\tilde{\Gamma}_k} \Delta ( {{\boldsymbol u}}^\mu
+ {\mathbf W} \boldsymbol{\mu} + {{\mathbf B}}{{\boldsymbol c}})
\cdot \boldsymbol{\tau} \, dS =
8\pi c_k \, .$$ Thus $c_k = 0$ for $k=1,2,\ldots N$.
The first row of the system \[eq:IntEqBiharm1new\] then reads $$\left(-\frac{1}{2}\mathbf{I}_{\mathcal{X}\times\mathcal{X}}
+ \mathbf{K}_\Gamma \right)
\boldsymbol{\mu} = 0 \, . \label{eq:IntEq1}$$ From the invertibility of $-\frac{1}{2}\mathbf{I}_{\mathcal{X}\times\mathcal{X}}
+ \mathbf{K}_\Gamma$, we conclude that $\boldsymbol{\mu} \equiv {{\boldsymbol 0}}$. Because $\boldsymbol{\mu}\equiv {{\boldsymbol 0}}$ and $c_k = 0$ for $k = 1, \ldots , N$, we get that $w \equiv c_0$. It then follows that $c_0 = 0$ as well, proving the injectivity of the system.
Results {#sec:results}
=======
We first review the existing numerical tools used to compute solutions of the integral equation \[eq:IntEqBiharm1new\]. To discretize the integral equations, we use the Nyström method. We divide the boundary into panels and represent the unknown density and the boundary data by their values at scaled Gauss-Legendre nodes on each panel. Let $n_{p}$ denote the number of Gauss-Legendre panels. We discretize each panel using 16 scaled Gauss-Legendre nodes. Then $n_{d} = 16 n_{p}$ is the number of discretization points on the boundary. Let ${{\boldsymbol x}}_{j}$ denote the discretization nodes, $w_{j}$ denote the appropriately scaled Gauss-Legendre quadrature weights for smooth functions, and ${{\boldsymbol \mu}}_{j}$ denote the unknown density at ${{\boldsymbol x}}_{j}$. When forming the linear system, we use scaled unknowns, ${{\boldsymbol \mu}}_{j} \sqrt{w_{j}}$, so that the spectral properties of the discrete system with respect to the $l_2$ norm are approximations of the spectral properties of the continuous system as on operator on $L_2$ (for more on this point of view, see [@bremer2012]). The integral kernels in this paper are either smooth or have a weak (logarithmic) singularity. For the smooth kernels in the integral representation, we use standard Gauss-Legendre weights appropriately scaled. For kernels with a logarithmic singularity, we use order $20$ generalized Gaussian quadrature rules [@bremer2010; @bremer2010u].
After applying the integral rule, we obtain a linear system for the unknowns. This system is typically well-conditioned, but dense. Let ${{\mathbf{A}}}$ denote the discretized linear system of size $2n_{d}+N+1$ corresponding to the integral equation . Let $\kappa({{\mathbf{A}}})$ denote the condition number of the discretized matrix ${{\mathbf{A}}}$. For our applications, the system size was modest and we computed the unknowns $\boldsymbol{\mu}$ and $c_{k}$ using Gaussian elimination. For larger applications, the system is amenable to solution by any of a variety of iterative or fast-direct solvers, which we will not review here.
For the visualizations in this section, we evaluate the layer potentials inside the domain, with some points being very close to the boundary. The value of the layer potential can be difficult to evaluate at such points because of the near-singularity in the integral kernel. We use a sixth order quadrature by expansion method [@klockner2013; @rachh2016] to evaluate these integrals efficiently and accurately.
In this section we consider two test cases. The first example is a convergence study for a simply connected domain to demonstrate the order of convergence for the discretized integral equation. We also compare the condition numbers for the discretized linear systems corresponding to our integral representation and the existing integral representation by Farkas [@Farkas89] for a family of simply connected domains with increasing curvature. For the second example, we demonstrate the order of convergence and compute the Green’s function for a multiply connected domain. For all examples except the computation of the Green’s function, the boundary data $f$ and $g$ are chosen corresponding to a known solution of the biharmonic equation in $D$ given by
$$\label{eq:sourceformula}
w({{\boldsymbol x}}) = \sum_{j=1}^{n_{s}} q_{j} |{{\boldsymbol x}}- {{\boldsymbol s}}_{j}|^2 \log{|{{\boldsymbol x}}-{{\boldsymbol s}}_{j}|} \, ,$$
where $q_{j}$ are uniformly chosen from $[0,1]$. Let ${w_{\text{comp}}}({{\boldsymbol t}})$ denote the computed solution at targets ${{\boldsymbol t}}$ in the interior of $D$, and let ${\varepsilon}$ denote an estimate for the error given by $${\varepsilon}= \frac{
\sqrt{\sum_{j=1}^{n_{t}} ({w_{\text{comp}}}({{\boldsymbol t}}_{j}) - w({{\boldsymbol t}}_{j}))^2} }{
\sqrt{\sum_{j=1}^{n_{t}}
w({{\boldsymbol t}}_{j})^2} } \, .$$
Simply connected domain examples
--------------------------------
Let $D$ denote the interior of a rounded rectangular bar with length $a=1$, height $b=0.5$, and vertices at $(0,0),(a,0),(a,b),(0,b)$. Following the procedure discussed in [@mikecornerrounding], the corners are rounded using the Gaussian kernel $$\phi(x) = \frac{1}{\sqrt{2\pi h}} e^{-x^2/(2 h^2)} \, ,$$ with $h=0.05$. The boundary data $f$ and $g$ are chosen corresponding to a known solution $w$, defined as in \[eq:sourceformula\], with four sources ${{\boldsymbol s}}_{j}$ located at $$\begin{aligned}
{{\boldsymbol s}}_{1} = \left(a+0.2+ \delta_{1},\frac{b}{2} + \delta_{2}\right) \, , \, \,
&{{\boldsymbol s}}_{2} = \left(\frac{a}{2}+\delta_{3},b+0.2+\delta_{4}\right) \, , \, \, \\
{{\boldsymbol s}}_{3} = \left(-0.2+\delta_{5},\frac{b}{2} + \delta_{6}\right) \, , \, \,
&{{\boldsymbol s}}_{4} = \left(\frac{a}{2} + \delta_{7},-0.2 + \delta_{8}\right) \, ,\end{aligned}$$ with $\delta_{i}$ chosen uniformly from $[-0.05,0.05]$.
The potential $w$ is evaluated at targets ${{\boldsymbol t}}_{j}$ in the interior of $D$, $$\begin{aligned}
{{\boldsymbol t}}_{1} = \left( \frac{a}{4}+\delta_{9}, \frac{b}{4}+\delta_{10} \right) \, , \, \,
&{{\boldsymbol t}}_{2} = \left( \frac{a}{4}+\delta_{11}, \frac{3b}{4}+\delta_{12} \right) \, , \, \, \\
{{\boldsymbol t}}_{3} = \left( \frac{3a}{4}+\delta_{13}, \frac{b}{4}+\delta_{14} \right) \, , \, \,
&{{\boldsymbol t}}_{4} = \left( \frac{3a}{4}+\delta_{15}, \frac{3b}{4}+\delta_{16} \right) \, , \end{aligned}$$ with $\delta_{i}$ again chosen uniformly from $[-0.05,0.05]$. A sample geometry with sources ${{\boldsymbol s}}_{j}$ and targets ${{\boldsymbol t}}_{j}$ and the error ${\varepsilon}$ as a function of $n_{d}$ are shown in \[fig:simplyconnectedfig\]. The convergence study shows that the error decays like a $20$th order convergent scheme.
The integral equation presented in this paper is significantly better conditioned than the existing integral equation discussed in [@Farkas89] — particularly when the boundary has regions with large curvature. We plot the condition number $\kappa({{\mathbf{A}}})$ of the discretized system of integral equations for the representations given by both \[eq:IntEqBiharm1new\] and \[eq:farkIntEq\] as a function of the rounding parameter $h$ for the rounded rectangular bar in \[fig:condcomp\]. The maximum curvature of the boundary is directly proportional to $1/h^2$. The condition number $\kappa({{\mathbf{A}}})$ increases linearly with the maximum curvature for integral equation \[eq:farkIntEq\], but is independent of the curvature for the integral equation presented in this paper.
Multiply connected domain - examples
------------------------------------
Let $D$ now denote the interior of a multiply connected domain, where the outer boundary $\Gamma_{0}$ is the boundary of the rounded rectangular bar discussed above with rounding parameter $h=0.05$ and the domain has ten circular obstacles $\Gamma_{i}$ with radii $r_{0} = 0.04$ and centers located at ${{\boldsymbol x}}_{i}$, $$\begin{aligned}
{{\boldsymbol x}}_{i} &= \left( 0.12 + (i-1)0.2, 0.15 \right) \quad i=1,2,\ldots 5 \, , \\
{{\boldsymbol x}}_{i} &= \left( 0.08 + (i-6)0.2, 0.35 \right) \quad i=6,7,\ldots 10 \, .\end{aligned}$$
We will first perform a convergence study, as above, with a known solution $w$ defined in terms of point sources according to \[eq:sourceformula\]. We create ten sources, one located inside each obstacle, whose locations are given by $${{\boldsymbol s}}_{i} = {{\boldsymbol x}}_{i} + (\delta_{2i-1},\delta_{2i}) \, ,$$ where $\delta_{i}$ are chosen uniformly from $[-0.5r_0,0.5r_{0}]$. The potential is then tested at twelve targets located at $$\begin{aligned}
{{\boldsymbol t}}_{i} &= \left( 0.22+(i-1)0.2,0.05 \right) + (\delta_{2i-1},\delta_{2i})
\quad i=1,2,3,4 \, , \nonumber\\
{{\boldsymbol t}}_{i} &= \left( 0.22+(i-5)0.2,0.25 \right) + (\delta_{2i-1},\delta_{2i})
\quad i=5,6,7,8 \, , \label{eq:targmultloc}\\
{{\boldsymbol t}}_{i} &= \left( 0.18+(i-9)0.2,0.45 \right) + (\delta_{2i-1},\delta_{2i})
\quad i=9,10,11,12 \, , \nonumber\end{aligned}$$ where $\delta_{i}$ are chosen uniformly from $[-0.5r_{0},0.5r_{0}]$.
We observe $20$th order convergence in the error even for this example. The error as a function of the number of discretization points along with a sample geometry are shown in \[fig:mult-connected\]. We also plot the field, and the error in evaluating the potential in the volume using a sixth order quadrature by expansion method in \[fig:pot-err\]. We note that the error observed near the boundary is larger than at the targets used for the convergence study; this is a result of the relatively low order of the quadrature by expansion method and could be improved by increasing the number of points on the boundary.
![(left): Known biharmonic potential due to sources at $\{{{\boldsymbol s}}_{j}\}$ and (right) absolute pointwise error $|{w_{\text{comp}}}({{\boldsymbol t}}) - w({{\boldsymbol t}})|$ for targets in the interior of $D$. []{data-label="fig:pot-err"}](greentestfld2.pdf "fig:"){width="6cm"} ![(left): Known biharmonic potential due to sources at $\{{{\boldsymbol s}}_{j}\}$ and (right) absolute pointwise error $|{w_{\text{comp}}}({{\boldsymbol t}}) - w({{\boldsymbol t}})|$ for targets in the interior of $D$. []{data-label="fig:pot-err"}](greentesterr2.pdf "fig:"){width="6cm"}
For the final example, we compute the function $w$ which satisfies the PDE $$\begin{aligned}
-\Delta^2 w = \sum_{j=1}^{12} \delta_{{{\boldsymbol x}}={{\boldsymbol t}}_{j}} &\quad {{\boldsymbol x}}\in D
\nonumber\\
w = 0 &\quad {{\boldsymbol x}}\in \Gamma \label{eq:domgreenfun}\\
\frac{\partial w}{\partial n} =0 &\quad {{\boldsymbol x}}\in \Gamma \nonumber \, ,\end{aligned}$$ where $\delta_{{{\boldsymbol x}}={{\boldsymbol y}}}$ is the two dimensional radially symmetric Dirac delta function centered at ${{\boldsymbol y}}$ and $\{{{\boldsymbol t}}_{j}\}$ are defined in \[eq:targmultloc\]. This function describes the vertical displacement of an isotropic and homogeneous thin clamped plate with a transverse load given by point forces at the points ${{\boldsymbol t}}_j$. It is also, by definition, a linear combination of the domain Green’s function $G^D$, as in $$w({{\boldsymbol x}}) = \sum_{j=1}^{12} G^{D}({{\boldsymbol x}},{{\boldsymbol t}}_{j}) \, .$$
To compute $w$, we first obtain a particular solution $w_{p}$ which satisfies the PDE in the volume and add to it the solution of a homogeneous problem $w_{h}$ to fix the boundary conditions. We have $$w({{\boldsymbol x}}) = w_{p}({{\boldsymbol x}}) + w_{h}({{\boldsymbol x}}) \, ,$$ where $$w_{p}({{\boldsymbol x}}) = \sum_{j=1}^{12} G^{B} ({{\boldsymbol x}},{{\boldsymbol t}}_{j}) \, ,$$ and $w_{h}$ satisfies the following homogeneous biharmonic equation, $$\begin{aligned}
-\Delta^2 w_{h} = 0 &\quad {{\boldsymbol x}}\in D
\nonumber\\
w_{h} = -w_{p} &\quad {{\boldsymbol x}}\in \Gamma \\
\frac{\partial w_{h}}{\partial n} = -\frac{\partial w_{p}}{\partial n} &\quad {{\boldsymbol x}}\in \Gamma \nonumber \,.\end{aligned}$$ We plot the computed solution in \[fig:greenfunplot\].
![Biharmonic domain green’s function satisfying equations \[eq:domgreenfun\][]{data-label="fig:greenfunplot"}](greensfun-crop.png){width=".8\textwidth"}
Conclusion {#sec:conclusion}
==========
We have presented an integral representation for the biharmonic Dirichlet problem which is stable for domains which have a boundary with high curvature and is applicable to domains which are multiply connected. The representation is based on converting the Dirichlet problem into a problem with velocity boundary conditions, so that classical representations for the velocity boundary value problem can be used. While the technique of [@Farkas89] — in which integral kernels are chosen by optimizing over the derivatives of an appropriate Green’s function — is general and powerful, the spectral properties of the resulting operator are undesirable for boundaries with high curvature or a corner. Indeed, it seems intuitive that all direct representations for the biharmonic Dirichlet problem should suffer in some way: such an approach asks that one of the integral kernels be singular enough to result in a second kind Fredholm equation for the value of the layer potential and smooth enough to result in a first kind Fredholm equation for the normal derivative of the layer potential.
While some of the above is specific to the biharmonic equation, in particular the use of Goursat functions, it is reasonable to expect the approach to generalize to other high order elliptic problems as well. In particular, there are representations for the modified Stokes equations which are analogues of the completed double layer representation used here [@pozrikidis1992boundary]. The extension of this method to three dimensions is a topic of ongoing research and will be reported at a later date.
Acknowledgments {#acknowledgments .unnumbered}
===============
M. Rachh’s work was supported by the U.S. Department of Energy under contract DEFG0288ER25053, the Office of the Assistant Secretary of Defense for Research and Engineering and AFOSR under NSSEFF Program Award FA9550-10-1-0180, and the Office of Naval Research under award N00014-14-1-0797/16-1-2123. T. Askham’s work was supported by the U.S. Department of Energy under contract DEFG0288ER25053, the Office of the Assistant Secretary of Defense for Research and Engineering and AFOSR under NSSEFF Program Award FA9550-10-1-0180, and the Office of the Assistant Secretary of Defense for Research and Engineering and AFOSR under award FA95550-15-1-0385. The authors would like to thank Leslie Greengard for suggesting this problem and both Leslie Greengard and Shidong Jiang for many useful discussions.
|
---
abstract: 'There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space. However, currently not many general tools to construct such extended formulations are available. In this paper, we develop a framework of polyhedral relations that generalizes inductive constructions of extended formulations via projections, and we particularly elaborate on the special case of reflection relations. The latter ones provide polynomial size extended formulations for several polytopes that can be constructed as convex hulls of the unions of (exponentially) many copies of an input polytope obtained via sequences of reflections at hyperplanes. We demonstrate the use of the framework by deriving small extended formulations for the $G$-permutahedra of all finite reflection groups $G$ (generalizing both Goeman’s [@Goe09] extended formulation of the permutahedron of size ${\operatorname{O}({n\log n})}$ and Ben-Tal and Nemirovski’s [@BN01] extended formulation with ${\operatorname{O}({k})}$ inequalities for the regular $2^k$-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).'
author:
- Volker Kaibel
- Kanstantsin Pashkovich
title: Constructing Extended Formulations from Reflection Relations
---
[^1]
Introduction
============
An *extension* of a polyhedron $P\subseteq{\mathbbm{R}}^n$ is some polyhedron $Q\subseteq{\mathbbm{R}}^d$ and a linear projection $\pi:{\mathbbm{R}}^d\rightarrow{\mathbbm{R}}^n$ with $\pi(Q)=P$. A description of $Q$ by linear inequalities (and equations) is called an *extended formulation* for $P$. Extended formulations have received quite some interest, as in several cases, one can describe polytopes associated with combinatorial optimization problems much easier by means of extended formulations than by linear descriptions in the original space. In particular, such extensions $Q$ can have way less facets than the polyhedron $P$ has. For a nice survey on extended formulations we refer to [@CCZ10].
Many fundamental questions on the existence of extended formulations with small numbers of inequalities are open. A particularly prominent one asks whether there are polynomial size extended formulations for the perfect matching polytopes of complete graphs (see [@Yan91; @KPT10]). In fact, we lack good techniques to bound the sizes of extended formulations from below, and we also need more tools to construct extended formulations. This paper makes a contribution into the latter direction.
There are several ways to build extended formulations of polytopes from linear decriptions or from extended formulations of other ones (see, e.g., [@MRC90; @KL10]). A particular simple way is to construct them inductively from extended formulations one has already constructed before. As for an example, let for a vector $p\in{\mathbbm{R}_+}^n$ of *processing times* and for some $\sigma\in{\mathfrak{S}({n})}$ (where ${\mathfrak{S}({n})}$ is the set of all bijections $\gamma:{[{n}]}\rightarrow{[{n}]}$ with ${[{n}]}=\{1,\dots,n\}$), the *completion time vector* be the vector ${\operatorname{ct}({p},{\sigma})}\in{\mathbbm{R}}^n$ with ${\operatorname{ct}({p},{\sigma})}_j=\sum_{i=1}^{\sigma(j)}p_{\sigma^{-1}(i)}$ for all $j\in{[{n}]}$. By some simple arguments (resembling the correctness proof of Smith’ rule), one can show that ${\operatorname{P}_{\operatorname{ct}}^{p}}$ is the image of the polytope $P={\operatorname{P}_{\operatorname{ct}}^{\tilde{p}}}\times [0,1]^{n-1}$ for $\tilde{p}=(p_1,\dots,p_{n-1})\in{\mathbbm{R}}^{n-1}$ under the affine map $f:{\mathbbm{R}}^{2n-2}\rightarrow{\mathbbm{R}}^n$ defined via $f(x)=(x'+p_nx'',{\langle{\tilde{p}},{{\mathbbm{1}_{}}-x''}\rangle}+p_n)$ with $x=(x',x'')$ and $x',x''\in{\mathbbm{R}}^{n-1}$.
Applying this inductively, one finds that ${\operatorname{P}_{\operatorname{ct}}^{p}}$ is a *zonotope*, i.e., an affine projection of a cube of dimension $n(n-1)/2$ (which had already been proved by Wolsey in the 1980’s [@Wol]). This may appear surprisingly simple viewing the fact that ${\operatorname{P}_{\operatorname{ct}}^{p}}$ has exponentially many facets (see [@Que93]). For the special case of the *permutahedron* ${\operatorname{P}_{\operatorname{perm}}^{n}}={\operatorname{P}_{\operatorname{ct}}^{{\mathbbm{1}_{n}}}}=\operatorname{conv}{\{{(\gamma(1),\dots,\gamma(n))\in{\mathbbm{R}}^n}:{\gamma\in{\mathfrak{S}({n})}}\}}$, Goemans [@Goe09] found an even smaller extended formulation of size ${\operatorname{O}({n\log n})}$, which we will come back to later.
Let us look again at one step in the inductive construction described above. With the polyhedron $$\label{eq:intro:R}
R={\{{(x,y)\in{\mathbbm{R}}^{2n-2}\times{\mathbbm{R}}^{n}}:{y=f(x)}\}}\,,$$ the extension derived in such a step reads $$\label{eq:intro:PpviaR}
{\operatorname{P}_{\operatorname{ct}}^{p}}={\{{y\in{\mathbbm{R}}^n}:{(x,y)\in R\text{ for some }x\in P}\}}\,.$$ Thus, we have derived the extended formulation for ${\operatorname{P}_{\operatorname{ct}}^{p}}$ by applying in the sense of the “polyhedral relation” defined in to a polytope $P$ of which we had found (inductively) an extended formulation before. The goal of this paper is to generalize this technique of deriving extended formulations by using other “polyhedral relations” than graphs of affine maps (which $R$ as defined in is). We will introduce the framework of such general polyhedral relations in Section \[sec:polyRel\], and we are going to elaborate on one particular type of those, called *reflection relations*, in Section \[sec:reflRel\]. Reflection relations provide, for affine halfspaces $H^{\le}\subseteq{\mathbbm{R}}^n$ and polyhedra $P\subseteq{\mathbbm{R}}^n$, small extended formulations of the convex hull of the union of $P\cap H^{\le}$ and the image of $P\cap H^{\le}$ under the orthogonal reflection at the boundary hyperplane of $H^{\le}$. They turn out to be quite useful building blocks in the construction of some extended formulations. We derive some general results on reflection relations (Theorem \[thm:seqReflRel\]) that allow to construct rather easily extended formulations for some particular applications (in particular, without explicitly dealing with the intermediate polyhedra of iterated constructions) .
In a first application, we show how to derive, for each polytope $P\subseteq{\mathbbm{R}}^n$ that is contained in (the topological closure of) a region of a finite reflection group $G$ on ${\mathbbm{R}}^n$, an extended formulation of the $G$-permutahedron of $P$, i.e., the convex hull of the union of the polytopes in the orbit of $P$ under the action of $G$ (Section \[subsec:reflGroups\]). These extended formulations have $f+{\operatorname{O}({n\log n})}+{\operatorname{O}({n\log m})}$ inequalities, where $m$ is the largest number such that $I_2(m)$ appears in the decomposition of $G$ into irreducible finite reflection groups, and provided that there is an extended formulation for $P$ with at most $f$ inequalities. In particular, this generalizes Goemans’ extended formulation of the permutahedron ${\operatorname{P}_{\operatorname{perm}}^{n}}$ with ${\operatorname{O}({n\log n})}$ inequalities [@Goe09]. In fact, the starting point of our research was to give an alternative proof for the correctness of Goeman’s extended formulation that we would be able to generalize to other constructions.
As a second application, we provide an extended formulation with ${\operatorname{O}({n\log n})}$ inequalities for the convex hull of all weight-vectors of Huffman-codes with $n$ words (Section \[subsec:huffman\]). This *Huffman-polytope* ${\operatorname{P}_{\operatorname{huff}}^{n}}$ is the convex hull of all vectors $(v_1,\dots,v_n)\in{\mathbbm{R}}^n$ for which there is a rooted binary tree with $n$ leaves labelled by $1, \dots, n$ such that the distance of leaf $i$ from the root equals $v_i$ for all $i\in{[{n}]}$. This provides another striking example of the power of extended formulations, as no linear descriptions of ${\operatorname{P}_{\operatorname{huff}}^{n}}$ in ${\mathbbm{R}}^n$ is known so far, and Nguyen, Nguyen, and Maurras [@NNM10] showed that ${\operatorname{P}_{\operatorname{huff}}^{n}}$ has $2^{\Omega(n\log n)}$ facets.
Two well-known results we obtain easily within the framework of reflection relations are extended formulations with $2\lceil \log(m)\rceil+2$ inequalities for regular $m$-gons (reproving a result of Ben-Tal and Nemirovski [@BN01], see Section \[subsubsec:I\]) and an extended formulation with $4n-1$ inequalities of the *parity polytope*, i.e., the convex hull of all $v\in\{0,1\}^n$ with an odd number of one-entries (reproving a result of Carr and Konjevod [@CK04], see Section \[subsubsec:D\]).
We conclude by briefly discussing (Section \[sec:concl\]) directions for future research on the further extension of the tools presented in this paper .
#### Acknowledgements
We thank Samuel Fiorini and Michel Goemans for valuable hints and discussions.
Polyhedral Relations {#sec:polyRel}
====================
A *polyhedral relation* of *type $(n,m)$* is a non-empty polyhedron $\varnothing\ne R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$. The *image* of a subset $X\subseteq{\mathbbm{R}}^n$ under such a polyhedral relation $R$ is denoted by $${{R}({X})}={\{{y\in{\mathbbm{R}}^m}:{(x,y)\in R \text{ for some }x\in X}\}}\,.$$ Clearly, we have the monotonicity relations ${{R}({X})}\subseteq {{R}({\tilde{X}})}$ for $X\subseteq\tilde{X}$. Furthermore, ${{R}({X})}$ is a linear projection of $R\cap(X\times{\mathbbm{R}}^m)$ . Thus, images of polyhedra and convex sets under polyhedral relations are polyhedra and convex sets, respectively.
A *sequential polyhedral relation* of *type $(k_0,\dots,k_r)$* is a sequence $(R_1,\dots,R_r)$, where $R_i$ is a polyhedral relation of type $(k_{i-1},k_i)$ for each $i\in{[{r}]}$; its *length* is $r$. For such a sequential polyhedral relation, we denote by $\mathcal{R}=\mathcal{R}_{(R_1,\dots,R_r)}$ the set of all $(z^{(0)},z^{(r)})\in{\mathbbm{R}}^{k_0}\times{\mathbbm{R}}^{k_r}$ for which there is some $(z^{(1)},\dots,z^{(r-1)})$ with $(z^{(i-1)},z^{(i)})\in R_i$ for all $i\in{[{r}]}$. Note that, since $\mathcal{R}$ is a linear projection of a polyhedron, $\mathcal{R}$ is a polyhedral relation of type $(k_0,k_r)$. We call $\mathcal{R}_{(R_1,\dots,R_r)}$ the polyhedral relation that is *induced* by the sequential polyhedral relation $(R_1,\dots,R_r)$.
For a polyhedron $P\subseteq{\mathbbm{R}}^{k_0}$, the polyhedron $Q\subseteq{\mathbbm{R}}^{k_0}\times\cdots\times{\mathbbm{R}}^{k_r}$ defined by $$\label{eq:polyRel:extension}
z^{(0)}\in P
\quad\text{and}\quad
(z^{(i-1)},z^{(i)})\in R_i\quad\text{for all }i\in{[{r}]}$$ satisfies $\pi(Q)={{\mathcal{R}}({P})}$, where $\pi$ is the projection defined via $\pi(z^{(0)},\dots,z^{(r)})=z^{(r)}$. Thus, provides an extended formulation of the polyhedron ${{\mathcal{R}}({P})}$ with $k_0+\cdots+k_r$ variables and $f_0+\cdots+f_r$ constraints, provided we have linear descriptions of the polyhedra $P$, $R_1$, …, $R_r$ with $f_0$, $f_1$, …, $f_r$ constraints, respectively. Of course, one can reduce the number of variables in this extended formulation to $\dim(Q)$. In order to obtain useful upper bounds on this number by means of the polyhedral relations $R_1$, …, $R_r$, let us denote, for any polyhedral relation $R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$, by $\delta_1(R)$ and $\delta_2(R)$ the dimension of the non-empty fibers of the orthogonal projection of ${\operatorname{aff}({R})}$ to the first and second factor of ${\mathbbm{R}}^n\times{\mathbbm{R}}^m$, respectively. If ${\operatorname{aff}({R})}={\{{(x,y)\in{\mathbbm{R}}^n\times{\mathbbm{R}}^m}:{Ax+By=c}\}}$, then $\delta_1(R)=\dim({\operatorname{ker}({B})})$ and $\delta_2(R)=\dim({\operatorname{ker}({A})})$. With these parameters, we can estimate $$\dim(Q)\le\min\{k_{0}+\sum_{i=1}^r\delta_1(R_i),k_r+\sum_{i=1}^r\delta_2(R_i)\}\,.$$
\[rem:polyRelEF\] Let $(R_1,\dots,R_{r})$ be a sequential polyhedral relation of type $(k_0,\dots,k_r)$ with induced polyhedral relation $\mathcal{R}$, let $\pi:{\mathbbm{R}}^{k_0}\times\cdots\times{\mathbbm{R}}^{k_{r}}\rightarrow{\mathbbm{R}}^{k_r}$ be the projection defined via $\pi(z^{(0)},\dots,z^{(r)})=z^{(r)}$, and let $f_i$ be the number of facets of $R_i$ for each $i\in{[{r}]}$. If the polyhedron $P\subseteq{\mathbbm{R}}^{k_0}$ has an extended formulation with $k'$ variables and $f'$ inequalities, then we can construct an extended formulation for ${{\mathcal{R}}({P})}$ with $\min\{k'+\sum_{i=1}^r\delta_1(R_i),k_r+\sum_{i=1}^r\delta_2(R_i)\}$ variables and $f'+f_1+\cdots+f_r$ constraints.
A particularly simple class of polyhedral relations is defined by polyhedra $R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$ with $R={\{{(x,y)\in{\mathbbm{R}}^n\times{\mathbbm{R}}^m}:{y=f(x)}\}}$ for some affine map $f:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^m$. For these polyhedral relations, a (linear description of a) polyhedron $P\subseteq{\mathbbm{R}}^n$ is just an extended formulation of the polyhedron ${{R}({P})}$ via projection $f$.
The *domain* of a polyhedral relation $R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$ is the polyhedron $$\operatorname{dom}(R)={\{{x\in{\mathbbm{R}}^n}:{(x,y)\in R\text{ for some }y\in{\mathbbm{R}}^m}\}}\,.$$ We clearly have ${{R}({X})}=\bigcup_{x\in X\cap\operatorname{dom}(R)}{{R}({x})}$ for all $X\subseteq{\mathbbm{R}}^n$. Note that, for a polytope $P={\operatorname{conv}({V})}$ with a finite set $V\subseteq{\mathbbm{R}}^n$ and a polyhedral relation $R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$, in general the inclusion $$\label{eq:inclRofVerts}
\operatorname{conv}\bigcup_{v\in V}{{R}({v})}\subseteq{{R}({P})}$$ holds without equality, even in case of $P\subseteq\operatorname{dom}(R)$; as for an example you may consider $P=\operatorname{conv}\{0,2\}\subseteq{\mathbbm{R}}^1$ and $R=\operatorname{conv}\{(0,0),(1,1),(2,0)\}$ with ${{R}({P})}=[0,1]$ and ${{R}({0})}={{R}({2})}=\{0\}$. Fortunately, one can guarantee equality in (which makes it much easier to analyze ${{R}({P})}$) for an important subclass of polyhedral relations.
We call a relation $R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$ *affinely generated* by the family $(\varrho^{(f)})_{f\in F}$, if $F$ is finite and every $\varrho^{(f)}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^m$ is an affine map such that ${{R}({x})}=\operatorname{conv}\bigcup_{f\in F}{\varrho^{(f)}(x)}$ holds for all $x\in\operatorname{dom}(R)$. The maps $\varrho^{(f)}$ ($f\in F$) are called *affine generators* of $R$ in this case. For such a polyhedral relation $R$ and a polytope $P\subseteq{\mathbbm{R}}^n$ with $P\cap\operatorname{dom}(R)={\operatorname{conv}({V})}$ for some $V\subseteq{\mathbbm{R}}^n$, we find $$\begin{gathered}
{{R}({P})}
= \bigcup_{x\in P\cap\operatorname{dom}(R)}{{R}({x})}
= \bigcup_{x\in P\cap\operatorname{dom}(R)}\operatorname{conv}\bigcup_{f\in F}{\varrho^{(f)}(x)}\\
\subseteq \operatorname{conv}\bigcup_{x\in P\cap\operatorname{dom}(R)}\bigcup_{f\in F}{\varrho^{(f)}(x)}
= \operatorname{conv}\bigcup_{v\in V}\bigcup_{f\in F}{\varrho^{(f)}(v)}
\subseteq \operatorname{conv}\bigcup_{v\in V}{{R}({v})}\,,\end{gathered}$$ where, due to , all inclusions are equations. In particular, we have established the following result.
\[prop:polyRel\] For every polyhedral relation $R\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^m$ that is affinely generated by a finite family $(\varrho^{(f)})_{f\in F}$, and for every polytope $P\subseteq{\mathbbm{R}}^n$, we have $$\label{eq:relimconvunionaffim}
{{R}({P})}=\operatorname{conv}\bigcup_{f\in F}\varrho^{(f)}(P\cap\operatorname{dom}(R))\,.$$
As we will often deal with polyhedral relations $\mathcal{R}=\mathcal{R}_{(R_1,\dots,R_r)}$ that are induced by a sequential polyhedral relation $(R_1,\dots,R_r)$, it would be convenient to be able to derive affine generators for $\mathcal{R}$ from affine generators for $R_1$,…,$R_r$. This, however, seems impossible in general, where the difficulties arise from the interplay between images and domains in a sequence of polyhedral relations. However, one still can derive a very useful analogue of the inclusion “$\subseteq$” in .
\[lem:seqAffRel\] If $(R_1,\dots,R_{r})$ is a sequential polyhedral relation such that, for each $i\in{[{r}]}$, the relation $R_i$ is affinely generated by the finite family $(\varrho^{(f_i)})_{f_i\in F_i}$, then the inclusion $${{\mathcal{R}}({P})}\subseteq\operatorname{conv}\bigcup_{f\in F}\varrho^{(f)}(P\cap\operatorname{dom}(\mathcal{R}))$$ holds for every polyhedron $P\subseteq{\mathbbm{R}}^n$, where $F=F_1\times\cdots\times F_r$ and $\varrho^{(f)}=\varrho^{(f_r)}\circ\cdots\circ\varrho^{(f_1)}$ for each $f=(f_1,\dots,f_r)\in F$.
We omit the straight-forward proof of Lemma \[lem:seqAffRel\] in this extended abstract.
Reflection Relations {#sec:reflRel}
====================
For $a\in{\mathbbm{R}}^n\setminus\{{\mathbb{O}_{}}\}$ and $\beta\in{\mathbbm{R}}$, we denote by ${\operatorname{H}^{=}({a},{\beta})}={\{{x\in{\mathbbm{R}}^n}:{{\langle{a},{x}\rangle}=\beta}\}}$ the hyperplane defined by the equation ${\langle{a},{x}\rangle}=\beta$ and by ${\operatorname{H}^{\le}({a},{\beta})}={\{{x\in{\mathbbm{R}}^n}:{{\langle{a},{x}\rangle}\le\beta}\}}$ the halfspace defined by the inequality ${\langle{a},{x}\rangle}\le\beta$ (with ${\langle{v},{w}\rangle}=\sum_{i=1}^nv_iw_i$ for all $v,w\in{\mathbbm{R}}^n$). The reflection at $H={\operatorname{H}^{=}({a},{\beta})}$ is ${\varrho^{({H})}}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^n$ where ${\varrho^{({H})}}(x)$ is the point with ${\varrho^{({H})}}(x)-x\in{{H}^{\perp}}$ lying in the one-dimensional linear subspace ${{H}^{\perp}}={\{{\lambda a}:{\lambda\in{\mathbbm{R}}}\}}$ that is orthogonal to $H$ and ${\langle{a},{{\varrho^{({H})}}(x)}\rangle}=2\beta-{\langle{a},{x}\rangle}$. The *reflection relation* defined by $(a,\beta)$ is $${\operatorname{R}_{{a},{\beta}}}={\{{(x,y)\in{\mathbbm{R}}^n\times{\mathbbm{R}}^n}:{y-x\in{{({\operatorname{H}^{=}({a},{\beta})})}^{\perp}}, {\langle{a},{x}\rangle}\le{\langle{a},{y}\rangle}\le 2\beta-{\langle{a},{x}\rangle}}\}}$$ (the definition is invariant against scaling $(a,\beta)$ by positive scalars). For the halfspace $H^{\le}={\operatorname{H}^{\le}({a},{\beta})}$, we also denote ${\operatorname{R}_{H^{\le}}}={\operatorname{R}_{{a},{\beta}}}$. The domain of the reflection relation is $\operatorname{dom}({\operatorname{R}_{{a},{\beta}}})=H^{\le}$, as $(x,y)\in{\operatorname{R}_{{a},{\beta}}}$ implies ${\langle{a},{x}\rangle}\le 2\beta-{\langle{a},{x}\rangle}$, thus ${\langle{a},{x}\rangle}\le \beta$, and furthermore, for each $x\in{\operatorname{H}^{\le}({a},{\beta})}$, we obviously have $(x,x)\in{\operatorname{R}_{{a},{\beta}}}$. Note that, although $(a,\beta)$ and $(-a,-\beta)$ define the same reflection, the reflection relations ${\operatorname{R}_{{a},{\beta}}}$ and ${\operatorname{R}_{{-a},{-\beta}}}$ have different domains.
From the constraint $y-x\in{{({\operatorname{H}^{=}({a},{\beta})})}^{\perp}}$ it follows that $\delta_1({\operatorname{R}_{{a},{\beta}}})=1$ holds. Thus, we can deduce the following from Remark \[rem:polyRelEF\].
\[rem:sizeSeqRefl\] If $\mathcal{R}$ is induced by a sequential polyhedral relation of type $(n,\dots,n)$ and length $r$ consisting of reflection relations only, then, for every polyhedron $P\subseteq{\mathbbm{R}}^n$, an extended formulation of ${{\mathcal{R}}({P})}$ with $n'+r$ variables and $f'+2r$ inequalities can be constructed, provided one has at hands an extended formulation for $P$ with $n'$ variables and $f'$ inequalities.
\[prop:polyRel:refl\] For $a\in{\mathbbm{R}}^n\setminus\{{\mathbb{O}_{}}\}$, $\beta\in{\mathbbm{R}}$ and the hyperplane $H={\operatorname{H}^{=}({a},{\beta})}$, the reflection relation ${\operatorname{R}_{{a},{\beta}}}$ is affinely generated by the identity map and the reflection ${\varrho^{({H})}}$.
We need to show ${{{\operatorname{R}_{{a},{\beta}}}}({x})}=\operatorname{conv}\{x,{\varrho^{({H})}}(x)\}$ for every $x\in \operatorname{dom}({\operatorname{R}_{{a},{\beta}}})={\operatorname{H}^{\le}({a},{\beta})}$. Since, for each such $x$, we have $(x,x)\in{{{\operatorname{R}_{{a},{\beta}}}}({x})}$ and $(x,{\varrho^{({H})}}(x))\in{{{\operatorname{R}_{{a},{\beta}}}}({x})}$, and due to the convexity of ${{{\operatorname{R}_{{a},{\beta}}}}({x})}$, it suffices to establish the inclusion “$\subseteq$”. Thus, let $y\in{{{\operatorname{R}_{{a},{\beta}}}}({x})}$ be an arbitrary point in ${{{\operatorname{R}_{{a},{\beta}}}}({x})}$. Due to ${\varrho^{({H})}}(x)-x\in{{H}^{\perp}}$ and $y-x\in{{H}^{\perp}}$, both $x$ and ${\varrho^{({H})}}(x)$ are contained in the line $y+{{H}^{\perp}}$. From $2\beta-{\langle{a},{x}\rangle}={\langle{a},{{\varrho^{({H})}}(x)}\rangle}$ and ${\langle{a},{x}\rangle}\le{\langle{a},{y}\rangle}\le 2\beta-{\langle{a},{x}\rangle}$ we hence conclude that $y$ is a convex combination of $x$ and ${\varrho^{({H})}}(x)$.
From Proposition \[prop:polyRel\] and Proposition \[prop:polyRel:refl\], one obtains the following result.
\[cor:polyRel:refl\] If $P\subseteq{\mathbbm{R}}^n$ is a polytope, then we have, for $a\in{\mathbbm{R}}^n\setminus\{{\mathbb{O}_{}}\}$ and $\beta\in{\mathbbm{R}}$ defining the hyperplane $H={\operatorname{H}^{=}({a},{\beta})}$ and the halfspace $H^{\le}={\operatorname{H}^{\le}({a},{\beta})}$, $${{{\operatorname{R}_{{a},{\beta}}}}({P})}=\operatorname{conv}\big((P\cap H^{\le})\cup{\varrho^{({H})}}(P\cap H^{\le})\big)\,.$$
While Corollary \[cor:polyRel:refl\] describes images under single reflection relations, for analyses of the images under sequences of reflection relations we define, for each $a\in{\mathbbm{R}}^n\setminus\{{\mathbb{O}_{}}\}$, $\beta\in{\mathbbm{R}}$, $H^{\le}={\operatorname{H}^{\le}({a},{\beta})}$, and $H={\operatorname{H}^{=}({a},{\beta})}$, the map ${\varrho^{\star({H^{\le}})}}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^n$ via $${\varrho^{\star({H^{\le}})}}(y)=
\begin{cases}
y & \text{if } y \in H^{\le}\\
{\varrho^{({H})}}(y) & \text{otherwise}
\end{cases}$$ for all $y\in{\mathbbm{R}}^n$, which assigns a canonical preimage to every $y\in{\mathbbm{R}}^n$. If $\mathcal{R}$ denotes the polyhedraöl relation induced by the sequential polyhedral relation $({\operatorname{R}_{H^{\le}_1}},\ldots,{\operatorname{R}_{H^{\le}_r}})$, for all $y\in{\mathbbm{R}}^n$, we have $$\label{eq:Hstar}
y\in{{\mathcal{R}}({{\varrho^{\star({H^{\le}_1})}}\circ\cdots\circ{\varrho^{\star({H^{\le}_r})}}(y)})}\,.$$
\[thm:seqReflRel\] Let the sequential polyhedral relation $({\operatorname{R}_{H^{\le}_1}},\ldots,{\operatorname{R}_{H^{\le}_r}})$ with halfspaces $H^{\le}_1, \dots,H^{\le}_r\subseteq{\mathbbm{R}}^n$ and boundary hyperplanes $H_1,\dots,H_r$ induce the polyhedral relation $\mathcal{R}$. For polytopes $P, Q \subseteq{\mathbbm{R}}^n$, with $Q={\operatorname{conv}({W})}$ for some $W\subseteq{\mathbbm{R}}^n$, we have $Q={{\mathcal{R}}({P})}$, whenever the following two conditions are satisfied:
1. \[cond::seqReflRel1\] We have $P\subseteq Q$ and ${\varrho^{({H_i})}}(Q)\subseteq Q$ for all $i\in{[{r}]}$.
2. \[cond::seqReflRel2\] We have ${\varrho^{\star({H^{\le}_1})}}\circ\cdots\circ{\varrho^{\star({H^{\le}_r})}}(w)\in P$ for all $w\in W$.
From the first condition it follows that the image of $P$ under every combination of maps ${\varrho^{({H_i})}}$ lies in $Q$. Thus, from Lemma \[lem:seqAffRel\] we have the inclusion ${{\mathcal{R}}({P})}\subseteq Q$. By the second condition and , we have $W\subseteq{{\mathcal{R}}({P})}$, and hence $Q={\operatorname{conv}({W})}\subseteq{{\mathcal{R}}({P})}$ due to the convexity of ${{\mathcal{R}}({P})}$.
In order to provide simple examples of extended formulations obtained from reflection relations, let us define the *signing* of a polyhedron $P\subseteq{\mathbbm{R}}^n$ to be $${\operatorname{sign}({P})}=\operatorname{conv}\bigcup_{\epsilon\in\{-,+\}^n}\epsilon.P\,,$$ where $\epsilon.x$ is the vector obtained from $x\in{\mathbbm{R}}^n$ by changing the signs of all coordinates $i$ with $\epsilon_i$ being minus. For $x\in{\mathbbm{R}}^n$, we denote by ${{x}^{(\text{abs})}}\in{\mathbbm{R}}^n$ the vector that is obtained from $x$ by changing every component to its absolute value.
For the construction below we use the reflection relations ${\operatorname{R}_{{-{\mathbbm{e}_{k}}},{0}}}$, denoted by ${\operatorname{S}_{{k}}}$, for all $k\in{[{n}]}$. The corresponding reflection ${\sigma_{{k}}}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^n$ is just the sign change of the $k$-th coordinate, given by $${\sigma_{{k}}}(x)_{i}=
\begin{cases}
-x_i & \text{if }i=k\\
x_i & \text{otherwise}
\end{cases}$$ for all $x\in{\mathbbm{R}}^n$. The map which defines the canonical preimage with respect to the relation ${\operatorname{S}_{{k}}}$ is given by $${\sigma^{\star}_{{k}}}(y)_i=
\begin{cases}
|y_i| & \text{if }i=k\\
y_i & \text{otherwise}
\end{cases}$$ for all $y\in{\mathbbm{R}}^n$.
\[prop:changingSigns\] If $\mathcal{R}$ is the polyhedral relation that is induced by the sequence $({\operatorname{S}_{{1}}}, \ldots, {\operatorname{S}_{{n}}})$ and $P\subseteq{\mathbbm{R}}^n$ is a polytope with ${{v}^{(\text{abs})}}\in P$ for each vertex $v$ of $P$, then we have $${{\mathcal{R}}({P})}={\operatorname{sign}({P})}\,.$$
With $Q={\operatorname{sign}({P})}$, the first condition of Theorem \[thm:seqReflRel\] is satisfied. Furthermore, we have $Q={\operatorname{conv}({W})}$ with $W={\{{\epsilon.v}:{\epsilon\in\{-,+\}^n,v\text{ vertex of }P}\}}$. As, for every $w\in W$ with $w=\epsilon.v$ for some vertex $v$ of $P$ and $\epsilon\in\{-,+\}^n$, we have ${\sigma^{\star}_{{1}}}\circ\dots\circ{\sigma^{\star}_{{n}}}(w)={{w}^{(\text{abs})}}={{v}^{(\text{abs})}}\in P$, also the second condition of Theorem \[thm:seqReflRel\] is satisfied. Hence the claim follows.
Proposition \[prop:changingSigns\] and Remark \[rem:sizeSeqRefl\] imply the following.
For each polytope $P\subseteq{\mathbbm{R}}^n$ with ${{v}^{(\text{abs})}}\in P$ for each vertex $v$ of $P$ that admits an extended formulation with $n'$ variables and $f'$ inequalities, there is an extended formulation of ${\operatorname{sign}({P})}$ with $n'+n$ variables and $f'+2n$ inequalities.
Applications
============
Reflection Groups {#subsec:reflGroups}
-----------------
A *finite reflection group* is a group $G$ of finite cardinality that is generated by a (finite) family ${\varrho^{({H_i})}}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^n$ ($i\in I$) of reflections at hyperplanes ${\mathbb{O}_{}}\in H_i\subseteq{\mathbbm{R}}^n$ containing the origin. We refer to [@Hum90; @FR07] for all results on reflection groups that we will mention. The set of *reflection hyperplanes* $H\subseteq{\mathbbm{R}}^n$ with ${\varrho^{({H})}}\in G$ (and thus ${\mathbb{O}_{}}\in H$) — called the *Coxeter arrangement* of $G$ — cuts ${\mathbbm{R}}^n$ into open connected components, which are called the *regions* of $G$. The group $G$ is in bijection with the set of its regions, and it acts transitively on these regions. If one distinguishes arbitrarily the topological closure of one of them as the *fundamental domain* ${\Phi_{G}}$ of $G$, then, for every point $x\in{\mathbbm{R}}^n$, there is a *unique* point ${{x}^{({\Phi_{G}})}}\in {\Phi_{G}}$ that belongs to the orbit of $x$ under the action of the group $G$ on ${\mathbbm{R}}^n$.
A finite reflection group $G$ is called *irreducible* if the set of reflection hyperplanes cannot be partitioned into two sets $\mathcal{H}_1$ and $\mathcal{H}_2$ such that the normal vectors of all hyperplanes in $\mathcal{H}_1$ are orthogonal to the normal vectors of all hyperplanes from $\mathcal{H}_2$. According to a central classification result, up to linear transformations, the family of irreducible finite reflection groups consists of the four infinite subfamilies $I_2(m)$ (on ${\mathbbm{R}}^2$), $A_{n-1}$, $B_n$, and $D_n$ (on ${\mathbbm{R}}^n$), as well as six special groups.
For a finite reflection group $G$ on ${\mathbbm{R}}^n$ and some polytope $P\subseteq{\mathbbm{R}}^n$ of $G$, the *$G$-permutahedron* ${\Pi_{G}({P})}$ of $P$ is the convex hull of the union of the orbit of $P$ under the action of $G$. In this subsection, we show for $G$ being one of $I_2(m)$, $A_{n-1}$, $B_n$, or $D_n$, how to construct an extended formulation for ${\Pi_{G}({P})}$ from an extended formulation for $P$. The numbers of inequalities in the constructed extended formulations will be bounded by $f+{\operatorname{O}({\log m})}$ in case of $G=I_2(m)$ and by $f+{\operatorname{O}({n\log n})}$ in the other cases, provided that we have at hands an extended formulation of $P$ with $f$ inequalities. By the decomposition into irreducible finite reflection groups, one can extend these constructions to arbitrary finite reflection groups $G$ on ${\mathbbm{R}}^n$, where the resulting extended formulations have $f+{\operatorname{O}({n\log m})}+{\operatorname{O}({n\log n})}$ inequalities, where $m$ is the largest number such that $I_2(m)$ appears in the decomposition of $G$ into irreducible finite reflection groups. Details on this will be in the full version of the paper.
### The reflection group $I_2(m)$ {#subsubsec:I}
For $\varphi\in{\mathbbm{R}}$, let us denote $H_{\varphi}={\operatorname{H}^{=}({(-\sin\varphi,\cos\varphi)},{0})}$ and $H^{\le}_{\varphi}={\operatorname{H}^{\le}({(-\sin\varphi,\cos\varphi)},{0})}$. The group $I_2(m)$ is generated by the reflections at $H_0$ and $H_{\pi/m}$. It is the symmetry group of the regular $m$-gon with its center at the origin and one of its vertices at $(1,0)$. The group $I_2(m)$ consists of the (finite) set of all reflections ${\varrho^{({H_{k\pi/m}})}}$ (for $k\in{\mathbbm{Z}}$) and the (finite) set of all rotations around the origin by angles $2k\pi/m$ (for $k\in{\mathbbm{Z}}$). We choose ${\Phi_{I_2(m)}}={\{{x\in{\mathbbm{R}}^2}:{x_2\ge 0,x\in H^{\le}_{\pi/m}}\}}$ as the fundamental domain.
\[prop:I\] Let $\mathcal{R}$ be induced by the sequence $({\operatorname{R}_{H^{\le}_{\pi/m}}},{\operatorname{R}_{H^{\le}_{2\pi/m}}},{\operatorname{R}_{H^{\le}_{4\pi/m}}},\dots,{\operatorname{R}_{H^{\le}_{2^r\pi/m}}})$ of reflection relations with $r=\lceil \log(m)\rceil$. If $P\subseteq{\mathbbm{R}}^2$ is a polytope with ${{v}^{({\Phi_{I_2(m)}})}}\in P$ for each vertex $v$ of $P$, then we have ${{\mathcal{R}}({P})}={\Pi_{I_2(m)}({P})}$.
With $Q={\Pi_{I_{2}(m)}({P})}$, the first condition of Theorem \[thm:seqReflRel\] is satisfied. Furthermore, we have $Q={\operatorname{conv}({W})}$ with $W={\{{\gamma.v}:{\gamma\in I_2(m),v\text{ vertex of }P}\}}$. Let $w\in W$ be some point with $w=\gamma.v$ for some vertex $v$ of $P$ and $\gamma\in I_2(m)$. Observing that $${\varrho^{\star({H^{\le}_{\pi/m}})}}\circ{\varrho^{\star({H^{\le}_{2\pi/m}})}}\circ\dots\circ
{\varrho^{\star({H^{\le}_{2^r\pi/m}})}}(w)$$ is contained in ${\Phi_{I_2(m)}}$, we conclude that it equals ${{w}^{({\Phi_{I_2(m)}})}}={{v}^{({\Phi_{I_2(m)}})}}\in P$. Therefore, also the second condition of Theorem \[thm:seqReflRel\] is satisfied. Hence the claim follows.
From Proposition \[prop:I\] and Remark \[rem:sizeSeqRefl\], we can conclude the following theorem.
\[thm:I\] For each polytope $P\subseteq{\mathbbm{R}}^2$ with ${{v}^{({\Phi_{I_2(m)}})}}\in P$ for each vertex $v$ of $P$ that admits an extended formulation with $n'$ variables and $f'$ inequalities, there is an extended formulation of ${\Pi_{I_2(m)}({P})}$ with $n'+\lceil \log(m)\rceil +1$ variables and $f'+2\lceil \log(m)\rceil+2$ inequalities.
In particular, we obtain an extended formulation of a regular $m$-gon with $\lceil \log(m)\rceil+1$ variables and $2\lceil \log(m)\rceil+2$ inequalities by choosing $P=\{(1,0)\}$ in Theorem \[thm:I\], thus reproving a result due to Ben-Tal and Nemirovski [@BN01].
### The reflection group $A_{n-1}$ {#subsubsec:An}
The group $A_{n-1}$ is generated by the reflections in ${\mathbbm{R}}^{n}$ at the hyperplanes ${\operatorname{H}^{=}({{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0})}$ for all pairwise distinct $k,\ell\in{[{n}]}$. It is the symmetry group of the $(n-1)$-dimensional (hence the index in the notation $A_{n-1}$) simplex $\operatorname{conv}\{{\mathbbm{e}_{1}},\dots,{\mathbbm{e}_{n}}\}\subseteq{\mathbbm{R}}^n$. We choose ${\Phi_{A_{n-1}}}={\{{x\in{\mathbbm{R}}^n}:{x_1 \le\dots\le x_n}\}}$ as the fundamental domain. The orbit of a point $x\in{\mathbbm{R}}^n$ under the action of $A_{n-1}$ consists of all points which can be obtained from $x$ by permuting coordinates. Thus the $A_{n-1}$-permutahedron of a polytope $P\subseteq{\mathbbm{R}}^n$ is $${\Pi_{A_{n-1}}({P})}=\operatorname{conv}\bigcup_{\gamma\in{\mathfrak{S}({n})}}\gamma.P\,,$$ where $\gamma.x$ is the vector obtained from $x\in{\mathbbm{R}}^n$ by permuting the coordinates according to $\gamma$.
Let us consider more closely the reflection relation ${\operatorname{T}_{{k},{\ell}}}={\operatorname{R}_{{{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0}}}\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^n$. The corresponding reflection ${\tau_{{k},{\ell}}}={\varrho^{({H_{k,\ell}})}}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^n$ with $H_{k,\ell}={\operatorname{H}^{=}({{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0})}$ is the transposition of coordinates $k$ and $\ell$, i.e., we have $${\tau_{{k},{\ell}}}(x)_{i}=
\begin{cases}
x_{\ell} & \text{if }i=k\\
x_k & \text{if }i={\ell}\\
x_i & \text{otherwise}
\end{cases}$$ for al $x\in{\mathbbm{R}}^n$. The map ${\tau^{\star}_{{k},{\ell}}}={\varrho^{\star({H_{k,\ell}})}}:{\mathbbm{R}}^n\rightarrow{\mathbbm{R}}^n$ (assigning canonical preimages) is given by $${\tau^{\star}_{{k},{\ell}}}(y)=
\begin{cases}
{\tau_{{k},{\ell}}}(y) & \text{if }y_k>y_{\ell}\\
y & \text{otherwise}
\end{cases}$$ for all $y\in{\mathbbm{R}}^n$.
A sequence $(k_1,\ell_1),\dots,(k_r,\ell_r)\in{[{n}]}\times{[{n}]}$ with $k_i\ne\ell_i$ for all $i\in{[{r}]}$ is called a *sorting network* if ${\tau^{\star}_{{k_1},{\ell_1}}}\circ\cdots\circ{\tau^{\star}_{{k_r},{\ell_r}}}(y)={{y}^{(\text{sort})}}$ holds for all $y\in{\mathbbm{R}}^n$, where we denote by ${{y}^{(\text{sort})}}\in{\mathbbm{R}}^n$ the vector that is obtained from $y$ by sorting the components in non-decreasing order. Note that we have ${{y}^{({\Phi_{A_{n-1}}})}}={{y}^{(\text{sort})}}$ for all $y\in{\mathbbm{R}}^n$.
\[prop:sortingNetworks\] Let $\mathcal{R}$ be induced by a sequence $({\operatorname{T}_{{k_1},{\ell_1}}}, \ldots, {\operatorname{T}_{{k_r},{\ell_r}}})$ of reflection relations, where $(k_1,\ell_1),\dots,(k_r,\ell_r)\in{[{n}]}\times{[{n}]}$ is a sorting network. If $P\subseteq{\mathbbm{R}}^n$ is a polytope with ${{v}^{(\text{sort})}}\in P$ for each vertex $v$ of $P$, then we have ${{\mathcal{R}}({P})}={\Pi_{A_{n-1}}({P})}$.
With $Q={\Pi_{A_{n-1}}({P})}$, the first condition of Theorem \[thm:seqReflRel\] is satisfied. Furthermore, we have $Q={\operatorname{conv}({W})}$ with $W={\{{\gamma.v}:{\gamma\in{\mathfrak{S}({n})},v\text{ vertex of }P}\}}$. As, for every $w\in W$ with $w=\gamma.v$ for some vertex $v$ of $P$ and $\gamma\in{\mathfrak{S}({n})}$, we have $${\tau^{\star}_{{k_1},{\ell_1}}}\circ\cdots\circ{\tau^{\star}_{{k_r},{\ell_r}}}(w)={{w}^{(\text{sort})}}={{v}^{(\text{sort})}}\in P\,,$$ also the second condition of Theorem \[thm:seqReflRel\] is satisfied. Hence the claim follows.
As there are sorting networks of size $r={\operatorname{O}({n\log n})}$ (see [@AKS83]), from Proposition \[prop:sortingNetworks\] and Remark \[rem:sizeSeqRefl\] we can conclude the following theorem
\[thm:sortingNetworks\] For each polytope $P\subseteq{\mathbbm{R}}^n$ with ${{v}^{(\text{sort})}}\in P$ for each vertex $v$ of $P$ that admits an extended formulation with $n'$ variables and $f'$ inequalities, there is an extended formulation of ${\Pi_{A_{n-1}}({P})}$ with $n'+{\operatorname{O}({n\log n})}$ variables and $f'+{\operatorname{O}({n\log n})}$ inequalities.
Choosing the one-point polytope $P=\{(1,2,\dots,n)\}\subseteq{\mathbbm{R}}^n$, Theorem \[thm:sortingNetworks\] yields basically the same extended formulation with ${\operatorname{O}({n\log n})}$ variables and inequalities of the permutahedron ${\operatorname{P}_{\operatorname{perm}}^{n}}={\Pi_{A_{n-1}}({P})}$ that has been constructed by Goemans [@Goe09] (see the remarks in the Introduction).
### The reflection group $B_n$
The group $B_{n}$ is generated by the reflections in ${\mathbbm{R}}^{n}$ at the hyperplanes ${\operatorname{H}^{=}({{\mathbbm{e}_{k}}+{\mathbbm{e}_{\ell}}},{0})}$, ${\operatorname{H}^{=}({{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0})}$ and ${\operatorname{H}^{=}({{\mathbbm{e}_{k}}},{0})}$ for all pairwise distinct $k,\ell\in{[{n}]}$. It is the symmetry group of both the $n$-dimensional cube $\operatorname{conv}\{-1,+1\}^n$ and the $n$-dimensional cross-polytope $\operatorname{conv}\{\pm{\mathbbm{e}_{1}},\dots,\pm{\mathbbm{e}_{n}}\}$. We choose ${\Phi_{B_{n}}}={\{{x\in{\mathbbm{R}}^n}:{0\le x_1 \le\dots\le x_n}\}}$ as the fundamental domain. The orbit of a point $x\in{\mathbbm{R}}^n$ under the action of $B_{n}$ consists of all points which can be obtained from $x$ by permuting its coordinates and changing the signs of some subset of its coordinates. Note that we have ${{y}^{({\Phi_{B_{n}}})}}={{y}^{(\text{sort-abs})}}$ for all $y\in{\mathbbm{R}}^n$, where ${{y}^{(\text{sort-abs})}}={{v'}^{(\text{sort})}}$ with $v'={{v}^{(\text{abs})}}$.
\[prop:Bn\] Let $\mathcal{R}$ be induced by a sequence $({\operatorname{T}_{{k_1},{\ell_1}}}, \ldots, {\operatorname{T}_{{k_r},{\ell_r}}},S_1, \ldots, S_n)$ of reflection relations, where $(k_1,\ell_1),\dots,(k_r,\ell_r)\in{[{n}]}\times{[{n}]}$ is a sorting network (and the $S_i$ are defined as at the end of Section \[sec:reflRel\]). If $P\subseteq{\mathbbm{R}}^n$ is a polytope with ${{v}^{(\text{sort-abs})}}\in P$ for each vertex $v$ of $P$, then we have ${{\mathcal{R}}({P})}={\Pi_{B_{n}}({P})}$.
With $Q={\Pi_{B_{n}}({P})}$, the first condition of Theorem \[thm:seqReflRel\] is satisfied. Furthermore, we have $Q={\operatorname{conv}({W})}$ with $W={\{{\gamma.\epsilon.v}:{\gamma\in{\mathfrak{S}({n})},\epsilon\in\{-,+\}^n,v\text{ vertex of }P}\}}$. As, for every $w\in W$ with $w=\gamma.\epsilon.v$ for some vertex $v$ of $P$ and $\gamma\in{\mathfrak{S}({n})}$, $\epsilon\in\{-,+\}^n$, we have $${\tau^{\star}_{{k_1},{\ell_1}}}\circ\cdots\circ{\tau^{\star}_{{k_r},{\ell_r}}}\circ{\sigma^{\star}_{{1}}}\circ\dots\circ{\sigma^{\star}_{{n}}}(w)={{w}^{(\text{sort-abs})}}={{v}^{(\text{sort-abs})}}\in P\,,$$ also the second condition of Theorem \[thm:seqReflRel\] is satisfied. Hence the claim follows.
As for $A_{n-1}$, we thus can conclude the following from Proposition \[prop:Bn\] and Remark \[rem:sizeSeqRefl\].
For each polytope $P\subseteq{\mathbbm{R}}^n$ with ${{v}^{(\text{sort-abs})}}\in P$ for each vertex $v$ of $P$ that admits an extended formulation with $n'$ variables and $f'$ inequalities, there is an extended formulation of ${\Pi_{B_{n}}({P})}$ with $n'+{\operatorname{O}({n\log n})}$ variables and $f'+{\operatorname{O}({n\log n})}$ inequalities.
### The reflection group $D_n$ {#subsubsec:D}
The group $D_{n}$ is generated by the reflections in ${\mathbbm{R}}^{n}$ at the hyperplanes ${\operatorname{H}^{=}({{\mathbbm{e}_{k}}+{\mathbbm{e}_{\ell}}},{0})}$ and ${\operatorname{H}^{=}({{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0})}$ for all pairwise distinct $k,\ell\in{[{n}]}$. Thus, $D_n$ is a proper subgroup of $B_n$. It is not the symmetry group of a polytope. We choose ${\Phi_{D_{n}}}={\{{x\in{\mathbbm{R}}^n}:{|x_1|\le x_2 \le\dots\le x_n}\}}$ as the fundamental domain. The orbit of a point $x\in{\mathbbm{R}}^n$ under the action of $D_{n}$ consists of all points which can be obtained from $x$ by permuting its coordinates and changing the signs of an *even* number of its coordinates. For every $x\in{\mathbbm{R}}^n$, the point ${{x}^{({\Phi_{D_n}})}}$ arises from ${{x}^{(\text{sort-abs})}}$ by multiplying the first component by $-1$ in case $x$ has an odd number of negative components. For $k,\ell\in{[{n}]}$ with $k\ne\ell$, we denote the ordered pair $({\operatorname{R}_{{{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0}}}, {\operatorname{R}_{{-{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0}}})$ of reflection relations by $E_{k,\ell}$.
\[prop:Dn\] Let $\mathcal{R}$ be induced by a sequence $({\operatorname{T}_{{k_1},{\ell_1}}}, \ldots, {\operatorname{T}_{{k_r},{\ell_r}}},E_{1,2},\dots,E_{n-1,n})$ of polyhedral relations, where $(k_1,\ell_1),\dots,(k_r,\ell_r)\in{[{n}]}\times{[{n}]}$ is a sorting network. If $P\subseteq{\mathbbm{R}}^n$ is a polytope with ${{x}^{({\Phi_{D_n}})}}\in P$ for each vertex $v$ of $P$, then we have ${{\mathcal{R}}({P})}={\Pi_{D_{n}}({P})}$.
With $Q={\Pi_{D_{n}}({P})}$, the first condition of Theorem \[thm:seqReflRel\] is satisfied. Let us denote by $\{-,+\}^n_{\text{even}}$ the set of all $\epsilon\in\{-,+\}^n$ with an even number of components equal to minus. Then, we have $Q={\operatorname{conv}({W})}$ with $W={\{{\gamma.\epsilon.v}:{\gamma\in{\mathfrak{S}({n})},\epsilon\in\{-,+\}^n_{\text{even}},v\text{ vertex of }P}\}}$. For $k,\ell\in{[{n}]}$ with $k\ne\ell$, we define $\eta^{\star}_{k,\ell}={\varrho^{\star({{\operatorname{H}^{\le}({{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0})}})}}\circ{\varrho^{\star({{\operatorname{H}^{\le}({-{\mathbbm{e}_{k}}-{\mathbbm{e}_{\ell}}},{0})}})}}$. For each $y\in{\mathbbm{R}}^n$, the vector $\eta^{\star}_{k,\ell}(y)$ is the vector $y'\in\{y,\tau_{k,\ell}(y),\rho_{k,\ell}(y),\rho_{k,\ell}(\tau_{k,\ell}(y))\}$ with $|y'_k|\le y'_{\ell}$, where $\rho_{k,\ell}(y)$ arises from $y$ by multiplying both components $k$ and $\ell$ by $-1$. As, for every $w\in W$ with $w=\gamma.\epsilon.v$ for some vertex $v$ of $P$ and $\gamma\in{\mathfrak{S}({n})}$, $\epsilon\in\{-,+\}^n_{\text{even}}$, we have $${\tau^{\star}_{{k_1},{\ell_1}}}\circ\cdots\circ{\tau^{\star}_{{k_r},{\ell_r}}}\circ\eta^{\star}_{1,2}\circ\dots\circ\eta^{\star}_{n-1,n}(w)={{w}^{({\Phi_{D_n}})}}={{v}^{({\Phi_{D_n}})}}\in P\,,$$ also the second condition of Theorem \[thm:seqReflRel\] is satisfied. Hence the claim follows.
And again, similarly to the cases $A_{n-1}$ and $B_n$, we derive the following result from Proposition \[prop:Dn\] and Remark \[rem:sizeSeqRefl\].
\[thm:Dn\] For each polytope $P\subseteq{\mathbbm{R}}^n$ with ${{v}^{({\Phi_{D_n}})}}(v)\in P$ for each vertex $v$ of $P$ that admits an extended formulation with $n'$ variables and $f'$ inequalities, there is an extended formulation of ${\Pi_{D_{n}}({P})}$ with $n'+{\operatorname{O}({n\log n})}$ variables and $f'+{\operatorname{O}({n\log n})}$ inequalities.
If we restrict attention to the polytopes $P=\{(-1,1,\dots,1)\}\subseteq{\mathbbm{R}}^n$ and $P=\{(1,1,\dots,1)\}\subseteq{\mathbbm{R}}^n$, then we can remove the reflection relations $T_{i_1,j_1},\dots,T_{i_r,j_r}$ from the construction in Proposition \[prop:Dn\]. Thus, we obtain extended formulations with $2(n-1)$ variables and $4(n-1)$ inequalities of the convex hulls of all vectors in $\{-1,+1\}^n$ with an odd respectively even number of ones. Thus, applying the affine transformation of ${\mathbbm{R}}^n$ given by $y\mapsto \frac12({\mathbbm{1}_{}}-y)$, we derive extended formulations with $2(n-1)$ variables and $4(n-1)$ inequalities for the *parity polytopes* $\operatorname{conv}{\{{v\in\{0,1\}^n}:{\sum_i v_i\text{ odd}}\}}$ and $\operatorname{conv}{\{{v\in\{0,1\}^n}:{\sum_i v_i\text{ even}}\}}$, respectively (reproving a result by Carr and Konjevod [@CK04]).
Huffman Polytopes {#subsec:huffman}
-----------------
A vector $v\in{\mathbbm{R}}^n$ (with $n\ge 2$) is a *Huffman-vector* if there is a rooted binary tree with $n$ leaves (all non-leaf nodes having two children) and a labeling of the leaves by $1,\dots,n$ such that, for each $i\in{[{n}]}$, the number of arcs on the path from the root to the leaf labelled $i$ equals $v_i$. Let us denote by ${\operatorname{V}_{\operatorname{huff}}^{n}}$ the set of all Huffman-vectors in ${\mathbbm{R}}^n$, and by ${\operatorname{P}_{\operatorname{huff}}^{n}}={\operatorname{conv}({{\operatorname{V}_{\operatorname{huff}}^{n}}})}$ the *Huffman polytope*. Note that currently no linear description of ${\operatorname{P}_{\operatorname{huff}}^{n}}$ in ${\mathbbm{R}}^n$ is known. In fact, it seems that such descriptions are extremely complicated. For instance, Nguyen, Nguyen, and Maurras [@NNM10] proved that ${\operatorname{P}_{\operatorname{huff}}^{n}}$ has $(\Omega(n))!$ facets. It is easy to see that Huffman-vectors and -polytopes have the following properties.
\[obs:huff\]
1. For each $\gamma\in{\mathfrak{S}({n})}$, we have $\gamma.{\operatorname{V}_{\operatorname{huff}}^{n}}={\operatorname{V}_{\operatorname{huff}}^{n}}$.
2. For each $v\in{\operatorname{V}_{\operatorname{huff}}^{n}}$ there are at least two components of $v$ equal to $\max{\{{v_k}:{k\in{[{n}]}}\}}$.
3. For each $v\in{\operatorname{V}_{\operatorname{huff}}^{n}}$ ($n\ge 3$) and $v_i=v_j=\max{\{{v_k}:{k\in{[{n}]}}\}}$ for some pair $i<j$, we have $$(v_1,\dots,v_{i-1},v_i-1,v_{i+1},\dots,v_{j-1},v_{j+1},\dots,v_n)\in{\operatorname{V}_{\operatorname{huff}}^{n-1}}\,.$$
4. For each $w'\in{\operatorname{V}_{\operatorname{huff}}^{n-1}}$ ($n\ge 3$), we have $(w'_1,\dots,w'_{n-2},w'_{n-1}+1,w'_{n-1}+1)\in{\operatorname{V}_{\operatorname{huff}}^{n}}$.
For $n\ge 3$, let us define the embedding $$P^{n-1}={\{{(x_1,\dots,x_{n-2},x_{n-1}+1,x_{n-1}+1)}:{(x_1,\dots,x_{n-1})\in{\operatorname{P}_{\operatorname{huff}}^{n-1}}}\}}$$ of ${\operatorname{P}_{\operatorname{huff}}^{n-1}}$ into ${\mathbbm{R}}^n$.
Let $\mathcal{R}\subseteq{\mathbbm{R}}^n\times{\mathbbm{R}}^n$ be the polyhedral relation that is induced by the following sequence of transposition relations: $$\label{eq:TNSquare}
{\operatorname{T}_{{n-2},{n-1}}},
{\operatorname{T}_{{n-3},{n-2}}},
\dots,
{\operatorname{T}_{{2},{3}}},
{\operatorname{T}_{{1},{2}}},
{\operatorname{T}_{{n-1},{n}}},
{\operatorname{T}_{{n-2},{n-1}}},
\dots,
{\operatorname{T}_{{2},{3}}},
{\operatorname{T}_{{1},{2}}}$$ Then we have $\mathcal{R}(P^{n-1})={\operatorname{P}_{\operatorname{huff}}^{n}}$.
With $P=P^{n-1}$ and $Q={\operatorname{P}_{\operatorname{huff}}^{n}}$, the first condition of Theorem \[thm:seqReflRel\] is obviously satisfied (due to parts (1) and (4) of Observation \[obs:huff\]). We have $Q={\operatorname{conv}({W})}$ with $W={\operatorname{V}_{\operatorname{huff}}^{n}}$. Furthermore, for every $w\in W$ and $x=\tau^{\star}(w)$ with $$\label{eq:tauStarNSquare}
\tau^{\star}=
{\tau^{\star}_{{n-2},{n-1}}}\circ
{\tau^{\star}_{{n-3},{n-2}}}\circ
\dots\circ
{\tau^{\star}_{{2},{3}}}\circ
{\tau^{\star}_{{1},{2}}}\circ
{\tau^{\star}_{{n-1},{n}}}\circ
{\tau^{\star}_{{n-2},{n-1}}}\circ
\dots\circ
{\tau^{\star}_{{2},{3}}}\circ
{\tau^{\star}_{{1},{2}}}\,,$$ we have $x_n=x_{n-1}=\max{\{{w_i}:{i\in{[{n}]}}\}}$, hence part (3) of Observation \[obs:huff\] (with $i=n-1$ and $j=n$) implies $\tau^{\star}(w)\in P^{n-1}$. Therefore, the claim follows by Theorem \[thm:seqReflRel\].
From Remark \[rem:sizeSeqRefl\] we thus obtain an extended formulation for ${\operatorname{P}_{\operatorname{huff}}^{n}}$ with $n'+2n-3$ variables and $f'+4n-6$ inequalities, provided we have an extended formulation for ${\operatorname{P}_{\operatorname{huff}}^{n-1}}$ with $n'$ variables and $f'$ inequalities. As ${\operatorname{P}_{\operatorname{huff}}^{2}}$ is a single point, we thus can establish inductively the following result.
There are extended formulations of ${\operatorname{P}_{\operatorname{huff}}^{n}}$ with ${\operatorname{O}({n^2})}$ variables and inequalities.
Actually, one can reduce the size of the extended formulation of ${\operatorname{P}_{\operatorname{huff}}^{n}}$ to ${\operatorname{O}({n\log n})}$. In order to indicate the necessary modifications, let us denote by $\Theta_k$ the sequence $$(k-2,k-1),(k-3,k-2),\dots,(2,3),(1,2),(k-1,k),(k-2,k-1),\dots,(2,3),(1,2)$$ of pairs of indices used (with $k=n$) in and . For every sequence $\Theta=((i_1,j_1),\dots,(i_r,j_r))$ of pairs of pairwise different indices, we define $\tau^{\star}_{\Theta}={\tau^{\star}_{{i_1},{j_1}}}\circ\cdots\circ{\tau^{\star}_{{i_r},{j_r}}}$ (thus, $\tau^{\star}$ in equals $\tau^{\star}_{\Theta_n}$). Furthermore, we denote by $\eta_k:{\mathbbm{R}}^k\rightarrow{\mathbbm{R}}^{k-1}$ (for $k\ge 3$) the linear map defined via $\eta_k(y)=(y_1,\dots,y_{k-2},y_{k-1}-1)$ for all $y\in{\mathbbm{R}}^k$. The crucial property for the above construction to work is that the following holds for every $v\in {\operatorname{V}_{\operatorname{huff}}^{n}}$ and $k\in\{3,\dots,n\}$: The vector $$x=\tau^{\star}_{\Theta_{k}}\circ\eta_{k+1}\circ\tau^{\star}_{\Theta_{k+1}}\circ\cdots\circ
\eta_n\circ\tau^{\star}_{\Theta_n}(v)$$ satisfies $x_{k-1}=x_{k}=\max{\{{x_i}:{i\in{[{k}]}}\}}$. It turns out that this property is preserved when replacing the sequence $\Theta_n$ by an arbitrary sorting network (e.g. of size ${\operatorname{O}({n\log n})}$, see Section \[subsubsec:An\]) and, for $k\in\{3,\dots,n-1\}$, the sequence $\Theta_k$ (of length $2k-3$) by the sequence $$\begin{gathered}
(i^k_2,i^k_1),(i^k_3,i^k_2),\dots,(i^k_{r(k)-1},i^k_{r(k)-2}),(i^k_{r(k)},i^k_{r(k)-1}),(i^k_{r(k)-1},i^k_{r(k)-2}),\dots,(i^k_3,i^k_2),(i^k_2,i^k_1)\end{gathered}$$ with $i^k_1=k$, $i^k_2=k-1$, $i^k_{\ell}=i^k_{\ell-1}-2^{\ell-3}$ for all $\ell\ge 3$, and $r(k)$ being the maximal $\ell$ with $i^k_{\ell}\ge 1$. As $r(k)$ is bounded by ${\operatorname{O}({\log k})}$ we obtain the following theorem, whose detailed proof will be included in the full version of the paper.
There are extended formulations of ${\operatorname{P}_{\operatorname{huff}}^{n}}$ with ${\operatorname{O}({n\log n})}$ variables and inequalities.
Conclusions {#sec:concl}
===========
We hope to have demonstrated that and how the framework of reflection relations extends the currently available toolbox for constructing extended formulations. We conclude with briefly mentioning two directions for future research.
One of the most interesting questions in this context seems to be that for other polyhedral relations that can be useful for constructing extended formulations. In particular, what other types of affinely generated polyhedral relations are there?
The reflections we referred to are reflections at hyperplanes. It would be of great interest to find tools to deal with reflections at lower dimensional subspaces as well. This, however, seems to be much harder. In particular, it is unclear whether some concept similar to that of polyhedral relations can help here.
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[^1]: Supported by the *Max Planck Institute for Dynamics of Complex Technical Systems*.
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---
abstract: 'Simulations of galaxy clusters have a difficult time reproducing the radial gas-property gradients and red central galaxies observed to exist in the cores of galaxy clusters. Thermal conduction has been suggested as a mechanism that can help bring simulations of cluster cores into better alignment with observations by stabilizing the feedback processes that regulate gas cooling, but this idea has not yet been well tested with cosmological numerical simulations. Here we present cosmological simulations of ten galaxy clusters performed with five different levels of isotropic Spitzer conduction, which alters both the cores and outskirts of clusters, but not dramatically. In the cores, conduction flattens central temperature gradients, making them nearly isothermal and slightly lowering the central density but failing to prevent a cooling catastrophe there. Conduction has little effect on temperature gradients outside of cluster cores because outward conductive heat flow tends to inflate the outer parts of the intracluster medium (ICM) instead of raising its temperature. In general, conduction tends reduce temperature inhomogeneity in the ICM, but our simulations indicate that those homogenizing effects would be extremely difficult to observe in $\sim 5$ keV clusters. Outside the virial radius, our conduction implementation lowers the gas densities and temperatures because it reduces the Mach numbers of accretion shocks. We conclude that despite the numerous small ways in which conduction alters the structure of galaxy clusters, none of these effects are significant enough to make the efficiency of conduction easily measurable unless its effects are more pronounced in clusters hotter than those we have simulated.'
author:
- 'Britton Smith, Brian W. O’Shea, G. Mark Voit, David Ventimiglia'
- 'Samuel W. Skillman'
title: Cosmological Simulations of Isotropic Conduction in Galaxy Clusters
---
Introduction
============
Numerical simulations of galaxy clusters have not yet succeeded in producing objects with properties identical to those of observed galaxy clusters. The most serious discrepancies with observations are in the cores [e.g, @2007ApJ...668....1N; @2011ASL.....4..204B; @2013ApJ...763...38S]. Within the central $\sim$100 kpc of clusters whose central cooling time is less than a Hubble time, radiative cooling of intracluster gas in the simulations tends to be too efficient, leading to overproduction of young stars and excessive entropy levels in the core gas, as higher-entropy material flows inward to replace the gas that has condensed. The resulting cluster cores in such simulations consequently have temperature profiles that decline from $\sim$10 kpc outward, in stark disagreement with the observed temperature profiles of cool-core clusters, which rise in the $\sim$10–100 kpc range.
Feedback from a central active galactic nucleus helps solve this problem because it slows the process of cooling. However, the core structures of clusters produced in cosmological simulations depend sensitively on the details of the implementation of feedback, the radiative cooling scheme, and numerical resolution. For example, @2011MNRAS.417.1853D simulated the same cosmological galaxy cluster with two different mechanisms for feedback and two different cooling algorithms, one with metal-line cooling and one without. The core structures of the resulting clusters were quite diverse, with kinetic jet feedback and metal-free cooling producing the most realistic-looking cores. But discouragingly, allowing cooling via metal lines had the effect of supercharging AGN feedback and producing a cluster core with excessively high entropy.
Another oft-proposed solution to the core-structure problem in galaxy clusters is thermal conduction. It has been invoked many times to mitigate the prodigious mass flows predicted for cool-core clusters lacking a central heat source [e.g., @1983ApJ...267..547T; @1986ApJ...306L...1B; @2001ApJ...562L.129N] but cannot be in stable balance with cooling [e.g., @1988ApJ...326..639B; @2003MNRAS.342..463S; @2008ApJ...688..859G] and does not completely compensate for radiative cooling in all cluster cores [@2002MNRAS.335L...7V; @2003ApJ...582..162Z]. Nevertheless, the temperature and density profiles of many cool-core clusters are tantalizingly close to being in conductive balance, suggesting that AGN feedback is triggered only when inward thermal conduction, perhaps assisted by turbulent heat transport, fails to compensate for radiative losses from the core [e.g., @2011ApJ...740...28V].
The nebular line emission observed in many cool-core clusters also suggests that thermal conduction may be important. Ultraviolet light from young stars accounts for some of the line emission but cannot explain all of it, implying that another heat source is present [e.g., @1987MNRAS.224...75J; @1997ApJ...486..242V; @2013ApJ...767..153W]. Observations of the optical and infrared line ratios are consistent with heat input by a population of suprathermal electrons, possibly entering the nebular gas from the ambient hot medium [@2009MNRAS.392.1475F]. The emission-line studies are not conclusive on this point, but recent observations of ultraviolet emission from a filament in the Virgo cluster have bolstered the case for conduction [@2009ApJ...704L..20S; @Sparks+11]. The C IV 1550 Å and He II 1640 Å emission lines from this filament are consistent with a model in which the filament is surrounded by a conductive sheath that channels heat into the filament.
If conduction is indeed important in determining the structure of cluster cores and mediating the heat flow into cool gas clouds, then it ought to be included in cosmological simulations of galaxy clusters. @2004ApJ...606L..97D presented the first simulations of conductive intracluster media in cosmological galaxy clusters, using the @2004MNRAS.351..423J implementation of isotropic conduction in the `GADGET` smooth-particle hydrodynamics code. These simulations were notable in that they vividly demonstrated the inability of conduction to prevent a cooling catastrophe. Even at 1/3 of the full Spitzer rate, thermal conduction could not prevent a large cooling flow from developing in the cluster core. Conduction delayed the cooling catastrophe but did not diminish the magnitude of the eventual mass inflow. An updated set of cluster simulations using the same conduction algorithm was recently performed by @2011MNRAS.416..801F. Isotropic conductivity was set to 1/3 of the full Spitzer value for simulations including cooling and star formation but without AGN feedback. As with the previous simulation set, there was not much difference between the global properties of the final states of clusters simulated with and without thermal conduction.
Conduction is also expected to smooth out temperature inhomogeneities in hot clusters [e.g., @2003ApJ...586L..19M]. One of the clusters simulated by @2004ApJ...606L..97D was quite hot, with a peak gas temperature of $\sim 12$ keV, and conduction in that cluster produced a much more homogeneous ICM, with far less azimuthal temperature variation than its conduction-free counterpart. Producing simulated clusters with small-scale temperature variations similar to those of observed clusters continues to be somewhat tricky [see, for example, @2012ApJ...747..123V]. In particular, some recent simulations of sloshing, magnetized cluster cores with anisotropic conduction are suggesting that conduction needs to be somewhat suppressed [*along*]{} the magnetic field lines into order to explain observations of cluster cores with cold fronts [@2013ApJ...762...69Z]
Implementations of anisotropic conduction directed along magnetic field lines are also of interest because the MHD instabilities that result may have important implications for the structure of cluster cores and the generation of turbulence in cluster outskirts [e.g., @2009ApJ...703...96P; @2012MNRAS.419L..29P]. @2011ApJ...740...81R recently simulated a cosmological cluster with anisotropic conduction using the `FLASH` adaptive mesh-refinement (AMR) code. Runs were performed with radiative cooling on and off, but no AGN feedback was included. In this case, anisotropic conduction significantly reduced mass accretion into the central galaxy, even though the magnetic field geometry significantly suppressed radial heat conduction. However, the $\sim 31$ kpc/$h$ effective resolution was insufficient to adequately resolve core structure.
In this paper we present a new set of cosmological cluster simulations performed with the AMR code `Enzo`, including isotropic Spitzer conduction. We simulate 10 clusters spanning a mass range of $2 - 8 \times 10^{14} \, M_\odot$. We also model a range of suppression factors in order to evaluate the effects of thermal conductivity on core structure. As with previous calculations, we do not include AGN feedback, and a cooling catastrophe therefore results, once again confirming that conduction alone cannot prevent strong cooling flows from developing in galaxy clusters. However, our focus here is on how conduction affects the radial profiles and homogeneity of gas density and temperature outside the central $\sim 20$ kpc.
Our presentation of these conductive cluster simulations proceeds as follows. In § 2 we describe the primary features of the `Enzo` code employed in this work and describe our recent implementation of isotropic Spitzer conduction, with the simulation setup given in § 3. Section 4 discusses the qualitative morphological properties of the simulated clusters, while § 5 compares their radial profiles. Section 6 focuses on conduction affects the thermal homogeneity of the ICM, § 7 looks at the star formation histories of the clusters, and § 8 summarizes our results.
Numerical methods
=================
In this work, we use the open-source cosmological, adaptive mesh-refinement (AMR) hydrodynamic + N-body code `Enzo`[^1] [@bryan97; @bryan99; @norman99; @oshea04; @2005ApJS..160....1O]. Below, we describe the primary features of `Enzo` used in this work, including newly-added machinery for solving thermal conduction. All analysis is performed with the open-source simulation analysis toolkit, `yt`[^2] [@2011ApJS..192....9T] using the Parallel-HOP halo finder [@2010ApJS..191...43S].
The Enzo code {#sec:enzo}
-------------
The `Enzo` code couples an N-body particle-mesh (PM) solver [@Efstathiou85; @Hockney88] used to follow the evolution of a collisionless dark matter component with an Eulerian AMR method for ideal gas dynamics by @Berger89, which allows high dynamic range in gravitational physics and hydrodynamics in an expanding universe. This AMR method (referred to as *structured* AMR) utilizes an adaptive hierarchy of grid patches at varying levels of resolution. Each rectangular grid patch (referred to as a “grid”) covers some region of space in its *parent grid* which requires higher resolution, and can itself become the parent grid to an even more highly resolved *child grid*. `Enzo`’s implementation of structured AMR places no fundamental restrictions on the number of grids at a given level of refinement, or on the number of levels of refinement. However, owing to limited computational resources it is practical to institute a maximum level of refinement, $\ell_{max}$. Additionally, the `Enzo` AMR implementation allows arbitrary integer ratios of parent and child grid resolution, though in general for cosmological simulations (including the work described in this paper) a refinement ratio of 2 is used.
Multiple hydrodynamic methods are implemented in `Enzo`. For this work, we use the method from the ZEUS code [@stone92a; @stone92b] for its robustness in conditions with steep internal energy gradients, which are common in cosmological structure formation when radiative cooling is used. We use the metallicity-dependent radiative cooling method described in @2008MNRAS.385.1443S and @2011ApJ...731....6S. This method solves the non-equilibrium chemistry and cooling for atomic H and He [@abel97; @anninos97] and calculates the cooling and heating from metals by interpolating from multi-dimensional tables created with the photo-ionization software, `Cloudy`[^3] [@1998PASP..110..761F]. Both the primordial and metal cooling also take into account heating from a time-dependent, spatially uniform metagalactic UV background [@2001cghr.confE..64H].
To model the effects of star formation and feedback, we use a modified version of the algorithm presented by @1992ApJ...399L.113C. A grid cell is capable of forming stars if the following criteria are met: the baryon overdensity is above some threshold (here 1000), the velocity divergence is negative (i.e., the flow is converging), the gas mass in the cell is greater than the Jeans mass, and the cooling time is less than the self-gravitational dynamical time. Because conduction can potentially balance radiative cooling, instead of the classical definition of the cooling time, $e/\dot{e}$, we use a modified cooling time that includes the change in energy from conduction, defined as $$t_{cool} = \frac{e}{\dot{e}_{rad} + \dot{e}_{cond}},$$ where $\dot{e}_{rad}$ is the cooling rate and $\dot{e}_{cond}$ is the conduction rate. In the original implementation, if all of the above criteria are satisfied, then a star particle representing a large, coeval group of stars with the following mass is formed: $$\label{eq:m_star}
m_{*} = f_{*} \ m_{\rm cell} \ \frac{\Delta t}{t_{\rm dyn}} \; ,$$ where $f_{*}$ is an efficiency parameter, $m_{\rm cell}$ is the baryon mass in the cell, $t_{\rm dyn}$ is the dynamical time of the combined baryon and dark matter fluid, and $\Delta t$ is the timestep taken by the grid containing the cell in question. The factor of $t_{dyn}$ is added to provide a connection between the physical conditions of the gas and the timescale over which star formation and feedback occur. When a star particle is created, feedback in the form of thermal energy, gas, and metals is returned to the grid at a rate given by $$\label{eq:m_star_form}
\Delta m_{\rm sf} = m_{*} \ \frac{\Delta t}{t_{\rm dyn}} \ \frac{(t - t_{*})}
{t_{\rm dyn}} \ e^{-(t - t_{*}) / t_{\rm dyn}} \; ,$$ where $t_{*}$ is the creation time for the particle, so that $\Delta m_{\rm
sf}$ rises linearly over one dynamical time, then falls off exponentially after that. This feature ensures that the feedback response unfolds over a dynamical time, even if star formation in the simulation is formally instantaneous. We also use a distributed feedback model, in which feedback material is distributed evenly over a $3\times3\times3$ cube of cells centered on the star particle’s location. This was shown by @2011ApJ...731....6S to more effectively transport hot, metal-enriched gas out of galaxies and to minimize overcooling issues that are common to simulations of structure formation. This method was also shown by @2013ApJ...763...38S to provide a better match to observed cluster properties than injecting all feedback into a single grid cell.
In practice, `Enzo` simulations often implement an additional restriction, allowing creation of a star particle only if $m_{*}$ is above some minimum mass. This prevents the formation of a large number of low-mass star particles whose presence can significantly slow down a simulation. However, the use of this algorithm in concert with conduction must be done with care. As we discuss in § \[sec:conduct\], explicit modeling of thermal conduction can require timesteps that are significantly shorter than the hydrodynamic Courant condition, which has the effect of considerably reducing the value of $m_{*}$. If the value of $m_{*}$ is only slightly less than the minimum particle size, this condition will likely be satisfied in a relatively short time as condensation proceeds, causing the gas density to increase and its dynamical time to decrease. When the conduction algorithm is active, the resulting time delay can allow cold, dense gas to persist while in thermal contact with the hot ICM. Since radiative cooling is proportional to the square of the density, heat conducted into high-density gas can be easily radiated away. This artificial time delay therefore produces a spurious heat sink in the ICM that can potentially boost the rates of cooling and star formation. In order to avoid this unphysical behavior, we remove the factor of $\Delta
t/t_{dyn}$ from Equation \[eq:m\_star\] and adopt a constant value of 10 Myr for $t_{dyn}$ for use in Equation \[eq:m\_star\_form\]. We have tested the effects of these changes by comparing two simulations run without conduction using both the original star-formation implementation and this modification and find the stellar masses at any given time differ by less than one percent.
Conduction Algorithm {#sec:conduct}
--------------------
We implement the equations of isotropic heat conduction in a manner similar to that of @2004MNRAS.351..423J, where the heat flux, $j$, for a temperature field, $T$, is given by $$j = -\kappa \nabla T,$$ where $\kappa$ is the conductivity coefficient. In an ionized plasma, heat transport is mediated by Coulomb interactions between electrons. In this scenario, referred to as Spitzer conduction [@1962pfig.book.....S], the conductivity coefficient is given by
$$\label{eq:kappa_sp}
\kappa_{sp} = 1.31 n_{e} \lambda_{e} k \left( \frac{k_BT_{e}}{m_{e}} \right)^{1/2},$$
where $m_{e}$, $n_{e}$, $\lambda_{e}$, and $T_{e}$ are the electron mass, number density, mean free path, and temperature, and $k_B$ is the Boltzmann constant. The electron mean free path is $$\lambda_{e} = \frac{3^{3/2}(k_BT)^{2}}{4\pi^{1/2}e^{4}n_{e} \ln \Lambda},$$ where $e$ is the electron charge and $\ln \Lambda$ is the Coulomb logarithm, defined to be the ratio of the maximum and minimum impact parameters over which Coulomb collisions yield significant momentum change in the interacting particles, which are electrons in this case. The Coulomb logarithm for electron-electron collisions is
$$\ln \Lambda = 23.5 - \ln\ (n_{e}^{1/2} T_{e}^{-5/4}) - \left( \frac{10^{-5} +
(\ln\ (T_{e}) - 2)^{2}}{16} \right)^{1/2}$$
[@Huba2011 pg. 34]. Because of its relative insensitivity to $n_{e}$ and $T_{e}$, we follow @2004MNRAS.351..423J and @1988xrec.book.....S and adopt a constant value of $\ln \Lambda
= 37.8$, corresponding to $n_{e} = 1$ cm$^{-3}$ and $T_{e} = 10^{6}$ K. For values more appropriate to the ICM ($n_{e} = 10^{-3}$ cm$^{-3}$, $T_{e} = 10^{7}$ K), $\ln \Lambda = 43.6$. At a constant density, $\ln \Lambda$ varies by $\sim$10-20% over the range of temperatures relevant here. The Spitzer conductivity then reduces to $$\kappa_{sp} = 4.9 \times 10^{-7}~T^{5/2}~{\rm erg~s^{-1}~cm^{-1}~K^{-1}}.$$ Since $\ln \Lambda$ increases with $T$ and $\kappa_{sp} \propto 1 /
\ln \Lambda$, including a direct calculation of $\ln \Lambda$ would serve to somewhat soften dependence of $\kappa_{sp}$ on $T$. In low density plasmas with large temperature gradients, the characteristic length scale of the temperature gradient, $\ell_T
\equiv T / |\nabla T|$, can be smaller than the electron mean free path, at which point Equation \[eq:kappa\_sp\] no longer applies. Instead, the maximum allowable heat flux is described by a saturation term [@1977ApJ...211..135C], given as $$j_{sat} \simeq 0.4 n_e k_B T \left( \frac{2 k T}{\pi m_e} \right)^{1/2}.$$ To smoothly connect the saturated and unsaturated regimes, we use an effective conductivity [@1988xrec.book.....S] given by $$\kappa_{eff} = \frac{\kappa_{sp}}{1 + 4.2 \lambda_e / \ell_T}.$$ The rate of change of the internal energy, $u$, due to conduction is then $$\label{eq:cond}
\frac{du}{dt} = - \frac{1}{\rho} \nabla \cdot j.$$
Our numerical implementation closely follows that of @2005ApJ...633..334P. We use an explicit, first-order, forward time, centered space algorithm, which is the most straightforward to implement in an AMR code. For each cell, Equation \[eq:cond\] is solved by summing the heat fluxes from all grid cell faces, and calculating the electron density and temperature on the grid cell face as the arithmetic mean of the cell and its neighbor sharing that face. The time step stability criterion for an explicit solution of the conduction equation is $$\label{eq:t_cond}
dt < 0.5 \frac{\Delta x^2}{\alpha}
\label{eq-dt}$$ where $\Delta x$ is the grid cell size and $\alpha$ is the thermal diffusivity, defined as $$\alpha = \frac{\kappa}{n_e k_B}.$$ Equation (\[eq-dt\]) has the potentially to be considerably more constraining than the hydrodyamical Courant condition, which is proportional to $T^{-1/2}\Delta x $, and so the conduction timestep in conditions typical of the ICM is often much shorter than the hydrodynamic timestep. The conduction timestep is calculated on a per-level basis and is taken to be the minimum of all such values on a given level. `Enzo` pads each AMR grid patch with three rows of ghost zones from neighboring grids. This allows the conduction routine to take three sub-cycled time steps for every hydrodynamic step, and thus allows the minimum grid timestep to be a factor of three larger than Equation \[eq:t\_cond\]. After three steps, temperature information from the outermost ghost zone has propagated to the edge of the active region of the grid, so performing additional conduction cycles would yield inconsistent solutions with neighboring grids. By default, `Enzo` rebuilds the adaptive mesh hierarchy for a given refinement level after each timestep, which is computationally expensive. However, because conduction does not explicitly change the density field (the value of which is the only quantity that determines mesh refinement), we modify this behavior such that the hierarchy is only rebuilt after an amount of time has passed equivalent to what the minimum timestep would be if conduction were not enabled.
This implementation models the case of isotropic Spitzer conduction, where heat flows unimpeded along temperature gradients. If magnetic fields are present, and strong enough that the electron gyroradius is small compared to the physical scales of interest (which is likely to be true in the intracluster medium), then heat is restricted to flow primarily along magnetic field lines, so the heat flux becomes the dot product of the temperature gradient with the magnetic field direction (i.e., $j = -\kappa {\mathbf{b}} {\mathbf{b}} \cdot \nabla T$, where ${\mathbf{b}}$ is the unit vector pointing in the direction of the magnetic field). Diffusion perpendicular to magnetic field lines is generally considered to be negligible. We do not consider magnetic fields in this work, but instead approximate the suppression of conduction by magnetic fields by adding a suppression factor, $\fsp$, varying from 0 to 1, to the heat flux calculation. In reality, the strength and orientation of magnetic fields in galaxy clusters is poorly understood. Hence, the degree to which conduction is suppressed from its maximum efficiency is not known. The extreme limit of tangled magnetic fields is equivalent to isotropic conduction with $\fsp =
1/3$. A number of works have shown that the presence of magnetic fields alongside conduction can create a variety of instabilities that greatly influence the effective level of isotropic heat transport [e.g., @2005ApJ...633..334P; @2007ApJ...664..135P; @2010ApJ...713.1332R; @2011MNRAS.413.1295M; @2012MNRAS.419.3319M]. For this reason, we simulate a large range of values of $\fsp$, from strongly suppressed ($\fsp = 0.01$) to fully unsuppressed ($\fsp = 1$).
Simulation Setup
================
The simulations described in this work are are initialized at $z=99$ assuming the WMAP Year 7 “best fit” cosmological model [@2011ApJS..192...16L; @2011ApJS..192...18K]: $\Omega_m = 0.268$, $\Omega_b = 0.0441$, $\Omega_{CDM} = 0.2239$, $\Omega_\Lambda =
0.732$, $h=0.704$ (in units of 100 km/s/Mpc), $\sigma_8 = 0.809$, and using an Eisenstein & Hu power spectrum [@eishu99] with a spectral index of $n = 0.96$. We use a single cosmological realization with a box size of 128 Mpc/$h$ (comoving) and a simulation resolution of 256$^{3}$ root grid cells and dark matter particles. From this realization, we select the 10 most massive clusters and resimulate each individually, allowing adaptive mesh refinement only in a region that minimally encloses the initial positions of all particles that end up in the halo in question at $z = 0$. We allow a maximum of 5 levels of refinement, each by a factor of 2, refining on baryon and dark matter overdensities of 8. This gives us a maximum comoving spatial resolution of 15.6 kpc/$h$. We set the `Enzo` parameter `MinimumMassForRefinementLevelExponent` to $-0.2$ for both dark matter and baryon-based refinement, resulting in slightly super-Lagrangian refinement behavior. For each halo in our sample, we perform a control simulation with $\fsp = 0$ and conductive simulations with $\fsp = 0.01$, 0.1, 0.3, and 1, evolving each to $z = 0$. All of the simulations employ the star formation/feedback and radiative cooling methods described above in §\[sec:enzo\]. Table \[tab:halos\] lists the masses and equivalent temperatures for all 10 clusters in the sample.
[lcc]{} 0 & 8.04 & 5.56\
1 & 8.26 & 5.66\
2 & 5.60 & 4.37\
3 & 5.56 & 4.35\
4 & 3.76 & 3.35\
5 & 4.64 & 3.85\
6 & 3.85 & 3.40\
7 & 3.52 & 3.20\
8 & 1.91 & 2.13\
9 & 3.41 & 3.14\
\[tab:halos\]
Qualitative Morphology
======================
![Projections of a 500 kpc thick slab centered on the most massive cluster at $z = 0$ ($M_{200} = 8.0\times10^{14}\ M\subsun,\ T_{200} = 5.6$ keV) for the simulations with $\fsp = 0$ (left) and 1. (right) The top and middle rows show average density and temperature, weighted by X-ray emissivity in the energy range of 0.5 to 2 keV. The bottom row shows the X-ray surface brightness in the same energy range. []{data-label="fig:thin_proj_0"}](thin_projections_halo_0000.pdf){width="45.00000%"}
![Projections of a 500 kpc thick slab centered on the third most massive cluster at $z = 0$ (M$_{200} = 5.6\times10^{14}\ M\subsun,\ T_{200} = 4.4$ keV) for the simulations with $\fsp = 0$ (left) and 1 (right). []{data-label="fig:thin_proj_2"}](thin_projections_halo_0002.pdf){width="45.00000%"}
![Projections of a 500 kpc thick slab centered on the sixth most massive halo at $z = 0$ (M$_{200} = 4.6\times10^{14}\ M\subsun,\ T_{200} = 3.9$ keV) for the simulations with $\fsp = 0$ (left) and 1 (right). []{data-label="fig:thin_proj_5"}](thin_projections_halo_0005.pdf){width="45.00000%"}
Perhaps the most surprising finding to emerge from our simulations is the insensitivity of the qualitative morphological features of the ICM to thermal conduction, even in clusters with typical gas temperatures $\sim 5$ keV. Figures \[fig:thin\_proj\_0\], \[fig:thin\_proj\_2\], and \[fig:thin\_proj\_5\] show gas density, temperature, and X-ray surface brightness projections for three clusters at the minimum and maximum conduction efficiency. We choose these clusters in order to capture a wide range of potential influence from conduction. Halo 0 is the most massive cluster in the sample (with a mass of $8 \times
10^{14}$ M$_\odot$), Halo 2 is a highly unrelaxed cluster, and Halo 5 is relatively relaxed and is close to the average mass for the sample (M$_{200}$ $\simeq 4.9 \times
10^{14}$ M$_\odot$) In order to highlight differences between values of $\fsp$, we limit the projected line of sight to 0.5 Mpc. The morphological similarity between these conductive and non-conductive clusters starkly contrasts with the obvious differences found by @2004ApJ...606L..97D for a $\sim 12$ keV cluster. In their work, it is immediately obvious which simulation included conduction.
There are, however, some subtle but noticeable morphological differences between our conductive and non-conductive clusters. The overall density structure is slightly smoother for $\fsp = 1$ in all three clusters, and the temperature field is visibly smoother in Halo 5. In all three clusters shown, the core appears to be marginally less dense with conduction on and is also markedly hotter in Halos 2 and 5. Nevertheless, while it is for the most part apparent that conduction is present in our cluster sample with $\fsp = 1$, its effects are not nearly as striking as in @2004ApJ...606L..97D. Quite possibly, this difference is due to the strong temperature dependence of Spitzer conductivity, suggesting that it may be possible to estimate the typical conductivity of the ICM by analyzing the dependence of azimuthal temperature homogeneity on cluster temperature (or lack thereof) in a large sample of galaxy clusters.
Radial Profiles {#sec:profiles}
===============
Systematic conductivity-dependent differences among our simulated clusters are hard to see in individual cases but are more apparent in comparisons of average properties of sample sets with differing conductivity. Therefore, in order to understand the systematic effects of increasing the conduction efficiency, we aggregate the radial profiles from all the clusters in our simulated sample for each value of $\fsp$, as shown in Figure \[fig:radial\_profiles\]. To minimize artifacts that result from the rebinning of profile data, we perform the initial profiling in physical distance units to calculate the virial radius for each cluster (which we take for convenience to be $r_{200}$ with respect to the critical density). We then create a second set of profiles in units of $\rnorm$ that are combined to make the aggregate profiles presented here. We refer to these as “averaged profiles” and employ this method for Figures \[fig:radial\_profiles\] and \[fig:potential\_profiles\] and all subsequent figures showing aggregate properties.
Cluster Interiors {#sec:cluster_interiors}
-----------------
The presence of conduction leads to higher gas temperatures both inside and outside of the conduction-free cluster’s characteristic temperature peak at $\rnorm \sim 0.07$. Within $\rnorm \sim 0.07$, gas temperatures in conductive clusters are greater than those in the non-conductive control simulation, with values of $\fsp \gtrsim 0.1$ leading to nearly isothermal cores and central temperatures $\sim 40$% greater than in non-conductive cluster simulations. The temperature profiles of our conductive clusters are in reasonably good agreement with those of @2011ApJ...740...81R for their cluster with isotropic conduction and $\fsp = 1/3$, although the core of their cluster is less isothermal than ours, with cooler temperatures near the center. This could be because the simulations of @2011ApJ...740...81R did not include a prescription for star formation and feedback in cluster galaxies, which can increase the core temperature both by consuming the cold gas and through thermal and mechanical feedback. Agreement with the temperature profiles of the simulated clusters from @2004ApJ...606L..97D is not nearly as good. Those clusters show a significant reduction in both the peak and central temperatures with conduction present, which we do not observe. The theoretical temperature profiles of @2013MNRAS.tmp.1104M for the $10^{14.5} M\subsun$ clusters show an elevation in the core temperatures in conductive clusters in rough agreement with our results. We also see a marginal inward movement of the location of the temperature peak as they predict, although not nearly to the same degree. The temperature peak for our clusters is also at a smaller radius to begin with.
Despite its ability to maintain approximate isothermality in the cluster cores, conduction at even maximal efficiency is unable to avert a cooling catastrophe at the very center of the halo. This cooling catastrophe and the resulting condensation and star-formation activity produce sharp peaks at $\rnorm \lesssim 0.03$ in all the gas density profiles in Figure \[fig:radial\_profiles\], as well as a slight temperature increase at the same location in the more highly conductive clusters. The elevated central temperatures at the centers of conductive clusters reduce the central gas density relative to the control clusters by $\sim$20-30%. Material that would have fallen into the center is displaced to larger radii, as can be seen by the elevated density at $0.04 \lesssim \rnorm \lesssim 0.6$ in the more conductive runs. Gas density is enhanced primarily in the range $0.04 \lesssim \rnorm \lesssim 0.2$, but the enhancement appears to saturate at $\sim$20% for $\fsp = 0.33$. For $\fsp =
1$, the density is actually lower than for $\fsp = 0.33$ in this range, but is then higher for $0.2 \lesssim \rnorm \lesssim 0.6$. It is unclear what causes the outward transport (or prevention of inflow) to stall just inside 0.2 $\rvir$, but it seems that it can be overcome for some value of $\fsp$ above 0.33, likely closer to 1.
Further evidence of conduction-driven inflation of the core can be seen in Figure \[fig:potential\_profiles\], where we plot averaged profiles of $M/r$, including the individual contributions from dark matter, stars, and gas. Despite the lack of perfect monotonicity in the density and temperature profiles, the gaseous component of the potential decreases monotonically with increasing $\fsp$, indicating that the clusters are indeed responding to the presence of conduction by puffing up their cores and redistributing gas to larger radii. As is the case for most simulated galaxy clusters, the stellar component is extremely centrally concentrated, dominating the gravitational potential inside 0.04 $\rvir$. The stellar and dark matter components of the potential are largely unaffected by conduction.
In the right panel of Figure \[fig:radial\_profiles\], we plot normalized entropy profiles, where the normalization term, $K_{200}$ [@2005RvMP...77..207V; @2005MNRAS.364..909V], is given by $$K_{200} = k_BT_{200}~\bar{n_{e}}^{-2/3},$$ where $T_{200}$ is $$\label{eq:T200}
k_BT_{200} = \frac{G M_{200}~\mu m_{p}}{2 r_{200}},$$ and $\bar{n_{e}}$ is the average electron number density within $\rvir$, assuming a fully ionized plasma of primordial composition. The decrease in density and increase in temperature in the cluster cores with conduction yield entropy profiles that are enhanced by 40-70%, but still show the steep decline in the very center that is typical of simulated clusters. This enhancement decreases out to $\rnorm \sim 0.07$, where the mean entropy values are approximately equivalent for all values of $\fsp$. The entropy profiles for the clusters with conduction then dip below the control sample by roughly 10% out to $\rnorm \sim 0.2$. In the range $0.2
\lesssim \rnorm \lesssim 0.6$, the combined temperature and density enhancements appear to be in perfect balance, producing nearly identical entropy profiles across the entire cluster sample with a very small variance.
@2005MNRAS.364..909V find that for a sample of non-radiative, non-conducting clusters simulated with `Enzo`, the entropy profiles in the range $0.2 \lesssim \rnorm \lesssim 1$ are best fit by the power-law $$\label{eq:VKB_fit}
K(r) / K_{200} = 1.51~(r / r_{200})^{1.24},$$ which we overplot on Figure \[fig:radial\_profiles\] with a black dashed line. This power law matches our sample well in normalization, but has a slightly steeper slope. We find that our sample is best matched by $$\label{eq:our_fit}
K(r) / K_{200} = 1.50~(r / r_{200})^{1.09}.$$ For comparison, we also plot the predicted entropy profile for a pure cooling model [@2001Natur.414..425V; @2002ApJ...576..601V] of a 5 keV cluster. This model assumes gas and dark matter density distributions that follow an NFW profile [@1997ApJ...490..493N] with concentration $c = 6$. The gas is initially in hydrostatic equilibrium and allowed to cool for a Hubble time. The pure cooling model agrees quite well with our clusters in the range $0.2 \lesssim \rnorm
\lesssim 1$, and especially well in the range $0.2 \lesssim \rnorm
\lesssim 0.5$. Outside of 0.5 $\rvir$, the slope of the pure-cooling model is slightly too steep and is closer in slope to the @2005MNRAS.364..909V fit.
{width="100.00000%"}
![Averaged profiles of the radial mass distribution in terms of $M/r$, including individual contributions from dark matter, stars, and gas for all clusters. The process by which the averaged profiles are created is described in §\[sec:profiles\] and in the caption of Figure \[fig:radial\_profiles\]. []{data-label="fig:potential_profiles"}](profiles_potential.pdf){width="45.00000%"}
Cluster Exteriors
-----------------
The outskirts of galaxy clusters, while having lower temperatures and hence lower conductivities, are not subject to intermittent heat injection from stellar feedback, and thus offer an intriguing laboratory for studying the effects of conduction. We identify two distinct regions in the cluster outskirts where conduction appears to have influence, at $\sim \rvir$ and at $\sim3~\rvir$. At $\rnorm
\gtrsim 0.6$, the density excess seen in the cluster interiors turns into a deficit for all values of conduction simultaneously, as can be see in Figure \[fig:radial\_profiles\]. From this point out to well past the virial radius, there exists a perfectly monotonic trend of lower gas densities for higher values of $\fsp$. For the maximum value of $\fsp$, the average gas density at the virial radius is 10% lower than that of the clusters simulated without conduction. In fact, the density is measurably lower for all values of $\fsp \ge 0.1$ out to a few virial radii. The temperature just inside $\rvir$ is marginally higher, while the temperature just outside $\rvir$ is reduced.
Conduction transports heat outward in these regions because the gravitational potential there produces a declining gas temperature gradient. This heat transfer causes the entropy of the outer gas to increase. However, because gas in the cluster outskirts is not pressure-confined like the gas in the core, it is free to expand outward and decrease in both density and temperature, while its temperature gradient remains determined by the gravitational potential. This happens because the timescale for conduction in the outer regions is substantially greater than the sound-crossing time. Consequently, conduction of heat outward causes the outer gas to expand without much change in the temperature gradient. Beyond $\rvir$, however, we see a slight steepening in the temperature profile as predicted for conductive clusters by @2013MNRAS.tmp.1104M, presumably because inflation of the ICM due to conduction pushes gas near the virial radius farther from the cluster center without adding much thermal energy. The fact that the entropy profiles of conductive clusters are lower than those of non-conductive clusters beyond the virial radius supports this interpretation.
Further out, at $\sim3~\rvir$, there is another systematic decrease in both density and temperature. Interestingly, as was pointed out by @2008ApJ...689.1063S, this is the typical location for a galaxy cluster’s accretion shocks, which are responsible for heating gas up to the virial temperature. This raises a critical question: *How can conduction affect a large halo’s accretion shocks?*
{width="100.00000%"}
![Profiles of the average Mach number as a function of gas density for the entire region where refinement is allowed in the simulations of the most massive galaxy cluster. The average Mach number is calculated as a weighted mean, where the weight is the rate of kinetic energy processed through the shock, given by $\onehalf \rho v^{3} A$, where $\rho$, $v$, and $A$ are the pre-shock density, shock velocity, and shock surface area. []{data-label="fig:shock_profiles"}](shock_profiles.pdf){width="45.00000%"}
To test this question in a more controlled environment than a cosmological simulation, we performed a pair of one-dimensional simulations designed to mimic the conditions of an accretion shock around a galaxy cluster. Following the shock properties characterized by @2008ApJ...689.1063S, we initiated a Mach 100 standing shockwave with preshock conditions of $n =
10^{-4}$ cm$^{-3}$ and $T = 10^{4}$ K, comparable to gas that is falling directly onto a galaxy cluster (i.e., through spherical accretion of the surrounding intergalactic medium, rather than being accreted via filaments). We ran simulations with $\fsp
= 0$ and 1, with profiles shown in Figure \[fig:shocks\_1d\]. Within tens of Myr, a conduction front in the $\fsp = 1$ simulation races ahead of the original shock. The conduction front continues to advance with ever-decreasing speed and settles into a near-steady state by $t = 200$ Myr, the time shown in the figure. Conduction of heat ahead of the main shock front therefore results in a separation of the density and temperature jumps, effectively creating two shocks, one produced by conductive preheating and a second nearly isothermal shock front some distance downstream. The lower-right panel of Figure \[fig:shocks\_1d\] shows the Mach numbers determined using the shock-finding algorithm described in @2008ApJ...689.1063S. Bifurcation of the original Mach 100 shock has produced a Mach $\sim 70$ conductive-precursor shock followed by a Mach $\sim 1.5$ shock where the main density jump occurs. This splitting is also evident in the plot showing the local Mach ratio, $v/c_{s}$ (lower-middle panel of Figure \[fig:shocks\_1d\]) on either side of the temperature jump. The heat transfer upstream also causes a slight drop in pressure at the original shock front, shifting the primary density jump downstream by about 5 kpc.
These results are independent of resolution. The simulations shown in Figure \[fig:shocks\_1d\] are for a grid 128 cells across, and we observe nearly identical behavior down to a resolution of 16 cells. For this configuration, the results are strongly dependent on $\fsp$. At $\fsp =
0.67$, the distance between the separated shocks is approximately half of that at $\fsp = 1$ and the Mach number of the primary shock is only reduced to just under 90. For $\fsp \lesssim 0.4$, the results are indistinguishable from those without conduction. However, we find that it is possible to produce significant shockwave alteration for lower values of $\fsp$ simply by lowering the initial Mach number and increasing the preshock temperature (resulting in gas in a thermodynamic regime comparable to gas that is being accreted from cosmological filaments). As long as the postshock gas is able to reach temperatures in the range of 10$^{7}$ K, where the Spitzer conductivity becomes considerable, conduction is capable of bifurcating the shock front.
Figure \[fig:shock\_profiles\] shows the average Mach number as a function of gas density in our simulations of the most massive halo for all shocks in the subvolume in which refinement is allowed. Conduction reduces the Mach numbers of the strongest shocks, which occur preferentially in the lowest density gas, by roughly 10% from $\fsp = 0$ to $\fsp = 1$. We therefore conclude that the density, temperature, and entropy deficits observed beyond the virial radius in our conductive clusters are indeed due to shock bifurcations similar to those seen in our idealized one-dimensional simulations of conductive shock fronts.
Yet, the physics of the actual accretion shocks around real galaxy clusters is undoubtedly more complex. In particular, it is important to note that the electron mean free path in that gas is several times greater than the intershock distance in our one-dimensional simulations, and furthermore that real accretion shocks are likely to be collisionless and magnetically-mediated. Nonetheless, the general qualitative point these simulations illustrate is interesting: Heating of preshock gas by a hot electron precursor has the potential to alter the expected relationships between the sizes and locations of the density and temperature jumps in accretion and merger shocks.
Progress in understanding the effects of conduction on accretion shocks will require modeling the gas as a fully ionized plasma [@Zeldovich1957; @Shafranov1957], which is beyond the scope of this work. The high Mach number of an accretion shock combined with a post-shock temperature high enough for significant heat flux create conditions similar to a supercritical radiative shock, as described by @2007ShWav..16..445L. When simulated with a two-fluid approach, the combination of preheating of the preshock medium via conduction in the electrons, electron-ion coupling, and compression of the ion fluid in the postshock region can produce a small region, known as a Zel’dovich spike, where the ion temperature is slightly higher than the equilibrium postshock temperature [@2007ShWav..16..445L; @2008ShWav..18..129L; @2011ShWav..21..367M]. Our single-fluid simulations, despite their limitations, produce shocks quite similar to the non-equilibrium results of @2008ShWav..18..129L, where a diffusion term proportional to $T^{5/2}$ (like Spitzer conductivity) is used. Nevertheless, a more detailed study of the characteristics of accretion shocks employing a two-fluid MHD treatment along with physically motivated conduction and cooling rates seems warranted.
Temperature Homogeneity
=======================
Conduction strong enough to alter the temperature gradients in cluster cores should also be effective at smoothing out small-scale thermal variations in the ICM. However, those effects turn out to be rather subtle, as shown in Figure \[fig:T\_var\], which plots the normalized variance of the temperature field as a function of radius, averaged over all clusters in the sample that have the same level of conductivity. It reveals an extremely weak trend of greater temperature homogeneity with increasing $\fsp$, but at all radii the difference in homogeneity among cluster sets with different values of $\fsp$ is less than the cluster-to-cluster variation. Furthermore, this trend reverses beyond the virial radius, at $2 \lesssim \rnorm \lesssim 3$, albeit at a nearly marginal level.
![Averaged radial profile of the mass-weighted temperature variance normalized to mean temperature within each radial bin. The process by which the averaged profiles are created is described in §\[sec:profiles\] and in the caption of Figure \[fig:radial\_profiles\]. []{data-label="fig:T_var"}](profiles_T_variance.pdf){width="45.00000%"}
The homogenizing effects of conduction are even harder to see in projection, which perhaps is not surprising given the scarcity of obvious conduction-dependent morphological differences in Figures \[fig:thin\_proj\_0\], \[fig:thin\_proj\_2\], and \[fig:thin\_proj\_5\]. Those differences are further diluted when projected over a 4 Mpc line of sight through each cluster, as in Figures \[fig:proj\_0\]-\[fig:proj\_5\]. These latter figures show mean temperature weighted by the X-ray emission in the 0.5-2 keV energy band, and X-ray emission is calculated by interpolating from density, temperature, and metallicity-dependent emissivity tables computed with the `Cloudy` code. Since X-ray emission is proportional to $n_e^{2} T_e^{1/2}$, this weighting should highlight clumpiness and differences in temperature. However, as stated previously, these results contrast considerably with the cluster map comparison of @2004ApJ...606L..97D, which shows a significantly hotter cluster in which conduction should be much more efficient.
![Similar projections to Figure \[fig:thin\_proj\_0\] for Halo 0, but with a projected depth of 4 Mpc. []{data-label="fig:proj_0"}](projections_halo_0000.pdf){width="45.00000%"}
![Similar projections to Figure \[fig:thin\_proj\_2\] for Halo 2, but with a projected depth of 4 Mpc. []{data-label="fig:proj_2"}](projections_halo_0002.pdf){width="45.00000%"}
![Similar projections to Figure \[fig:thin\_proj\_5\] for Halo 5, but with a projected depth of 4 Mpc. []{data-label="fig:proj_5"}](projections_halo_0005.pdf){width="45.00000%"}
The change in temperature homogeneity due to conduction is quite small in our simulated cluster sample, but important insights into the physics of the ICM could be gained if the effective value of $\fsp$ could be measured observationally. To evaluate this possibility, we created X-ray-weighted temperature maps of the central 300 kpc for each of the clusters in the sample, masking out the pixels within 40 kpc of the cluster centers to remove features that would be considered part of the central galaxy. We then quantified the amount of azimuthal temperature structure by dividing each temperature map into azimuthal bins and calculating both the mean temperature in each bin and the temperature variance among all azimuthal bins in the map. We performed this calculation multiple times for each map while rotating the azimuthal bins, and took the maximum variance calculated as the value for that map. Finally, we averaged the values together for all clusters in the sample and performed the entire exercise over a range in the total number of azimuthal bins, from 2 to 9. Figure \[fig:azimuthal\_variance\] plots the maximum variance as a function of the number of bins for each value of $\fsp$. We find that, in general, the maximum variance is lower for the simulations with conduction, but only by approximately 10%. There does not appear to be any sort of monotonic trend with increasing $\fsp$. We repeated this experiment, varying the inner and outer radius for the temperature maps, but were unable to find conditions that produce a better trend than can be seen in Figure \[fig:azimuthal\_variance\]. Thus, we conclude that the efficiency of conduction is difficult to determine solely from the observable temperature homogeneity of the ICM, at least for clusters of temperature $\lesssim 6$ keV.
![The maximum variance in X-ray-weighted temperature projections within a region 40 kpc $\le r \le$ 150 kpc as a function of the number of azimuthal bins. []{data-label="fig:azimuthal_variance"}](azimuthal_variance.pdf){width="45.00000%"}
Star Formation
==============
Figure \[fig:stellar\_mass\] shows the average difference in stellar mass between the clusters with conduction and those without as a function of time. The difference in stellar mass at $z = 0$ for all levels of conduction is less than 5%. Surprisingly, the clusters with higher levels of conduction form [*more*]{} stars in our calculations [see also @2004ApJ...606L..97D]. However, the shaded regions in Figure \[fig:stellar\_mass\] give an indication of just how much variation there is between clusters. As we have shown, even the highest level of conduction is unable to prevent a cooling catastrophe in the center of a cluster. Therefore, one should not expect a large difference in the amount of star formation. Given the coarseness with which the star forming regions are resolved, it is possible that the enhancement in star formation with increasing $\fsp$ is numerical and not physical. We propose two potential numerical explanations. First, the short timesteps required for the stability of the conduction algorithm may not provide enough time for a single star particle to sufficiently heat up a grid cell and quench star formation in a given cell in the following timestep. Second, conduction may transport thermal energy too quickly out of regions heated by recent star formation. This would allow a star-forming region to recool and form additional stars too rapidly.
Because conduction is able to create a nearly isothermal core for $\fsp \ge 0.1$, yet the final stellar mass from the clusters with $\fsp = 0.1$ is consistent with no change, we find it reasonable to conclude that the isothermality of the core in a conductive cluster has no influence on the star formation rate `Enzo` calculates for the central galaxy. However, because we do not resolve the interface between the ICM and the interstellar medium (ISM), we cannot definitively state the effect of conduction on star-forming gas in a cosmological simulation.
![The ratio of total stellar mass in the simulations with $\fsp > 0$ to the control run as a function of cosmic time for all clusters in the sample, with the shaded regions denoting the cluster to cluster variance. The process by which the averaged quantities are calculated is described in §\[sec:profiles\] and in the caption of Figure \[fig:radial\_profiles\]. []{data-label="fig:stellar_mass"}](stellar_mass_enhancement.pdf){width="45.00000%"}
Conclusions
===========
We have performed cosmological simulations of 10 galaxy clusters using isotropic thermal conduction with five values of the conductive suppression factor in order to study the effects of conduction on galaxy cluster cores and the intracluster medium. By studying the aggregate properties of the clusters in our sample, we find that the presence of conduction even at its maximum possible efficiency induces changes to the density and temperature structure on the order of only 20-30%. For $\fsp \ge 0.1$, the cluster cores become roughly isothermal. However, conduction at any level is incapable of stopping the cooling catastrophe at the very centers of our clusters, where the density profile is always very sharply peaked. To some extent, this is due to our limited spatial resolution, since the temperature gradients on which heat conduction depends are limited by the scale of the smallest grid cell, which at $\sim15$ kpc/$h$ is still quite large compared to the scales of galaxies and the Field length.
However, the extremely well-resolved study of @2012ApJ...747...26L also finds that conduction can at best slightly delay the cooling catastrophe. While conduction is unable to prevent the cooling catastrophe, the elevation of gas entropy in a conductive, isothermal core displaces some of the core gas, moving it out to larger radii. For values of $\fsp$ up to 0.33, this material is redistributed mostly within $\sim0.2~\rvir$. For higher values of $\fsp$, it is transported out even further, up to $\sim0.6~\rvir$. A similar phenomenon occurs around $\rvir$, where the negative temperature gradient allows outward heat conduction to inflate the outer parts of the cluster. However, because this material is not deep in the potential well, it is free to expand and cool, leading to slightly lower temperatures just outside the virial radius.
More surprisingly, we observe a systematic decrease in both the density and temperature with increasing $\fsp$ at large radii, out to $\sim3~\rvir$. We hypothesize that this is due to alteration of the accretion shocks by conduction. To test this, we perform one-dimensional “shock tube” simulations with conditions characterizing an accretion shock around a galaxy cluster, with the level of conduction treated as the sole free parameter. As long as the post-shock temperature is high enough for the Spitzer conductivity to be efficient ($T \gtrsim 10^{7}$ K), conduction moves the temperature jump upstream and the density jump downstream of the original shock face. This creates two distinct shocks, both with Mach numbers less than the original shock. We conclude that conduction is responsible for the systematic decrease in density and temperature in the outskirts of our simulated clusters, because it acts to weaken the shocks. We acknowledge that our modeling of this problem is not totally accurate, and instead requires a two-fluid MHD approach, which is beyond the current capabilities of our simulation tool. However, more rigorous two-fluid simulations of shockwaves in fully ionized plasmas show qualitatively similar behavior, save a tiny feature in the ion temperature that cannot be achieved in a single-fluid approach. Unfortunately, because the effect on clusters is only at the 10% level and at very large radii, where the X-ray surface brightness of the plasma is extremely low, observing the effects of conduction on accretion shocks may never be possible.
We also find that in addition to altering temperature gradients, conduction is able to make the intracluster medium more thermally uniform. This effect, while measurable in spherically-averaged radial profiles, is almost totally lost in projection. Our results contrast with the temperature maps of @2004ApJ...606L..97D, wherein the effect of conduction is instantly recognizable. The cluster shown in @2004ApJ...606L..97D is significantly more massive than our most massive cluster, so it is possible that a hotter ICM, with a higher thermal conductivity, is made more homogeneous, suggesting that the temperature dependence of temperature inhomogeneity in a large cluster sample could help reveal the typical conductivity of the ICM. We attempted to find a means of distinguishing the level of conduction observationally by measuring the variance in our projected temperature maps, but without success.
Finally, conduction appears to have very little influence on the star formation rate within our simulated clusters. When determining whether a grid cell should form a star particle, we include the energy change from conduction in the calculation of the cooling time, but this seems to have very little influence. This is likely because star-forming grid cells are surrounded mostly by cells that are also quite cool. Somewhat surprisingly, we observe a marginal increase in the total stellar mass with increasing conduction, such that the sample with $\fsp = 1$ shows an enhancement in star formation rate of $\sim5\%$. The fact that conduction cannot suppress star formation is directly related to its inability to prevent the cooling catastrophe in the very center of the cluster. However, the reasons for the slight increase in star formation may be more numerical than physical. Further progress on understanding the effects of thermal conduction on star formation in cluster cores will require properly resolving the interface between ISM and the ICM, which at present is impractical in cosmological galaxy cluster simulations.
This work was supported by NASA through grant NNX09AD80G and NNX12AC98G, and by the NSF through AST grant 0908819. The simulations presented here were performed and analyzed on the NICS Kraken and Nautilus supercomputing resources under XSEDE allocations TG-AST090040 and TG-AST120009. We thank Greg Bryan, Gus Evrard, Eric Hallman, Andrey Kravtsov, Jack Burns, Matthew Turk, and Stephen Skory for helpful discussions during the course of preparing this manuscript. SWS has been supported by a DOE Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308. BWO was supported in part by the MSU Institute for Cyber-Enabled Research. `Enzo` and `yt` are developed by a large number of independent research from numerous institutions around the world. Their committment to open science has helped make this work possible.
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[^1]: http://enzo-project.org/
[^2]: http://yt-project.org/
[^3]: http://nublado.org/
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---
abstract: |
In this work we present results about the rate of (relative) information loss induced by passing a real-valued, stationary stochastic process through a memoryless system. We show that for a special class of systems the information loss rate is closely related to the difference of differential entropy rates of the input and output processes. It is further shown that the rate of (relative) information loss is bounded from above by the (relative) information loss the system induces on a random variable distributed according to the process’s marginal distribution.
As a side result, in this work we present conditions such that for a Markovian input process also the output process possesses the Markov property.
author:
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title: On the Rate of Information Loss in Memoryless Systems
---
data processing inequality, information loss, entropy rate, Rényi information dimension, system theory, lumpability
Introduction {#sec:Intro}
============
Signal processing, as defined by many textbooks, is related to the “representation, transformation, and manipulation of signals and the *information* the signals contain” [@Oppenheim_Discrete3 emphasis added]. Yet, most of these textbooks leave the notion of *information* completely aside and focus, instead, on purely energetic aspects or second-order statistics: transfer functions for linear filters, their effect on the auto-correlation function of its output signal, and similar results for nonlinear, memoryless systems (e.g., [@Price_NLGaussianInput]) are popular characterizations. However, except for the purely Gaussian case, energy (or second-order statistics) and information show an inherently different behavior. It is therefore desirable to extend current system theory by information-theoretic aspects.
While the data processing inequality (e.g., [@Cover_Information2 p. 35]) captures the fact that deterministic functions of random variables (RVs) destroy information, relatively little has been done to quantify this information loss. Pinsker showed that the entropy rate of a function of a stationary stochastic process on a finite alphabet is bounded from above by the entropy rate of the original process [@Pinsker_InfoEngl Ch. 6.3]. Similarily, Watanabe and Abraham analyzed the rate of information loss for functions of stationary stochastic processes, introducing also a relative version of information loss in [@Watanabe_InfoLoss]. Results on the information loss rate in dynamical systems, together with an upper bound, were presented in [@Geiger_NLDyn1starXiv].
While these works focus on finite or countable alphabets, [@Geiger_LossLong] analyzes the absolute and, in case the latter is infinite, relative information loss induced by passing a real-valued RV through a memoryless system. In this work we extend [@Geiger_LossLong] to real-valued, stationary stochastic processes. In particular we show that the information loss for RVs distributed according to the marginal distribution of the process is an upper bound on the information loss rate (Section \[sec:LossRate\]). A similar result is shown also for the relative information loss rate, although there the bound is tight in many more cases (Section \[sec:RelativeLossRate\]): While redundancy helps to reduce the rate of information loss, it often fails to reduce the rate of *relative* informtion loss. The connection between the rate of information loss and the differential entropy rates of the input and output processes shown in Section \[sec:LossRate\] is remarkably similar to the corresponding result for information loss presented in [@Geiger_LossLong].
In search for processes which are simple to analyze, we found a set of sufficient conditions such that for a Markovian input process also the output process has the Markov property (Section \[sec:lumpability\]). This extends the notion of *lumpability* (cf. [@Kemeny_FMC]) from discrete-time and continuous-time, homogeneous Markov chains to discrete-time, homogeneous, real-valued Markov processes. These conditions, together with our other theoretical findings, are illustrated with the help of examples in Section \[sec:Examples\].
Preliminaries & Notation {#sec:prelim}
========================
Throughout this work we consider discrete-time, stationary stochastic processes $\Xvec$ with alphabet $\dom{X}\subseteq\mathbb{R}$. Let $X_n$ be the $n$-th sample of the process, and let $X_i^j=\{X_i,X_{i+1},\dots,X_j\}$. By stationarity, the distribution of $X_n$, $P_{X_n}$, equals the marginal distribution $P_X$. We assume that $P_X$ is absolutely continuous w.r.t. the Lebesgue measure, and that it thus possesses a probability density function (PDF) $f_X$. Similarily, we assume that for all $n$, the joint PDF $f_{X_1^n}$ and the conditional PDF $f_{X_n|X_1^{n-1}}$ exist. Let $\ent{\cdot}$, $\diffent{\cdot}$, $\entrate{\cdot}$, and $\derate{\cdot}$ denote the entropy, the differential entropy, the entropy rate, and the differential entropy rate of the RVs and stochastic processes in the argument (see [@Cover_Information2] or [@Papoulis_Probability] for definitions). We assume that the joint differential entropy of an arbitrary collection of RVs exists and is finite, and that also the entropy rate [@Papoulis_Probability Thm. 14.7] $$\derate{X} := \limn \diffent{X_n|X_1^{n-1}} = \limn\frac{1}{n}\diffent{X_1^n}\label{eq:derate}$$ exists and is finite. The logarithm for the entropies is taken to the base 2.
Information Loss Rate Piecewise Bijective Functions {#sec:LossRate}
===================================================
In this section we devote our attention to a specific class of functions for which the preimage of every point of its range is an at most countable set:
\[def:PBF\] A piecewise bijective function $g{:}\ \dom{X}\to\dom{Y}$, $\dom{X},\dom{Y}\subseteq\mathbb{R}^N$, is a surjective, measurable function defined piecewise on an at most countable partition $\{\dom{X}_i\}$ of its domain: $$g(x) = \begin{cases}
g_1(x), & \text{if } x \in\dom{X}_1\\
g_2(x), & \text{if } x \in\dom{X}_2\\
\vdots
\end{cases}\label{eq:defg}$$ where each $g_i{:}\ \dom{X}_i\to\dom{Y}_i$ is bijective. Furthermore, the derivative $g'$ exists on the closures of $\dom{X}_i$, and its magnitude is non-zero $P_X$-a.s.
Feeding the stationary stochastic process $\Xvec$ through a memoryless system described by such a function $g$ gives rise to another stationary stochastic process $\Yvec$ defined by $Y_n:=g(X_n)$, which, intuitively, conveys less information. In order to analyze the amount of information lost *per sample* we introduce
\[def:lossrate\] The information loss rate is $$\lossrate{X\to Y} := \limn\frac{1}{n} \loss{X_1^n\to Y_1^n} = \limn\frac{1}{n} \ent{X_1^n|Y_1^n}$$ i.e., the average of the block information loss.
We showed in [@Geiger_LossLong] that the information loss in systems described by functions satisfying Definition \[def:PBF\] can be computed as $$\loss{X\to Y} = \ent{X|Y}=\diffent{X}-\diffent{Y} + \expec{\log|g'(X)|}\label{eq:lossdiffent}$$ where $Y=g(X)$ and where the expectation is taken w.r.t. $X$. We now present a corresponding result for stationary stochastic processes:
\[prop:lossratePBFs\] The information loss rate induced by feeding a stationary stochastic process $\Xvec$ through a PBF $g$ is $$\lossrate{X \to Y} = \derate{X}-\derate{Y} + \expec{\log|g'(X)|}.\label{eq:lossdiffentRate}$$
For the proof we note that the $n$ RVs $X_1^n$ can be interpreted as a single, $n$-dimensional RV; similarily, we can define an extended function $g^n{:}\ \dom{X}^n\to\dom{Y}^n$, applying $g$ coordinate-wise. The Jacobian matrix of $g^n$ is a diagonal matrix constituted of the elements $g'(x_i)$. With the extension of to multivariate functions we thus obtain [@Geiger_LossLong] $$\begin{gathered}
\loss{X_1^n\to Y_1^n} = \diffent{X_1^n}-\diffent{Y_1^n}+\expec{\log\left|\prod_{i=1}^n g'(X_i)\right|}\\
=\diffent{X_1^n}-\diffent{Y_1^n}+n\expec{\log\left|g'(X)\right|}\end{gathered}$$ where the first line is because the determinant of a diagonal matrix is the product of its diagonal elements, and where we employed stationarity of $\Xvec$ to obtain the second line. Dividing by $n$ and taking the limit completes the proof.
In [@Geiger_LossLong] we showed that the information loss of a cascade of systems equals the sum of the information losses induced in the systems constituting the cascade. Indeed, this result can be carried over to the information loss rate as well:
\[prop:rateCascade\] Let $\Xvec$ be fed through a PBF $g$ to obtain $\Yvec$, and let $\Yvec$ be fed through a PBF $h$ to obtain $\Zvec$. The information loss rate of the cascade is given as the sum of the individual information loss rates: $$\lossrate{X\to Z}=\lossrate{X\to Y}+\lossrate{Y\to Z}.$$
The proof follows from the fact that the cascade is described by the function $h\circ g$, and that $$\begin{gathered}
\expec{\log|(h\circ g)'(X)|} = \expec{\log|g'(X)h'(g(X))|}\\
=\expec{\log|g'(X)|}+\expec{\log|h'(Y)|}.\end{gathered}$$
It is often not possible to obtain closed-form expressions for the information loss rate induced by a system. Moreover, estimating the information loss rate by simulations soon suffers the curse of dimensionality, as, in principle, infinitely long random sequences have to be drawn and averaged. Much simpler is an estimation of the information loss, since a single realized, sufficiently long sequence allows for an estimation of the latter. As the next proposition shows, this relatively simple estimation delivers an upper bound on the information loss rate:
\[prop:lossgelossrate\] Let $\Xvec$ be a stationary stochastic process and $X$ an RV distributed according to the process’s marginal distribution. The information loss induced by feeding $X$ through a PBF $g$ is an upper bound on the information loss rate induced by passing $\Xvec$ through $g$, i.e., $$\lossrate{X\to Y} \le \loss{X\to Y}.$$
Clearly, this bound is tight whenever the input process $\Xvec$ is an iid process. Moreover, it is trivially tight whenever the function is bijective, i.e., when $\loss{X\to Y}=0$. In we present an example which renders this bound tight in the general case.
Intuitively, this bound suggests that redundancy of a process, i.e., the statistical dependence of its samples, reduces the amount of information lost *per sample* when fed through a deterministic system. The same connection between information loss and information loss rate has already been observed in [@Watanabe_InfoLoss] for stationary stochastic processes with finite alphabets.
The next bound again extends a result from [@Geiger_LossLong], bounding the information loss rate by the entropy rate of a stationary stochastic process on an at most countable alphabet. As such, it presents a different way to estimate the information loss rate efficiently using numerical simulations.
\[prop:WBound\] Let $\Wvec$ be a stationary stochastic process defined by $W_n:=i$ if $X_n\in\dom{X}_i$. Then, $$\lossrate{X \to Y} = \entrate{W|Y} \le \entrate{W}.$$
We again treat $X_1^n$ as an $n$-dimensional RV; $g^n$ induces a partition of its domain $\dom{X}^n$, which is equivalent to the $n$-fold product of the partition $\{\dom{X}_i\}$. Letting $\tilde{W}$ be the RV obtained by quantizing $X_1^n$ according to this partition, it is easy to see that $W_1^n$ is equivalent to $\tilde{W}$. Thus, with [@Geiger_LossLong], $$\ent{X_1^n|Y_1^n} = \ent{\tilde{W}|Y_1^n} = \ent{W_1^n|Y_1^n}$$ for all $n$. This, together with the fact that conditioning reduces entropy, completes the proof.
For the case that the input process is a Markov process, i.e., if $f_{X_n|X_1^{n-1}}=f_{X_n|X_{n-1}}$ for all $n$, an additional, sharper, upper bound can be presented:
\[prop:UBMarkov\] Let $\Xvec$ be a Markov process, and let $\Wvec$ be as in Proposition \[prop:WBound\]. Then, for finite $\loss{X \to Y}$, $$\lossrate{X \to Y} \le \ent{W_2|X_1}.$$
We again apply the chain rule, Markovity of $\Xvec$, and the fact that conditioning reduces entropy to arrive at $$\ent{X_1^n|Y_1^n} \le\sum_{i=1}^n \ent{X_i|X_{i-1},Y_i}.$$ By stationarity we obtain
[RCL]{} && \[eq:Markovity\]\
&&\
&& \[eq:prop52ndeq\]
where $(a)$ holds since, for all $x\in\dom{X}$, $\ent{X_2|Y_2,X_1=x}=\ent{W_2|Y_2,X_1=x}$ [@Geiger_LossLong]. The last inequality is due conditioning [@Cover_Information2 Thm. 2.6.5] and completes the proof.
That the bound is sharper than the one of Proposition \[prop:WBound\] follows from observing that $$\begin{gathered}
\ent{W_n|X_{n-1}}=\limn\ent{W_n|X_1^{n-1}}\\\le\limn\ent{W_n|W_1^{n-1}}=\entrate{W}.\end{gathered}$$
The interpretation of this result is that a function destroys little information if the process is such that, given the current sample $X_{n-1}$, the next sample $X_n$ falls within some element of the partition with a high probability.
Lumpability for Continuous-Valued Markov Processes {#sec:lumpability}
==================================================
It is well-known that the function of a Markov process need not possess the Markov property itself. However, as it is known for Markov chains, there exist conditions on the function and/or the chain such that the output is Markov. In [@Kemeny_FMC] this has been termed *lumpability* and subsequently investigated by numerous researchers. While most results are given for finite Markov chains (e.g., [@Burke_MarkovFunction; @Rogers_MarkovFunct]) relatively little is known in the general case of an uncountable alphabet (see [@Rosenblat_Markov] for an exception). Our small contribution to this field of research lies in presenting sufficient conditions for lumpability of continuous-valued Markov processes.
Let $f_{X_n|X_1^{n-1}}=f_{X_n|X_{n-1}}$ for all $n$, i.e., let $\Xvec$ be a Markov process. We maintain
\[prop:equalityThenLump\] If $$\begin{gathered}
\forall y_1^2\in\dom{Y}^2: \forall x\in\preim{y_1}:\\
f_{Y_2,X_1}(y_2,x)>0 \Rightarrow f_{Y_2|X_1}(y_2|x) = f_{Y_2|Y_1}(y_2|y_1)\end{gathered}$$ then $\Xvec$ is lumpable w.r.t. $g$, i.e., $\Yvec$ is Markov.
\[cor:sufficientConditions\] If for all $y_1^2\in\dom{Y}^2$ and all $x,x'\in\preim{y_1}$ such that $f_{X}(x)>0$ and $f_{X}(x')>0$ the following holds $$\begin{aligned}
\sum_{x_2\in\preim{y_2}} \frac{f_{X_2|X_1}(x_2|x)}{|g'(x_2)|} &= \sum_{x_2\in\preim{y_2}} \frac{f_{X_2|X_1}(x_2|x')}{|g'(x_2)|} \label{eq:condEq}\end{aligned}$$ then the condition of Proposition \[prop:equalityThenLump\] is fulfilled and $\Yvec$ is Markov.
Relative Information Loss Rate for Functions which Reduce Dimensionality {#sec:RelativeLossRate}
========================================================================
Not all systems can be described by functions satisfying Definition \[def:PBF\]. In particular, a simple quantizer already violates this definition and suffers from infinite information loss. To analyze the information processing characteristics of a broader class of systems, in [@Geiger_LossLong] the notion of relative information loss was introduced, capturing the *percentage* of information available at the input lost in the system. To extend this notion to stochastic processes, we introduce
\[def:RelLossRate\] The relative information loss rate is $$\relLossrate{X\to Y} := \limn \relLoss{X_1^n\to Y_1^n} = \limn \lim_{\hat{\Xvec}\to\Xvec}\frac{\ent{\hat{X}_1^n|Y_1^n}}{\ent{\hat{X}_1^n}}$$ whenever the limit exists.
The limit $\hat{\Xvec}\to\Xvec$ is equivalent to $\lim_{k\to\infty} \lfloor 2^k \Xvec\rfloor/2^k$, where flooring and scalar multiplication are applied element-wise (cf. [@Geiger_LossLong]).
However, in many cases this inequality is an equality, as we show in
\[prop:RelLossgeRelLossRate\] Let $\Xvec$ be a stationary stochastic process and $X$ an RV distributed according to the process’s marginal distribution. Let further $g$ be defined on a finite partition $\{\dom{X}_i\}$ of $\dom{X}$ into non-empty sets as in , where $g_i\in\mathcal{C}^\infty$ is either injective or constant (i.e., $g_i(x)=c_i$ for all $x\in\dom{X}_i$). Then, $$\relLossrate{X\to Y} = \relLoss{X\to Y} = P_X(\dom{X}_c)$$ where $\dom{X}_c$ is the union of all elements $\dom{X}_i$ of the partition on which $g$ is constant.
Indeed, we conjecture that equality is indeed the “usual” case, prevailing in most practical scenarios. Thus, while redundancy can help reduce information loss, it may be useless when it comes to relative information loss. Applications of this result may be the scalar quantization of a stochastic process (leading to a relative information loss rate of 1, i.e., 100% of the information is lost [@Geiger_LossLong]) and system blocks for multirate signal processing (see the example in Section \[ssec:multirate\]).
Examples {#sec:Examples}
========
AR-Process and Magnitude Function {#ssec:ARProcess}
---------------------------------
(1.5,1)(9,2.5) (1.5,2)[z]{}[$Z$]{} (3,2)[oplus]{} (5,2)[dot]{} (7,2)[c]{}[$|\cdot|$]{} (4,1)[del]{}[$z^{-1}$]{} (9,2)[y]{}[$\Yvec$]{} \[naput\][z,oplus,dot,c $\Xvec$ ,y]{}
In this example we assume that a first-order, zero-mean, Gaussian auto-regressive process $\Xvec$ is fed through a magnitude function (see Fig. \[fig:ARSysModel\]). Let the AR process be generated by the following difference equation: $$X_n=aX_{n-1}+Z_n$$ where $a\in (0,1)$ and where $Z_n$ are samples drawn independently from a Gaussian distribution with zero mean and variance $\sigma^2$. It follows immediately that the process $\Xvec$ is also zero mean and has variance $\sigma_X^2=\frac{\sigma^2}{1-a^2}$ [@Oppenheim_Discrete3 Ex. 6.11]. Let $\Yvec$ be defined by $Y_n=|X_n|$.
For the sake of brevity we define ${\phi(\mu,\sigma^2;x)}$ as the PDF of a Gaussian RV with mean $\mu$ and variance $\sigma^2$, evaluated at $x$. Thus, we get $$f_X(x) = {\phi(0,\sigma_X^2;x)}$$ and $$f_{X_2|X_1}(x_2|x_1) = {\phi(ax_1,\sigma^2;x_2)}.$$ It follows that is satisfied with $|g'(x)|\equiv 1$ and since ${\phi(ax_1,\sigma^2;x_2)}={\phi(-ax_1,\sigma^2;-x_2)}$, $$\begin{gathered}
\sum_{x_2\in\preim{y_2}} \frac{f_{X_2|X_1}(x_2|y_1)}{|g'(x_2)|}\\
={\phi(ay_1,\sigma^2;y_2)}+{\phi(ay_1,\sigma^2;{-}y_2)}\\
={\phi({-}ay_1,\sigma^2;{-}y_2)}+{\phi({-}ay_1,\sigma^2;y_2)}\\
=\sum_{x_2\in\preim{y_2}} \frac{f_{X_2|X_1}(x_2|{-}y_1)}{|g'(x_2)|}.\end{gathered}$$ As a consequence, the output process $\Yvec$ is Markov.
We performed a series of simulations, as the information loss rate for this example cannot be expressed in closed form. Rewriting, e.g., the lower bound on the information loss rate as $$\begin{aligned}
& \lossrate{X\to Y}\notag\\ &\ge \diffent{X_2|X_1}-\diffent{Y_2|X_1}+\expec{\log|g'(X)|}\\
&= \diffent{X}-\mutinf{X_1;X_2}-\diffent{Y}+\mutinf{X_1;Y_2}\notag\\&\quad{}+\expec{\log|g'(X)|}\\
&= \loss{X\to Y}-\mutinf{X_1;X_2}+\mutinf{X_1;Y_2}\end{aligned}$$ allowed us to employ the histogram-based mutual information estimation from [@Moddemeijer_Matlab] together with $\loss{X\to Y}=1$, as shown in [@Geiger_LossLong]. The upper bound $H(W_2|X_1)$ from Proposition \[prop:UBMarkov\] was computed using numerical integration. In Fig. \[fig:ARInfoLoss\] one can see that the first-order upper and lower bounds on the information loss rate are indistinguishable, which suggests that the output process is indeed Markov. Moreover, it can be seen that a higher value for the magnitude $a$ of the pole leads to a smaller information loss rate. This can be explained by the fact that the redundancy[^1] of the process $\Xvec$ increases with increasing $a$, which helps preventing information loss.
(-1,-0.5)(8,3.6) (0,0)(7.5,3.5)\[$a$,-90\]\[$\lossrate{X\to Y}$,0\] (0.2,0.2)
[ll]{}
------------------------------------------------------------------------
& Upper Bound\
------------------------------------------------------------------------
& Lower Bound\
------------------------------------------------------------------------
& $\ent{W_2|X_1}$
Generally, while redundancy reduces the information loss rate compared to an iid process (cf. Proposition \[prop:lossgelossrate\]), it is not necessarily true that *more* redundancy leads to a smaller information loss rate than *less* redundancy. Indeed, one can generate examples where a process with a higher redundancy suffers from a higher information loss rate than a process with less redundancy. This suggests that the redundancy of a process has to be matched to the function $g$ in order to efficiently prevent information from being lost; in that sense, this parallels the field of channel coding, where the code needs to be matched to the characteristics of the channel (noise, fading, burst errors) in order to successfully reduce the bit error rate.
Multirate Systems {#ssec:multirate}
-----------------
Although strictly speaking not time-invariant, also multirate systems can be analyzed with the proposed relative information loss rates. In particular, we will show that for an $M$-fold downsampler, which is described by the input-output relation $Y_n = X_{nM}$, the information loss rate equals $$\relLossrate{X\to Y} = \frac{M-1}{M}.$$ To this end, note that the stationary output process $\Yvec$ is equivalent to the cyclo-stationary process $\tilde{\Yvec}$, whose samples are defined as $$\tilde{Y}_n=
\begin{cases}
X_n, & \text{if } n/M \in\mathbb{Z}\\
0,& \text{else }
\end{cases}.\label{eq:downsampler}$$ In essence, the function in implements a projection on a subspace of lower dimensionality. For these type we showed in [@Geiger_LossLong] that the relative information loss is related to the information dimension of the output, which in our case is given by the number of its non-zero entries, i.e., by $$\infodim{\tilde{Y}_1^n} = \left\lfloor\frac{n}{M}\right\rfloor.$$ With $\infodim{X_1^n}=n$ and by the fact that $\lfloor n/M\rfloor = n/M+\{n/M\}$, where $\{\cdot\}$ denotes the fractional part, we obtain $$\begin{gathered}
\limn \relLoss{X_1^n\to \tilde{Y}_1^n} = 1-\limn\frac{n/M+\{n/M\}}{n}\\
= 1-\frac{1}{M} = \frac{M-1}{M}.\end{gathered}$$ The second equality follows because the magnitude of the fractional part is bounded by unity and that, thus, this term vanishes in the limit.
Conclusion
==========
In this work we extended previous results about the information loss induced by deterministic, memoryless input-output systems from random variables to stationary stochastic processes with continuous distribution. Notably, we showed a connection between the rate of information loss and the differential entropy rates of the input and output processes for a special class of functions. While redundancy decreases the information loss rate for this class of systems, systems which destroy an infinite amount of information do not benefit from redundancy of the process in most practical cases. Future investigations shall focus on the extension to systems with memory and on the problem of reconstructing the input process.
As side results, we presented sufficient conditions for the Markovian input process and the system function such that the output process is Markov.
[^1]: The redundancy is defined as the difference between the entropy of the marginal distribution and the entropy rate of the process. The former increases due to increasing variance $\sigma^2_X$, while the latter remains constant and equal to $\diffent{Z}$ (cf. [@Dumitrescu_EntropyInvariance]).
|
---
abstract: 'We propose a simple algorithm for generating normally distributed pseudo random numbers. The algorithm simulates $N$ molecules that exchange energy among themselves following a simple stochastic rule. We prove that the system is ergodic, and that a Maxwell like distribution that may be used as a source of normally distributed random deviates follows in the $N\rightarrow\infty$ limit. The algorithm passes various performance tests, including Monte Carlo simulation of a finite 2D Ising model using Wolff’s algorithm. It only requires four simple lines of computer code, and is approximately ten times faster than the Box-Muller algorithm.'
address: |
$^{(1)}$Instituto de Ciencia de Materiales de Aragón\
Consejo Superior de Investigaciones Científicas\
and Universidad de Zaragoza, 50009-Zaragoza, Spain\
$^{(2)}$Departamento de Física Aplicada I, Universidad de Málaga\
29071-Málaga, Spain\
author:
- 'Julio F. Fernández$^{1}$ and Carlos Criado$^2$'
title: Algorithm for normal random numbers
---
Pseudo random number (PRN) generation is a subject of considerable current interest[@reviews]. Deterministic algorithms lead to undesirable correlations, and some of them have been shown to give rise to erroneous results for random walk simulations [@rw], Monte Carlo (MC) calculations [@ising1; @clock], and growth models [@growth]. Most of the interest has been focused on PRN’s with uniform distributions. Less attention has been paid to non-uniform PRN generation.
Sequences of random numbers with Gaussian probability distribution functions (pdf’s) are needed to simulate on computers gaussian noise that is inherent to a wide variety of natural phenomena [@noise]. Their usefulness transcends physics. For instance, numerical simulations of economic systems that make use of so called [*geometric*]{} Brownian models (in which noise is multiplicative) also need a source of normally distributed PRN’s [@econo]. There are several algorithms available for PRN’s with Gaussian pdf’s[@devroye]. Some, such as Box-Muller’s algorithm, require an input of uniform PRN’s, and their output often suffers from the pitfalls of the latter [@BM]. Robustness is therefore a relevant issue. In addition, Box-Muller’s algorithm is slow and can consequently consume significant fractions of computer simulation times [@jffjr]. The comparison method demands several uniform PRN’s per normal PRN, and is therefore also slow [@comparison]. Use of tables [@toral] is not a very accurate method. Algorithms that are related, but not equivalent, to the one we propose here have been published [@jffjr; @wallace], but they are somewhat cumbersome to use. In addition, no proof of their validity has been given.
We propose here a new algorithm for the generation of normally distributed PRN’s that is quite simple and fast. It is a stochastic caricature of a closed classical system of $N$ particles. Their velocities provide a source of PRN’s. We prove that, for any initial state, their pdf becomes Maxwellian in the $N\rightarrow\infty$ limit, after an infinite number of two-particle “collisions” take place. To this end, we first prove that our system is ergodic [@lebow; @arnold]. The proof is not exceedingly difficult because our system is not deterministic. We also study its output as a function of $N$, and establish useful criteria for its implementation. Correlation test results are also reported.
For the motivation, consider numbers $v_1,v_2 \ldots v_N$, placed in $N$ computer registers, analogous to velocities of $N$ particles that make up a closed classical system in 1D. Pairs of registers $\imath$ and $\jmath$, say, selected at random without bias, are to “interact” somehow, conserving quantity $v_i^2+v_j^2$. By analogy with the approach to equilibrium (i.e., to Maxwell’s velocities distribution) that is believed to take place in Statistical Physics, we expect that sufficient number of iterations will lead to an approximately Gaussian pdf of register values, from which the desired PRN’s may be drawn. (See also Ref. [@jffjr].) We define below the simplest interaction we can think of in order that (1) implementation on a computer be very fast, and (2) that we may be able to prove that a Gaussian pdf does indeed ensue.
Before the algorithm is implemented, all $N$ registers must be initialized to, say, $v_\imath=1$ for all $\imath$ satisfying $1\leq \imath\leq N$, or all $v_\imath$ may be read from a set of $N$ register values saved from a previous computer run, which we assume to fulfill $\sum v_\imath^2=N$. Let $U(1,N)$, $U_\imath(1,N)$ be unbiased integer random variables, both in the interval $[1,N]$, except that $U_\imath$ cannot equal $\imath$. The algorithm follows: $$\begin{aligned}
\imath=U(1,N);\;\jmath=U_i(1,N);\\
\label{1}
v_\imath\leftarrow (v_\imath+v_\jmath)/\sqrt{2};\\
\label{2}
v_\jmath\leftarrow -v_\imath+\sqrt{2}v_\jmath
\label{3}\end{aligned}$$ The updated value of $v_\imath$, from Eq. (2), is used in Eq. (3). After an initial [*warm up*]{} phase (see below), $v_\imath$ and $v_\jmath$ may be drawn each time transformation (1-3) is applied. They are two independent PRN’s, each one with an approximately Gaussian pdf, with $\langle v_\imath\rangle =0$ and $\langle v^2_\imath\rangle =1$ for all $\imath$, if $N$ is sufficiently large (see below). Transformation (1-3) may be thought of as a rotation of $\pm\pi/4$ with respect to a randomly chosen $\imath \jmath$ plane ($+$ and $-$ signs are for the two possible index orderings, $\imath \jmath$ and $\jmath \imath$). Thus, quantity $\sum v_{\ell}^2$ is conserved. Frequencies of events from sequences of $10^6$, $10^8$ and $10^{10}$ PRN’s generated with transformation (1-3), with $N=1024$, are exhibited in Fig.\[Fig1\].
We first explain why PRN’s genered by transformation (1-3) are expected to be normally distributed. Let ${\bf P}_n({\bf v})$ be the probability density at ${\bf v}=(v_1, v_2,\ldots,v_N)$, after transformation (1-3) has been applied $n$ times, on the $(N-1)$-dimensional spherical surface ${\cal S}_{N-1}$, of radius $\sqrt N$ given by $N=\sum_{\ell=1,\ldots,N} v_\ell^2$. Let the single register pdf $p(v)$ be the $n\rightarrow\infty$ limit of $p_n(v)$, where $p_n(v_1)=\int {\bf P}_n({\bf v})\:dv_2dv_3\ldots dv_N$. We show further below that ${\bf P}_n({\bf v})\rightarrow constant$ over ${\cal S}_{N-1}$ as $n\rightarrow\infty$. It then follows by integration that, $$p(v)\propto \left(1-{{v^2}\over N}\right)^{(N-3)/2}.
\label{max}$$ Clearly, $p(v)\rightarrow C\exp(-v^2/2)$ in the $N\rightarrow\infty$ limit, which is the desired result.
We prove below, in three stages, that $P_n({\bf v})$ does indeed become homogeneous over spherical surface ${\cal S}_{N-1}$, if $N\geq 3$, in the $n\rightarrow\infty$ limit. We first prove $P_n({\bf v})\leftrightarrow
P_n({\bf u})$ as $n\rightarrow\infty$ if ${\bf v}$ and ${\bf u}$ are related. \[From here on, we say that points ${\bf v}$ and ${\bf u}$ are [*related*]{} if succesive transformations (1-3) of ${\bf v}$ can lead to ${\bf u}$.\] We then prove that the system’s “orbit” covers ${\cal S}_{N-1}$ densely \[that is, that any point ${\bf v}\in{\cal S}_{N-1}$ can be brought arbitrarily close to any other point ${\bf u}\in{\cal S}_{N-1}$ by applying transformations (1-3) to ${\bf v}$ a sufficient number of times\]. Then, the desired result follows easily. It may help to place the significance of the proof that follows into proper perspective to note that if in Eq. (1) $\jmath\leftarrow U_i[1,N]$ is replaced by $\jmath=\imath +1\; mod N$, the system becomes then non-ergodic, as can be easily checked numerically.
To start the proof, let kernel $K({\bf v},{\bf v^\prime})$ be defined by $P_{n+1}({\bf v})=\int K({\bf v},{\bf v^\prime})P_n({\bf v^\prime})\;d{\bf v^\prime}$, and let $$F_n\equiv \int \{P^2_{n+1}({\bf v})-P^2_n({\bf v})\}\;d{\bf v}.$$ Note first that $F_n<0$ implies that $P_{n+1}({\bf v})$ is more uniform than $P_n({\bf v})$, in the sense that $\int d{\bf v}\;[P_{n+1}({\bf v})-{\overline P}]^2
<\int d{\bf v}\;[P_{n}({\bf v})-{\overline P}]^2$, where ${\overline P}=1/\int dv $. It follows from the definition of $K({\bf v},{\bf v^\prime})$ that $$F_n=
\int d{\bf v}\{[\int d{\bf v_1}K({\bf v},{\bf v}_1)
P_n({\bf v}_1)]^2-P^2_n({\bf v})\}.
\label{Fn}$$ Making use of the detailed balance condition, $K({\bf v},{\bf v^\prime})=K({\bf v^\prime},{\bf v})$, which our system satisfies, and the relation $\int\;d{\bf v}\;K({\bf v},{\bf v^\prime })=1$, Eq. (\[Fn\]) can be cast into, $$F_n=
-{1\over 2}\int d{\bf v}\int d{\bf v_1}\int d{\bf v_2}\;
Q({\bf v},{\bf v}_1,{\bf v}_2),$$ where, $Q=K({\bf v},{\bf v}_1)
K({\bf v},{\bf v}_2)[P_n({\bf v}_1)-P_n({\bf v}_2)]^2$. Therefore, in the $n\rightarrow\infty$ limit, $P_n({\bf v})$ becomes constant over each set in ${\cal S}_{N-1}$ within which any two points ${\bf v,u}$ are related.
We now prove that the system’s orbit covers ${\cal S}_{N-1}$ densely. Let $H_N$ be the group of transformations in $N$ dimensions defined by Eqs. (1-3). We first show that any rotation in $3$D \[that is, any element of $SO(3)$\] can be approximated arbitrarily close by elements of $H_3$. The proof is extended to higher dimensions by induction. Note first that $H_3$ does not belong to the set of [*finite*]{} rotation groups in $3$D[@coxeter], and is therefore an infinite group. Let group $SO(3)$ be covered by spheres of radius $\epsilon/2$ each. A finite number of them is sufficient, since the volume of $SO(3)$ is finite [@gilmore]. It follows that there must be at least one sphere with two elements of $H_3$ in it, since $H_3$ has an infinite number of elements. Let these two elements be $r$ and $s$, and let $g({\bf u},\epsilon )$ be element $r s^{-1}$ of $H_3$, which is a rotation by angle $\epsilon$ about some undetermined ${\bf u}$ axis. We will build elements of $H_3$ that are as near as desired to any given rotation. To this end, it is sufficient to show that it can be done for a set of infinitesimal generators of rotations [@goldstein]. One such set is made up of infinitesimal rotations about three linearly independent axes. Consider axes ${\bf u}_1$, ${\bf u}_2$, and ${\bf u}_3$ that are obtained from ${\bf u}$ by rotations $g(1,\pi /2)$, $g(2,\pi /2)$, and $g(3,\pi /2)$ about each one of the coordinate axes by angle $\pi /2$. The correspondng infinitesimal rotations are given by [@gilmore], $g({\bf u}_\imath,\epsilon )=g(\imath ,\pi /2) g({\bf u},\epsilon)
g^{-1}(\imath ,\pi /2).$ This concludes the proof for 3 dimensions.
We now prove by induction that any element $g(\imath \jmath ,\alpha )$, for the rotation about plane $\imath \jmath$, by angle $\alpha$ of the rotation group $SO(N)$ can be approximated as nearly as desired by an element $g$ of group $H_N$, for $N>3$. By hypothesis, any $g(\imath \jmath ,\alpha )$, for $\imath ,\jmath =1,2,\ldots ,N$ can be approximated by an element $g$ of $H_N$. We show now that $g(\imath\; N+1 ,\alpha )$, for $\imath =1,2,\ldots ,N$, can also be approximated by elements of $H_{N+1}$. We take $g\in H_N$ within distance $\epsilon$ of $g_{\imath \jmath}(\alpha )$. Now, since rotations preserve distances, it follows that $g(\imath\;N+1,\alpha )\in SO(N+1)$, given by $g(\imath\;N+1,\alpha )=g(\imath\;N+1,\pi /2)
g(\imath \jmath ,\alpha )g^{-1}(\imath\;N+1,\pi /2)$ is within distance $\epsilon$ of $g^\prime\in H_{N+1}$, given by $g^\prime =g(\imath\;N+1,\pi /2)
gg^{-1}(\imath\;N+1,\pi /2)$. This proves dense coverage in $N\geq 3$ dimensions. This is a stochastic generalization of Jacobi’s theorem [@arnold] to more than two dimensions.
To conclude the proof that $P_n({\bf v})\rightarrow constant$ in the $n\rightarrow \infty$ limit, consider any two points ${\bf V}$ and ${\bf U}^\prime$ as centers of disks ${\cal D}_{\bf V}$ and ${\cal D}_{{\bf U}^\prime}$, both of radius $r$, in ${\cal S}_{N-1}$. Since the system’s orbit covers ${\cal S}_{N-1}$ densely for $N\geq 3$, it follows that a point ${\bf U}$ that is related to ${\bf V}$ exists arbitrarily close to ${\bf U}^\prime$. Consider now disks ${\cal D}_{\bf V}$ and ${\cal D}_{\bf U}$. The fact that there exists at least one sequence of rotations in $H_N$ that take ${\bf V}$ into ${\bf U}$ implies that there exists at least one single rotation $g$ in $H_N$ that transforms ${\bf V}$ into ${\bf U}$. Since $g$ is a rotation, it transforms ${\cal D}_{\bf V}$ [*rigidly*]{} into ${\cal D}_{\bf U}$. It follows that $\int d{\bf v}\; P_N({\bf v})$ over ${\cal D}_{\bf V}$ equals $\int d{\bf u}\;P_N({\bf u})$ over ${\cal D}_{\bf U}$. Since $r$ is arbitrary, and ${\bf V}$ and ${\bf U}^\prime$ are any two points in ${\cal S}_{N-1}$, it follows that $P({\bf v})$ is constant over ${\cal S}_{N-1}$ (except, perhaps, on a set of measure zero). This is the desired result. Ergodicity follows [@arnold].
We next address the following practical issues: (1) how good an approximation to a Gaussian pdf of PRN’s is achieved with a necessarily [*finite*]{} set of $N$ registers; (2) how long must the warm up phase be.
It is convenient to rewrite Eq. (\[max\]) as follows, $$p(v)\propto e^{-v^2/2}e^{{g_N(v)}/{N}}.
\label{gofx}$$ where $g_N(v)=v^2(3-v^2/2)/2+{\cal O}(1/N)$. $N^{-1}g_N(v)$ is approximately the fractional deviation, $\delta p(v)/p(v)$, from Gaussian form if $\delta p(v)/p(v)\ll 1$. We have checked this behavior numerically. Clearly, the [*number of registers*]{} $N$ that must be used increases with the number $M$ of PRN’s one intends to generate. This is because the value of the largest PRN generated increases, on the average, with $M$. More precisely, the value of $v$ beyond which PRN’s are only generated with probability $q$ is approximately given by $v^2\approx 2\ln(M/vq)$. Now, it follows from Eq. (\[gofx\]) that the fractional error $\delta P/P$ in the probability density at $v$ is approximately $N^{-1}v^2(3-v^2/2)/2$ for very large $N$. (It is pointless to require this error to be too small since a PRN is expected to be generated beyond $x$ with a small probability $q$.) It then follows that $[\ln(M/qv)]^2\lesssim N\delta P/P$ must be satisfied by $N$. Thus, approximately $10^4$ registers are sufficient in order to generate as many as $10^{15}$ PRN’s, with a roughly $10\%$ error in the probability for the [*largest*]{} PRN in the sequence. For results obtained from a sequence of $10^{10}$ PRN’s generated with $1024$ registers, see Fig. 1.
Our algorithm must be applied a number $n_pN$ of times before it is ready for use unless all $v_\imath$ are initialized with“equilibrium” values (stored from some previous computer run). The distribution of all register values then evolves towards equilibrium, as illustrated in Fig. 2. Deviations from equilibrium are statistically insignificant for $n_p\gtrsim 2$ and $N=1024$, and for $n_p\gtrsim 4$ and $N=1\:048\:576$. Since $n_p$ is expected to increase as $\ln N$, $n_p=8$ should provide ample warm up for any forseeable applications.
The number of PRN’s that must be generated before each PRN in sequence $v_1,v_2,\ldots,v_N$ returns within distance $r$ from its initial value is exponential in $N$. More specifically, we estimate it to be $(\tau /\sqrt{N})(1/r)^N$ for $N\gg 1$, where $\tau$ is the period of the algorithm used to select $\imath$ and $\jmath$ in Eq. (1). The estimation is based on $P_n({\bf v})\rightarrow constant$ over ${\cal S}_{N-1}$ as $n\rightarrow\infty$. Thus, an effectively infinite recurrence time follows for any reasonable value of $N$.
Correlations between a finite number of PRN’s clearly vanish as $N\rightarrow\infty$, since $\imath$ and $\jmath$ in Eq. (1) are supposedly independent PRN’s. We have searched for correlations in $m$ succesively generated PRN’s $v_1,v_2,\ldots v_m$, for $m=3,4,\ldots ,6$, performing a chi-square isotropy test over the corresponding $m$-dimensional space. An $m$-tuple ${\bf v}=v_1,v_2,\ldots,v_m$ was said to belong to the $i-th$ cone, of $1024$ randomly oriented cones with axes ${\bf w}_1,{\bf w}_2,\ldots,{\bf w}_{1024}$, if $0.99\leq {\bf v.w_\imath}\leq 1$. No significant deviations from isotropy were observed for $10^6$ generated $m$-tuples.
Implementation of Wolff’s algorithm [@wolff] in MC calculations of the Ising model’s critical behavior is a demanding test that some well known uniform PRN generators have failed [@ising1]. Large clusters are then flipped as a whole, and this tests correlations in very long sequences. We have used normal PRN’s generated by our algorithm as input into a MC simulation of an Ising system of $16\times 16$ spins at the critical temperature. \[For that, we note that $v^2_\imath +v^2_\jmath >2x$ as often as $u>\exp (-x)$ if $v_\imath$ and $v_\jmath$ ($u$) are PRN’s with Gaussian (uniform) pdf’s, respectively.\] The energy obtained is shown in Fig. 3 as a function of the number of registers $N$. The following uniform PRN algorithms were used to select $\imath$ and $\jmath$ in Eq. (1): ggl [@ising1], R(250, 103,xor) [@rw; @ising1], and RAN3 [@nr]. We tried the latter two algorithms, which have been shown to lead by themselves to unacceptable results for the Ising model [@ising1], in order to test our algorithm’s robustness. The results shown in Fig. 3 are gratifying.
Similarly, the specific heat $c$ and magnetization $m$ fluctuations data points obtained follow approximately the relations $c\simeq c_0+8.4/N$, and $\langle (\delta m)^2\rangle \simeq \chi_0+33/N$, respectively, where $c_0=1.497(1)$ and $\chi_0=0.5454(2)$, in agreement with the known exact values[@ising1; @fisher].
Double precision is recommended. It prevents excesive drift of the sum $\sum v_\imath^2$ away from its assigned value. Even then, single precision accuracy is to be expected at the end of a sequence of some $10^{16}$ PRN’s, unless the sum is normalized several times during the run.
In summary, we have shown that implementation of Eqs. (1-3) provides a source of PRN’s with an approximately Gaussian pdf. Some $10^4$ registers (molecules) are sufficient for some purposes, but up to $10^5$ or more may be necessary for more demanding tasks. (Having to make a decision about the number of registers to be used may sometimes be an unwelcomed task. On the other hand, it is a virtue of the algorithm, that one can control, through the value of $N$, how close the output is to be from sequences of truly independent random numbers with Gaussian pdf’s.) Initial warm ups for arbitrary initial conditions are necessary; it is sufficient to let each register initially interact an average number of, say, 8 times. The system’s recurrence time was shown to be exponential in $N$, and therefore effectively infinite. Its behavior appears to be robust. The proposed algorithm runs an order of magnitude faster on computers than the most often used Box-Muller method [@devroye; @BM]. For a fortran code of our algorithm or other questions, please write [email protected].
Continuous help from Dr. Pedro Martínez with computer systems is deeply appreciated by JFF. We are indebted to Prof. P. Grassberger for an important suggestion. JFF and CC are grateful for partial financial support from DGICYT of Spain, through grants No. PB95-0797 and PB97-1080, respectively.
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---
abstract: 'We present a new and simple technique for selecting extensive, complete and pure quasar samples, based on their intrinsic variability. We parametrize the single-band variability by a power-law model for the light-curve structure function, with amplitude $A$ and power-law index $\gamma$. We show that quasars can be efficiently separated from other non-variable and variable sources by the location of the individual sources in the $A$-$\gamma$ plane. We use $\sim$60 epochs of imaging data, taken over $\sim $5 years, from the SDSS stripe 82 (S82) survey, where extensive spectroscopy provides a reference sample of quasars, to demonstrate the power of variability as a quasar classifier in multi-epoch surveys. For UV-excess selected objects, variability performs just as well as the standard SDSS color selection, identifying quasars with a completeness of 90% and a purity of 95%. In the redshift range $2.5<z<3$, where color selection is known to be problematic, variability can select quasars with a completeness of 90% and a purity of 96%. This is a factor of 5-10 times more pure than existing color-selection of quasars in this redshift range. Selecting objects from a broad $griz$ color box *without* $u$-band information, variability selection in S82 can afford completeness and purity of 92%, despite a factor of 30 more contaminants than quasars in the color-selected feeder sample. This confirms that the fraction of quasars hidden in the “stellar locus” of color-space is small. To test variability selection in the context of Pan-STARRS 1 (PS1) we created mock PS1 data by down-sampling the S82 data to just 6 epochs over 3 years. Even with this much sparser time sampling, variability is an encouragingly efficient classifier. For instance, a 92% pure and 44% complete quasar candidate sample is attainable from the above $griz$-selected catalog. Finally, we show that the presented $A$-$\gamma$ technique, besides selecting clean and pure samples of quasars (which are stochastically varying objects), is also efficient at selecting (periodic) variable objects such as RR Lyrae.'
author:
- |
Kasper B. Schmidt$^{1}$, Philip J. Marshall$^{2,3}$, Hans-Walter Rix$^{1}$,\
Sebastian Jester$^{1}$, Joseph F. Hennawi$^{1}$, & Gregory Dobler$^{4,5}$
title: Selecting Quasars by their Intrinsic Variability
---
Introduction {#sec:intro}
============
Large, complete and pure samples of quasars have proven crucial for observational cosmology. Quasars serve as probes of galaxy evolution, map black hole growth and probe (and affect) the intergalactic medium. Quasar clustering is a tracer of mass clustering on both large and small scales [@croom05; @croom09b; @shen07; @shen09; @ross09], and the large samples provide precise measurements of the evolution and spectral properties of the quasars themselves [@vandenberk01; @richards02b; @richards04; @richards06; @richards09; @boyle00; @croom09b]. Furthermore, huge quasar samples are required to find a large number of gravitationally lensed quasars [@oguri06; @inada08]. Through the gravitationally magnified quasars the quasar samples indirectly contribute to the understanding of the molecular gas content in distant galaxies [@yun97; @riechers07a; @riechers07b], mapping of the intergalactic medium and structures [e.g. @metcalf05; @hennawi06; @hennawi07] and exploration of the dark matter (halo) content of galaxies [@dalal02; @bradac02; @dobler06; @maccio08]. In short, large well-defined quasar samples are a cornerstone of observational cosmology.
Photometric quasar samples have recently grown to nearly a million objects [850,000 actual quasars; @richards09]. Despite these impressive catalog sizes the number statistics still limit the achievable science in various cases; especially those where particular and hence rare geometric constellations of quasars are needed. For instance a 3$\sigma$ detection of a luminosity-dependent quasar bias above $z\gtrsim$1.9 when analyzing the angular clustering of quasars, needs an estimated sample size of at least 1,200,000 actual quasars [@myers07a; @myers07b; @myers09]. Searches for binary quasars [@hennawi06; @hennawi09; @myers08] - which provide interesting knowledge about small scale clustering and hence shed light on quasar triggering mechanisms and the nature of quasar progenitors - also should be based on samples with $>10^6$ actual quasars in order to obtain reasonably sized statistical samples of possible quasar pairs. Also quasar-galaxy clustering [e.g. @scranton05; @lopez08; @padmanabhan08; @burbidge09], i.e. exploring the statistics of quasars behind the foreground galaxies, calls for larger (relatively low-$z$) quasar samples than exist to date. Furthermore, exploring the “transverse proximity effect” in the Ly$\alpha$ forest of quasars, with foreground quasars near the sight line of background quasars [e.g. @hennawi06a; @hennawi07] is presently limited by quasar sample sizes. Obtaining larger photometric quasar catalogs to boost possible candidates for spectroscopic follow-up is needed. The $\sim$3$\sigma$ detections of the integrated Sachs-Wolfe effect by cross correlating quasars with the CMB to estimate the size of cosmological parameters and the dark energy equation of state [e.g. @giannantonio08; @xia09; @scranton05] will also be improved by larger photometric samples of 1$<$$z$$<$5 quasars. Last but not least, larger photometric quasar catalogs will enhance the number of known gravitationally lensed quasars [e.g. @oguri10]. At present $\sim$100 quasar lenses are known and an even larger sample of the relatively rare gravitationally lensed quasar systems will among other things improve our knowledge about cosmology, galaxy mass distributions, quasar hosts and the growth of the host’s central black holes [@schneider06 and references therein]. These few examples serve as a scientific justification for pursuing even larger photometric samples of low as well as of high redshift quasars.
Most existing large quasar samples haven been selected on the basis of their UV/optical colors or radio flux. However, quasars are known to vary intrinsically on timescales of months to years and can therefore be selected alternatively on the basis of their variability (alone). Several physical processes are discussed as important causes of this variability: foremost are accretion disc instabilities [@rees84; @kawaguchi98; @pereyra06] but also large-scale changes in the amount of in-falling material may be important [e.g. @hopkins06 and references therein]. Also, starbursts in the host galaxy [@aretxaga97; @cidfernandes97], micro lensing by the host galaxy and compact dark matter objects [@hawkins96; @zackrisson03], and stochasticity of multiple supernovae [@terlevich92] have all been proposed. Irrespective of the physics behind the variability, quasars are observed to exhibit brightness variations, of typically $\gtrsim10\%$ over several years [e.g. @giveon99; @vandenberk04; @rengstorf04; @sesar07; @macleod08; @bramich08; @wilhite08; @kozlowski09; @bauer09; @kelly09]. This variability has been exploited for several purposes, e.g. to estimate Eddington ratios and black hole masses [@bauer09; @wilhite08], or simply to identify them [@geha03].
With SDSS, QUEST and OGLE [see e.g. @abazajian09; @rengstorf04; @udalski97 respectively], large-scale, multi-epoch and multi-band surveys have emerged, and have been used to search for quasars. The Panoramic Survey Telescope & Rapid Response System 1 [Pan-STARRS 1, @kaiser02] and 4 [Pan-STARRS 4, @morgan08], and the Large Synoptic Survey Telescope [LSST, @ivezic08; @LSST] will take such surveys to the next level. In all these surveys, the largest quasar samples stem from color selection in imaging [e.g. @richards02; @richards04; @richards09; @atlee07; @d'abrusco09]. The characteristic so-called “UV excess” of quasars, their bright blue $u-g$ color, is capable of separating the quasars from their stellar contaminants in color-color space, allowing for efficient selection of targets for spectroscopic follow-up [e.g. @strauss02]. Such UV excess color selection is, however, only efficient for low ($z\lesssim2.5$) and high ($z\gtrsim3$) redshift quasars, since the quasar and stellar loci overlap in the $u-g$ color for $2.5<z<3.0$ objects, causing the selection efficiency (or purity) in that region to drop below 50%. For quasars with $2.6<z<2.8$ the efficiency is close to 10% [@richards06]. This confusion reigns until the Ly-break of high-$z$ quasars moves into the $g$-band and again makes for unusual colors [e.g. @fan01; @fan06].
Moreover, $u$-band imaging is expensive: the area and depth of an optical imaging survey can be greatly increased by focusing on redder filters, where atmospheric attenuation is lower and detectors more efficient. For example, the Pan-STARRS 1 telescope offers the possibility of creating the largest sample of quasars to date with its multi-epoch 30,000 deg$^2$, 3/4 sky, “3$\pi$” [*grizY*]{} imaging survey. For the purposes of identifying quasars in this data set, the question remains, “can we compensate for the lack of $u$-band data by exploiting the multi-epoch nature of the $g$-band imaging instead?” With one eye on the potential of Pan-STARRS 1, we therefore explore here the possibilities of creating large, complete and pure samples of quasars based on limited color information, but with light curves spanning several years. We use Stripe 82 of SDSS [@abazajian09] as a testbed, both for the method in general and for making mock PS1 data sets.
This paper is organized as follows. In Section \[sec:var\] we briefly review previous attempts to characterize quasar variability in optical imaging surveys, and then introduce the SDSS Stripe 82 data sets in Section \[sec:data\]. We introduce our methodology for quantifying the variability of various objects via their individual power-law structure functions in Section \[sec:pwrfit\], and show results of selection experiments in Stripe 82 in Section \[sec:results\]. After a brief discussion in Section \[sec:disc\], we conclude in Section \[sec:conc\]. All magnitudes are given in the AB system.
Variability Characterization of Sources and the Structure Function {#sec:var}
==================================================================
In this section we briefly review the strengths and weaknesses of optical color selection, of particular sources, focusing on quasars. We then present our approach to quantifying source variability, which we will then explore as an additional approach to selecting quasars, of other sources.
Color Selection {#sec:COL}
---------------
The most common way to generate large samples of optical quasar candidates for follow-up is by specifying a particular region of interest in color space, as was done e.g. in SDSS [@richards02; @richards06; @richards09]. For quasars at $z<3$ the $u-g$ color is crucial in this approach since it enables a photometric separation of the quasar candidates from the stellar locus, reducing the number of contaminating objects to a point where spectroscopic follow-up is feasible. This is illustrated in Figure \[fig:coloroverlap\], where we have plotted the median color of $\sim$9000 spectroscopically confirmed quasars, as well as an illustrative comparison sample of 5000 F/G and 483 RR Lyrae stars (see Section \[sec:data\]), all drawn from the SDSS Stripe 82 photometric catalog DR7 [@abazajian09]. The top panels and the bottom left panel of Figure \[fig:coloroverlap\] shows the distribution of the samples in the color-color planes of the SDSS $ugriz$ color cube. This clearly shows the power of the $u-g$ color (upper left panel) compared to the $g-r$, $r-i$ and $i-z$ colors in separating the quasars from their contaminants, especially for low redshift quasars (i.e. $z\lesssim2.5$ shown as light blue points). The color magnitude diagram in the lower right panel illustrates that a cut in magnitude will also eliminate contaminants. The contours indicate the stellar locus of Stripe 82 point sources with $15<r<18$
For higher redshift objects ($z\gtrsim 2.5$, shown as magenta points in Figure \[fig:coloroverlap\]) the quasars intersect the stellar locus (top left panel). The purity for $2.5<z<3.0$ quasar candidate samples is around 10-50% in the color-selected SDSS quasar target sample [@richards06]. In general the color selection method is efficient for low and high redshift quasars, but for the intermediate redshift objects contamination becomes a severe problem. We refer to Figures 13 and 14 in @richards02 for a more complete version of Figure \[fig:coloroverlap\].
With (henceforth PS1) the contamination problem is even more pronounced when using the color selection method only. has a 5-filter system consisting of SDSS-like $g$, $r$, $i$, $z$ bands (albeit with significantly higher red sensitivity) and a $Y$ filter. The crucial $u-g$ color used in the SDSS color selection method is not available: the contamination of a color-selected quasar sample will be a problem for $z>2.5$ as well as for $z<2.5$. It is therefore necessary to find a way of separating the majority of quasars from the contaminating stellar locus in order to obtain a pure quasar sample. The intrinsic variability of the quasars (and their contaminants) is a very promising tool for doing this.
Source variability: power-law structure functions {#sec:SF}
-------------------------------------------------
The structure function characterizes the variability of quasars (and the other sources) by quantifying the variability amplitude as a function of the time lag between compared observations [@cristiani96; @giveon99; @eyer02; @vandenberk04; @devries05; @rengstorf06]. For any object the observables for estimating the structure function are the $\frac{N(N-1)}{2}$ data pairs, assuming $N$ light curve data points, describing the variability as the magnitude difference between two epochs $i$ and $j$, corrected for measurement errors, i.e., $$\label{eqn:datapairs}
V_{i,j}(\Delta t_{i,j}) = \Delta m_{i,j} - \sqrt{\sigma_i^2+\sigma_j^2} \: .$$ Here $\Delta m_{i,j}$ is the measured magnitude difference between observation $i$ and $j$. The $\sigma_i$ and $\sigma_j$ are the photometric errors on the measurements and $\Delta t_{i,j}$ is the time difference between the two observations. The quantity $V$ is defined like this so that its average, over a large number of data pairs, is an [*estimator for the intrinsic standard deviation of the source magnitude.*]{}
At this point we note that $\Delta t_{i,j}$ usually refers to the time lag in the quasar rest frame. However, computing this requires [*a priori*]{} knowledge of the quasar redshift, and when selecting objects in imaging-only surveys, we do not know the object redshift. Therefore, we work with time lags in the [*observed frame.*]{} This necessary convention differs from most of the quasar variability literature; we will make some comparisons in Section \[sec:pwrfit\].
In previous analyses, the average $V$ has been calculated in $n$ time lag bins using data pairs from many quasars, thus “stacking” the variability signal to allow the properties of the quasar population to be probed. Then $$\label{eqn:SFbin}
V(\Delta t) = \left\langle
\sqrt\frac{\pi}{2}
|\Delta m_{i,j}|-\sqrt{\sigma_i^2+\sigma_j^2}
\right\rangle_{\Delta t} \, ,$$ where the average, $\left\langle \; \right\rangle_{\Delta t}$, is taken over all epoch pairs $i,j$, whose lag falls in the bin $\Delta t$. The same approach can be taken in estimating the structure function of classes of objects (e.g. quasars at a given luminosity and redshift bin) if, say, only two epochs are available per object, but large samples exist (e.g. [@richards06; @richards09]); in that case the $\left\langle \; \right\rangle$ in Equation \[eqn:SFbin\] becomes and ensemble average. @vandenberk04 and others find that the ensemble average quasar structure function appears to follow an increasing power law with time lag.
On the other hand, for the case where the light curve sampling of each object is high, we can compute the average $V(\Delta t)$ for an [*individual object*]{} [@eyer02]. Binning the $\frac{N(N-1)}{2}$ data pairs from an object’s $N$-point light curve gives an estimate of $V(\Delta t)$ defined at each bin center. This approach is computationally efficient, and provides a free-form view of the object’s structure function. However, in the case of relatively sparse sampled data (6 epochs over 3 years in the $3\pi$ survey, see Section \[sec:DS\]), binning the data pairs to obtain the structure function from Equation \[eqn:SFbin\] to estimate the variability may not be the optimal approach.
In Equation \[eqn:SFbin\], both the noise and the intrinsic photometric variability are assumed (implicitly) to have a Gaussian distribution [@rengstorf06]. We can then extend this simple model of quasar variability to include a power law increase in variability with time lag. Drawing on the results from [@richards06] we propose a power-law model for the structure function given by $$\label{eqn:powerlaw}
V_\textrm{mod}(\Delta t_{i,j} | A,\gamma) =
A \left( \frac{\Delta t_{i,j}}{1 \textrm{yr}}\right)^\gamma.$$ We can then fit this model to a given set of data, $(\Delta m_{i,j},\Delta t_{i,j})$, as follows. We write the likelihood for the power law parameters $A$ and $\gamma$ as $$\label{eqn:L}
\mathcal{L}(A,\gamma) = \prod_{i,j} L_{i,j} \; ,$$ assuming a set of independent magnitude differences as our data. Here $L_{i,j}$ is the likelihood of observing one particular magnitude difference $\Delta m_{i,j}$ between two light curve points separated by $\Delta t_{i,j}$. Following the ensemble analyses referred to above, we assume an underlying Gaussian distribution of $\Delta m$ values and Gaussian photometric errors: $$L_{ij} = \frac{1}{\sqrt{2\pi \textrm{V}_{{\rm eff},ij}^2}}
\exp\left( -\frac{\Delta m_{ij}^2}{2\textrm{V}_{{\rm eff},ij}^2} \right),$$ Here, the effective (observed) variability $\textrm{V}_\textrm{eff}$ is $$\label{eqn:Veff}
\textrm{V}_{{\rm eff},ij}^2 =
V_{\rm mod}(\Delta t_{ij}|A,\gamma)^2 + \left(\sigma_i^2+\sigma_j^2\right),$$ i.e., we propagate the photometric errors $\sigma_i$ and $\sigma_j$ by adding them in quadrature to the variability “error” $V_{\rm mod}(\Delta
t_{ij}|A,\gamma)$
This approach can yield posterior probability distributions on the two model parameters, $A$ and $\gamma$. The amplitude $A$ quantifies the root-mean-square magnitude difference on a 1 year timescale, while $\gamma$ is the logarithmic gradient of this mean change in magnitude. We assign uninformative priors for the parameters (uniform in the logarithm of $A$, and uniform in the arctangent of $\gamma$ – since $\gamma$ represents the slope of a straight line), and then explore the posterior probability distribution for these two power law parameters via a simple Markov chain Monte Carlo (MCMC) code [@metropolis53; @hastings70] as described in Appendix \[sec:mcmc1\]. All we are doing is replacing $n$ binned structure function parameters (the values of $V$ in each of the $n$ bins) with two parameters that define a power law structure function, and then inferring these parameters instead of constructing estimators for them. We will show in Section \[sec:pwrfit\] that using a power law model for the variability actually provides a good fit.
SDSS Stripe 82: a Testbed for Variability Studies {#sec:data}
=================================================
Anticipating the results of Section \[sec:pwrfit\], we note that to detect and quantify intrinsic quasar variability will likely require multi-epoch data spanning several years. Before surveys with facilities such as Pan-STARRS and LSST become available, SDSS Stripe 82 [@abazajian09] forms an excellent training set and methodological test bed [e.g. @sullivan05; @sesar07; @bramich08; @frieman08]. In this section we will describe the various Stripe 82 data sets that we have created in order to test and illustrate the prospects of our algorithm.
The SDSS Stripe 82 region (henceforth S82) covers approximately 320 deg$^2$, from right ascension around 290$^\circ$ to 60$^\circ$ in a $2.5^\circ$ wide band on the celestial equator. Over eight observing seasons it has been repeatedly observed in the fall months, resulting in many epochs (typically $\sim$60) in each of the 5 SDSS bands. As the $3\pi$ survey will contain fewer epochs, we can “down-sample” the S82 object light curves to simulate observations taken with (albeit ones at lower angular resolution and depth).
Relative to PS1, S82 does have the advantage of $u-g$ color coverage, and extensive bright object spectroscopy. One can therefore construct quite pure samples of quasars, RR Lyrae and so on, that may serve as ground truth for our variability selection.
In the following subsections we describe these various subsamples in some detail, and provide a brief overview here. We have selected all the spectroscopically confirmed quasars in S82 together with a representative set of (stellar locus) contaminants, which contains non-varying (type F/G stars) as well as varying (RR Lyrae) point sources to illustrate our method and algorithm prospects. These objects’ photometry data are plotted in $ugriz$ color space in Figure \[fig:coloroverlap\]. To investigate the selection of quasars by their colors, we define three color selection boxes and explore the objects returned by each. One of these mimics the more limited color selection possible with PS1: quantifying how the variability information then improves the quasar selection is one of the main goals of this paper.
In the following subsections, we describe two preparatory steps for turning the $\sim$ 60 epoch S82 data into a testbed for color plus variability based quasar selection in SDSS (S82) and PS1: first, we describe the definition of various sub-sets of candidate objects; then we describe some technical steps “cleaning” the light curves and down-sampling the S82 data to mimic PS1.
Spectroscopically-confirmed quasars in Stripe 82 {#sec:qsos}
------------------------------------------------
Key to designing a quasar variability selection algorithm is an understanding of the variability properties of objects that are indeed spectroscopically confirmed quasars. We have selected all of these (both point sources and extended objects) published in the SDSS DR5 quasar catalog [@schneider07] that fall within S82. There are 9157 spectroscopically confirmed DR5 quasars in S82, spanning a redshift range from 0.08 to 5.09. These quasars have $15.4<i<22.0$ with a mean of 19.5. See [@schneider07] for the corresponding numbers for the whole DR5 quasar catalog.
To get the multi-epoch photometry (light curves) for the 9157 quasars we performed an SQL neighbor search in the S82 DR7 database, choosing a search radius of 0.5” to minimize the light curve contamination from misidentified (spatial) neighbors. This search on average yielded 60 epochs per object, after selecting only entries with good `BRIGHT`, `EDGE`, `BLENDED`, `NODEBLEND`, `SATUR`, `PEAKCENTER`, `NOTCHECKED`, `INTERP_CENTER` and `DEBLEND_NOPEAK` flags (of which the first 5 are referred to as fatal and the rest as non-fatal flags by @richards02 – see their Table 2 or @stoughton02 Table 9 for a description of the flags). For consistency we applied these same flag checks on [*all*]{} object samples we drew from the S82 catalog. We describe these other samples below.
Stellar locus “contaminants” in Stripe 82 {#sec:stelcont}
-----------------------------------------
To get a sample of typical non-variable stellar contaminants we used the SDSS standard star catalog of 1.01 million non-variable point-source objects in S82 published in [@ivezic07]. From that we created a set of F/G-star colored objects, presumably non-varying, by applying a color-magnitude cut on the standard star catalog so that $0.2 < g-r
< 0.48$ and $14.0 < g < 20.2$ for all the objects. This is a suitable cut for F/G-stars according to the SEGUE team [@yanny09] and makes them potential quasar sample contaminants because of their $g-r$ color (see Figure \[fig:coloroverlap\]). We took a randomly selected subsample of 5000 objects from this catalog and did a neighbor search in S82 to get multi-epoch observations of these contaminants. We again used a search radius of 0.5” and again made sure that none of the flags listed in Section \[sec:qsos\] were set.
To be able to test whether our algorithm is able to separate quasars from known variable contaminants, we used the largest available sample of securely identified RR Lyrae within S82 [@sesar09], which consists of 483 RR Lyrae.
We will refer to the F/G stars and RR Lyrae catalogs collectively as the “contaminants” in the remainder of the paper.
UV-excess (UVX) objects {#sec:UVXdata}
-----------------------
We would also like to test our ability to detect quasars in the absence of spectroscopic data. To this end, we defined three photometrically-selected samples of S82 objects, whose variability properties we will explore.
The first of these is defined by a three-dimensional $ugri$ color box in which the SDSS quasar sample is complete for extinction-corrected $i$ magnitudes brighter than 19.1 [@richards02]. This color box is given in Table \[tab:colorbox\], and is shown in black lines in Figure \[fig:coloroverlap\]. Note that this selection uses the SDSS $u$-band data: we extracted all point sources within S82 that obeyed these “UV excess” (UVX) criteria. This returned a catalog of 2912 UVX point sources.
Non-UV excess (nUVX) objects {#sec:nUVXdata}
----------------------------
As a compliment to the UVX object sample defined above, where the color selection is known to efficiently return quasars at high completeness, a catalog of “non-UVX” (nUVX) objects was created from a region of $ugriz$ color space where color selection of quasars is known to have problems. The color box from which we selected these nUVX point sources is given in Table \[tab:colorboxNUV\], and shown as a green box in Figure \[fig:coloroverlap\] [and also Figure 7 of @richards02]. In this particular color box, the quasar locus, containing mostly intermediate redshift ($2.5<z<3$) quasars, crosses the stellar locus. The color selection therefore has efficiency as low as 10% in this region of color space [@richards06]. In the nUVX color box we find 3258 objects in S82.
Quasar Candidate Color Selection without $u$-band Data {#sec:grizbox}
------------------------------------------------------
To simulate approximately the anticipated $3\pi$ survey light curves, we defined a third color box suitable for a first cut of the catalog. The main purpose of this selection (where no $u$-band information is used) is to excise the quasar locus as it threads through the 3-dimensional $griz$ color space. However, part of the stellar locus also lies in this box. We restrict ourselves to right ascensions between 0 and 20 degrees (enclosing a sixth of the S82 area, $\sim$50 deg$^2$) in order to return a manageable catalog of 12,714 objects. The $griz$ box is indicated by the blue solid lines in Figure \[fig:coloroverlap\] and is defined in Table \[tab:grizbox\]. The magnitude cut of 19.1 is chosen to allow straightforward comparison with the UVX and nUVX boxes.
Eliminating light curve “outliers” in Stripe 82 {#sec:LC}
-----------------------------------------------
Plotting the complete S82 multi-epoch photometry output for the various objects revealed some outlying points that were several magnitudes fainter than adjacent flux points (see Figure \[fig:LC\]); only some of these outliers were found to be caused by image defects. However, we assume that such a significant decrease in magnitude in a single observation must be non-physical. We therefore removed the outliers (irrespective of their origin) by running a median filter on the photometric measurements. Measurements with a residual between the medianized light curve and the photometric data larger than 0.25 magnitudes were removed. In Figure \[fig:LC\] we show the $g$, $r$ and $i$-band multi-epoch photometric measurements (open symbols indicating the removed measurements) with the corresponding medianized light curves over-plotted for quasar SDSS J203817.37+003029.8. The bottom panel shows the residuals, with the limit of 0.25 magnitudes indicated by the dashed lines. As is the case here, in general, only a small fraction of the epochs is removed.
It is these cleaned multi-epoch measurements, where the outlying observations have been removed (i.e. the filled symbols in Figure \[fig:LC\]), we use in the determination and exploration of the objects’ variability.
Down-sampling S82 light curves to the cadence {#sec:DS}
---------------------------------------------
In order to explore quasar selection in the context of the $3\pi$ survey, we down-sampled the S82 data to mimic the planned observations [@PS1 shown schematically in Table \[tab:Pan-STARRS1obs\]]. We assumed 3 observing seasons for PS1, with a duration of 155 days (covering all filters) each. Only S82 objects with more than 7 epochs in each (SDSS) season were passed to the actual down-sampling routine: $\sim$1% of the quasars, $<0.1\%$ of the F/G stars and $\sim$20% of the RR Lyrae did not satisfy this criteria.
We down-sampled the S82 light curve data by matching each season of observations for the $g$, $r$, $i$ and $z$-band with the (approximately) correct time intervals between consecutive observations in each band. No color information went into the down-sampling. After identifying 6 suitable S82 epochs in each band we removed all other observations, providing a set of mock data.
Power-Law Structure Functions for Sources in Stripe 82 {#sec:pwrfit}
======================================================
In Figure \[fig:SF\] we show the binned structure functions (Equation \[eqn:SFbin\]) from the $g$, $r$ and $i$-band light curves of quasar SDSS J203817.37+003029.8 (Figure \[fig:LC\]). Calculating binned structure functions for individual S82 objects is a simple way of quantifying the variability of each object if there is a large number of epochs available. However, Figure \[fig:SF\] suggests that we might indeed be justified in further compressing the structure function into a two parameters power-law fit, as proposed in Section \[sec:SF\].
Before doing so, we explore wether this power law behavior is present in an average sense. In Figure \[fig:meanStrAll\] we show the median sample structure function, created by median-combining all the individual binned structure functions calculated with Equation \[eqn:SFbin\] separately for the well sampled S82 quasars (Section \[sec:qsos\]), F/G-stars (Section \[sec:stelcont\]) and RR Lyrae [Section \[sec:stelcont\] and @sesar09]. The shaded regions in Figure \[fig:meanStrAll\] around the medians enclose 68% and 95% of the individual structure functions. Figure \[fig:meanStrAll\] shows that the quasar sample median of the binned structure function closely resembles a power law with $A=0.093\pm0.0002$ and $\gamma=0.43\pm0.002$, in agreement with findings elsewhere in the literature [@vandenberk04; @rengstorf06; @wilhite08; @bauer09]. In particular, a value of the slope of the sample median structure function of $\gamma = 0.43$ agrees well with most of the literature (see e.g. Table 4 in [@bauer09] for a brief overview). Rough estimates of the 1-year observed frame power law amplitudes in the literature give amplitudes between 0.10 and 0.14 (depending on the assumed mean redshift of the samples), in good agreement with our estimate for $A$ of 0.093 mag.
Figure \[fig:meanStrAll\] also shows that the sample structure functions of contaminants are also well described by power laws, but with small values of $\gamma$ (i.e. they show no long-term growth in their variability). Note that the F/G-stars, chosen to be non-variable, have a variability amplitude of $\sim$0.04 mag. The RR Lyrae variability, when sparsely and randomly sampled in S82, looks like white noise ($|\gamma| \ll 1$) with an amplitude $\sim$0.2 mag. Thus, the use of a power-law model of the form given in Equation \[eqn:powerlaw\] would seem to be a fairly good assumption, and different types of objects may differ both in amplitude and in slope of their structure function.
In Figure \[fig:SF\] the power law fits to the three (binned) structure functions are shown as dashed lines. The $A$ and $\gamma$ with their estimated errors are given in the lower right corner of the plot.
Results {#sec:results}
=======
Having defined different sub-samples of sources, and having shown that we can sensibly quantify their light curve characteristics by a power-law structure function model, we now proceed to characterize each of these sources by their best-fit parameters $A$ and $\gamma$.
As opposed to the earlier SDSS analysis [@richards02; @richards06; @richards09] the improved time sampling of the S82 (and even PS1) surveys, enables us to investigate the [*distributions*]{} of the $A$ and $\gamma$ parameters for the individual sources, not only for ensembles.
The A-$\gamma$ Distribution {#sec:AGspace}
---------------------------
Figure \[fig:AvsGallNOBIN\] shows the distribution of the variability characteristics, quantified by the best-fit $A$ and $\gamma$ for all the spectroscopically confirmed quasars, and for the “contaminant” F/G stars and RR Lyrae, as described in Section \[sec:data\], based on their $r$-band S82 light curves. The histograms along the axes of the two dimensional scatter plot show the projected parameter distribution of the quasars and of the contaminants. Visual inspection of Figure \[fig:AvsGallNOBIN\] alone shows how well the spectroscopically confirmed quasars separate from the (stellar locus) contaminants in this space, demonstrating that the power-law structure function fit from a single band is an efficient classifier for data of this quality ($\sim$60 epochs).
The analogous $A$-$\gamma$ distributions for the much sparser PS1-like sampling of the $r$-band measurements (Section \[sec:DS\]) are shown in Figure \[fig:AvsGdsNOBIN\]: these parameter estimates are based on only 6 epochs of photometry over 3 years, rather than the $\sim 60$ in the full S82 survey. The separation of the quasars and the contaminants is less clean with the sampling, but one nevertheless clearly sees a quasar-dominated region with rather low contamination. Plots analogous to the ones shown in Figures \[fig:AvsGallNOBIN\] and \[fig:AvsGdsNOBIN\], but for the $g$, $i$ and $z$-band measurements, show that on average $A$ decreases by 30-50% going from $g$ to $z$ band, whereas $\gamma$ is unchanged with varying wavelength. Thus, the separation of the quasars from the contaminants via their $\gamma$ values appears to work comparably well in all 4 bands (with somewhat more scatter in the $z$-band). In agreement with [@kozlowski09], no clear difference in the ratio of the amplitudes at different wavelengths between RR Lyrae and quasars is detected.
Since most F/G-stars should not vary, but RR Lyrae do, they should have different $A$ distributions. This is seen in Figure \[fig:AvsGallNOBIN\] and \[fig:AvsGdsNOBIN\]: RR Lyrae have magnitude amplitudes above $\sim$0.1 [e.g. @soszynski03; @sesar09], while the F/G-stars have characteristic values of $A\sim0.01$. It is therefore clear that our approach can also separate RR Lyrae from stellar (non-varying) contaminants without doing a full fit of a periodic light curve.
Completeness and Purity {#sec:compur}
-----------------------
To move beyond a merely qualitative assessment of the separability of the quasars from contaminants we now estimate the achievable completeness and purity of the resulting sample. Purity is the more difficult quantity to estimate as it requires appropriate abundances for the contaminants. We do *not* have these for the training set parent quasar, F/G star and RR Lyrae samples, where the ratio of quasars to contaminants is 2:1, instead of a more realistic $\sim$1:25, and therefore can only estimate completeness when working with these samples. However, by design the UVX, nUVX and $griz$ selection boxes (Sections \[sec:UVXdata\]–\[sec:grizbox\] and \[sec:UVX\]–\[sec:grizboxres\]) give parent samples with the correct quasar-contaminants ratio: we use these to explore the purity of our quasar selection algorithm.
For the present we divide the $A$-$\gamma$ plane by simple cuts that define a quasar selection box, and then quantify its performance. Specifically, our fiducial quasar selection region is bounded by the following three straight lines: $$\begin{aligned}
\gamma(A) &=& 0.5*\log(A)+0.50 \label{eqn:cutDSone} \\
\gamma(A) &=& -2*\log( A)-2.25 \\
\gamma(A) &=& 0.055 \label{eqn:cutDSthree}\end{aligned}$$ These cuts are shown as black solid lines in Figures \[fig:AvsGallNOBIN\]–\[fig:AGcolorPS1\].
This can be thought of as a way of providing lower limits on the available completeness and purity, that a more sophisticated selection procedure would improve upon. One could of course tweak the cuts in Equations \[eqn:cutDSone\]–\[eqn:cutDSthree\] to explore the trade-off between purity and completeness, which we have only done here “by eye.”
Applying these cuts to the data leads to the completeness given in Table \[tab:compur\] for the “QSO+contam” catalog. The completeness is calculated as the fraction of spectroscopically confirmed quasars in the sample that fall within the cuts. In our simple illustrative setup we have a completeness of 93% for the quasars in the case where the time sampling is equal to Stripe 82 ($\sim$60 epochs). In the case of a PS1-like time sampling the completeness drops to 76%.
Of the RR Lyrae 97% and 83% (for the S82 and sampling respectively) fall in the high-$A$ low-$\gamma$ corner below the line given by Equation \[eqn:cutDSone\]. In this region $<$0.5% of the 5000 F/G-stars lie, illustrating quite a clean separation between RR Lyrae and non-varying stellar contaminants.
Besides calculating the overall completeness, we also split our data into redshift bins. The completeness of the quasars is rather constant as a function of redshift, with a minor loss of about 5-10% for redshifts above 4. Since our training set only contains 52 quasars at $z>4$, we were not able to investigate this decreased completeness further.
In the next three subsections we proceed to explore the $A$-$\gamma$ distributions for the samples of objects that were selected only on the basis of their colors (Table \[tab:colorbox\], \[tab:colorboxNUV\] and \[tab:grizbox\]). For those samples, we estimate the completeness (as above) and also the purity, defined as the fraction of known quasars compared to the total number of objects inside the $A$-$\gamma$ selection regions.
### Quasars in the UVX catalog {#sec:UVX}
For the UVX object sample (Section \[sec:UVXdata\]) there is enough spectroscopy in S82 to define a spectroscopically confirmed quasar subsample, a reference catalog of quasars which is complete in S82. By combining the spectroscopically confirmed quasars in S82 (Section \[sec:qsos\]), with the objects from the 2SLAQ (2-degree field SDSS luminous red galaxies and QSO) survey[@croom09], our final quasar reference catalog contains 11216 individual quasars (9157 from SDSS and 2059 from 2SLAQ). We only selected objects flagged as “QSO” in the publicly available 2SLAQ data.[^1]
Matching this quasar reference catalog with the catalog of UVX S82 point sources returned 2140 quasars out of the 2912 objects in the UVX catalog. Thus, 73% percent of the point sources in the UVX color box are known quasars. It is not surprising that the fraction is so large, since we used the powerful $u-g$ color in the definition of the color box. This simply re-affirms that the UVX box is a region of color space where the quasars are in the majority, as they are well separated from the stellar locus by the $u-g$ color; it is exactly this separation to which we are trying to find an alternative.
We estimated $A$ and $\gamma$ for the entire UVX catalog, using both the full (S82) sampling and the sparser PS1-like version of it. The result is shown in the top row of the $A$-$\gamma$ plots in Figure \[fig:AGs\]. Applying our simple variability selection cuts (Equations \[eqn:cutDSone\]–\[eqn:cutDSthree\]) returned 2033 and 1734 quasar candidates for the S82 and PS1 sampling, respectively. Matching these objects with the 2140 know SDSS+2SLAQ quasars in the UVX catalog returned 1935 and 1573 matches. Thus we are able to detect the UVX quasars with a completeness of 90% and a purity of 95% (1935 matches/2033 candidates) when using the S82 time sampled data. For the PS1-like sampling of the data we get a completeness of 73% and a purity of 92% (Table \[tab:compur\]).
### Quasars in the nUVX catalog {#sec:greenbox}
The catalog of the UVX point sources was deliberately chosen from a region of color space where the color selection already does a superb job finding quasars, as confirmed by the complete SDSS and 2SLAQ quasar catalog and the completeness and purity of our $A$-$\gamma$ approach. However, one might argue that in this case (of UVX quasars) a light curve analysis adds little. To explore the $A$-$\gamma$ approach further we applied it to the non-UV excess objects described in Section \[sec:nUVXdata\].
In the nUVX color box (Table \[tab:colorboxNUV\]) there is no simple way to quantify the completeness of the parent sample of color-selected objects, since we do not know how many quasars were missed in this region of color space during the SDSS survey. To try to quantify this, we extracted the spectra from the 973 objects in this catalog that were targeted for SDSS spectroscopy. By visually inspecting these spectra we were able to compile a catalog of 77 quasars among the S82 nUVX objects. (Of these 77, 6 quasars are not in the SDSS+2SLAQ catalog). This means that if the SDSS fibers had been allocated to nUVX objects randomly, then the purity of the nUVX quasar sample would be $77/973 =
8$%. However, in practice the fibers were placed according to a Bayesian ranking that made fuller use of the color information, so that this 8% is likely an upper limit on the purity of the nUVX quasar sample. (The fraction of quasars hidden in the un-targeted nUVX objects is likely lower than the fraction of quasars found in the nUVX objects with spectra.)
Applying our $A$-$\gamma$ analysis and our variability selection criteria to the spectroscopic sub-catalog of 973 nUVX objects returned 72 (178) quasar candidates for the S82 (PS1-like) time sampling. The nUVX objects’ variability parameters are plotted in the bottom row of Figure \[fig:AGs\]. Estimating the completeness and purity in the S82 sampling case, assuming that the 973 objects with spectra are a random subset of all the nUVX objects, gives that we are 90% complete and 96% pure (Table \[tab:compur\]). By the same argument as above, the purity is an upper limit on the overall purity of the nUVX quasar sample whereas the completeness is exact. The purity and the completeness stand on their own in quantifying our ability to recover the [*spectroscopically confirmed quasars.*]{} Thus, the addition of the variability information enhances the purity to 96% instead of 8%, as is the case for the pure color selected sample. In the case of the sparser PS1-like sampling we still have a completeness of 65%, and a purity of 32%. This clearly demonstrates that a variability-based approach is very efficient at selecting nUVX quasars when the data is well sampled in time. Even with the sparse sampling, the purity increases by a factor of four when variability information is used.
The plot of the S82-sampled nUVX objects (lower left corner of Figure \[fig:AGs\]) shows a clear bimodality of the $A$ parameter distribution of the unknown (red) objects. As seen with the object training set, this bimodality is a probable separation between the non-varying contaminants and the varying (possible RR Lyrae) contaminants. Thus the nUVX objects have been separated into quasar candidates (high $\gamma$; intermediate $A$), RR Lyrae candidates (low $\gamma$; high $A$) and non-varying stars (low $\gamma$; low $A$).
### Quasars in the $griz$ Color Box {#sec:grizboxres}
To simulate quasar candidate selection without $u$-band photometry, we applied our variability analysis to the objects lying in a fairly large multi-color region in $griz$ space, the so-called $griz$ box defined in Section \[sec:grizbox\]. This color box fully contains an important part of the stellar locus. Estimating the completeness and purity of our algorithm for the $griz$ box is difficult, as no clear estimates of the abundance of quasars exist for such a color cut. We therefore checked the quasar candidates against a catalog of the SDSS+2SLAQ quasars, plus the 6 extra quasars found via the visual inspection of the nUVX spectra. This may fall considerably short of a complete sample of quasars, but at the moment it is the best we can do; the purities calculated in this section are therefore lower limits. Matching this quasar reference catalog to the 12,714 objects in the $griz$ box we found 443 known quasars. Estimating $A$-$\gamma$ for all these sources, returned 442 (1,118) variability-based quasar candidates, when considering the S82 (PS1) sampling and when applying the $A$-$\gamma$ cuts of Equations \[eqn:cutDSone\]–\[eqn:cutDSthree\]. Of these candidates 407 (333) were found to be known quasars. Thus, for the broad $griz$ color pre-selection, variability selection achieves 92%(75%) completeness and a purity of 92%(30%) for the S82 (PS1) sampled data, respectively. The $A$ and $\gamma$ distributions for the $griz$ box selected objects are shown in the top panel of Figures \[fig:AGcolor\] (S82 sampling) and \[fig:AGcolorPS1\] (PS1 sampling).
In the bottom panel of Figure \[fig:AGcolor\] and \[fig:AGcolorPS1\] we have projected our variability selected quasar candidates back into $ugr$ and $gri$ color space, in order to understand the color distributions of variability selected quasar candidates. The figures show that 86% and 98% of the not spectroscopically confirmed candidates fall on the $gri$ stellar locus, defined as the (blue) contour level containing 95% of the stellar locus objects in S82. This suggests that many, if not most, of these unconfirmed quasar candidates are stars scattered into our variability-selection region. However, there are some possible quasars among the unknowns judging from their colors. For instance, a few of the unknown objects (6% and 1% in the S82 and PS1 sampled case respectively) fall in the $ugr$ UVX selection box. This illustrates that our purity estimates are lower limits but close to the likely truth. It also shows that the $ugriz$ quasar color selection in SDSS [@richards02; @richards06; @richards09] has done an excellent job, implying that $<10\%$ of the quasars with $i<19.1$ are “hiding” in the stellar locus and have been missed by the SDSS selection.
When defining the $griz$ box in Section \[sec:grizbox\] we included the stellar locus to achieve as high a completeness as possible and to illustrate the prospects of our approach. However, removing objects falling within the stellar locus, could greatly enhance the purity at a modest reduction of the completeness. Thus, selecting quasar candidates without $u$-band information can be put into four scenarios:
- A “naive” $griz$ box ($i<19.1$) including the stellar locus and not considering variability information will have a quasar selection completeness above 90%, but a purity of only 4%.
- Taking a $griz$ box color selection, but cutting out the stellar locus (defined by the (blue) contour in Figures \[fig:AGcolor\] and \[fig:AGcolorPS1\]) and still not considering variability information improves the purity to $\sim$48% but lowers the completeness to $\sim$59%. This is what would be easily achievable in a single-epoch $griz$ survey.
- Combining a $griz$ color selection including the stellar locus, but considering variability information (illustrated in Figures \[fig:AGcolor\] and \[fig:AGcolorPS1\]) leads to a completeness of 92%(75%) and a purity of 92%(30%) for the S82 (PS1) sampled data, respectively. This means that PS1 can provide a 75% complete quasar sample, if 30% purity were acceptable.
- Finally, removing the stellar locus from the $griz$ selection box and combining this with variability information lowers the completeness to 54%(44%), but returns a purity of 97%(92%) with S82 (PS1) time-sampled data. PS1 can provide a high purity quasar sample that is $\sim$50% complete.
This illustrates how a variability selection cut greatly improves the candidate sample as compared to quasar candidate selection based on colors alone when $u$-band information is not available. Keeping in mind that the purities calculated here are lower limits, these results suggests that the variability selection of quasar candidates in or other multi-epoch surveys will produce high quality, extensive samples.
Discussion {#sec:disc}
==========
In this paper we have explored the selection of quasars by their variability in lieu of their UV excess in the context of the PS1 $3\pi$ survey as a test case for various upcoming multi-epoch surveys. Given its extensive, multi-year time sampling, SDSS Stripe 82 is an excellent test bed, that allows us to address variability selection of quasars (and other sources) in general. Besides and SDSS, LSST [@ivezic08; @LSST] and Gaia [@jordi06] will be able to employ variability selection similar to that which we have presented.
For any source, we have characterized its variability by a simple power law model for its light curve structure function, which is similar to (but not the same as) [@macleod08; @sumi05; @eyer02; @cristiani96]. For each object the amplitude $A$ is the typical variability within 1 year, and the exponent $\gamma$ describes how the expectation value for the magnitude differences changes with the time between measurements. Applying this simple variability characterization to spectroscopically confirmed quasars, to presumably non-variable sources (F/G stars) and to known RR Lyrae, we have found that the $A$-$\gamma$ space separates these source classes nicely. Quasars have $0.07<A<0.25$ and $0.15<\gamma<0.5$; RR Lyrae have $A\sim0.2$ and $\gamma\sim0$, as the rapid periodic variations in their light curves appear as “white noise” with no secular trend when coarsely sampled on year-long time scales; finally, non-variable sources have $A<0.05$. The variability properties for quasars derived here for individual objects with $\sim$60-epoch light curves, are consistent with those derived by for instance [@vandenberk04] and [@bauer09] using ensemble averaging. The results allowed the definition of a simple variability-based quasar selection in multi-epoch data, using cuts in the $A$-$\gamma$ plane. As we summarize quantitatively below, this variability selection works very well when considering 60-epoch light curves in S82, and still works quite well for the sparser sampling expected for PS1.
The presented algorithm for variability selection is simple enough that it can be applied to large samples. Our IDL code (not optimized for speed) running on two dual-core CPUs takes 2 hours to characterize the $\sim$12,000 objects of the $griz$ color box with the sampling.
The extensive spectroscopy of quasars in S82 has enabled stringent completeness and purity estimates for variability selection, at least for UVX quasars. Our analysis has shown that variability-based quasar selection with only 6 epochs over 3 years (as for PS1) is effective, producing either complete or pure samples. However, something more like the $\sim$60 epochs of S82 is needed to produce variability-selected quasar samples that are both complete and pure (with only weak color pre-selection).
Therefore, it is paramount to eventually combine the variability information in different filters, in order to boost the number of epochs in PS1. We can attempt to do this by predicting time-offset synthetic $r$-band magnitudes from the $g$, $i$, $z$ and $Y$ bands which could reduce the scatter in the $A$-$\gamma$ plane, and so enhance the completeness and the purity of the sampled catalogs. We leave this important extension of our method to further work.
Considerably further in the future, LSST [@ivezic08; @LSST] will start operating and is planned to ultimately have $\sim$200 epochs spread over 10 years. This makes LSST optimal for creating complete and pure variability selected quasar samples. Considering a UVX box like the one used here, a quasar candidate variability selection with the LSST cadence and a 10 year baseline would definitely push both the completeness and purity above the (lower) S82 limits of 90% and 95% shown in the top left plot of Figure \[fig:AGs\] and in Table \[tab:compur\]. Such samples are expected to be obtainable at least down to LSST’s 10$\sigma$ limiting $i_\textrm{AB}$ magnitude of around 23.3 [@oguri10], or even down to 5$\sigma$ ($i_\textrm{AB}\lesssim24$) with the 200 epoch variability selection. If we further consider the fact that LSST will have a $u$-band, a LSST color-variability quasar selection will presumably be able to create (UVX) quasar catalogs almost as pure and complete as spectroscopic samples. Hence, LSST will be superior to PS1 in the 20,000 deg$^2$ LSST will observe. However, until LSST happens, PS1 is excellent for improving and further developing variability selection of quasars, and it will provide the first large variability (and $grizy$-color) selected quasar samples. The LSST (and PS1) quasar catalogs will with estimated sizes of 100 quasars per square degree, or even more, continue the wild growth of quasar catalog sizes seen the last 40-50 years [@richards09], and thereby stress that we are on the verge of an era where confirming the quasar nature of all objects in the catalogs by spectroscopic follow-up will be unfeasible. This makes the purity of the photometric samples crucial for doing statistical analyses of the quasars. As demonstrated variability provides purities above 90% for well sampled data at all redshifts and not only for UVX object and will therefore be an important step in achieving very pure photometric quasar catalogs in the future.
Conclusions {#sec:conc}
===========
We have presented a simple, parameterized variability characterization for sources with many epoch photometry. We do this by fitting a power-law model for the structure function of each source’s light curve; the model is specified by an amplitude A and a power-law index $\gamma$.
We have applied this approach to understanding variability-selection of quasars in multi-epoch, multi-color surveys, either augmenting or supplanting the more common color-selection. Specifically, we have analyzed data in SDSS Stripe 82 (S82) both to understand variability selection [*per se*]{} and as a testbed for ongoing and upcoming surveys, such as PS1 and LSST. To predict PS1’s ability to identify quasars, we have down-sampled S82 data to a set of 6 epochs, resembling the expected single-band time sampling in the PS1 3$\pi$ survey.
For all sources in various sub-samples we calculated the parameters $A$ and $\gamma$, and found that for sufficiently many epochs over a multi-year interval, quasars separate well in the $A$-$\gamma$ parameter plane from non-variable sources and e.g. RR Lyrae. Quasar variability as typically characterized by $0.07<A<0.25$ and $0.15<\gamma<0.5$, consistent with earlier ensemble analyses.
Drawing on the nearly complete spectral identification of quasars with $i<19.1$ in S82, we have explored both the completeness and the purity of single-band, variability-selected quasar samples with both S82 and PS1 time-sampling, and with or without the benefit of $u$-band photometry (which PS1 will not have).
Specifically, we found the following:
- Among the complete, spectroscopically confirmed sample of UV-excess quasars in S82, we can identify 90% of them on the basis of $\sim$60 $r$-band epochs over $\sim$5 years. This variability selected sample only has a 5% contamination from other objects.
- Repeating the same exercise but with the 10-times sparser PS1 sampling, reduces the completeness to 73% but with still high purity (top right of Figure \[fig:AGs\]).
- In the redshift range $2.5<z<3$, where quasars overlap with the stellar locus in color-color space, variability can find 90% (65%) of all spectroscopically confirmed quasars for S82 (PS1) sampling. This is a factor of 5-10 more complete than existing color-selection in this particular regime (bottom panel of Figure \[fig:AGs\]).
- To understand our ability to select quasars through their variability when no $u$-band data are available, we selected all sources in a broad [*griz*]{} color box, known to contain almost all spectroscopically confirmed quasars. In this color box, stars and other contaminants outnumber quasars by a factor of 30 for $17<i<19.1$. Nonetheless, variability selection encompasses 92% (75%) of known quasars for S82 (PS1) sampling. For S82 sampling (Figure \[fig:AGcolor\]) the purity of this sample is still very high (92%), but it is rather lower with PS1-like sampling (Figure \[fig:AGcolorPS1\]), 30%.
- If in the case of PS1 (6 epochs, no $u$-band) a more pure quasar sample is desired, this can be done by omitting the stellar locus in the $griz$ box and then looking at the variability of the remaining sources. This yields a completeness of only 44%, but with a purity of 92%.
- The high purity of the variability selected quasar sample in the [*griz*]{} box (with S82 sampling) confirms that the fraction of overlooked quasars in S82 must be small ($<10\%$); this inference is predicted on the assumption that the quasars “buried” in the stellar locus have a variability behavior similar to the others.
- The same ($A$,$\gamma$) analysis is also very effective at identifying RR Lyrae. With S82 sampling 97% of the RR Lyrae from [@sesar09] are found, with PS1 sampling still 83%; they appear as objects with $(A,\gamma)\approx (0.2,0)$ (red points in Figure \[fig:AvsGallNOBIN\] and \[fig:AvsGdsNOBIN\]).
As mentioned in the introduction, new and larger quasar samples have many and varied interesting applications. One of these is the possibility of finding lensed quasars. In this work we have only focused on point source selection, with the purpose of finding quasar candidates. Such an approach would be directly applicable to a search for wide separation lenses, where (by definition) the multiple images are well-resolved. Another very interesting use of variability selection, again with quasar lens finding in mind, is to look at spatially extended objects rather than point sources. Applying the algorithm to a set of extended, yet quasar-colored, objects will return a list of small separation quasar lens candidates. We have initiated such a search in Stripe 82 with promising preliminary results, which we will present in a forthcoming paper. This search is a pilot for the lensed quasar search, which will be able to be started after about a year of $3\pi$ survey imaging has been built up. We expect to find as many as 2000 lenses in this database [@oguri10]. Our analysis illustrates that combining color-selection of quasars with variability selection is a powerful approach to take.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Gordon Richards, Xiao-hui Fan and David Hogg for useful discussions, and Branimir Sesar for assistance with the RR Lyrae sample. KBS is funded by and would like to thank the Marie Curie Initial Training Network ELIXIR, which is funded by the Seventh Framework Programme (FP7) of the European Commission. PJM was supported in part by research fellowships from the TABASGO and Kavli foundations, and by the US Department of Energy under contract number DE-AC02-76SF00515.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is `http://www.sdss.org/`.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
Individual structure function parameter inference by MCMC {#sec:mcmc1}
=========================================================
We use a simple Markov chain Monte Carlo (MCMC) approach [e.g. @metropolis53; @hastings70; @press92; @hansen03] to infer the parameters $A$ and $\gamma$ (Equation \[eqn:powerlaw\]) and their confidence regions. Our MCMC procedure takes as input a catalog of magnitudes, photometric errors and observed frame MJDs, converted to $N(N-1)/2$ pairs of observations (Equation \[eqn:datapairs\]). We then initialize it as follows:
- Pick a starting point for $A$ and $\gamma$: we chose 0.1 for both
- Define an initial Gaussian proposal distribution (PD)
- Set an initial temperature $\beta$ for the chain
The PD width sets the mobility of the Markov chain. Based on several tests we set the initial PD width to 0.05 in both the $\log{A}$ and $\gamma$ directions. During a “burn-in” period at the start of sampling, we draw samples from a modified posterior PDF for the parameters, given by the product of the prior PDF, and the likelihood raised to the power of $\lambda$. This parameter is an inverse temperature; we start from $\lambda_0=\frac{1}{\beta_0}=10^{-3}$. The $\lambda$ parameter is then increased geometrically to unity as 500 samples are drawn, at which point burn-in is declared over, the inverse temperature is fixed at $\lambda=1$, and the subsequent samples are stored and used to compute various statistics. For the post burn-in sampling, we use an updated Gaussian proposal distribution, whose widths are set to 10% of the standard deviations of the parameter ($\log{A}$ and $\gamma$) values sampled during burn-in.
At each point in parameter space proposed, we calculate the (un-normalized) log posterior probability distribution of the step, which we define as $$\label{eqn:logP}
\log P = \log P(A) + \log P(\gamma) - \lambda \frac{\chi^2}{2} \; .$$ Here, $\chi^2$ is related to the logarithm of the likelihood defined in Equation \[eqn:L\] $$\label{eqn:logL}
\chi^2 = -2 \log \mathcal{L} =
\left( \sum_{ij} \log( 2\pi V_{{\rm eff},ij}^2) +
\sum_{ij} \frac{\Delta m_{ij}^2}{V_{{\rm eff},ij}^2} \right) \; ,$$ with the effective variability defined as in Equation \[eqn:Veff\]: $$V_{{\rm eff},ij}^2 =
V_{\rm mod}(A,\gamma | \Delta t_{ij})^2 + \delta \Delta m_{ij}^2
= \left(A \Delta t_{ij}^{\gamma}\right)^2 + \delta \Delta m_{ij}^2 \; .$$ The sums in Equation \[eqn:logL\] are over the data pairs derived from the light curve of the given object, and $\delta\Delta m_{i,j}$ is the photometric error on the $ij^{\rm th}$ magnitude pair. In Equation \[eqn:logP\] $\log P(A)$ and $\log P(\gamma)$ represent the (log) prior PDFs for the parameters $A$ and $\gamma$, which we chose to be uninformative. We assigned the following functional forms: $$\begin{aligned}
P(A) &\propto& \frac{1}{A} \\
P(\gamma) &\propto& \frac{1}{1+\gamma^2} \; .\end{aligned}$$ If $\gamma$ is negative or $A$ lies outside the range $[0,1]$, the log prior density for that parameter is set to $-10^{32}$, lowering the overall posterior probability for that particular iteration step to effectively zero. In this way we enforce our assumption that the power law exponent is positive and that the average variability on a 1 year timescale is less than 1 magnitude . Samples are accepted or rejected via the Metropolis-Hastings algorithm [@metropolis53; @hastings70]. Care is taken to make the comparison of the log posterior values at constant inverse temperature.
For our final $A$ and $\gamma$ values, we choose to take the position of the global peak of the likelihood, an approximation of the “best-fit” point. We approximate this by keeping track of the value of the likelihood as we sample, and then using the sample with the highest value as our estimate. In practice the posterior PDF is not dominated by the prior, such that the peaks of the likelihood and the posterior PDF are usually quite close together.
The uncertainties on the parameters are estimated by considering the $16^{\rm th}$ and $84^{\rm th}$ percentiles of the 1-dimensional marginalized distributions. In the case of a (symmetric) Gaussian distribution, this would correspond to the $1\sigma$ error bar.
To confirm our choice of initial conditions, cooling schedule and PD evolution as sensible, we tested our sampler on simulated light curves for both sine wave and Gaussian white noise sources, and recovered the correct parameters (zero $\gamma$ and analytically calculated $A$).
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[^1]: http://www.2slaq.info/
|
---
abstract: 'We introduce a version of the chained Bell inequality for an arbitrary number of measurement outcomes, and use it to give a simple proof that the maximally entangled state of two $d$ dimensional quantum systems has no local component. That is, if we write its quantum correlations as a mixture of local correlations and general (not necessarily quantum) correlations, the coefficient of the local correlations must be zero. This suggests an experimental programme to obtain as good an upper bound as possible on the fraction of local states, and provides a lower bound on the amount of classical communication needed to simulate a maximally entangled state in $d\times d$ dimensions. We also prove that the quantum correlations violating the inequality are monogamous among non-signalling correlations, and hence can be used for quantum key distribution secure against post-quantum (but non-signalling) eavesdroppers.'
author:
- Jonathan Barrett
- Adrian Kent
- Stefano Pironio
date: 'May 19, 2006'
title: 'Maximally Non-Local and Monogamous Quantum Correlations'
---
Quantum theory predicts that measurements on separated entangled systems will produce outcome correlations that, in Bell’s terminology [@bellbook], are not [ *locally causal*]{} or, in what has become standard terminology, are [ *non-local*]{}. In particular, if a Bell inequality is violated then one cannot consistently assume that the outcomes of measurements on each system are predetermined and independent of the measurements carried out on the other system(s). Violation of Bell inequalities has been confirmed in numerous experiments [@exp].
Violation of Bell inequalities not only tells us something fundamental about Nature, but also has practical applications. For example, the nonlocality of quantum correlations allows communication complexity problems to be solved using an amount of communication that is smaller than is possible classically [@commcompl]. Barrett-Hardy-Kent (BHK) recently showed that testing particular nonlocal quantum correlations allows two parties to distribute a secret key securely, in such a way that the security is guaranteed by the no-signalling principle alone [@qkd] (i.e. without relying on the validity of quantum theory).
In this paper, we extend the chained Bell inequality to an inequality with an arbitrary number of measurement outcomes, which can thus be applied to states in arbitrary dimensions. In the limit of a large number of measurement settings, quantum mechanics predicts correlations for a maximally entangled bipartite state that resemble those of the tripartite Greenberger-Horne-Zeilinger (GHZ) state in that, with probability tending to one, the predictions of any local hidden variable model contradict those of quantum mechanics for at least one pair of measurements. We use this to give a constructive proof of a result originally due to Elitzur, Popescu and Rohrlich (EPR) [@epr]: if the quantum correlations of a maximally entangled state of two qubits are written as a convex combination of local and nonlocal correlations, then the local fraction must be zero. Our proof extends EPR’s result to maximally entangled states in any dimension and also removes the need for a technical assumption required for EPR’s original proof [@footnote1]. Moreover, because our proof is constructive, it motivates an experimental programme to establish the best possible upper bound on the fraction of local states in a maximally entangled state. More generally, our proof method works, and motivates experimental tests, for any example of GHZ-type correlations.
Next, we give a rigorous proof of the *monogamy* of the correlations obtained from a $d\times d$-dimensional maximally entangled state. Here, monogamy means that a third party cannot get any information about the measurement outcomes, so long as a no-signalling condition holds, even if quantum theory is incorrect. This property has a particular significance in the context of secret key distribution, and our results here generalise those of BHK [@qkd], which proved monogamy in the $2 \times 2$-dimensional case in order to demonstrate the security of BHK’s scheme against general non-signalling eavesdroppers.
Finally, as a corollary, we derive a lower bound on the classical communication needed to simulate measurements on a maximally entangled state in $d\times d$ dimensions.
#### Some chained Bell inequalities.
Consider a standard Bell-type experiment: two parties, Alice and Bob, share a joint system in an entangled quantum state, and perform measurements on their local subsystems. Each party may choose one out of $N$ different measurements, and each measurement $A_k$ of Alice and $B_l$ of Bob ($k,l=1,\ldots,N$) may have $d$ possible outcomes: $A_k,B_l=0,\ldots,d-1$. Quantum theory predicts the joint probabilities $P^{\scriptstyle QM}(A_k=a,B_l=b)$ that Alice’s and Bob’s measurements, $A_k$ and $B_l$, have respective outcomes $a$ and $b$. The correlations predicted by quantum theory are [*local*]{} if and only if they can be written in the form $$\nonumber
P^{\scriptstyle QM}(A_k=a,B_l=b)=
\sum_{\lambda} q_{\lambda} P_{\lambda}(A_k=a)\times P_{\lambda}(B_l=b) \, ,$$ with $0\leq q_{\lambda}\leq 1$ and $\sum_{\lambda} q_{\lambda}=1$. Without loss of generality, we can assume that the terms $P_{\lambda}(A_k=a)$ and $P_{\lambda}(B_l=b)$ are deterministic, that is, that they only take the values $0$ or $1$ [@ww]. The correlations are thus local if and only if they can be reproduced by a mixture of hidden states assigning definite values to each measurement. Violation of a Bell inequality implies that the correlations cannot be written in this form.
In general one can write the correlations as $$\begin{gathered}
\label{mixture}
%\begin{split}
P^{\scriptstyle QM}(A_k=a,B_l=b)=p\,P^{\scriptstyle L}(A_k=a,B_l=b)\\
+(1-p)\,P^{\scriptstyle NL}(A_k=a,B_l=b)\,,
%\end{split}\end{gathered}$$ where $P^{\scriptstyle L}(A_k=a,B_l=b)$ and $P^{\scriptstyle NL}(A_k=a,B_l=b)$ represent local and non-local joint distributions, respectively, and $p$ and $(1-p)$ their corresponding weights in the mixture, with $0 \leq p \leq 1$. Suppose now that we have a set of quantum correlations which satisfy some relation with certainty, and we can show that any local model must fail to satisfy the relation at least some of the time. It follows that the correlations cannot be written in the form of Eq. except with $p=0$. (Note that this is true even if the term $P^{\scriptstyle NL}(A_k=a,B_l=b)$ is allowed to describe signalling correlations.) A particularly well-known example of such correlations was that produced by GHZ [@ghz] and simplified by Mermin [@mermin]; hence we refer to correlations with these properties as GHZ-type correlations. Bipartite examples have been given by Heywood-Redhead [@heywoodredhead] and Cabello [@cabello]. A set of quantum correlations is of GHZ-type if and only if it has an associated Bell inequality which is violated right up to the algebraic limit of the expression.
We now derive a Bell inequality and show that in the limit of a large number of measurement settings, the quantum correlations from a maximally entangled state violate the inequality up to the algebraic limit. Thus the quantum correlations tend to GHZ-type as the number of measurement settings becomes large. Hence, we show, maximally entangled states in any dimension have zero local component.
Consider first the case where Alice and Bob each have a choice between two different measurements ($N=2$). Local correlations satisfy the following inequality [@cglmp], $$\begin{gathered}
\label{cglmp}
I_2=\langle[A_1-B_1]\rangle+\langle[B_1-A_2]\rangle+\langle[A_2-B_2]\rangle\\
+\langle[B_2-A_1-1]\rangle \geq d-1\, ,\end{gathered}$$ where $\langle X \rangle= \sum_{i=1}^{d-1} i P\,(X=i)$ is the average value of the random variable $X\in\{0,\ldots,d-1\}$ and $[X]$ denotes $X$ modulo $d$. This follows from the identity $$[A_1 - B_1 + B_1 - A_2 + A_2 - B_2 + B_2 - A_1 - 1]=d-1\,,$$ and the relation $[X]+[Y]\geq [X+Y]$. When $d=2$, it corresponds to the CHSH inequality [@chsh].
We can extend the above inequality to an arbitrary number $N$ of measurement choices: $$\begin{gathered}
\label{chained}
I_N=\langle[A_1-B_1]\rangle+\langle[B_1-A_2]\rangle+\langle[A_2-B_2]+\cdots\\
\cdots+\langle[A_N-B_N]\rangle+\langle[B_N-A_1-1]\rangle\geq d-1\, .\end{gathered}$$ This extended inequality can be viewed as a chained version of inequality , and follows by a similar argument. It is equivalent, when $d=2$, to the chained inequality introduced by Pearle [@pearle] and Braunstein and Caves [@bc]. The relation between the Pearle-Braunstein-Caves inequality and GHZ-type correlations was first pointed out by Hardy [@hardy].
We now show that if Alice and Bob share the maximally entangled state $$\label{mes}
|\psi_d\rangle=1/\sqrt{d}\,\sum_{q=0}^{d-1}|q\rangle_A|q\rangle_B \, ,$$ there exist measurement settings such that for large $N$, $I_N({\scriptstyle
QM})$ tends to zero. The maximally entangled state $|\psi_d\rangle$ has the property that if Alice measures an observable with eigenvectors $|r\rangle$ ($r=0,\ldots,d-1$) and Bob measures the observable with complex conjugate eigenvectors $|r\rangle^*$, they get perfectly correlated outcomes. We define the eigenvectors characterizing Alice’s measurement $A_k$ as $$\label{ka}
|r\rangle_{A_k}=\frac{1}{\sqrt{d}}\sum_{q=0}^{d-1}
\exp\left[\frac{2\pi i}{d}q(r-\alpha_k)\right]|q\rangle_A\,,$$ and those characterizing Bob’s measurement $B_l$ as $$\label{kb}
|r\rangle_{B_l}=\frac{1}{\sqrt{d}}\sum_{q=0}^{d-1}
\exp\left[-\frac{2\pi i}{d}q(r-\beta_l)\right]|q\rangle_B\,,$$ where $\alpha_k=(k-1/2)/N$ and $\beta_l=l/N$. A straightforward calculation shows that each term in is equal to $\gamma/N^2+O(1/N^3)$, where $\gamma=\pi^2/(4d^2)\sum_{j=1}^{d-1}
j/\sin^2(\pi j/d)$. As there are $2N$ such terms in the inequality , it follows that $$\label{qmin}
I_N({\scriptstyle QM})= 2\gamma/N+O(1/N^2),$$ which can be made arbitrarily small for sufficiently large $N$.
For any particular finite value of $N$, the quantity $I_N({\scriptstyle QM})$ implies an upper bound on the fraction of local states in any model that reproduces the correlations, i.e., an upper bound on the $p$ of Eq. . This is because although the term $P^{\scriptstyle NL}(A_k=a,B_l=b)$ can violate the Bell inequality , it must satisfy $I_N({\scriptstyle NL})\geq 0$, since each term in is a positive quantity. Thus we can write $I_N({\scriptstyle
QM})=p\,I_N({\scriptstyle L})+(1-p)\,I_N({\scriptstyle NL})$, and, since $I_N({\scriptstyle L})\geq d-1$ and $I_N({\scriptstyle NL})\geq 0$, $I_N({\scriptstyle QM})\geq p\,(d-1)$, or $$\label{bound}
p \leq \frac{I_N({\scriptstyle QM})}{d-1} \, .$$ Of course, in a real experiment, the state prepared will not be precisely (\[mes\]), the measurements will not be precisely defined by projections onto the vectors (\[ka\]) and (\[kb\]), and so on, and thus the experimentally determined value, $I_N ({\scriptstyle EXP})$, will generally be greater than $I_N({\scriptstyle QM})$. Nonetheless, given a value for $I_N ({\scriptstyle EXP})$, we can obtain a bound of the form $$\label{expbound}
p \leq \frac{I_N({\scriptstyle EXP})}{d-1} \, .$$ Hence we propose an experimental challenge: to obtain the lowest possible bound on $p$ (for any $d$ and $N$) for bipartite maximally entangled states.
Our proof extends to any example of GHZ-type correlations; thus another natural challenge is to obtain the lowest possible bound on $p$ for any set of quantum correlations on any bipartite or multipartite entangled state.
#### Indeterminacy of the measurement outcomes and Monogamy.
The correlations that we just introduced have a particular significance in the context of key distribution. BHK showed [@qkd] how non-local correlations can be used as the basis of a key distribution scheme that is secure against non-signalling eavesdroppers. Non-local correlations can also be used to give at least partial security against non-signalling eavesdroppers in more practical QKD schemes [@bhkunpub; @acin]. In these discussions, it is not assumed that such eavesdroppers are constrained by the laws of quantum mechanics, but it is assumed that they can only prepare systems in states whose correlations are [*non-signalling*]{}, in the following sense. Suppose that Alice and Bob share a bipartite system characterized by correlations $P(A_k=a, B_l=b)$. The correlations are non-signalling if they satisfy $$\label{nosig}
\sum_a P(A_k=a, B_l=b) = \sum_{a'} P(A_{k'}=a', B_l=b) \, ,$$ for all $k,k',l,b$, and a similar set of conditions obtained by summing over Bob’s input. These no-signalling conditions ensure that the marginal distributions $P(A_k=a)$ and $P(B_l=b)$ are well defined quantities. The definition of no-signalling can be extended to more than two parties, by requiring a condition similar to for every possible grouping of the parties into two subsets.
In the protocols of Refs. [@qkd] and [@bhkunpub; @acin], maximally entangled states are prepared by a source, which is situated between Alice and Bob and assumed to be under control of the eavesdropper Eve. On each pair of particles, Alice and Bob perform measurements $A_k$ and $B_l$, chosen independently and randomly, and use the corresponding measurement outcomes to establish a shared secret key.
If the correlations used to distribute the key admit a model with fractions $p$ and $1-p$ of local and non-signalling non-local states respectively, then for a fraction $p$ of pairs, Eve could prepare a deterministic local state that would give her complete information about Alice’s and Bob’s measurement outcomes. This strategy is clearly not significantly useful to Eve if the quantum correlations imply $p\approx 0$, which is the case in BHK’s protocol [@qkd], as noted above.
However, a zero fraction of local states does not necessarily imply that Eve’s information is zero, as it does not exclude the possibility that she could prepare a non-local state that has definite values for a non-empty proper subset of the measurement inputs. For instance, in the case of Cabello’s example [@cabello], there is a model reproducing the quantum correlations with the following properties: i) every hidden state is non-signalling, ii) for every hidden state, at least one measurement has a predetermined outcome, iii) each measurement has a predetermined outcome for at least some finite fraction of the hidden states. A non-signalling Eve exploiting this model could obtain some knowledge of Alice’s and Bob’s measurement outcomes. Similar remarks apply to the GHZ and Mermin examples.
The correlations defined by , and , however, are stronger in the sense that any non-signalling model that reproduces the correlations has the following property: for every hidden state, the outcomes of any measurement are completely undetermined and occur all with the same probability $1/d$. This property is implied by the following theorem.\
**Theorem.**[ *Any no-signalling distribution for which ${I}_N\leq I_N^*$ satisfies*]{} $$\nonumber
P(A_k=a)\leq \frac{1}{d}+\frac{d}{4} I_N^* \quad \mbox{and} \quad P(B_l=b)
\leq \frac{1}{d}+\frac{d}{4} I_N^*$$ [*for all measurements $A_k$ and $B_l$ and for all outcomes $a$ and $b$.*]{}
In the limit $I_N^* \rightarrow 0$, Eve cannot therefore gain any knowledge about Alice and Bob’s measurement outcomes. In other words, any tripartite distribution describing the joint systems of Alice, Bob and Eve is of the form $P(A_k=a, B_l=b)\times P(E_m=e)$. We say that Alice’s and Bob’s correlations are [*monogamous*]{}, in obvious analogy with the familiar monogamy of entanglement.
*Proof of the Theorem.* Suppose that $P(A_k=a)>1/d+d/4\,I^*_N$ for some measurement $A_k$ and outcome $a$. We will then show that $I_N>I_N^*$. (The same argument applies if we suppose that $P(B_l=b)>1/d+d/4\,I^*_N$ for some measurement $B_l$ and outcome $b$.)
Defining $A_{N+1}=A_1+1$ (modulo $d$), we can write $$\begin{split}
I_N&=\sum_{j=1}^N ( \langle[A_j-B_j]\rangle+\langle[B_j-A_{j+1}]\rangle ) \\
&\geq 2N-\sum_{j=1}^N ( P(A_j=B_j)+P(A_{j+1}=B_j) ) \,,
\end{split}$$ since $\langle[X]\rangle= \sum_{i=1}^{d-1} i \, P([X]=i) \geq 1-P([X]=0)$. Now $$\begin{split}
P(A_i=B_j)&=\sum_{r=0}^{d-1} P(A_i=r, B_j=r)\\
&\leq \min(P(A_i=q),P(B_j=q))\\
&\quad + \min(1-P(A_i=q),1-P(B_j=q))\\
&=1-|P(A_i=q)-P(B_j=q)|\,,
\end{split}$$ for any $q\in\{0,\ldots,d-1\}$. Using this expression in the above inequality for $I_N$ and defining $N$ arbitrary different values $q_j$, we get $$\begin{split}\label{int}
I_N&\geq\sum_{j=1}^N ( |P(A_j=q_j)-P(B_j=q_j)|\\
&\qquad+|P(A_{j+1}=q_j)-P(B_j=q_j)| ) \\
&\geq \sum_{j=1}^N|P(A_j=q_j)-P(A_{j+1}=q_j)|\,,
\end{split}$$ where the second line follows from the triangle inequality. The hypothesis $P(A_k=a)> 1/d+d/4\,I_N^*$ implies that $$|P(A_k=a')-P(A_k=a'+1)|>I_N^*$$ for some $a'$. If we define the values $q_j$ used in by $q_1=\ldots=q_{k-1}=a'$ and $q_k=\ldots=q_{N}=a'+1$ (modulo $d$), we obtain the inequality $$\begin{split}
I_N&\geq|P(A_1=a')-P(A_k=a')\\
&\qquad+P(A_{k}=a'+1)-P(A_{N+1}=a'+1)|\\
&= |P(A_k=a')-P(A_{k}=a'+1)|\\
&>I_N^* \, ,
\end{split}$$ since $A_{N+1}=A_1+1$ (modulo $d$), by definition. $\square$
#### Classical simulation of the correlations.
Finally, we note that the bound can be interpreted, in the spirit of [@pironio], as a bound on the average communication necessary to simulate classically non-local correlations. The fact that every hidden state must be non-local in any model that reproduces measurements on a maximally entangled state implies that in any run of a communication-assisted classical protocol some communication must be exchanged between the parties. Since the minimal amount of communication that the parties can exchange in any such run is one bit, it follows that at least one bit of communication is necessary to simulate these correlations. For qubits, i.e, when $d=2$, one bit is known to be sufficient [@tb]. For other values of $d$, it is possible to extend our analysis by letting the range of measuring settings go from 1 to $(d-1)\,N$, while keeping the measurements as defined in and . Using inequalities similar to , one can then show that at least $\log_2 d$ bits of communication are necessary. This bound is not optimal asymptotically since for large $d$ a bound of $O(d)$ bits is known [@brassard]. It may, however, be useful for small $d$.
#### Summary and conclusions.
The quantum correlations introduced imply that maximally entangled quantum states in arbitrary dimensions have zero local component. They motivate experiments that could bound the weight of the local component as close to zero as possible. In a non-signalling context, the correlations are also provably monogamous, which gives them an immediate application in key distribution. It would be interesting to characterise the sets of quantum correlations that are monogamous in this sense; in particular, it would be interesting to know if there are monogamous quantum correlations that can be obtained in an experiment with only a finite number of measurement settings at each site, rather than in the limit in which the number of settings tends to infinity.
#### Acknowledgments.
AK thanks Jonathan Oppenheim for helpful conversations. SP acknowledges support by the David and Alice Van Buuren fellowship of the Belgian American Educational Foundation, by the National Science Foundation under Grant No. EIA-0086038, and by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848.
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abstract: 'Pressure-imposed rheometry is used to examine the influence of surface roughness on the rheology of immersed and dry frictional spheres in the dense regime. The quasi-static value of the effective friction coefficient is not significantly affected by particle roughness while the critical volume fraction at jamming decreases with increasing roughness. These values are found to be similar in immersed and dry conditions. Rescaling the volume fraction by the maximum volume fraction leads to collapses of rheological data on master curves. The asymptotic behaviors are examined close to the jamming transition.'
author:
- 'Franco Tapia, Olivier Pouliquen, and Élisabeth Guazzelli$^{\ast}$'
bibliography:
- 'References.bib'
title: The influence of surface roughness on the rheology of immersed and dry frictional spheres
---
Introduction {#sec:intro}
============
While being quite different particulate systems, viscous non-colloidal suspensions and dry granular materials can be described by rheological laws which use a common framework [@BoyeretalPRL2011]. When a dense collection of dry hard spheres (having diameter $d$ and density $\rho_p$) is sheared at a given shear rate, $\dot{\gamma}$, under an imposed particle pressure, $P$, the rheology is determined by the knowledge of two dimensionless quantities: the packing faction, $\phi$, and the effective friction coefficient (or stress ratio), $\mu=\tau/P$, where $\tau$ is the shear stress. Dimensional analysis implies that these two quantities only depend on a single inertial dimensionless number, $I= d \dot{\gamma} \sqrt{\rho_p/P}$, and are thus written as $\mu(I)$ and $\phi(I)$, see e.g. [@ForterrePouliquen2008]. A similar approach can be applied to the viscous flow of suspensions of hard non-Brownian spheres. The rheological laws adopt a similar form, $\mu(J)$ and $\phi(J)$, but with the inertial number, $I$, replaced by a viscous number, $J=\eta_f \dot{\gamma}/P$, where $\eta_f$ is the suspending fluid viscosity [@BoyeretalPRL2011]. This frictional formulation is equivalent to the more classical description of the rheology of suspensions in terms of effective viscosities: the shear, $\eta_s(\phi)=\tau/\eta_f \dot{\gamma}=\mu/J(\phi)$, and normal, $\eta_n(\phi)=P/\eta_f \dot{\gamma}=1/J(\phi)$, relative viscosities where $J(\phi)$ is the inverse function of $\phi(J)$ which is perfectly defined since $\phi(J)$ is monotonic.
These rheological properties become singular in the dense regime when reaching the jamming transition for which the particulate system ceases to flow, both in the viscous (suspensions) and inertial (dry granular materials) cases. There is not yet a complete understanding of these singular behaviors and the current description remains rather empirical. The major problem lies in relating the mechanics at the grain scale to these macroscopic properties. As the jamming transition is approached, particles form an extended network of contacts, see e.g. [@Catesetal1998; @Lerneretal2012; @Andreottietal2012], and the rheology is then dominated by contact forces, even in the case of viscous suspensions for which the hydrodynamics interactions between the particles become of lesser importance and are overshadowed by direct contact interactions, see e.g. [@GuazzelliPouliquen2018]. Numerical simulations and scaling arguments [@DeGiulietal2016; @Trulssonetal2017] have recently pointed out the role of friction between the grains, and in particular the effect of varying the interparticle sliding friction coefficient, $\mu_{sf}$. For low $\mu_{sf}$, the dissipation mainly occurs in the interstitial fluid between the grains for viscous flow of hard spheres while it is due to inelastic particle collisions for inertial flow. Similar physical mechanisms occur at large $\mu_{sf}$ in a rolling regime wherein the spheres roll relative to each other. The dissipation due to the sliding contacts is dominant in the intermediate range of $\mu_{sf}$ which has been expected to be most relevant experimentally in particular close to the jamming transition.
Only a few experiments have examined the impact of interparticle friction, and in particular surface roughness, on the rheological properties of these particulate systems. The earliest study of Lootens *et al.* [@Lootensetal2005] showed that increasing surface roughness of colloidal particles shifts the shear-thickening transition to lower critical stresses. For non-colloidal viscous suspensions, the available experimental results show that increasing roughness leads to higher viscosities [@Moonetal2015; @TannerDai2016] in qualitative agreement with numerical simulations [@Gallieretal2014; @Marietal2014] using rather large value of $\mu_{sf}$ ($\approx0.5-1$) to match the observations. The influence of the interparticle friction has been mostly studied in numerical simulations for dry granular media [@SunSundaresan2011; @Chialvoetal2012; @DeGiulietal2016]. No systematic experimental study has been undertaken in the very dense regime close to the jamming transition. The present work aims at filling this gap by using pressure-imposed rheometry which is particularly suitable to infer the singular rheological properties of particulate systems at the jamming transition [@BoyeretalPRL2011; @Kuwanoetal2013; @Dagois-Bohyetal2015; @Falletal2015; @Tapiaetal2017]. These rheological measurements are reported in section \[sec:results\], both for dry and immersed hard spheres, after presenting the materials and techniques in section \[sec:exp\]; conclusions are drawn in section \[sec:conclusion\].
Experiments {#sec:exp}
===========
Rheometry
---------
![(a) Sketch of the experimental apparatus. (b) Microscopic image of slightly and highly roughened polystyrene spheres. (c) Image of the top permeable plate showing the corrugated strips.[]{data-label="fig:Rheo"}](Fig1Tapia){width="80.00000%"}
The experimental apparatus, depicted in Fig.\[fig:Rheo\](a), is a custom-made rheometer enabling pressure- and volume-imposed rheological measurements of dry or immersed granular materials. The granular sample is subjected to a simple shear in a plane-plane geometry consisting of a cylindrical annulus (of internal and external radii $R_1 = 43.95$mm and $R_2 = 90.28$mm, respectively) that is attached to a bottom plate and is covered by a top plate. In order to obtain a linear shearing of the granular sample, the cylindrical annulus rotates at a constant angular velocity controlled by an asynchronous motor (Parvalux SD18) regulated by a frequency controller (OMRON MX2 0.4 kW) while the top plate does not rotate. A wide range of shear rate, $\dot{\gamma}$, can be achieved, spanning between $0.02$ and $130$s$^{-1}$. The important feature of this rheometer is that the top plate is permeable to the fluid but not to the particles, see a blowup of a picture of this plate in Fig.\[fig:Rheo\](c). It is manufactured with holes of size $2-5$mm enabling fluid to flow through it but is also covered by a $0.24$mm nylon mesh designed to stop particles. This top plate can be moved vertically by using a linear positioning stage (Physics Instrumente M-521) and fits into the cylindrical annulus with a precision of $280$$\mu$m. Both the top and bottom plates are also roughened by positioning corrugated strips of height and width 0.5mm onto their surfaces, as seen in Fig.\[fig:Rheo\](c). This apparatus was initially built to study suspension rheology [@Dagois-Bohyetal2015; @Tapiaetal2017] and has been adapted to make possible the investigation of dry granular material. An important ingredient was to create appropriate roughnesses on the top and bottom plates to enable bulk granular motion. Another important point was to operate the rheometer in an air-conditioned room (at $25^{\circ}$ Celcius) with a high level of humidity (at a relative humidity of 80%) to avoid electrostatic effects between the dry polystyrene particles.
The shear stress, $\tau$, is deduced from the torque exerted on the top plate measured by a torque transducer (TEI - CFF401) connected to the top plate. The component of the normal stress perpendicular to the top plate, simply referred as the particle pressure $P$, is given by a precision scale (Mettler-Toledo XS6002S) attached to the translation stage. The bulk packing fraction of the sample, $\phi$, can be adjusted by displacing the top plate. The plate position, $h$, is continuously measured by a position sensor (Novotechnik T-50). A feedback control system connects the positioning stage and the precision scale in order to perform pressure-imposed experiments on the sample. In this pressure-imposed mode, the resulting shear stress, $\tau$, and packing fraction, $\phi$, are measured as functions of shear rate, $\dot{\gamma}$, for a set particle pressure, $P$, once steady state is established. Classical volume-imposed rheometry can also be performed by fixing the top plate position, i.e. fixing the volume fraction $\phi$. In this volume-imposed mode, the shear stress, $\tau$, and particle pressure, $P$, are measured as functions of shear rate, $\dot{\gamma}$, for a given volume fraction $\phi$. Note that a soft spring is positioned between the top plate and the torque sensor to avoid blockage during highly dense experiments. A series of calibration experiments with a pure fluid is also performed to infer the undesired friction from the central axis as well as the viscous contribution from the gap between top plate and cell walls. These effects are subtracted to the torque measurements. Buoyancy effects are also accounted for in the measurements of the normal force that the particles exert on the porous plate in the gradient direction.
In immersed experiments performed in pressure- or volume-imposed modes, polystyrene particles, shown in Fig.\[fig:Rheo\](b), are suspended in a Newtonian fluid (polyethylene glycol-ran-propylene glycol monobutylether) which has a matching density with the particles ($\rho_f = 1056$kg/m$^3$ at $25^{\circ}$ Celcius) and a large viscosity ($\eta_f=2.01$Pas at $25^{\circ}$ Celcius). Air bubbles are removed by vacuum extraction and then by a long period of pre-shearing. In experiments with dry particles undertaken in the pressure-imposed mode, the cell is filled up to a given height by a rain-like procedure which provides a good homogeneous distribution of the particles within the cell. A small dispersion in size is required to avoid ordering. Ranges of imposed pressure ($P = 60-380$Pa) and sample height ($h<16$mm) are chosen to avoid shear banding nucleation. It is important to stress that the smallest confining pressure has to be larger than the hydrostatic pressure to avoid large inhomogeneity in the bulk. This results in a more limited range of $\phi$ in the dry case.
Manufacturing rough spheres
---------------------------
![(a) Top: Sketch of the particle roughening apparatus. Bottom: Confocal image of the upper rough surface. (b) Microscopic images of typical slightly (right) and highly (left) roughened spheres. Bottom: Corresponding confocal images of the particles surface.[]{data-label="fig:roughening"}](Fig2Tapia){width="80.00000%"}
Two set of spheres having different surface roughnesses but approximatively the same size are used in the experiments, see Figs.\[fig:Rheo\](b) and \[fig:roughening\](b). The particles labeled “Slighly Roughened” (SR) correspond to rigid polystyrene particles (Dynoseeds TS 500) manufactured by Microbeads SA, having a density $\rho_p=1050$kg/m$^3$ and an approximately Gaussian distribution in size with a mean diameter of $d=580$$\mu$m. As can be seen in the right panel of Fig.\[fig:roughening\](b), these spheres are not perfectly smooth but possess some small surface roughnesses. The particles labeled “Highly Roughened” (HR) are produced by further roughening these polystyrene spheres, see the left panel of Fig.\[fig:roughening\](b).
The roughening procedure consists in forcing a continuous motion of the particles between two parallel rough plates resulting in a mechanical erosion of the particle surfaces. The particle-roughening apparatus is sketched in Fig.\[fig:roughening\](a). Sandpapers (Walfcraft 80) cover both the top and bottom circular plates ($20$ cm in diameter) to avoid slippage. The bottom circular plate is fixed while the displacement of its top counterpart is driven by an electrical stirring device having a shifted rotational-axis ($\delta=5$mm) with respect to the bottom-plate axis. Two rolling bearings transfer rotation into translation resulting in a circular movement. Circular translation instead of rotation ensures that the particle trajectory is independent of its location. Each particle thus experiences the same mechanical erosion. In addition, the top plate exerts a static load (1200g) in order to amplify the impact on the particle surface. A toothed soft foam is also used to transfer the motion of the stirring device to the top plate while avoiding particle fracturing.
A typical protocol consists in spreading $1$g of polystyrene spheres onto the bottom plate. Fixing the stirring frequency at $1.2$Hz results in a relative velocity between the plate of $37$mm/s. Fig.\[fig:roughening\](b) shows the effect of one hour of erosion for a representative individual sphere. A slightly reduction in particle size is observed resulting in a slightly smaller mean diameter for the HR spheres, $d=540$$\mu$m.
Characterizing rough spheres
----------------------------
$R_a$ ($\mu$m) $R_q$ ($\mu$m) $R_z$ ($\mu$m) $\mu^d_{sf}$ $\mu^i_{sf}$ $\mu^d_{rf}$ $d$ ($\mu$m)
---- ------------------ ------------------ ------------------ ---------------- ---------------- ------------------ --------------
SR $0.287\pm 0.008$ $0.387\pm 0.008$ $2.073\pm 0.008$ $0.23\pm 0.03$ $0.25\pm 0.03$ $0.004\pm 0.001$ $580\pm20$
HR $1.896\pm 0.008$ $2.410\pm 0.008$ $9.808\pm 0.008$ $0.37\pm 0.03$ $0.35\pm 0.03$ $0.007\pm 0.001$ $540\pm20$
: Properties of the “Slightly Roughened” (SR) and “Highly Roughened” (HR) spheres: Roughness coefficients ($R_a, R_q, R_z$), sliding friction coefficients ($\mu^d_{sf}$ and $\mu^i_{sf}$ in the dry and immersed cases, respectively), rolling friction coefficients ($\mu^d_{rf}$ in the dry case), and diameters ($d$).[]{data-label="tab:surface"}
To provide some indications regarding the particle geometry, typical surface-roughness characteristics have been measured, see Fig.\[fig:roughening\](b). Confocal scanning microscopy of specimens of SR and HR spheres has been performed on a surface region of $170\times170\,\mu$m$^2$. Average roughness, $R_a$, standard deviation, $R_q$, and ten-point mean roughness, $R_z = (\mid Z_{+}\mid + \mid Z_{-}\mid)$ (where $Z_{+}$ and $Z_{-}$ are the average of the 5 tallest peaks and the 5 lowest valleys, respectively), are computed after fitting and then subtracting the spherical shape. Surface properties of both batches of spherical particles are summarized in Table\[tab:surface\].
![Interparticle sliding friction experiment. (a) Lateral view of the setup (b) Top view showing the static-weight distribution on the top plate. (c) Bottom plate: the sample is a ring of effective radius $r=(r_2+r_1)/2 =18.5$mm ($r_1=25$mm and $r_2=12$mm). (d) Plot of the shear stress $\tau$ versus the normal pressure $P$ and corresponding linear fits for SR ($\bigcirc$) and HR ($\square$) spheres in the dry (magenta and red, respectively) and immersed (cyan and blue, respectively) cases.[]{data-label="fig:slidingfriction"}](Fig3Tapia){width="70.00000%"}
To characterize the interparticle sliding friction coefficients, we perform controlled sliding-experiments using two parallels plates coated with a monolayer of polystyrene particles identical to those used in the rheological experiments. The beads are glued on both the top and bottom plates in a circular annulus geometry, as shown in Fig.\[fig:slidingfriction\](a),(c). Distributing spatially the glued particles in a random manner ensures to avoid any geometrical match between the facing surfaces during sliding. Frictional characteristics are explored by changing the sliding velocity ($8 \leqslant v \leqslant160$ mm/s) and the static load ($40 \leqslant P \leqslant 180$ Pa) imposed onto the sample by adding weights on the top plate. The resulting shear stress, $\tau$, is obtained by measuring the torque using a rheometer measuring head (Anton Paar DSR 502). This head is coupled to an horizontal rail which transfers the angular velocity to the top plate, see Fig.\[fig:slidingfriction\](a)-(b). The bottom plate is fixed while top plate rotates. Controlled experiments with a misaligned axis have been performed in order to verify that the contact area does not have any influence on the torque measurements. We also check that there is no shear-rate dependence in $\tau$. The measured shear stress, $\tau$, is plotted as a function of the applied normal stress, $P$, in Fig.\[fig:slidingfriction\](d) for two batches of SR and HR spheres. Clearly, a linear dependence is observed. A linear fit yields the slope, i.e. the sliding friction coefficient, for each sample. The interparticle sliding friction coefficients for SR and HR spheres are given in Table\[tab:surface\] in both dry and immersed situations, $\mu^d_{sf}$ and $\mu^i_{sf}$ respectively. The sliding friction coefficient increases with increasing roughness but does not change significantly between dry or immersed conditions.
![Interparticle rolling friction experiment. (a) Lateral view of the setup (b) Plot of the rolling force, $F_R$, versus the applied normal force, $F_N$, and corresponding linear fits for SR ($\bigcirc$) and HR ($\square$) spheres in a dry situation.[]{data-label="fig:rollingfriction"}](Fig4Tapia){width="90.00000%"}
In order to estimate the interparticle rolling friction coefficients, we also conducted controlled rolling-experiments using a device with a design similar to the particle-roughening apparatus of Fig.\[fig:roughening\](a). The bottom plate is fixed while the top plate performs a translational motion with a shifted rotational-axis ($\delta=5$mm) with respect to the bottom-plate axis, see Fig.\[fig:rollingfriction\](a). The particles which are sandwiched between these two plates experience the same rolling motion. The surfaces of the plates have been chosen to be rather smooth to avoid any impact on the measurements. The rolling force, $F_R$, exerted on the particles is obtained by measuring the torque using a rheometer measuring head (Anton Paar DSR 502). This force is plotted as a function of the applied normal force, $F_N$, in Fig.\[fig:rollingfriction\](b) for the SR and HR spheres. The interparticle rolling friction coefficient, $\mu^d_{rf}$, increases with roughness but is found to be always much smaller than the sliding friction coefficient, see Table\[tab:surface\].
Experimental Results {#sec:results}
====================
![(a) Effective friction coefficient, $\mu=\tau/P$, and (b) volume fraction, $\phi$, versus the viscous number, $J=\eta_f \dot{\gamma}/P$, for the immersed case (blue and cyan colors) and versus the inertial number, $I= d \dot{\gamma} \sqrt{P/\rho_p}$, for the dry case (red and magenta colors), for SR ($\bigcirc$) and HR ($\square$) spheres. The dashed lines are the linear regressions in $\sqrt{J}$ (immersed case) and in $I$ (dry case), see Table\[tab:coeff\]. (c) Effective friction coefficient, $\mu=\tau/P$, versus volume fraction, $\phi$. (d) Rescaled friction coefficient, $\mu-\mu_c$, versus rescaled volume fraction, $1-\phi/\phi_c$. The inset shows the same data in linear scales. The black dashed lines are the linear regressions in $1-\phi/\phi_c$ for the dry and the immersed cases.[]{data-label="fig:MuPhi"}](Fig5Tapia){width="95.00000%"}
models label $\mu_c$ $\phi_c$
-- ------------------------------- ------- ---------------- -------------------
$\mu- \mu_c \propto J^{1/2}$ SR $0.37\pm 0.01$ $0.584 \pm 0.002$
$\phi_c-\phi \propto J^{1/2}$ HR $0.36\pm 0.01$ $0.565 \pm 0.002$
$\mu - \mu_c \propto I$ SR $0.36\pm 0.01$ $0.587 \pm 0.002$
$\phi_c-\phi \propto I$ HR $0.36\pm 0.01$ $0.563 \pm 0.002$
: Values of $\mu_c$ and $\phi_c$ for the SR and HR spheres in the immersed and dry cases.[]{data-label="tab:coeff"}
We display first the rheological measurements in Fig.\[fig:MuPhi\](a) and (b) by plotting the effective friction coefficient, $\mu=\tau/P$, and the bulk volume fraction, $\phi$, as a function of the viscous number, $J=\eta_f \dot{\gamma}/P$, for the immersed particles and as function of the inertial number, $I= d \dot{\gamma} \sqrt{\rho_p/P}$, for the dry particles. A remarkable result is that $\mu$ is not significantly affected by an increase in particle roughness (or in interparticle friction). Conversely, there is a conspicuous shift of $\phi(J)$ for the immersed case and $\phi(I)$ for the dry case toward lower values of $\phi$ with increasing surface roughness (or interparticle friction).
The semi-logarithmic plots of Fig.\[fig:MuPhi\](a) and (b) are particularly amenable to a close examination of the jamming transition, and in particular show that both $\phi$ and $\mu$ tend to finite measurable values, $\phi_c$ and $\mu_c$ respectively, at the jamming point. The critical (or maximum flowable) volume fraction, $\phi_c$, and friction coefficient, $\mu_c$, can be measured by fitting the data using a linear regression in $\sqrt{J}$ [@BoyeretalPRL2011] or in $I$ [@ForterrePouliquen2008] in the immersed or dry cases, respectively, as summarized in Table\[tab:coeff\] and shown by the dashed lines of Fig.\[fig:MuPhi\](a) and (b). The critical volume fraction, $\phi_c$, happens to be similar within error bars in the immersed and dry cases and shows a decrease with increasing roughness (or interparticle friction). The value of $\phi_c$ measured for the SR spheres is found to be similar to those obtained ($\phi_c \approx 0.585$) in previous immersed experiments by [@BoyeretalPRL2011; @Dagois-Bohyetal2015] but differs with that obtained ($\phi_c \approx 0.625$) in the dry case by [@Falletal2015]. The values of $\mu_c$ for the immersed and dry cases are similar within error bars and do not significantly differ for the SR and HR spheres. They are slightly larger than those found previously ($\mu_c \approx 0.30-0.32$) in the immersed case by [@BoyeretalPRL2011; @Dagois-Bohyetal2015] and much larger than that obtained ($\mu_c \approx 0.26-0.27$) in the dry case by [@Falletal2015]. The present value of 0.36 corresponds to a friction angle of $20$ degree which is close to the typical pile angles observed for spherical particles.
Fig.\[fig:MuPhi\](c) shows that $\mu$ presents a quasi-linear decrease in $\phi$ but with different slopes for the dry and immersed cases. By rescaling $\mu$ by $\mu-\mu_c$ and $\phi$ by $1-\phi/\phi_c$, good collapses of the data for the two different roughnesses are obtained in Fig.\[fig:MuPhi\](d) for the dry and immersed cases, respectively. A linear regression, $\mu(\phi) - \mu_c \propto ( 1 - \phi/\phi_c)$, yields a decent fit of all the data with coefficients of proportionality $a^i \approx 3.52$ in the (dry) inertial case and $a^v \approx 8.05$ in the (immersed) viscous case, see the inset of Fig.\[fig:MuPhi\](d) and the inset of Fig. \[fig:dryRheo\](c).
![Rheological data in the immersed case: (a) $\eta_s = \tau/\eta\dot{\gamma}$ and (b) $\eta_n = P/\eta\dot{\gamma}$ versus $\phi/\phi_c$ as well as (c) $\mu=\tau/P=\eta_s/\eta_n$ and (d) $\phi$ versus $J$. The insets of graphs (a,b) and (c,d) are log-log and semi-log plots, respectively. The dashed lines correspond to the best fits using the models summarized in Table \[tab:coeff\] and the linear variation $\mu(\phi) = \mu_c + a^v ( 1 - \phi/\phi_c)$ as a leading expansion.[]{data-label="fig:wetRheo"}](Fig6Tapia){width="95.00000%"}
We now turn to a detailed examination of the rheological data obtained in the viscous (immersed) case in Fig. \[fig:wetRheo\]. Rescaling $\phi$ by $\phi_c$ provides a complete collapse of all the data for both the SR and HR spheres. In addition to the frictional description displayed in Fig. \[fig:wetRheo\](c) and (d), we provide the classical viscosity description in Fig. \[fig:wetRheo\](a) and (b). The shear, $\eta_s = \tau/\eta\dot\gamma$, and normal, $\eta_n = P/\eta\dot\gamma$, relative viscosities increase with increasing $\phi$ and diverge asymptotically at the maximum flowable volume fraction, $\phi_c$. The log-log plot shown in the inset of Fig. \[fig:wetRheo\](b) reveals that $\eta_n$ diverges as $(1 - \phi/\phi_c)^{-2}$ near jamming. Note that since by construction $J=1/\eta_n$, the relation $\eta_n \propto (1 - \phi/\phi_c)^{-2}$ confirms the relation $(\phi_c - \phi) \propto J^{1/2}$ evidenced in Fig.\[fig:MuPhi\](b) for the immersed case. The shear viscosity, $\eta_s$, is determined by the rheological law $\eta_s(\phi)=\mu(\phi)\eta_n(\phi)$, where $\mu(\phi)$ is given by the leading expansion $\mu(\phi) = \mu_c + a^v ( 1 - \phi/\phi_c)$ introduced in Fig.\[fig:MuPhi\](d). The divergence in $(1 - \phi/\phi_c)^{-2}$ dominates close to jamming.
![Rheological data in the dry case: (a) $\eta_I = \tau/\rho_p d^2 \dot{\gamma}^2$ and (b) $\eta_{II} = P/\rho_p d^2 \dot{\gamma}^2$ versus $\phi/\phi_c$ as well as (c) $\mu=\tau/P=\eta_s/\eta_n$ and (d) $\phi$ versus $I$. The insets of graphs (a,b) and (d) are log-log and linear plots, respectively. The dashed lines correspond to the best fits using the models summarized in Table \[tab:coeff\] and the linear variation $\mu(\phi) = \mu_c + a^i ( 1 - \phi/\phi_c)$ as a leading expansion. The inset of graph (c) shows the rescaled friction coefficient, $\mu-\mu_c$, versus the rescaled volume fraction, $1-\phi/\phi_c$, together with this leading expansion.[]{data-label="fig:dryRheo"}](Fig7Tapia){width="95.00000%"}
An analogous data representation for the inertial (dry) case is displayed in Fig.\[fig:dryRheo\] and shows a decent collapse of the data when rescaling $\phi$ by $\phi_c$. While the collapse is excellent in the quasi-static regime, discrepancies arise as $I \gtrsim 10^{-2}$. The shear and normal stresses are normalized by the Bagnold scaling, $\rho_p d^2 \dot{\gamma}^2$ [@Bagnold1954]. This scaling defines two dimensionless functions of the volume fraction, $\eta_I= \tau/\rho_p d^2 \dot{\gamma}^2$ and $\eta_{II}= P/\rho_p d^2 \dot{\gamma}^2$, which diverge when approaching the maximum volume fraction, $\phi_c$. The function $\eta_{II}$ is seen to diverge as $(1 - \phi/\phi_c)^{-2}$ near jamming as evidenced in the inset of Fig. \[fig:dryRheo\](b). This is consistent with the relation $(\phi_c - \phi) \propto I$ observed in Fig.\[fig:MuPhi\](b) for the dry case since $\eta_{II}=1/I^2$, see also the inset of Fig. \[fig:dryRheo\](d). The other dimensionless function is given by $\eta_I(\phi)=\mu(\phi)\eta_{II}(\phi)$ and has again a leading divergence in $(1 - \phi/\phi_c)^{-2}$ near jamming. A detailed display of $\mu(I)$, $\phi(I)$, and $\mu(\phi)$ is also provided in Fig. \[fig:dryRheo\](c) and (d).
The rheological data of Figs.\[fig:MuPhi\], \[fig:wetRheo\], \[fig:dryRheo\] are given as Supplemental Material.
Discussion and Conclusions {#sec:conclusion}
==========================
In this work, we have examined how particle roughness affects the rheological properties of particulate systems in both immersed and dry conditions. We have used two batches of spherical particles, a batch of “slightly roughened” regular spheres (SR) and another batch of “highly roughened” spheres (HR) produced by further roughening these regular spheres. We have provided a detailed characterization of these particles, and more precisely determined their surface-roughness characteristics and their interparticle sliding and rolling friction coefficients using custom-made experimental devices. We have then performed pressure- and volume-imposed measurements of the rheology of theses two batches of spheres in the dense regime.
An important finding of the present study is the examination of the rheological behavior in the vicinity of the jamming transition in the dry and immersed cases. In the limit of vanishing shear rate, the critical values for the effective friction coefficient, $\mu_c$, and for the volume fraction, $\phi_c$, are found to be similar for suspensions and dry granular media. This seems to imply that hydrodynamic interactions are inconsequential and contact forces are prevailing in this quasi-static limit. A striking result is that $\mu_c$ is not significantly affected by particle roughness while $\phi_c$ decreases with increasing roughness. This later finding shows that the granular system needs to dilate more in order to flow when roughness is increased. Rescaling the volume fraction, $\phi$, by this maximum volume fraction, $\phi_c$, leads to an excellent collapse of all the data on master curves for the effective friction coefficient (or stress ratio), $\mu$, as well as for the shear and normal stresses, $\tau$ and $P$ respectively, normalized by a viscous scaling ($\eta\dot{\gamma}$) in the immersed case and by an inertial scaling ($\rho_p d^2 \dot{\gamma}^2$) in the dry case.
The excellent collapse of the experimental data by using the reduced volume fraction, $\phi/\phi_c$, has been previously observed for the shear viscosity in the suspension case [@GuazzelliPouliquen2018], but without a controlled study of the influence of roughness and friction. Similar behavior has been also observed in numerical simulations when interparticle friction is increased [@Gallieretal2014; @Marietal2014]. However these simulations match the present experimental $\phi_c$ for interparticle sliding friction coefficients ($\mu_{sf} \approx 0.5-1$) quite larger than those experimentally measured ($\mu_{sf}= 0.23$ for the SR spheres and $=0.37$ for the HR spheres). It is likely that rolling friction (with experimental coefficients $\mu_{sf}= 0.0043$ for the SR spheres and $=0.0070$ for the HR spheres) is somehow captured by using larger sliding friction coefficients as studied in granular media by [@Estradaetal2008].
The present work not only yields the quasi-static values but also the asymptotic behaviors of $\mu$ and $\phi$ in the vicinity of the jamming transition. The departures of the effective friction coefficient and of the volume fraction from their static limits, $\mu-\mu_c$ and $\phi_c-\phi$, are found to vary as the square root of the viscous number in the immersed case, i.e. $\propto \sqrt{J}$, and linearly with the inertial number in the dry case, i.e. $\propto I$. The present experimental data for the suspensions of the two batches of SR and HR spheres also reveal that both relative shear and normal viscosities, $\eta_s$ and $\eta_n$, present a similar algebraic divergence in $\approx (\phi_c-\phi)^{-2}$ as previously observed for a variety of diverse suspensions of frictional spheres [@BoyeretalPRL2011; @GuazzelliPouliquen2018]. Interestingly, the analogous dimensionless functions obtained by normalizing the shear and normal stresses with an inertial scaling in the dry case, $\eta_I$ and $\eta_{II}$, present the same dominant algebraic divergence.
-------------------------------- ------------------------------------------------------------ ---------------- --------- -------------------------- ------------------------------------------ --------------------------------------------
relations exponent present predictions simulations simulations
\[-10pt\] work (frictionless) (frictionless) (frictional)
$\mu- \mu_c \propto J^{\gamma_{\mu}}$ $\gamma_{\mu}$ $0.5$ $0.35$[@DeGiulietal2015] 0.32[@DeGiulietal2015] $0.5$[@Trulssonetal2012; @Amarsidetal2017]
[[**(immersed)**]{.nodecor}]{} $\phi_c-\phi \propto J^{\gamma_{\phi}}$
$\eta_{s,n} \propto (\phi_c-\phi)^{-\gamma_{\phi}^{-1}}$ 1.5-1.6[@Marietal2014; @Gallieretal2018] 2.3[@Marietal2014],1.9[@Gallieretal2018]
$\mu - \mu_c \propto I^{\alpha_{\mu}}$ $\alpha_{\mu}$ $1$ $0.35$[@DeGiulietal2015] 0.38[@Peyneauetal2008] 1[@AzemaRadjai2014]
[[**(dry)**]{.nodecor}]{} $\phi_c-\phi \propto I^{\alpha_{\phi}}$
$\eta_{I,II} \propto (\phi_c-\phi)^{-2\alpha_{\phi}^{-1}}$
-------------------------------- ------------------------------------------------------------ ---------------- --------- -------------------------- ------------------------------------------ --------------------------------------------
: Critical exponents found in the present work versus predicted and numerical values.[]{data-label="tab:exponents"}
The critical exponents found in the present experiments are compared in Table \[tab:exponents\] to the theoretical predictions [@DeGiulietal2015] which have been obtained for frictionless spheres and to results of various numerical simulations performed in two dimensions [@Trulssonetal2012; @DeGiulietal2015] as well as in three dimensions [@Marietal2014; @Gallieretal2018; @Peyneauetal2008; @AzemaRadjai2014; @Amarsidetal2017] for dry and immersed spheres with and without frictional interactions. They are in good agreement with the frictional simulations, in particular those performed in three dimensions. It is nonetheless difficult to assess what is the prevailing dissipation regime. The $J-\mu_{sf}$ diagram proposed by [@Trulssonetal2017] suggests that the immersed experiments lies between the rolling and frictional-sliding regimes while the $I-\mu_{sf}$ diagram proposed by [@DeGiulietal2016] seems to indicate that the dry experiments belong to the frictional-sliding regime. However, the rolling friction is not accounted for in these studies.
Finally, the present work may shed some light on the transition between the viscous and inertial regimes which is far from being fully understood and still a subject of debate [@Trulssonetal2012; @DeGiulietal2015; @Amarsidetal2017]. Assuming that this transition arises when the viscous and inertial stresses are matched, the crossover shear-rate is then $\dot{\gamma}_{v \rightarrow i} = (\eta_f/\rho_pd^2)\, \eta_s(\phi)/\eta_I(\phi)$. The similar divergence of the functions $\eta_s(\phi)$ and $\eta_I(\phi)$ found here seems to suggest that $\dot{\gamma}_{v \rightarrow i}$ is independent of the volume fraction close to the jamming transition. Further investigations are clearly necessary to understand thoroughly this viscous-inertial transition for dense frictional particles.
This work has been carried out thanks to the support of the ANR project ‘Dense Particulate Systems’ (ANR-13-IS09-0005-01), the ‘Laboratoire d’Excellence Mécanique et Complexité’ (ANR-11-LABX-0092), the Excellence Initiative of Aix-Marseille University - A$^{\ast}$MIDEX (ANR-11-IDEX-0001-02) funded by the French Government ‘Investissements d’Avenir programme’. FT benefited from a fellowship of CONICYT (Engagement: 74170026).
|
---
author:
- 'Yutao Tang [^1]'
bibliography:
- 'opt\_adaptive.bib'
title: 'Distributed Optimization for a Class of High-order Nonlinear Multi-agent Systems with Unknown Dynamics'
---
[**Abstract**]{}: In this paper, we study a distributed optimization problem for a class of high-order multi-agent systems with unknown dynamics. In comparison with existing results for integrators or linear agents, we need to overcome the difficulties brought by the unknown nonlinearities and also the optimization requirement. For this purpose, we employ an embedded control based design and first convert this problem into an output stabilization problem. Then, two kinds of adaptive controllers are given for these agents to drive their outputs to the global optimal solution under some mild conditions. Finally, we show that the estimated parameter vector converges to the true parameter vector under some well-known persistence of excitation condition. The efficacy of these algorithms was verified by a simulation example.
[**Keywords**]{}: Distributed optimization, unknown dynamics, embedded design, adaptive control, parameter convergence.
Introduction
============
Over the past few years, distributed optimization of multi-agent systems has become a hot topic due to the fast development of multi-robot networks, machine learning and big data technologies [@derenick2007convex; @nedic2010constrained; @boyd2011distributed]. In a typical setting of this problem, each agent is assigned with a local cost function and the control objective is to propose distributed controls that guarantee a consensus on the optimal solution of the sum of all local cost functions. Many effective algorithms have been proposed to achieve this goal in different situations [@shi2013reaching; @zhu2012distributed; @yuan2015gradient; @kia2015distributed; @yang2016PI; @yang2017distributed].
Here, we follow this technical line but consider high-order continuous-time nonlinear agents unknown dynamics. While most of the existing works were only devoted to single-integrator agents, there are many distributed optimization tasks implemented by or depending on physical plants of continuous dynamics in practice, e.g. source seeking in multi-robot systems [@zhang2011extremum], attitude formation control of rigid bodies[@song2017relative], optimal power dispatch over power networks [@stegink2017unifying]. Note that these physical dynamics can hardly be described well by single integrators. In fact, an example was given to show that the algorithms designed for single integrators might fail to achieve a desired performance for these high-order agents [@tang2018distributed]. Thus, we have to take the high-order and possible nonlinear agents’ dynamics into account to achieve the distributed optimization goal.
In light of the optimization requirement for agents, the gradient-based closed-loop systems are basically nonlinear. Furthermore, the high-order feature of these agents brings many new difficulties to the associated analysis and design, which makes this problem much more challenging. In fact, very few optimization results have been obtained on this topic with continuous-time agents in the form of high-order dynamics. For example, a distributed algorithm for double integrators was proposed in [@zhang2017distributed] with an integral control idea and then extended to Euler-Lagrange agents. A class of nonlinear minimum phase agents in output feedback form with unity relative degree was considered by an internal-model design [@wang2016cyber]. Distributed optimization problem with bounded controls was also explored for both single and double integrators [@xie2017global] . However, distributed optimization of general high-order multi-agent systems is still far from being solved.
Recently, we proposed an embedded control scheme to solve this problem for general linear systems having well-defined vector relative degrees [@tang2017cyb]. To overcome the difficulties brought by both high-order dynamics and possible nonlinearities from gradients, we divided the design into two main parts: optimal signal generator construction and reference tracker design. This embedded technique makes the control design carried out in a “separative” way, so as to simplify the whole design by almost independently tackling the optimal consensus problem for single integrators and output tracking problem of high-order agents.
Note that exact information of system matrices is required in this embedded design, which may be not available or with measurement errors in applications. This motivates us to investigate the distributed optimization problem for high-order nonlinear agents with unknown time-varying dynamics. Furthermore, we focus on the case when this unknown time-varying nonlinearity can be linearly parameterized. The objective of this paper is to extend existing embedded control results to this class of uncertain high-order nonlinear agents and achieve the distributed optimization goal.
In view of the aforementioned observations, the contribution of this paper is at least two-fold. Firstly, a distributed optimization problem was formulated and solved for a group of high-order nonlinear agents, which can be taken as an extended version of existing results for single integrators [@nedic2010constrained; @kia2015distributed]. Secondly, the embedded control technique proposed in [@tang2017cyb] was further explored and extended to solve the distributed optimization problem of high-order multi-agent systems with unknown dynamics. By removing the requirement of knowing agents’ exact dynamics, this work can be taken as an adaptive extension to existing results for linear multi-agent systems [@kia2015distributed; @xie2017global; @tang2017cyb] and nonlinear ones with unity relative degree [@wang2016cyber]. Moreover, the obtained conclusions can be applied to achieve an output average consensus for these uncertain nonlinear agents, while only integrators or linear agents were considered in existing works [@ren2008distributed; @rezaee2015average; @tang2017kyb]. The remainder of this paper is organized as follows. Problem formulation is presented in Section \[sec:formulation\]. Then the main result is presented in Sections \[sec:main\] along with both stability analysis and parameter convergence. Following that, an example is given to illustrate the effectiveness of our algorithms in Section \[sec:simu\]. Finally, concluding remarks are given in Section \[sec:con\].
*Notations:* Let ${\mathbb{R}}^n$ be the $n$-dimensional Euclidean space. For a vector $x$, $||x||$ denotes its Euclidean norm. ${\bf 1}_N$ (and ${\bf 0}_N$) denotes an $N$-dimensional all-one (and all-zero) column vector. $\mbox{diag}\{b_1,\,{\dots},\,b_n\}$ denotes an $n\times n$ diagonal matrix with diagonal elements $b_i,\,(i=1,\,{\dots},\,n)$. $\mbox{col}(a_1,\,{\dots},\,a_n) = [a_1^{{\mathrm{T}}},\,{\dots},\,a_n^{{\mathrm{T}}}]^{{\mathrm{T}}}$ for column vectors $a_i\; (i=1,\,{\dots},\,n)$. A weighted directed graph (or digraph) $\mathcal {G}=(\mathcal {N}, \mathcal {E}, \mathcal{A})$ is defined as follows, where $\mathcal{N}=\{1,{\dots},n\}$ is the set of nodes, $\mathcal {E}\subset \mathcal{N}\times \mathcal{N}$ is the set of edges, and $\mathcal{A}\in \mathbb{R}^{n\times n}$ is a weighted adjacency matrix [@godsil2001algebraic]. $(i,j)\in \mathcal{E}$ denotes an edge leaving from node $i$ and entering node $j$. The weighted adjacency matrix of this digraph $\mathcal {G}$ is described by $A=[a_{ij}]\in {\mathbb{R}}^{n\times n}$, where $a_{ii}=0$ and $a_{ij}\geq 0$ ($a_{ij}>0$ if and only if there is an edge from agent $j$ to agent $i$). A path in graph $\mathcal {G}$ is an alternating sequence $i_{1}e_{1}i_{2}e_{2}{\cdots}e_{k-1}i_{k}$ of nodes $i_{l}$ and edges $e_{m}=(i_{m},\,i_{m+1})\in\mathcal {E}$ for $l=1,\,2,\,{\dots},\,k$. If there exists a path from node $i$ to node $j$ then node $i$ is said to be reachable from node $j$. The neighbor set of agent $i$ is defined as $\mathcal{N}_i=\{j\colon (j,\,i)\in \mathcal {E} \}$ for $i=1,\,\dots,\,n$. A graph is said to be undirected if $a_{ij}=a_{ji}$ ($i,\,j=1,\,{\dots},\,n$). An undirected graph is said to be connected if there is a path between any two vertices. The weighted Laplacian $L=[l_{ij}]\in \mathbb{R}^{n\times n}$ of graph $\mathcal{G}$ is defined as $l_{ii}=\sum\nolimits_{j\neq i}a_{ij}$ and $l_{ij}=-a_{ij} (j\neq i)$.
Problem formulation {#sec:formulation}
===================
In this paper, we consider a collection of heterogeneous high-order nonlinear systems described by: $$\begin{aligned}
\label{sys:agent}
\begin{split}
\dot{x}_{j,i}&=x_{j+1,i}\\
\dot{x}_{n_i,i}&=\Delta_i(x_i,\,\theta_i,\,t)+u_i\\
y_i&=x_{1,i},\quad i=1,\,\dots,\,N,\,j=1,\,\dots,\,n_i-1
\end{split}
\end{aligned}$$ where $x_{j,i} \in {\mathbb{R}}$ is the $j$-th state variable of agent $i$, $x_i\triangleq \mbox{col}(x_{1,i},\,\dots,\,x_{n_i,i}) \in {\mathbb{R}}^{n_i}$, $y_i \in \mathbb{R}$ and $u_i \in \mathbb{R}$ are respectively the state, output, and input of agent $i$. The function $\Delta_i(x_i,\, \theta_i,\,t)$ represents the unknown time-varying nonlinearities with uncertain parameter $\theta_i=\mbox{col}(\theta_{1,i},\,\dots,\,\theta_{n_{\theta_{n_i}},i})\in {\mathbb{R}}^{n_{\theta_i}}$, which might result from modeling errors or external perturbations.
Associated with this multi-agent system, each agent is endowed with a differentiable local cost function $f_i\colon {\mathbb{R}}\to {\mathbb{R}}$. The global cost function is defined as the sum of local costs, i.e., $f(y)=\sum\nolimits_{i=1}^{N} f_i(y)$. Moreover, we assume the local cost function $f_i(\cdot)$ is only known to agent $i$ itself and cannot be shared globally in the multi-agent network. Coupled with these nonlinear agents, we aim to design proper controllers such that the outputs of these agents asymptotically minimize the global cost function.
To clarify our following design, we focus on a class of systems in the form of as follows.
\[ass:system\] For any $i=1,\,\dots,\,N$, there exist a known basis function vector ${\bm p}_i(x_i,\,t)$ and an unknown parameter vector $\theta_i\in {\mathbb{R}}^{n_{\theta_i}}$ satisfying $\Delta_i(x_i,\,\theta_i,\,t)=\theta_i^{{\mathrm{T}}}{\bm p}_{i}(x_{i},\,t)$ for all $x_i\in {\mathbb{R}}^{n_i}$, $t\geq 0$ and $\theta_i\in {\mathbb{R}}^{n_{\theta_i}}$. Furthermore, ${\bm p}_i(x_i,\,t)$ can be uniformly bounded by smooth functions of $x_i$.
The unknown nonlinear dynamics of all agents are assumed to be linearly parameterized, which have been widely studied in both classical adaptive control and multi-agent coordination literature [@krstic1995nonlinear; @yu2012adaptive; @hu2014adaptive; @wen2017neuro]. In fact, a plenty of practical systems can be put into this form, which is general enough to cover integrators, Van der Pol systems, Duffing equations and many mechanical systems. Moreover, this time-varying feature can further be used to represent many typical external disturbances, e.g. constants and sinusoidal signals.
The following assumption is often made in convex optimization literature [@bertsekas2003convex; @kia2015distributed; @qiu2016distributed], which guarantees the existence and uniqueness of the optimal solution to problem .
\[ass:convexity-strong\] For $i=1,\,\dots,\,N$, the function $f_i(\cdot)$ is $\underline{l}$-strongly convex and its gradient is $\overline l$-Lipschitz for constants $\underline{l},\, \overline l>0$.
As usual, we assume this optimal solution is finite and denote it as $y^*$, i.e. $$\begin{aligned}
\label{opt:main}
y^*={\arg}\min_{ y \in {\mathbb{R}}} \; f(y)=\sum\nolimits_{i=1}^{N} f_i(y)
\end{aligned}$$
Due to the privacy of local cost function $f_i(\cdot)$, no agent can gather enough information to determine the global optimal solution $y^*$ by itself. Hence, our problem cannot be solved without cooperation and information sharing among these agents.
For this purpose, we use a weighted undirected graph $\mathcal {G}=(\mathcal {N}, \mathcal {E}, \mathcal{A})$ to describe the information sharing topology with note set $\mathcal{N}=\{1,\,{\dots},\,N\}$. An edge $(i,\,j)\in \mathcal{E}$ between nodes $i$ and $j$ means that agent $i$ and agent $j$ can share information with each other. Suppose the following assumption is fulfilled as in many publications [@kia2015distributed; @tang2015distributed; @zhang2017distributed; @tang2017cyb; @yang2017distributed].
\[ass:graph\] The graph $\mathcal{G}$ is connected.
This assumption is about the connectivity of information sharing graph $\mathcal{G}$, which guarantees that any agent’s information can reach any other agents. Under this assumption, it is well-known that the associated Laplacian $L$ is positive semidefinite with null space spanned by ${\bm 1}_N$ [@godsil2001algebraic].
The distributed optimization problem considered in this paper is readily formulated as follows.
For given agents of the form , information sharing graph $\mathcal{G}$ and local cost function $f_i(\cdot)$, if possible, determine a distributed protocol for each agent by using its own local data and exchanged information with its neighbors such that
- all the trajectories of agents are bounded over the time interval $[0,\,+\infty)$.
- their outputs of agents satisfy $$\begin{aligned}
\label{def:ooc}
\lim\limits_{t\to +\infty}||y_i(t)-y^*||=0,\quad i=1,\,\dots,\,N
\end{aligned}$$
The formulated problem can be taken as a combination of the well-studied topics: distributed optimization [@nedic2010constrained; @boyd2011distributed; @kia2015distributed; @yi2015distributed] and output consensus [@ren2008distributed; @fax2004information; @xi2012output]. Since an output consensus of the whole high-order multi-agent system must be achieved as the solution of a convex optimization problem in the formulated problem, it is certainly more challenging than the existing output consensus results for high-order agents.
In contrast with existing distributed optimization works, the agents considered here are high-order and heterogeneous, while only single integrators are considered in many publications [@shi2013reaching; @yang2016PI; @lin2016distributed; @qiu2016distributed]. Moreover, we study the case when these agents are with unknown dynamics, while the exact information of agents’ dynamics is required in [@zhang2017distributed] and [@tang2017cyb].
Particularly, when the local cost function is chosen as $f_i(s)=(s-y_i(0))^2$, this formulation can solve an output average consensus problem for these high-order uncertain nonlinear agents and thus includes exiting results for integrators and linear systems as special cases [@ren2008distributed; @rezaee2015average; @tang2017kyb].
As mentioned above, the main difficulty to solve the distributed optimization for these agents lies in the coupling of the high-order structure associated with agents’ dynamics and the global optimization requirement. To overcome this point, we adopt the embedded control scheme [@tang2017cyb] and propose adaptive control laws to solve the distributed optimization problem for agent in the following section.
Main Results {#sec:main}
============
In this section, we first employ an embedded control approach to convert our problem into an output stabilization problem, and then propose distributed adaptive algorithms for these high-order nonlinear agents with unknown dynamics to achieve the optimization goal along with parameter convergence analysis.
Embedded Control Design
-----------------------
The embedded control approach was first proposed by Tang et al [@tang2017cyb] to solve the distributed optimization problem for high-order linear agents. In this approach, an optimal signal generator must be constructed first by considering the same optimization problem for a group of single-integrator agents, in order to asymptotically reproduce the optimal solution $y^*$ by a signal $r_i$. Then, by taking $r_i$ as an output reference signal for agent $i$, this generator is embedded in the feedback loop via some proper interfaces for original agents. In this way, the distributed optimization problem for general linear agents is divided into two simpler subproblems, i.e., construction of an optimal signal generator for single integrators and design of proper trackers for linear agents with output references, which can be independently solved in a modular way.
The first subproblem is essentially a conventional distributed optimization problem for single integrators and has been well-studied in existing literature [@kia2015distributed; @yang2016PI; @yi2015distributed]. To solve this problem, the following optimal signal generator was proposed for problem in [@tang2017cyb]. $$\begin{aligned}
\label{sys:generator}
\begin{split}
\dot{r}_i&=-\nabla f_i(r_i)-\sum\nolimits_{j=1}^N a_{ij}({\lambda}_i-{\lambda}_j)\\
\dot{{\lambda}}_i&=\sum\nolimits_{j=1}^N a_{ij}(r_i-r_j)
\end{split}
\end{aligned}$$
The effectiveness of was already proven in [@tang2017cyb]. We repeat it as follows for completeness.
\[lem:generator\] Suppose Assumptions \[ass:convexity-strong\] and \[ass:graph\] hold. Then, along the trajectory of system , $r_i(t)$ approaches the optimal solution $y^*$ exponentially as $t\to \infty$ for $i=1,\,\ldots,\,N$.
Suppose we know the analytical form of $f_i(\cdot)$ or at least $\nabla f_i(\cdot)$, the optimal signal generator can be implemented independently to generate the minimizer of problem as showed in this lemma. However, this requirement may cost too much in applications and even impossible in many practical circumstances. Thus, we follow an oracle-based description[@nesterov2013introductory] of $f_i(\cdot)$ and consider the case when only the real-time gradient $\nabla f_i(y_i)$ is available upon requests in our following design.
Furthermore, exact information of system matrices is required for the proposed controllers in [@tang2017cyb]. This implies that the corresponding designs can not be directly used to solve our problem for uncertain agents in the form of . Thus, the distributed optimization problem considered in this paper is much more challenging than those addressed for integrators or linear agents in existing works [@kia2015distributed; @shi2013reaching; @qiu2016distributed; @yang2016PI; @tang2017cyb].
To deal with these two issues, we adopt a certainty-equivalence design and propose an algorithm as follows: $$\begin{aligned}
\label{ctr:online}
\begin{split}
u_i&=-\hat \theta_i^{{\mathrm{T}}}{\bm p}_{i}(x_{i},\,t)+\frac{1}{{\varepsilon}^{n_i}}[k_{1i}(x_{1,i}-r_i)+\sum\nolimits_{j=2}^{n_i}{\varepsilon}^{j-1}k_{ji}x_{j,\,i}]\\
\dot{\hat \theta}_i&=\phi_i(x_i,\,\hat \theta_i,\, r_i,\,t)\\
\dot{r}_i&=-\nabla f_i(y_i)-\sum\nolimits_{j=1}^N a_{ij}({\lambda}_i-{\lambda}_j)\\
\dot{{\lambda}}_i&=\sum\nolimits_{j=1}^N a_{ij}(r_i-r_j)
\end{split}
\end{aligned}$$ where $\hat \theta_i $ is the estimation of uncertain vector $\theta_i$, and the constants $k_{1i},\,\dots,\,k_{n_i i}$ and function $\phi_i(\cdot)$ to be specified later. Here the constant ${\varepsilon}>0$ is a tunable high-gain parameter to deal with the real-time gradient issue.
Here, the variables $\hat\theta_i,\,r_i,\,{\lambda}_i$ constitute the local compensator of agent $i$. It can be verified that this control is indeed distributed in the sense of only using the agents’ own local data and exchanged information with their neighbors. Moreover, from its nominal form , the above control is composed of two parts, where the last two subsystems can be taken as a modified version of optimal signal generator with real-time gradients and the rest compose an adaptive tracking controller such that $y_i(t)$ can track $r_i(t)$ as $t$ goes to infinity in spite of those unknown dynamics, which confirms the embedded design methodology.
Under the above control law, the composite system is then: $$\begin{aligned}
\begin{split}
\dot{x}_{1,\,i}&=x_{2, i} \\
\vdots&\\
\dot{x}_{n_i,i}&=(\theta_i^{{\mathrm{T}}}-\hat \theta_i^{{\mathrm{T}}}) {\bm p}_{i}(x_{i},\,t)+\frac{1}{{\varepsilon}^{n_i}}[k_{1i}(x_{1,i}-r_i)+\sum\nolimits_{j=2}^{n_i}{\varepsilon}^{j-1}k_{ji}x_{j,\,i}]\\
\dot{\hat \theta}_i&=\phi_i(x_i,\,\hat \theta_i,\,r_i,\,t)\\
\dot{r}_i&=-\nabla f_i(y_i)-\sum\nolimits_{j=1}^N a_{ij}({\lambda}_i-{\lambda}_j)\\
\dot{{\lambda}}_i&=\sum\nolimits_{j=1}^N a_{ij}(r_i-r_j)
\end{split}
\end{aligned}$$
By letting $\hat x_i=\mbox{col}(x_{1,i}-r_i,\,{\varepsilon}x_{2,i},\,\dots,\,{\varepsilon}^{n_i-1}x_{n_i,i})$, it can be further rewritten as: $$\begin{aligned}
\label{sys:closed-spform}
\begin{split}
{\varepsilon}\dot{\hat x}_i&=A_i\hat x_i-{\varepsilon}b_{1i} \dot{r}_i+{{\varepsilon}^{n_i}}b_{2i}(\theta_i^{{\mathrm{T}}}-\hat \theta_i^{{\mathrm{T}}}) {\bm p}_{i}(x_{i},\,t) \\
\dot{\hat {\theta}}_i&=\phi_i(x_i,\,\hat \theta_i,\,r_i,\,t)\\
\dot{r}_i&=-\nabla f_i(y_i)-\sum\nolimits_{j=1}^N a_{ij}({\lambda}_i-{\lambda}_j)\\
\dot{{\lambda}}_i&=\sum\nolimits_{j=1}^N a_{ij}(r_i-r_j)
\end{split}
\end{aligned}$$ where $b_{1i}=\mbox{col}(1,\,0,\,\dots,\,0)$, $b_{2i}=\mbox{col}(0,\,\dots,\,0,\,1)$ and $A_i=\left[\begin{array}{c|c}
0&I_{n_{x_i}-1}\\\hline
k_{1i}&[k_{2i}\,\dots\,k_{n_i\,i}]
\end{array}\right]$.
Note that the above system is almost in a singularly perturbed form except the adaption dynamics $\hat \theta_i$ [@khalil2002nonlinear]. By letting ${\varepsilon}=0$, we have $\hat x_i=0$ and $x_{1,i}=r_i$. The resultant quasi-state-state model of the above composite system is exactly the optimal signal generator , which in turn guarantees $x_{1,\,i}(t)\to y^*$ as $t$ goes to infinity by Lemma 1.
Based on these observations, our formulated distributed optimization problem for agent is converted to a decentralized output stabilization problem for the above system with output $\hat x_i$. In other words, we need to determine proper function $\phi_i(\cdot)$ and constant ${\varepsilon}>0$ such that all trajectories of is bounded over $[0,\,+\infty)$ and satisfying $\hat x_{i}(t)\to 0$ as $t$ goes to infinity.
Solvability Analysis
--------------------
To solve our problem, we denote $\hat x=\mbox{col}(\hat x_1,\,\dots,\,\hat x_N)$, $\theta=\mbox{col}(\theta_1,\,\dots,\,\theta_N)$, $\hat \theta=\mbox{col}(\hat \theta_1,\,\dots,\,\hat \theta_N)$, $r=\mbox{col}(r_1,\,\dots,\,r_N)$, ${\lambda}=\mbox{col}({\lambda}_1,\,\dots,\,{\lambda}_N)$ and $\bar \theta=\theta-\hat \theta$. The whole composite system can be put into a compact form as follows. $$\begin{aligned}
\label{sys:composite-compact}
\begin{split}
\dot{\hat x}&=\frac{1}{{\varepsilon}}A\hat x-B_{1} \dot{r}+EB_{2} {\bm p}^{{\mathrm{T}}}(x,\,t) \bar \theta\\
\dot{\bar {\theta}}&=\phi(x,\,\hat \theta,\,r,\,t) \\
\dot{r}&=-\nabla \tilde f(y)- L {\lambda}\\
\dot{{\lambda}}&=L r
\end{split}
\end{aligned}$$ where $\tilde f(y)=\sum\nolimits_{i=1}^N f_i(y_i)$, $A=\mbox{blockdiag}(A_1,\,\dots,\,A_N)$, $B_1=\mbox{blockdiag}(b_{11},\,\dots,\,b_{1N})$,$B_2=\mbox{blockdiag}(b_{21},\,\dots,\,b_{2N})$,$E=\mbox{blockdiag}({\varepsilon}^{n_1-1}I_{n_1},\,\dots,\,{\varepsilon}^{n_N-1}I_{n_N})$, ${\bm p}(x,\,t)\triangleq \mbox{blockdiag}({\bm p}_1(x_1,\,t),\,\dots,\,{\bm p}_N(x_N,\,t))$, $L$ is the Laplacian of $\mathcal{G}$, and the function $\phi(x,\,\hat \theta,\,r,\,t)$ is determined by $\phi_1(\cdot),\,\dots,\,\phi_N(\cdot)$.
We first choose gain constants $k_{1i},\,\dots,\,k_{n_i\,i}$ such that the polynomial $s^{n_i}-k_{n_i\, i}s^{n_i-1}-k_{2i}s-k_{1i}$ is Hurwitz for any $1\leq i\leq N$. This implies that the following Lyapunov equation $$\begin{aligned}
A_i^{{\mathrm{T}}}P_i+P_iA_i=-2I_{n_i}
\end{aligned}$$ has a unique positive definite solution $P_i$ with compatible dimensions.
Inspired by existing Lyapunov-based designs [@krstic1995nonlinear; @hu2014adaptive], we present the main result of this paper as follows.
\[thm:main:online\] Suppose Assumptions \[ass:system\]–\[ass:graph\] hold. Then, the distributed optimization problem determined by and can be solved by controllers of the form with $\phi_i(x_i,\,\hat \theta_i,\,r_i,\,t)={\bm p}_{i}(x_{i},\,t) b_{2i}^{{\mathrm{T}}}P_i \hat x_i$ for a small enough ${\varepsilon}>0$, where the constant $k_{ji}$ is chosen as above for $i=1,\,\dots,\,N,\,j=1,\,\dots,\,n_i$.
[**Proof.** ]{}The proof is mainly based on system composition techniques.
[*Step 1*]{}: consider the first two subsystems of . Let $\hat V_i=\hat W_i+{{\varepsilon}^{n_i-1}}\bar \theta_i^{{\mathrm{T}}}\bar \theta_i$ with $\hat W_i=\hat x_i^{{\mathrm{T}}}P_i\hat x_i$. Its time derivative along the trajectory of the composite system satisfies $$\begin{aligned}
\dot{\hat V}_i= &2\hat x_i^{{\mathrm{T}}}P_i [\frac{1}{{\varepsilon}}A_i\hat x_i-b_{1i} \dot{r}_i+{{\varepsilon}^{n_i-1}}b_{2i}\bar \theta_i^{{\mathrm{T}}}{\bm p}_{i}(x_{i},\,t)]\\
&-2{\varepsilon}^{n_i-1}\bar \theta_i^{{\mathrm{T}}}\phi_i(x_i,\,\hat \theta_i,\,r_i,\,t)\\
= & -\frac{2}{{\varepsilon}}\hat x_i^{{\mathrm{T}}}\hat x_i-2\hat x_i^{{\mathrm{T}}}P_i b_{1i} \dot{r}_i
\end{aligned}$$
By Young’s inequality, it holds that $$\begin{aligned}
\dot{\hat V}_i \leq &-\frac{c_1}{{\varepsilon}}||\hat x_i||^2+c_2||\dot{r}_i||^2
\end{aligned}$$ for some known positive constants $c_1$ and $c_2$.
Choose $\hat V=\sum\nolimits_{i=1}^N \hat V_i$. We further have $$\begin{aligned}
\label{eq:thm1-eq1}
\dot{\hat V} \leq &-\frac{c_1}{{\varepsilon}}||\hat x||^2+c_2||\dot{r}||^2
\end{aligned}$$
[*Step 2*]{}: consider the last two subsystems of , which can be rewritten as $$\begin{aligned}
\dot{r}&=-\nabla \tilde f(r)- L {\lambda}+ {\bm h}(r,\,y)\\
\dot{{\lambda}}&=L r
\end{aligned}$$ where the function ${\bm h}(r,\,y)\triangleq \nabla \tilde f(r)-\nabla \tilde f(y)$ is $\overline l$-Lipschitz in $r-y$ by Assumption \[ass:convexity-strong\].
By taking ${\bm h}(r,\,y)$ as perturbations, we let $\mbox{col}(r^*, {\lambda}^*)$ be the equilibrium point of unperturbed $(r,\,{\lambda})$-system (i.e. when ${\bm h}(r,\,y)\equiv {\bm 0}$). Note that the unperturbed system is exactly the optimal signal generator, it implies $r^*={\bm 1}_N y^*$ under Assumptions \[ass:convexity-strong\] and \[ass:graph\] by Lemma \[lem:generator\]. In fact, the equilibrium point satisfies $-\nabla f(r^*)-L{\lambda}^*={\bm 0}_N$ and $Lr^*={\bm 0}_{N}$. As a result, there exists a constant $\theta$ such that $r_1^*=\dots=r_N^*=\theta$ since the null space of $L$ is spanned by ${\bm 1}_N$ under Assumption \[ass:graph\]. Then, one can obtain $\sum\nolimits_{i=1}^N \nabla f_i(\theta)=0$ by ${\bm 1}_N^{{\mathrm{T}}}L={\bm 0}_N$, which implies that $\theta$ is an optimal solution of . By Assumption \[ass:convexity-strong\], this implies $\theta=y^*$ and $r^*={\bm 1}_N y^*$.
Letting $\bar r=r-r^*$ and $\bar {\lambda}={\lambda}-{\lambda}^*$ gives $$\begin{aligned}
\dot{\bar r}&=-{\bm h}(r,\,r^*) -L \bar {\lambda}+ {\bm h}(r,\,y)\\
\dot{\bar {\lambda}}&=L \bar r
\end{aligned}$$
Inspired by the proof of Lemma \[lem:generator\] in [@tang2017cyb] , we let $R \in \mathbb{R}^{N \times (N-1)}$ be a matrix satisfying $R^{{\mathrm{T}}}{\bm 1}_N={\bm 0}_N$, $R^{{\mathrm{T}}}R=I_{N-1}$, and $RR^{{\mathrm{T}}}=I_{N}-\frac{1}{N}{\bm 1}_N{\bm 1}_N^{{\mathrm{T}}}$. Apparently, the matrix $R$ has a full column rank. Denote $T=[\frac{{\bm 1}_N}{\sqrt{N}}~R]^{{\mathrm{T}}}$ and perform a coordinate transform $\hat {\lambda}=T \bar {\lambda}$. The above system is equivalent to $\dot{\hat {\lambda}}_1=0$ and $$\begin{aligned}
\label{sys:composite-osg}
\dot{\bar r}=-{\bm h}(r,\,r^*)-LR \hat {\lambda}_2+{\bm h}(r,\,y),\quad \dot{\hat {\lambda}}_2=R^{{\mathrm{T}}}L\bar r
\end{aligned}$$ Furthermore, the unperturbed system is globally exponential stable at the origin under Assumptions \[ass:convexity-strong\] and \[ass:graph\] by Lemma \[lem:generator\].
To apply the system composition arguments, one has to investigate the robustness of system . For this purpose, we let $\hat r\triangleq\mbox{col}(\bar r,\,\hat {\lambda}_2)$ and apply the converse Lyapunov theorem (Theorem 4.15 in [@khalil2002nonlinear]) to the unperturbed subsystem, that is, there is a continuously differentiable Lyapunov function $\bar V(\cdot)$ such that $$\begin{gathered}
c_3||\hat r||^2\leq \bar V(\hat r)\leq c_4 ||\hat r||^2 \\
\frac{\partial \bar V}{\partial \bar r}[-{\bm h}(r,\,r^*)-LR\hat {\lambda}_2]+\frac{\partial\bar V}{\partial \bar {\lambda}_2}R^{{\mathrm{T}}}L\bar r \leq - c_5 ||\hat r||^2 \\
||\frac{\partial \bar V}{\partial \hat r}||\leq c_6||\hat r||
\end{gathered}$$ for some positive constants $c_3,\,\dots,\, c_6$.
Along the trajectory of perturbed system , one can obtain: $$\begin{aligned}
\label{eq:thm1-eq2}
\begin{split}
\dot{\bar V}= &\frac{\partial \bar V}{\partial \bar r}[-{\bm h}(r,\,r^*)-LR\hat {\lambda}_2]+\frac{\partial\bar V}{\partial \bar {\lambda}_2}R^{{\mathrm{T}}}L\bar r + \frac{\partial \bar V}{\partial \hat r} {\bm h}(r,\,y)\\
&\leq - c_5||\hat r||^2+ c_6\bar l||\hat r||||\hat x||
\end{split}
\end{aligned}$$ where we use the Lipschitzness of ${\bm h}(r,\,y)$ in $r-y$ and thus in $\hat x$.
[*Step 3*]{}: consider the stability of the composite system composed by the first two subsystems of and system . Let $V=\hat V+ c \bar V$ with $c>0$ to be specified later. By using equalities and , the derivative of $V$ with respect to $t$ along the whole composite system satisfies $$\begin{aligned}
\dot{V}\leq &-\frac{c_1}{{\varepsilon}}||\hat x||^2+c_2||\dot{r}||^2- c_5||\hat r||^2+ c_6\bar l||\hat r||||\hat x||
\end{aligned}$$ Note that $-\nabla \tilde f(r)- L {\lambda}$ and ${\bm h}(r,\,y)$ are both globally Lipschitz in their arguments $\hat r$ and $\hat x$ under Assumption \[ass:convexity-strong\]. From , one can determine a known constant $c_7>0$ satisfying that $||\dot{r}||^2=||\dot{\bar r}||^2 \leq c_7(||\hat r||^2+||\hat x||^2)$
By Young’s inequality, we have $$\begin{aligned}
\dot{V}\leq &-\frac{c_1}{{\varepsilon}}||\hat x||^2+c_2c_7(||\hat {r}||^2+||\hat x^2||)- cc_5||\hat r||^2 + c^2||\hat x||^2+ c_6^2\bar l^2||\hat r||^2\\
&\leq -(\frac{c_1}{{\varepsilon}}-c_2c_7-c^2)||\hat x||^2-(cc_5-c_2c_7-c_6^2\bar l^2)||\hat r||^2
\end{aligned}$$ Letting $c>\frac{c_2c_7+c_6^2\bar l^2+1}{c_5}$ and ${\varepsilon}<\frac{c_1}{c_2c_7+c^2+1}$ gives that $$\begin{aligned}
\dot{V}\leq & -||\hat x||^2-||\hat{r}||^2
\end{aligned}$$ According to the LaSalle-Yoshizawa theorem (Theorem 2.1 in [@krstic1995nonlinear]), we have that all trajectories of the closed-loop system composed of and are bounded over the time interval $[0,\,+\infty)$ and satisfy that $\lim_{t\to\infty} ||\hat x_i(t)||+||\hat r(t)||=0$. As immediate results, one can conclude the boundedness of $x(t)$, $r(t)$, ${\lambda}(t)$ and $\hat \theta(t)$. Moreover, we can obtain that $\lim_{t\to\infty} \hat x_{1,i}(t)=0$ and $\lim_{t\to\infty} r_i(t)=y^*$, which implies that $$||y_i(t)-y^*||\leq||x_{1,i}-r_i(t)||+||r_i(t)-y^*||\to 0$$ as $t\to +\infty$. The proof is thus complete.
------------------------------------------------------------------------
Note that we consider uncertain high-order agents in the form of , which includes integrators as its special cases. Thus, the theorems can be taken as adaptive extensions of existing results with exact known dynamics[@zhang2017distributed; @rezaee2015average; @tang2017kyb; @tang2017cyb]. Moreover, many actuating disturbances can be represented in the form of , thus we provide different methods to achieve disturbance rejection from the internal model-based approach used in existing works [@tang2015distributed; @wang2016cyber].
Particularly, when the analytical form of $f_i(\cdot)$ or $\nabla f_i(\cdot)$ is known to us, the optimal signal generator can be implemented independently. Following a similar proof, we can choose the gain parameter ${\varepsilon}$ as any positive constant to solve our problem.
\[thm:main:offline\] Suppose Assumptions \[ass:system\]–\[ass:graph\] hold. Then, the distributed optimization problem determined by and can be solved by the following control $$\begin{aligned}
\label{ctr:offline}
\begin{split}
u_i&=-\hat \theta_i^{{\mathrm{T}}}{\bm p}_{i}(x_{i},\,t)+\frac{1}{{\varepsilon}^{n_i}}[k_{1i}(x_{1,i}-r_i)+\sum\nolimits_{j=2}^{n_i}{\varepsilon}^{j-1}k_{ji}x_{j,\,i}]\\
\dot{\hat \theta}_i&={\bm p}_{i}(x_{i},\,t) b_{2i}^{{\mathrm{T}}}P_i \hat x_i\\
\dot{r}_i&=-\nabla f_i(r_i)-\sum\nolimits_{j=1}^N a_{ij}({\lambda}_i-{\lambda}_j)\\
\dot{{\lambda}}_i&=\sum\nolimits_{j=1}^N a_{ij}(r_i-r_j)
\end{split}
\end{aligned}$$ where the constant $k_{ji}$ is chosen as above and ${\varepsilon}>0$ is arbitrary for $i=1,\,\dots,\,N,\,j=1,\,\dots,\,n_i$.
In some circumstances, we may further let $\phi_i(x_i,\,\hat \theta_i,\,r_i,\,t)=\Lambda_i{\bm p}_{i}(x_{i},\,t) b_{2i}^{{\mathrm{T}}}P_i \hat x_i$ with a positive definite matrix $\Lambda_{i}$. This matrix is called an adaption gain in literature[@ioannou1995robust]. It can be used to achieve a fast adaption and then improve the transient performance of our controllers to solve the distributed optimization problem.
Without further information of the unknown dynamics, the two controllers may fail in practical applications if there are external disturbances or noises in measurements of $x_i$, although it is theoretically proved to achieve the optimization goal as $t$ goes to infinity. To tackle this problem, we can employ a $\sigma$-modification [@ioannou1995robust] for $\hat \theta_i$ with sacrificing some accuracy in control performance as follows: $$\begin{aligned}
\label{ctr:sigma}
\begin{split}
u_i&=-\hat \theta_i^{{\mathrm{T}}}{\bm p}_{i}(x_{i},\,t)+\frac{1}{{\varepsilon}^{n_i}}[k_{1i}(x_{1,i}-r_i)+\sum\nolimits_{j=2}^{n_i}{\varepsilon}^{j-1}k_{ji}x_{j,\,i}]\\
\dot{\hat \theta}_i&=-\sigma_{\theta_i}\hat\theta_i+{\bm p}_{i}(x_{i},\,t) b_{2i}^{{\mathrm{T}}}P_i \hat x_i\\
\dot{r}_i&=-\nabla f_i(y_i)-\sum\nolimits_{j=1}^N a_{ij}({\lambda}_i-{\lambda}_j)\\
\dot{{\lambda}}_i&=\sum\nolimits_{j=1}^N a_{ij}(r_i-r_j)
\end{split}
\end{aligned}$$ where $\sigma_{\theta_i}>0$ is a tunable parameter such that $\lim_{t\to+\infty}||y_i(t)-y^*||$ can be smaller than any desired positive constant.
Parameter Convergence
---------------------
It has been shown that parameter convergence is essential in achieving robustness of the proposed adaptive controllers[@ioannou1995robust; @krstic1995nonlinear; @mazenc2009uniform]. From the proof of Theorem \[thm:main:online\], one can merely conclude that the estimator $\hat \theta_i$ converges to some constant. In this subsection, we assert conditions under which $\hat \theta_i(t)$ will converge to its true value $\theta_i$ as $t$ tends to infinity.
For this purpose, we further assume the basis function ${\bm p}_i(x_i,\,t)$ satisfying the following condition.
\[ass:PE\] For any $i=1,\,\dots,\,N$, along the trajectory of the closed-loop system composed of and or , there exist positive constants $m$,$t_0$,$T_0$ such that the function ${{\bm p}}_i(x_i(t),\,t)$ is uniformly bounded and the following inequality is satisfied: $$\begin{aligned}
\label{eq:PE}
\frac{1}{T_0}\int_{t}^{t+T_0}{\bm p}_i(x_i(\tau),\,\tau){\bm p}^{{\mathrm{T}}}(x_i(\tau),\,\tau){\rm d}\tau \geq mI_{n_{\theta_i}},\quad \forall t\geq t_0
\end{aligned}$$
Note that $x_i(t)$ is ultimately bounded by Theorem \[thm:main:online\], the boundedness of ${\bm p}_i(x_i(t),\,t)$ is not too strict. The inequality is often known as the persistence of excitation (PE) condition, which has widely used in adaptive control literature [@krstic1995nonlinear; @hu2014adaptive].
\[thm:parameter\] Suppose Assumptions \[ass:system\]–\[ass:PE\] hold. Then, along the trajectory of system under the controllers proposed in Theorems \[thm:main:online\] and \[thm:main:offline\] , it holds that $\lim_{t\to+\infty}\hat \theta_i(t)=\theta_i$ for $i=1,\,\dots,\,N$.
[**Proof.** ]{}To show this theorem, we first claim that $\lim_{t\to+\infty}\bar \theta_i^{{\mathrm{T}}}(t){\bm p}_i(x_i(t),\,t)=0$. By the proof of Theorem \[thm:main:online\], we have $\hat x_i(\infty)=\int_{0}^{+\infty}\dot{\hat x}_i(\tau){\rm d}\tau=0$. From the uniform boundedness of associated variables and Assumption \[ass:PE\], it follows that $\ddot{\hat x}_i(t)$ is also bounded. Using Barbalat’s lemma (Lemma 8.2 in [@khalil2002nonlinear]) to $\dot{\hat x}_i(t)$ implies that $\lim_{t\to+\infty}\dot{\hat x}_i(t)=0$, which confirms this claim.
Next, noting $\dot{\bar \theta}_i=\dot{\hat \theta}_i={\bm p}_{i}(x_{i},\,t) b_{2i}^{{\mathrm{T}}}P_i \hat x_i$ gives $\lim_{t\to+\infty}\dot{\bar \theta}_i(t)=0$ by Assumption \[ass:system\]. According to Lemma 1 in [@ortega1993asymptotic] or its proof, the two facts $\lim_{t\to+\infty}\dot{\bar \theta}_i(t)=0$ and $\lim_{t\to+\infty}\bar \theta_i^{{\mathrm{T}}}(t){\bm p}_i(x_i(t),\,t)=0$ provide us that $\lim_{t\to+\infty}\bar \theta_i(t)=0$ under Assumption \[ass:PE\]. The proof is thus complete.
------------------------------------------------------------------------
Since the unknown dynamics is linearly parameterized by Assumption \[ass:system\], this theorem can be further modified and applied to any number of components in ${\bm p}_i(x_i,\,t)$ satisfying such a PE condition, and then used to address the parameter convergence problem in a more precise way. Specially, when the basis function is time-invariant, the $j$-th component ${\bm p}_{j,i}(x_i)$ of ${\bm p}_i(x_i)$ is persistently excited if $\lim_{x_i\to \mbox{col}(y^*,\,0,\,\dots,\,0)}{\bm p}_{j,i}(x_i)\neq 0$ and guarantees the convergence of $\hat \theta_{j,i}(t)$ to $\theta_{j,i}$ as $t$ goes to infinity.
Simulations {#sec:simu}
===========
In this section, we present a numerical example to illustrate our problem and the effectiveness of our designs.
Consider a multi-agent system including four controlled Van der Pol systems as follows. $$\begin{aligned}
\dot{x}_{1,i}&=x_{2, i}\\
\dot{x}_{2,i}&=\Xi_i+u_i\\
y_i&=x_{1,i},\quad i=1,\,2,\,3,\,4
\end{aligned}$$ where $\Xi_i\triangleq -a_i x_{1,i}+b_i (1-x_{1,i}^2)x_{2,i}$ with $a_i,\,b_i>0$ but unknown. The trajectories of the unforced system with different initial conditions when $a_i=b_i=1$ are depicted in Figure \[fig:phase\].
![Phase portraits of the unforced Van der Pol system.[]{data-label="fig:phase"}](phase.eps){width="80.00000%"}
(node1) [1]{}; (node2)\[right of=node1\][2]{}; (node3)\[right of=node2\][3]{}; (node4)\[right of=node3\][4]{}; (node1) edge (node2) (node2) edge (node3) (node3) edge (node4) (node1) edge \[bend left=60\] (node3) ;
To make it more interesting, we further assume that agent $i$ is subject to an actuating disturbance $d_i(t)$ described by $$d_i(t)=D_i v_i(t),\quad \dot{v}_i=S_iv_i$$ where $D_i=[1~0]$ and $S_i=[0~1;\,-1~0]$.
The information sharing graph of this multi-agent system is depicted in Figure \[fig:graph\] with unity edge weights. The local cost functions are as follows. $${f_1}(y) = (y-8)^2,\, {f_2}(y) = \frac{y^2}{{20\sqrt {y^2 + 1} }} + y^2,\, {f_3}(y) = \frac{y^2}{80\ln {({y^2} + 2} )} + (y -5)^2,\, {f_4}(y) = \ln \left( {{e^{ - 0.05{y}}} + {e^{0.05{y}}}} \right) + y^2$$
Denote $\Delta_i(x_i,\,t)=\Xi_i+d_i(t)$. The agents are of the form . Note that $d_i(t)=A_{1i}\sin(t)+A_{2i}\cos(t)$ for some constants $A_{1i}$, $A_{2i}$ depending upon $v_i(0)$. By letting $\theta_i=\mbox{col}(\theta_{1,i},\,\dots,\,\theta_{4,i})=\mbox{col}(a_i,\,b_i,\,A_{1i},\,A_{2i})$ and ${\bm p}_i(x_i,\,t)=\mbox{col}(-x_{1,i},\,(1-x_{1,i}^2)x_{2,i},\,\sin(t),\,\cos(t))$, one can find that Assumption \[ass:system\] is fulfilled. Moreover, both Assumptions \[ass:convexity-strong\] and \[ass:graph\] are also verified. Additionally, the optimal solution is $y^*\approx 3.24$ by numerically minimizing the global cost function $f(y)=\sum\nolimits_{i=1}^4 f_i(y)$. According to Theorem \[thm:main:online\], the distributed optimization problem for these agents can be solved by a controller of the form .
![Profiles of $y_i(t)$ under the controller .[]{data-label="fig:simu"}](opt-adaptive-online.eps){width="90.00000%"}
For simulations, we choose the parameters as $a_i=b_i=1$, $v_i(0)=\mbox{col}(1,\,0)$, $k_{1i}=-4$, $k_{2i}=-4$, $\Lambda_{i}=10 I_4$ and ${\varepsilon}=0.2$. The profiles of agents’ outputs are shown in Figure \[fig:simu\]. Satisfactory performance is observed. For parameter convergence, we have ${\bm p}_{1,i}({\bm x}_i,\,t)=-x_{1,i}$, which is time-invariant and satisfies that $\lim_{x_i\to \mbox{col}(y^*,\,0,\,\dots,\,0)}{\bm p}_{1,i}({\bm x}_i)=-y^*\neq 0$. By some calculations, one can also determine that $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}[{\bm p}_{3,i}(\tau)~~{\bm p}_{4,i}(\tau)]^{{\mathrm{T}}}[{\bm p}_{3,i}(\tau)~~{\bm p}_{4,i}(\tau)]{\rm d}\tau=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}[\sin^2(\tau)~\sin(\tau)\cos(\tau); \sin(\tau)\cos(\tau)~\cos^2(\tau)]{\rm d}\tau=[\pi/2~~0;\,0~~{\pi/2}].$$ Using Theorem \[thm:parameter\], we can conclude that the estimators $\hat \theta_{1,i}$,$\hat \theta_{3,i}$, $\hat \theta_{4,i}$ will converge to their true values, while $\hat \theta_{2,i}$ may fail. This conclusion is confirmed by Figures \[fig:parameter-1\] and \[fig:parameter-2\].
Conclusions {#sec:con}
===========
A distributed optimization problem was formulated for a class of high-order nonlinear systems with unknown dynamics. By using an embedded control scheme, we proposed distributed adaptive controls to solve this problem under standard assumptions and parameter convergence was also addressed. Output feedback control with directed information sharing graphs will be our future work.
[^1]: This work was supported by National Natural Science Foundation of China under Grant 61503033. Y. Tang is with the School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]).
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---
abstract: 'Presentism is, roughly, the metaphysical doctrine that maintains that whatever exists, exists in the present. The compatibility of presentism with the theories of special and general relativity was much debated in recent years. It has been argued that at least some versions of presentism are consistent with time-orientable models of general relativity. In this paper we confront the thesis of presentism with relativistic physics, in the strong gravitational limit where black holes are formed. We conclude that the presentist position is at odds with the existence of black holes and other compact objects in the universe. A revision of the thesis is necessary, if it is intended to be consistent with the current scientific view of the universe.'
author:
- 'Gustavo E. Romero'
- Daniela Pérez
date: 'Received: date / Accepted: date'
title: Presentism meets black holes
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction
============
Presentism is a metaphysical thesis about what there is. It can be expressed as (e.g. Crisp 2003):
> [*Presentism*]{}. It is always the case that, for every $x$, $x$ is present.
The quantification is unrestricted, it ranges over all existents. In order to make this definition meaningful, the presentist must provide a specification of the term ‘present’. Crisp, in the cited paper, offers the following definition:
> [*Present*]{}. The mereological sum of all objects with null temporal distance.
The notion of temporal distance is defined loosely, but in such a way that it accords with common sense and the physical time interval between two events. From these definitions it follows that the present is a thing, not a concept. The present is the ontological aggregation of all present things. Hence, to say that ‘$x$ is present’, actually means “$x$ is part of the present".
The opposite thesis of presentism is eternalism, also called four-dimensionalism. Eternalists believe in the existence of past and future objects. The temporal distance between these objects is non-zero. The name four-dimensionalism comes form the fact that in the eternalist view, objects are extended through time, and then they have a 4-dimensional volume, with 3 spatial dimensions and 1 time dimension. There are different versions of eternalism. The reader is referred to Rea (2003) and references therein for a discussion of eternalism.
Several philosophers have defended presentism from various critiques, notably Chisholm (1990), Bigelow (1996), Zimmerman (1996, 2011), Merricks (1999), Hinchliff (2000), Crisp (2007), and Markosian (2004)[^1]. Some of these authors have considered criticisms arisen from the alleged incompatibility of presentism with relativity theory. In what follows we review the main objections posed to presentism in the framework of both relativities, special and general, and then we move on to consider a new type of argument based on the existence of compact objects in the universe. We show that the presentist cannot accept the standard physical understanding of these objects, without introducing changes in his ontological views or rejecting current astrophysics[^2].
Presentism and special relativity
=================================
Special relativity is the theory of moving bodies formulated by Albert Einstein in 1905 (Einstein 1905). It postulates the Lorentz-invariance of all physical law statements that hold in a special type of reference systems, called [*inertial frames*]{}. Hence the ‘restricted’ or ‘special’ character of the theory. The equations of Maxwell electrodynamics are Lorentz-invariant, but those of classical mechanics are not. When classical mechanics is revised to accommodate invariance under Lorentz transformations between inertial reference frames, several modifications appear. The most notorious is the impossibility of defining an absolute simultaneity relation among events. The simultaneity relation results to be frame-dependent. Then, some events can be future events in some reference system, and present or past in others. Since what there is cannot depend on the reference frame adopted for the description of nature, it is concluded that past, present, and future events exist. Then, presentism is false.
This argument has been formulated with different levels of sophistication by philosophers such as Smart (1963), Rietdijk (1966), and Putnam (1967). In particular, both Rietdijk and Putnam argued independently that special relativity implies determinism. This position was revisited by Maxwell (1985) in the eighties. He maintained that probabilism[^3] and special relativity were incompatible. Strong objections to Rietdijk, Putnam, and Maxwell can be found in Stein (1968, 1991)[^4]. In his 1991 paper Stein showed that “the openness of the future relative to the past for any region of space-time is, given special relativity, logically incompatible with the objective global temporal structure” (Harrington 2009). In later years Harrington (2008, 2009) defended a theory of local temporality, also called point-present theory, which assuming special relativity allows for the possibility of an open future, i.e not fully determined (see also Ellis (2006) and Ellis and Rothman 2010).
The argument for eternalism from special relativity seems particularly strong in the reformulation of this theory proposed by Hermann Minkowski. Minkowski (1907, 1909) introduced the concept of space-time: the mereological sum of all events. Space-time can be represented by a 4-dimensional manifold endowed with a metric that allows to compute distances between events. These are spatio-temporal distances or [*intervals*]{}. The differential (arbitrarily small) interval for Minkowski space-time, which is invariant under Lorentz transformations, is: $$ds^{2}=\eta_{\mu\nu}dx^{\mu} dx^{\nu}=(dx^{0})^{2}-(dx^{1})^{2}-(dx^{3})^{2}-(dx^{3})^{2},
\label{eq:eta}$$ where the Minkowskian metric tensor $\eta_{\mu\nu}$ is pseudo-Euclidean, with rank 2 and trace $-2$. The coordinates with the same sign are called [*spatial*]{} (the usual convention in notation is $x^{1}=x$, $x^{2}=y$, and $x^{3}=z$) and the coordinate $x^{0}=ct$ is called [*time coordinate*]{}. The constant $c$ is introduced to make the physical units uniform, and can be shown to be equal to the speed of light in empty space. Minkowski’s interpretation allows to separate space-time, at each point (every point represents an event), in three regions according to $ds^{2}<0$ (space-like region), $ds^{2}=0$ (light-like or null region), and $ds^{2}>0$ (time-like region). Particles that go through the origin can only reach time-like regions. The null surface $ds^{2}=0$ can be inhabited only by particles moving at the speed of light, like photons. Points in the space-like region cannot be reached by material objects from the origin of the [*light cone*]{} that can be formed at any space-time point (see Fig. \[Fig1\]).
![Light cone. Causal future, past, and the present plane of the central event are shown.[]{data-label="Fig1"}](Figura1.jpg){width="30pc"}
The introduction of the metric allows to define the future and the past [of a given event]{}. Once this is done, all time-like events can be classified by the relation ‘earlier than’ or ‘later than’. The selection of a ‘present’ event - or ‘now’ - is entirely conventional. To be present is not an intrinsic property of any event. Rather, it is a secondary, relational property that requires interaction with a conscious being. The extinction of the dinosaurs will always be earlier than the beginning of the Second World War. But the latter was present only to some human beings in some physical state. The present is a property like a scent or a color. It emerges from the interaction of self-conscious individuals with changing things and has not existence independently of them (for more about this point of view, see Grünbaum 1973, Chapter X).
This sounds convincing, but there is a strong ontological assumption: the system of all events is assumed. This provides a simplification that pleases the physicist, but can terrify the philosopher. Presentists, surely, will plainly reject this assumption as outrageous. But do they have an alternative interpretation of special relativity with the same explanatory and predictive power as that of Minkowski’s?
Yes, they have. The alternative was already suggested by Poincaré (1902) in the form of the conventionalist thesis. Poincaré remarked that it is always possible to save an interpretation based on an Euclidean physical geometry if additional fields are introduced in the theory to adjust all physical quantities (lengths, masses, speeds) to the adopted geometry. In the case of special relativity, this yields Lorentz’s theory of moving bodies. Physicist will always prefer the simplest theory, but the presentist can argue that from an ontological point of view, Minkowski’s interpretation is far more costly than the presentist one: a 3-dimensional Euclidean and Lorentz-invariant theory. An Euclidean space-time is consistent with presentism since the interval can be written as:
$$ds^{2}=(cdt)^{2}+dx^{2}+dy^{2}+dz^{2}.
\label{eq:eucl}$$
Hence, since in a proper system $ds=cd\tau$, and using the invariance of the interval, we get:
$$d\tau^{2}=dt^{2}+\frac{1}{c^{2}}d\vec{x}^{2}=dt^{2}\left(1+\vec{\beta}^{2}\right),
\label{eq:eucl2}$$
where $$\vec{\beta}=\frac{\vec{v}}{c}.$$ From Eq. (\[eq:eucl2\]) follows that $d\tau=0$ iff $dt=0$. All events have zero temporal distance in the 3-dimensional Euclidean plane. A full discussion of such an alternative is presented by Craig (2001).
An Euclidean geometry can be preserved (to a high price, i.e. introducing distorting fields) in special relativity since both Euclidean and Minkowskian geometries have zero intrinsic curvature. The curvature of a given manifold is given by the Riemann tensor $R^{\alpha\beta\gamma\delta}$, formed with second derivatives of the metric. This tensor is identically zero for any space-time with a constant metric. Since the intrinsic curvature is the same for both Euclidean and Minkowskian geometries, it is always possible to find a [*global*]{} transformation between them. In the case of more general space-times, represented by manifolds with intrinsic curvature, such transformations can only be local. This leads us to general relativity.
Presentism and general relativity
=================================
A basic characteristic of Minkowski space-time is that it is ‘flat’: all light cones point in the same direction, i.e. the local direction of the future does not depend on the coefficients of the metric since these are constants. More general space-times, however, are possible. If we want to describe gravity in the framework of space-time, we introduce a pseudo-Riemannian manifold, whose metric can be flexible, i.e. a function of the material properties (mass-energy and momentum) of the physical systems that produce the events represented by space-time points.
The key to relate space-time to gravitation is the [*equivalence principle*]{} introduced by Einstein (1907): at every point $P$ of the manifold that represents space-time, there is a flat tangent surface.
In order to introduce gravitation in a general space-time we define a metric tensor $g_{\mu\nu}$, such that its components can be related to those of a locally Minkowskian space-time defined by $ds^{2}=\eta_{\alpha\beta}d\xi^{\alpha} d\xi^{\beta}$ through a general transformation:
$$ds^{2}=\eta_{\alpha\beta}\frac{\partial \xi^{\alpha}}{\partial x^{\mu}}\frac{\partial \xi^{\beta}}{\partial x^{\nu}}dx^{\mu}dx^{\nu}=g_{\mu\nu}dx^{\mu}dx^{\nu}.$$
In the absence of gravity we can always find a global coordinate system ($\xi^{\alpha}$) for which the metric takes the form given by Eq. (\[eq:eta\]) everywhere. With gravity, conversely, such a coordinate system can represent space-time only in an infinitesimal neighborhood of a given point. The curvature of space-time means that it is not possible to find coordinates in which $g_{\mu\nu}=\eta_{\mu\nu}$ at [*all points on the manifold*]{}. It is always possible, however, to represent the event (point) $P$ in a system such that $g_{\mu\nu}(P)=\eta_{\mu\nu}$ and $(\partial g_{\mu\nu}/\partial x^{\sigma})_{P}=0$.
The key issue to determine the geometric structure of space-time, and hence to specify the effects of gravity, is to find the law that fixes the metric once the source of the gravitational field is given. The source of the gravitational field is represented by the energy-momentum tensor $T_{\mu\nu}$ that describes the physical properties of material things. This was Einstein’s fundamental intuition: the curvature of space-time at any event is related to the energy-momentum content at that event.
The Einstein field equations can be written in the simple form (Einstein 1915):
$$G_{\mu\nu} = -\frac{8\pi G}{c^4}\, T_{\mu\nu},
\label{einstein}$$
where $G_{\mu\nu}$ is the so-called Einstein’s tensor. It contains all the geometric information on space-time. The constants $G$ and $c$ are the gravitational constant and the speed of light in vacuum, respectively.
Einstein’s field equations are a set of ten non-linear partial differential equations for the metric coefficients. In Newtonian gravity, otherwise, there is only one gravitational field equation. General relativity involves numerous non-linear differential equations. In this fact lays its complexity, and its richness.
A crucial aspect of general relativity is that 4-dimensional space-time with non-zero curvature is not dispensable anymore. Is this implying that presentism should be abandoned? Not necessarily.
Thomas Crisp (2007) has proposed a “presentist-friendly” model of general relativity. He suggests that the world is represented by a 3-dimensional space-like hypersurface that evolves in time. This interpretation requires to introduce a preferred foliation of space-time, and to consider the $3+1$ usual decomposition for the dynamics of space-time in such a way that ‘the present’ is identified with the evolving hypersurface (see Fig. \[Fig2\]).
![A ‘presentist-friendly’ space-time: Evolving 3-dimensional space-like surfaces in a space-time with a preferred time-direction.[]{data-label="Fig2"}](Figura2.jpg){width="25pc"}
In order to formulate such a model of space-time, some global constraints must be imposed: there should be a possible foliation into Cauchy hypersurfaces in order to allow for global time-like continuous vector fields that can be used to introduce a “global time coordinate”. This is the case, for instance, of the Friedmann-Robertson-Walker-Lemaître metric. General conditions for such a metric requires the absence of Cauchy horizons and the fulfillment of the so-called [*energy conditions*]{} (Hawking and Ellis 1973).
Although a case can be made for general inhomogeneous metrics and massive violations of the energy conditions on the basis of recent cosmological data (e.g. Plebański and Krasiński 2006), in what follows we focus on local aspects that should be accommodated in any cosmological model compatible with current astrophysics.
Black holes and the present {#BH}
===========================
We shall now provide a general definition of a black hole, independently of the coordinate system adopted in the description of space-time, and even of the exact form of the field equations. A space-time is represented by an order pair, formed by a manifold $M$ and a metric $g_{\mu\nu}$, i.e. $(M, \; g_{\mu\nu})$.
Let us consider a space-time where all null geodesics that start in a region $\cal{J}^{-}$ end at $\cal{J}^{+}$. Then, such a space-time, $(M,\; g_{\mu\nu})$, is said to contain a [*black hole*]{} if $M$ [*is not*]{} contained in the causal past[^5] $J^{-}({\cal{J^+}})$. In other words, there is a region from where no null geodesic can reach the [*asymptotic flat*]{}[^6] future space-time, or, equivalently, there is a region of $M$ that is causally disconnected from the global future. The [*black hole region*]{}, $BH$, of such space-time is $BH=[M-J^{-}({\cal{J^+}})]$, and the boundary of $BH$ in $M$, $H=J^{-}({\cal{J^+}}) \bigcap M$, is the [*event horizon*]{}.
A black hole can be conceived as a space-time [*region*]{}, i.e. what characterizes the black hole is the metric and, consequently, space-time curvature. What is peculiar of this space-time region is that it is causally disconnected from the rest of the space-time: no events inside this region can make any influence on events outside. Hence the name of the boundary, event horizon: events inside the black hole are separated from events in the global external future of space-time. The events in the black hole, nonetheless, as all events, are causally determined by past events. A black hole does not represent a breakdown of classical causality.
The simplest type of black hole is described by the Schwarzschild metric; this metric characterizes the geometry of space-time outside a spherically symmetric matter distribution. The Schwarzschild metric for a static mass $M$ can be written in spherical coordinates $(t,r,\theta,\phi)$ as[^7]: $$ds^{2}= \left(1-\frac{2GM}{rc^{2}}\right) c^{2}dt^2- \left(1-\frac{2GM}{rc^{2}}\right)^{-1} dr^{2} -r^{2} (d\theta^{2}+\sin^{2}\theta d\phi^{2}).\label{Schw}$$
The radius $$r_{\rm Schw}=\frac{2GM}{c^{2}}, \label{rS}$$ is known as the [*Schwarzschild radius*]{}. It corresponds to the event horizon if all the matter that generates the curvature is located at $r<r_{\rm Schw}$.
The light cones can be calculated from the metric (\[Schw\]) imposing the null condition $ds^{2}=0$. Then: $$\frac{dr}{dt}=\pm\left(1-\frac{2GM}{r}\right), \label{cones-Schw}$$ where we made $c=1$. Notice that when $r\rightarrow \infty$, $dr/dt \rightarrow \pm 1$, as in Minkowski space-time. When $r\rightarrow 2GM$, $dr/dt \rightarrow 0$, and light moves along the surface $r=2GM$. The horizon is therefore a [*null surface*]{}. For $r<2GM$, the sign of the derivative is inverted. The inward region of $r=2GM$ is time-like for any physical system that has crossed the boundary surface. In Fig. \[Fig3\] we show the behavior of light cones in Schwarzschild coordinates. Similar figures are given in classical textbooks such as Misner et al. (1973, p. 848) and Carroll (2004, p. 219). As we approach to the horizon from the flat space-time region, the light cones become thinner and thinner indicating the restriction to the possible trajectories imposed by the increasing curvature. On the inner side of the horizon the local direction of time is ‘inverted’ in the sense that null or time-like trajectories have in their future the singularity at the center of the black hole.
![Space-time diagram in Schwarzschild coordinates showing the light cones close to and inside a black hole.[]{data-label="Fig3"}](Figura3.jpg){width="25pc"}
There is a very interesting consequence of all this: an observer on the horizon will have her present [*along*]{} the horizon. All events occurring on the horizon are simultaneous. The temporal distance from the observer at any point on the horizon to any event occurring on the horizon is zero (the observer is on a null surface $ds=0$ so the proper time interval is necessarily zero[^8]). If the black hole has existed during the whole history of the universe, all events on the horizon during such history (for example the emission of photons on the horizon by infalling matter) are [*present*]{} to any observer crossing the horizon. These events are certainly not all present to an observer outside the black hole. If the outer observer is a presentist, she surely will think that some of these events do not exist because they occurred or will occur either in the remote past or the remote future. But if we accept that what there is cannot depend on the reference frame adopted for the description of the events, it seems we have an argument against presentism here. Before going further into the ontological implications, let us clarify a few physical points.
What happens to an object when it crosses the event horizon? According to Eq. (\[Schw\]), there is a singularity at $r=r_{\rm Schw}$. However, the metric coefficients can be made regular by a change of coordinates. For instance we can consider Eddington-Finkelstein coordinates. Let us define a new radial coordinate $r_{*}$ such that radial null rays satisfy $d(c t\pm r_{*})=0$. Using Eq. (\[Schw\]) it can be shown that: $$r_{*}=r + \frac{2GM}{c^{2}} \log \left|\frac{r-2GM/c^{2}}{2GM/c^{2}}\right|.$$ Then, we introduce: $$v=ct+r_{*}.$$ The new coordinate $v$ can be used as a time coordinate replacing $t$ in Eq. (\[Schw\]). This yields: $$ds^{2}=\left(1-\frac{2GM}{rc^{2}}\right) (c^2dt^{2}-dr^{2}_{*})-r^{2} d\Omega^{2},$$ or $$ds^{2}=\left(1-\frac{2GM}{rc^{2}}\right) dv^{2}-2drdv -r^{2} d\Omega^{2}, \label{EF}$$ where $$d\Omega^{2}=d\theta^{2}+\sin^{2} \theta d\phi^{2}.$$
![Space-time diagram in Eddington-Finkelstein coordinates showing the light cones close to and inside a black hole. The event $u$ occurs at the event horizon, represented by the surface of the cylinder. The global time direction is in the direction of the axis of the cylinder.[]{data-label="Fig4"}](Figura4.jpg){width="15pc"}
Notice that in Eq. (\[EF\]) the metric is non-singular at $r=2GM/c^{2}$. The only real singularity is at $r=0$, since the Riemann tensor diverges there. In order to plot the space-time in a $(t,\; r)$-plane, we can introduce a new time coordinate $t_{*}=v-r$. From the metric (\[EF\]) or from Fig. \[Fig4\] we see that the line $r=r_{\rm Schw}$, $\theta=$constant, and $\phi=$ constant is a null ray, and hence, the surface at $r=r_{\rm Schw}$ is a null surface. This null surface is an event horizon because inside $r=r_{\rm Schw}$ all cones have $r=0$ in their future. In Figure \[Fig4\] we see how the light cones tilt in Eddington-Finkelstein coordinates. [*All*]{} light cones of events at $r=r_{\rm Schw}$ have their null surfaces coincident with the horizon. The horizon itself is a null surface common to all light cones associated with physical systems that are entering the black hole.
We remark, then, that the horizon 1) does not depend on the choice of the coordinate system adopted to describe the black hole, 2) it is an absolute null surface, in the sense that this property is intrinsic and not frame-dependent, and 3) is non-singular (or ‘well-behaved’, i.e. space-time is regular on the horizon).
More complex black holes can be considered. For instance, rotating (Kerr), charged (Reissner-Nordström), or both rotating and charged (Kerr-Newman) black holes. In all of them the outer event horizons are null surfaces, so the above considerations remain valid. Rotating black holes introduce additional problems to presentism, related to the existence of Cauchy horizons and closed timelike curves, but we do not discuss these issues here. Rather, we focus on the implications of the existence of null surfaces that can be crossed by massive particles or observers for presentism.
Ontological implications
========================
In a world described by special relativity, the only way to cross a null surface is by moving faster than the speed of light. As we have seen, this is not the case in a universe with black holes. We can then argue against presentism along the following lines.\
Argument $A1$:
- [$P1$: There are black holes in the universe.]{}
- [$P2$: Black holes are correctly described by general relativity.]{}
- [$P3$: Black holes have closed null surfaces (horizons).]{}
- [Therefore, there are closed null surfaces in the universe.]{}
Argument $A2$:
- [$P4$: All events on a closed null surface are simultaneous with any event on the same surface.]{}
- [$P4i$: All events on the closed null surface are simultaneous with the birth of the black hole.]{}
- [$P5$: Some distant events are simultaneous with the birth of the black hole, but not with other events related to the black hole.]{}
- [Therefore, there are events that are simultaneous in one reference frame, and not in another.]{}
Simultaneity is frame-dependent. Since what there exist cannot depend on the reference frame, we conclude that there are non-simultaneous events. Therefore, presentism is false.
Let us see which assumptions are open to criticism by the presentist.
An irreducible presentist might plainly reject $P1$. Although there is significant astronomical evidence supporting the existence of black holes (e.g. Casares 2006, Camenzind 2007, Paredes 2009, Romero and Vila 2013), the very elusive nature of these objects still leaves room for some speculations like gravastars, and other exotic compact objects. The price of rejecting $P1$, however, is very high: black holes are now a basic component of most mechanisms that explain extreme events in astrophysics, from quasars to the so-called gamma-ray bursts, from the formation of galaxies to the production of jets in binary systems. The presentist rejecting black holes should reformulate the bulk of contemporary high-energy astrophysics in terms of new mechanisms. In any case, $P1$ is susceptible of empirical validation through direct imagining of the super-massive black hole “shadow” in the center of our galaxy by sub-mm interferometric techniques in the next decade (e.g. Falcke et al. 2011). In the meanwhile, the cumulative case for the existence of black holes is overwhelming, and very few scientists would reject them on the basis of metaphysical considerations only.
The presentist might, instead, reject $P2$. After all, we [*know*]{} that general relativity fails at the Planck scale. Why should it provide a correct description of black holes? The reason is that the horizon of a black hole is quite far from the region where the theory fails (the singularity). The distance, in the case of a Schwarzschild black hole, is $r_{\rm Schw}$ (see Eq. \[rS\]). For a black hole of 10 solar masses, as the one suspected to form part of the binary system Cygnus X-1, this means $30$ km. And for the black hole in the center of the galaxy, the Schwarzschild radius is of about 12 million km. Any theory of gravitation must yield the same results as general relativity at such distances. So, even if general relativity is not the right theory for the classical gravitational field, the correct theory should predict the formation of black holes under the same conditions.
![Light cones aperture angles at different distances from the horizon of a Schwarzschild black hole. On the horizon the null surface is coincident with the hyperplane of present.[]{data-label="Fig5"}](Figura5.jpg){width="30pc"}
There is not much to do with $P4$, since it follows from the condition that defines the null surface: $ds=0$[^9]; similarly $P4i$ only specifies one of the events on the null surface. A presentist might refuse to identify ‘the present’ with a null surface. After all, in Minksowskian space-time or even in a globally time-orientable pseudo-Riemannian space-time the present is usually taken as the hyperplane perpendicular to the local time (see Figs. \[Fig1\] and \[Fig2\]). But in space-times with black holes, the horizon is not only a null surface, it is also a surface locally normal to the time direction. This can be appreciated in Figure \[Fig5\], where the angle $\theta$ is the angle between the null surface and the hyperplane of the present. In a Minkowskian space-time such an angle is 45$^\circ$ when the speed of light is measured in natural units ($c=1$). In such a space-time, certainly the plane of the present is not coincident with a null surface. However, close to the event horizon of a black hole, things change, as indicated by Eq. (\[cones-Schw\]). As we approach the horizon, $\theta$ goes to zero, i.e. the null surface matches the plane of the present. On the horizon, both surfaces are exactly coincident: $\theta\rightarrow 0$ when $r\rightarrow r_{\rm Schw}$. A presentist rejecting the identification of the present with a [*closed*]{} null surface on an event horizon should abandon what is perhaps her most cherished belief: the identification of ‘the present’ with hypersurfaces that are normal to a local time-like direction.
The result mentioned above is not a consequence of any particular choice of coordinates but, as mentioned in Sect. \[BH\], an intrinsic property of a black hole horizon. This statement can be easily proved. The symmetries of Schwarzschild space-time imply the existence of a preferred radial function, $r$, which serves as an affine parameter along both null directions. The gradient of this function, $r_{a}=\nabla_{a} r$ satisfies ($c=G=1$):
$$r^{a}r_{a}=\left(1-\frac{2M}{r}\right). \label{ra}$$
Thus, $r^{a}$ is space-like for $r>2M$, null for $r=2M$, and time-like for $r<2M$. The 3-surface given by $r=2M$ is the horizon $H$ of the black hole in Schwarzschild space-time. From Eq. (\[ra\]) it follows that $r^{a}r_{a}=0$ over $H$, and hence $H$ is a null surface[^10].
Premise $P5$, perhaps, looks more promising for a last line of presentist defense. It might be argued that events on the horizon are not simultaneous with any event in the external universe. They are, in a very precise sense, cut off from the universe, and hence cannot be simultaneous with any distant event. Let us work out a counterexample.
The so-called long gamma-ray bursts are thought to be the result of the implosion of a very massive and rapidly rotating star. The core of the star becomes a black hole, which accretes material from the remaining stellar crust. This produces a growth of the black hole mass and the ejection of matter from the magnetized central region in the form of relativistic jets (e.g. Woosley 1993). Approximately, one of these events occur in the universe per day. They are detected by satellites such as [*Swift*]{} (e.g. Piran and Fan 2007), with durations of a few tens of seconds. This is the time that takes for the black hole to swallow the collapsing star. Let us consider a gamma-ray burst of, say, 10 seconds. Before these 10 seconds, the black hole did not exist for a distant observer $O1$. Afterwards, there is a black hole in the universe that will last more than the life span of any human observer. Let us now consider an observer $O2$ collapsing with the star. At some instant she will cross the null surface of the horizon. This will occur within the 10 seconds that the collapse lasts for $O1$. But for $O2$ all photons that cross the horizon are simultaneous, including those that left $O1$ long after the 10 seconds of the event and crossed the horizon after traveling a long way. For instance, photons leaving the planet of $O1$ one million years after the gamma-ray burst, might cross the horizon, and then can interact with $O2$. So, the formation of the black hole is simultaneous with events in $O1$ and $O2$, but these very same events of $O2$ are simultaneous with events that are in the distant future of $O1$. The situation is illustrated in Fig. \[Fig6\].
![Foliation of a temporally-oriented space-time with a black hole. The time span from the birth of the black hole to the event $E_{1}$ is $\normalsize\Delta t$.[]{data-label="Fig6"}](Figura6.jpg){width="25pc"}
The reader used to work with Schwarzschild coordinates perhaps will object that $O2$ never reaches the horizon, since the approaching process takes an infinite time in a distant reference frame, like that of $O1$. This is, however, an effect of the choice of the coordinate system and the test-particle approximation (see, for instance, Hoyng 2006, p.116). If the process is represented in Eddington-Finkelstein coordinates, it takes a finite time for the whole star to disappear, as shown by the fact that the gamma-ray burst are quite short events. Accretion/ejection processes, well-documented in active galactic nuclei and microquasars (e.g. Mirabel et al. 1998) also show that the time taken to reach the horizon is finite in the asymptotically flat region of space-time.
Our conclusion is that presentism, as usually stated, provides a defective picture of the ontological substratum of the world.
Final remarks
=============
What kind of ontological view is compatible with black hole astrophysics? We suggest that one where what we call ‘present’ has a local rather than a global character. The detachment between local and global time occurring in a black hole results, as we have seen, in a checkmate for the presentist position. There is no way to re-identify the present to give it back a global meaning in a universe with black holes.
The intuitive ontology adopted by most practising astrophysicists is one where there are things, and these things change relative to each other. One can speculate that space-time is an emergent property of the system of all things (e.g. Bunge 1977, Perez-Bergliaffa et al. 1998). The exact formulation of such an ontological theory to encompass a relativistic view of the world, taking into account the peculiarities of non-local effects in quantum mechanics, is an open problem. A problem that cries for attention from both philosophers and scientists alike.
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[^1]: See also the recent review by Mozersky (2011).
[^2]: We note that some authors such as Savitt (2006), Dorato (2006), Dolev (2006), and more recently Norton (2013) have argued that the dispute between presentism and eternalism is not a genuine one. We are not concerned with this dispute, but with the consistency of presentism with general relativity.
[^3]: Probabilism is the thesis that the universe is such that, at any instant, there is only one past but many alternative futures (Maxwell 1985).
[^4]: The controversy between Putman and Stein is reviewed by Saunders (2002).
[^5]: The notions of causal past and future of a space-time region can be found, for instance, in Hawking and Ellis (1973) and Wald (1984).
[^6]: Asymptotic flatness is a property of the geometry of space-time which means that in appropriate coordinates, the limit of the metric at infinity approaches the metric of the flat (Minkowskian) space-time.
[^7]: These coordinates are usually referred as ‘Schwarzschild coordinates’.
[^8]: Notice that this can never occur in Minkowski space-time, since there only photons can exist on a null surface. The black hole horizon, a null surface, can be crossed, conversely, by massive particles. The fact that the event horizon is a null surface is demonstrated in most textbook on relativity, see, e.g. Hartle (2003, p. 273) and d’Inverno (2002, p. 215).
[^9]: $ds=cd\tau=0 \rightarrow d\tau=0$, where $d\tau$ is the proper temporal separation.
[^10]: An interesting case is Schwarzschild space-time in the so-called Painlevé-Gullstrand coordinates. In these coordinates the interval reads: $$ds^{2}=dT^{2}-\left(dr + \sqrt{\frac{2M}{r}} dT\right)^{2} - r^{2}d\Omega^{2},\label{PG}$$ with $$T=t + 4M \left(\sqrt{\frac{2M}{r}} + \frac{1}{2} \ln \left| \frac{\sqrt{\frac{2M}{r}}-1}{\sqrt{\frac{2M}{r}}+1} \right|\right).$$
If a presentist makes the choice of identifying the present with the surfaces of $T=$constant, from Eq. (\[PG\]): $ds^{2}= - dr^{2} - r^{2}d\Omega^{2}$. Notice that for $r=2M$ this is the event horizon, which in turn, is a null surface. Hence, with such a choice, the presentist is considering that the event horizon is the hypersurface of the present, for all values of $T$. This choice of coordinates makes particularly clear that the usual presentist approach to define the present in general relativity self-defeats her position if space-time allows for black holes.
|
---
abstract: 'In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this generalizes a result of Bump–Nakasuji.'
address:
- 'Department of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A.'
- 'Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY 12222, U.S.A.'
- 'School of Mathematical Science, Zhejiang University, Hangzhou 310027, Zhejiang, P.R. China '
- 'Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1'
- 'Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1'
author:
- 'Kyu-Hwan Lee$^{\star}$'
- 'Cristian Lenart$^{\dagger}$'
- |
Dongwen Liu,\
with Appendix by Dinakar Muthiah and Anna Puskás
title: Whittaker functions and Demazure characters
---
[^1]
[^2]
Introduction {#sec:intro}
============
The Casselman–Shalika formula describes a spherical Whittaker function using the root system and the character of an irreducible representation of the dual group. The formula not only plays a fundamental role in the theory of $p$-adic groups and automorphic forms, but also connects many different constructions in mathematics, such as Schubert varieties, crystal bases and Macdonald polynomials. For example, see [@BBL].
In this paper, we study a generalization of the Casselman–Shalika formula to the case of Iwahori–Whittaker functions through Demazure characters. To be precise, let $\mathfrak g$ be a finite-dimensional simple Lie algebra over $\mathbb C$, which should be considered as the Lie algebra of the dual group. Let $P$ be the weight lattice of $\mathfrak g$, and $\mathbb C[P]$ the group algebra of $P$, with basis $e^\lambda$, $\lambda \in P$. The subset of dominant weights will be denoted by $P_+$. We also denote by $\Phi\supset\Phi^+$ the set of roots and positive roots, by $\Pi=\{a_i \}_{i \in I}$ the set of simple roots, and by $S=\{ \sigma_i \}_{i \in I}$ the set of simple reflections, which generates the Weyl group $W$. Let $v$ be an indeterminate, and set $\mathcal O_v= \mathbb C(v) \otimes \mathbb C[P]$.
Consider the Demazure character $\partial_{w, \lambda}$ for $w\in W$ and $\lambda \in P_+$, which is the formal character of the Demazure module associated with the weight $w\lambda$. When $w =w_\circ$, the longest element, the character $\partial_{w_\circ, \lambda}$ is nothing but the character of the irreducible representation of $\mathfrak g$ with highest weight $\lambda$. Now the Casselman–Shalika formula is given by $$\label{CS} \widetilde{W}_{w_\circ, \lambda} = \left ( \prod_{\alpha \in \Phi_+} (1-v e^{-\alpha}) \right ) \partial_{w_\circ, \lambda},$$ where $\widetilde{W}_{w_\circ, \lambda}$ is the spherical Whittaker function.
As mentioned above, this paper is concerned with generalizing the formula to the case involving the Iwahori–Whittaker functions ${W}_{w, \lambda}$ (to be defined in the next section) and the Demazure characters $\partial_{x, \lambda}$, for $w, x \in W$. That is to say, we would like to compute the coefficients ${C}_{w,x}\in {\mathcal}O_v$, $x\leq w$, in the expansion $${W}_{w,\lambda}=\sum_{x\leq w} {C}_{w,x}\partial_{x,\lambda}.$$
To make the problem more tractable, we consider the *Demazure atoms* $D_{x, \lambda}$ (see Section \[sect2\]), instead of working with the Demazure characters directly. We write $${W}_{w,\lambda}=\sum_{x\leq w} c_{w,x} D_{x, \lambda},$$ and study how to compute $c_{w,x}\in \mathcal O_v$, $x\leq w$. The coefficients $C_{w,x}$ and $c_{w,x}$ are related in a simple way (Corollary \[simple\]): $$c_{w,x}=\sum_{x\leq y\leq w}C_{w,y} \quad \text{ and } \quad C_{w,x}=\sum_{x\leq y\leq w}(-1)^{\ell(y)-\ell(x)}c_{w,y}.$$
Still, in general, it would be difficult to obtain a complete description of the coefficients $c_{w,x}$. However, the main result of this paper shows how to compute the coefficients $c_{w,x}$ under some interesting conditions involving [*good words*]{} and [*shellability*]{}. More precisely, under Condition (A) or (B) at the beginning of Section \[sec-main\], we obtain:
Let $w=s_1 \cdots s_n$ be a reduced word with $s_i=s_{\alpha_i}$ for some $\alpha_i\in \Pi$, $i=1\ldots, n$, and $$\beta_i=s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots \alpha_i,\quad i=1,\ldots, n,$$ where the indices $i_1< \dots <i_d$ between $1$ and $n$ are determined by condition [(A)]{} or [(B)]{}. Then we have $$c_{w,x}=(1-ve^{-\beta_1})\cdots {\mathcal T}_{\beta_{i_1}}\left(\cdots
{\mathcal T}_{\beta_{i_d}}\left(\cdots (1-ve^{-\beta_n})\right)\cdots\right),$$ where ${\mathcal T}_{\beta}=(1-ve^{-\beta})\partial_{\beta}-1$ and $\partial_\beta$ is the Demazure operator corresponding to the root $\beta$.
Conditions (A) and (B) are intriguing. In fact, based on thorough computer tests, in Section \[cond-a-b\] we conjecture that they are equivalent in a strong sense. Shortly after posting our paper, D. Muthiah and A. Puskás proved our conjecture; their proof is included as an Appendix. As discussed in Section \[goodword\], Condition (A) is closely related to smoothness of Schubert varieties in flag varieties $G/B$. We also present some statistical information regarding the frequency with which these conditions are satisfied.
We establish an application of Conditions (A) and (B) to the problem of computing the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group. The same problem was studied by Bump and Nakasuji [@BN]. They showed that, in the simply-laced case, when $w$ admits a good word for $x$, the entry $m(x,w)$ of the transition matrix is given by $$\label{eqn-m}
m(x,w)=\prod_{\alpha\in S(x,w)}\frac{1-q^{-1}{\bf z}^\alpha}{1-{\bf z}^\alpha},$$ where $S(x,w)$ is the set of roots determined by the good word condition. However, it seems that there is a gap in the proof of [@BN], which we do not know how to fix at the present. In Section \[sec-Cass\], we assume Condition (B) and prove the formula with $S(x,w)$ replaced by a set determined by Condition (B). The main idea of the proof is similar to that of [@BN]. Given the equivalence of Conditions (A) and (B), the Bump-Nakasuji result in full root system generality follows. This provides another evidence that Conditions (A) and (B) are natural ones to be considered in representation theory.
Related to the above mentioned coefficients $m(w,x)$, it is also worth noting the recent paper of Nakasuji and Naruse [@NN]. By using a change of basis in the Hecke algebra, they express all of these coefficients in a completely different way compared to , namely as sums over combinatorial sets. The mentioned change of basis in the Hecke algebra generalizes the theory of so-called [*root polynomials*]{}, which provides similar combinatorial formulas for localizations of Schubert classes in the equivariant cohomology and $K$-theory of flag varieties, see [@LZ] and the references therein, as well as [@NN Remark 1].
The fact that there are two types of formulas for the coefficients $m(w,x)$, namely the general formula in [@NN] and the simpler formula if Conditions (A) and (B) hold, is very similar to the existence of a general summation formula for Schubert classes (via root polynomials), versus a much simpler product formula in the smooth case, see [@BL Chapter 7]. It turns out that the latter formula is hard to derive from the former, so completely separate proofs are needed. In this context, it is not surprising that Conditions (A) and (B) are related to smoothness of Schubert varieties, as noted above.
Description of the problem {#sect2}
==========================
In this section, we present the main question of this paper, introduced in the previous section, in more detail. We keep the notions fixed in the previous section.
Recall that the Hecke algebra ${\mathcal}H_v$ is the algebra over $\mathbb C(v)$ defined by the generators $T_i$, $i \in I$, subject to the quadratic relations $$T_i^2=(v-1)T_i +v , \qquad i\in I ,$$ and the braid relations corresponding to $W$. The algebra ${\mathcal}H_v$ acts on ${\mathcal}O_v$ by $$T_i \mapsto {\mathcal}T_i :=(1-v e^{-a_i})\partial_i-1,\quad i\in I,$$ where $\partial_i$, $i\in I$, are the Demazure operators defined by $$\partial_i =\frac{1-e^{-a_i}\sigma_i}{1-e^{-a_i}}.$$ In particular the operators ${\mathcal}T_i$, $i\in I$, satisfy the braid relations. Hence one may define $$T_w\mapsto {\mathcal}T_w={\mathcal}T_{i_1}\cdots {\mathcal}T_{i_l}$$ for an arbitrary choice of reduced expression $w=\sigma_{i_1}\cdots \sigma_{i_l}$. For a dominant weight $\lambda\in P_+$, define $$W_{w,\lambda}={\mathcal}T_w e^\lambda \quad \text{ and } \quad \widetilde{W}_{w,\lambda}=\sum_{x\leq w} W_{x,\lambda},
\quad w\in W.$$ As shown in [@BBL], the expression $W_{w, \lambda}$ corresponds to the Iwahori–Whittaker function, and the sum $\widetilde{W}_{w_\circ, \lambda}$ corresponds to the spherical Whittaker function where $w_\circ \in W$ is the longest element.
It is well-known that the Demazure operators $\partial_i$, $i\in I$, satisfy the braid relations as well so that the operator $\partial_w$ is well-defined for $w \in W$ using any reduced expression of $w$. Then the Demazure character is given by $$\partial_{w,\lambda}=\partial_w e^\lambda, \quad \lambda \in P_+,$$ which is the formal character of the Demazure module associated with the weight $w\lambda$. Recall the Casselman–Shalika formula: $$\label{CS-1} \widetilde{W}_{w_\circ, \lambda} = \left ( \prod_{\alpha \in \Phi_+} (1-v e^{-\alpha}) \right ) \partial_{w_\circ, \lambda}.$$
As mentioned above, we are interested in generalizing the formula to the cases involving $\widetilde{W}_{w, \lambda}$ (or $W_{w, \lambda}$) and $\partial_{x, \lambda}$ for $w, x \in W$. Precisely, we would like to compute the coefficients $\widetilde{C}_{w,x}\in {\mathcal}O_v$, $x\leq w$, in the expansion $$\widetilde{W}_{w,\lambda}=\sum_{x\leq w} \widetilde{C}_{w,x}\partial_{x,\lambda}.$$ Alternatively, if we write $$W_{w,\lambda}=\sum_{x\leq w} C_{w, x} \partial_{x,\lambda},$$ we have $$\widetilde{C}_{w,x}=\sum_{x\leq y\leq w} C_{y,x}$$ and $$C_{w,x}=\sum_{x\leq y\leq w}(-1)^{\ell(w)-\ell(y)}\widetilde{C}_{y,x}.$$ by the Möbius inversion [@D1 Theorem 1.2]. However, we found it more convenient to work with [*Demazure atoms*]{}. We define $$D_i=\partial_i-1=e^{-a_i}\frac{1-\sigma_i}{1-e^{-a_i}}, \quad i \in I,$$ which is the specialization of ${\mathcal}T_i$ at $v\to 0$. Then $D_i$, $i\in I$ satisfy the braid relations, and we define $D_w$, $w\in W$ in the obvious way. Now the Demazure atoms are defined to be $$D_{w,\lambda}=D_w e^\lambda \quad \text{ for } w \in W \text{ and } \lambda\in P_+.$$
\[prob\] Consider the transition between ${\mathcal}T_w$ and $D_w$, $${\mathcal}T_w=\sum_{x\leq w} c_{w,x} D_x,$$ and study how to compute $c_{w,x}\in {\bf Z}[v]\otimes{\bf Z}[P]$, $x\leq w$.
The coefficients $C_{w,x}$ and $c_{w,x}$ can be related in a simple way, using the fact that the Demazure character is the sum of all the lower Demazure atoms. We give a proof of this fact below using a result in [@BBL].
$\partial_w=\sum_{x\leq w}D_x$ and $D_w=\sum_{x\leq w} (-1)^{\ell(w)-\ell(x)} \partial_x$.
$\partial_i$, $i\in I$ are the specialization of $$\mathfrak{D}_i:={\mathcal}T_i +1=(1-ve^{-a_i})\partial_i$$ at $v\to 0$. Let $\mathfrak{w}$ be a reduced expression of $w$, and define $\mathfrak{D}_\mathfrak{w}$ in the obvious way. By [@BBL Theorem 6] one has $$\mathfrak{D}_\mathfrak{w}=\sum_{x\leq w}P_{\mathfrak{w}, x}(v){\mathcal}T_x,$$ where $P_{x,\mathfrak{w}}$ is the Poincaré polynomial of fibre of the Bott–Samelson resolution $Z_\mathfrak{w}\to X_w$ over the open cell $Y_x=BxB/B$. Specializing $v\to 0$ gives that $$\partial_w=\sum_{x\leq w}P_{x,\mathfrak{w}}(0)D_x=\sum_{x\leq w}D_x.$$
\[simple\] $c_{w,x}=\sum_{x\leq y\leq w}C_{w,y}$ and $C_{w,x}=\sum_{x\leq y\leq w}(-1)^{\ell(y)-\ell(x)}c_{w,y}.$
By the reduction made above, the computation of the coefficients $\widetilde{C}_{w,x}$ or $C_{w,x}$ is equivalent to the computation of the coefficients $c_{w,x}$, for $x\leq w$. Hence we will focus on Problem \[prob\] from now on.
Note that the operators $D_i$ are twisted derivations in the sense that $$\label{der}
D_i(fg)=D_i(f)\cdot g +\sigma_i(f)\cdot D_i(g),\quad f, g\in {\bf Z}[P].$$ In fact the last equation is the specialization at $v\to 0$ of $$\label{der2}
{\mathcal}T_i(fg)=(1-v)D_i(f)\cdot g +\sigma_i(f)\cdot {\mathcal}T_i(g),\quad f, g\in {\bf Z}[P].$$
It is also known that $ T_w$, $w\in W$ satisfy the relation $$\label{hecke}
T_i \cdot T_w=\left\{\begin{array}{ll} T_{\sigma_i w}& \textrm{if } \sigma_iw>w,\\
(v-1)T_w+ v T_{\sigma_i w}& \textrm{if }\sigma_iw <w.\end{array}\right.$$ For example one has the quadratic relation $ T_i^2=(v-1)T_i +v$, $i\in I$. Specializing (\[hecke\]) at $v\to 0$ gives $$\label{dem}
D_i \cdot D_w=\left\{\begin{array}{l} D_{\sigma_i w}\quad \textrm{if } \sigma_iw>w,\\
-D_w\quad \textrm{if }\sigma_iw <w.\end{array}\right.$$
Induction steps
===============
In this section we give some general inductive steps for later use. We recall a well-known lemma from [@D1], which is called $Z(s, w_1, w_2)$ property of the Bruhat order, and it will be used frequently in this paper.
\[lem\] Let $s\in S$ be a simple reflection and $w_1, w_2\in W$. Assume that $w_1< sw_1$, $w_2< sw_2$. Then $$w_1\leq w_2 \Longleftrightarrow w_1\leq sw_2 \Longleftrightarrow sw_1\leq sw_2.$$
This lemma can be visualized using the diamond square in Figure \[fig\], where the validities of the three dashed lines are all equivalent.
$$\xymatrix{ & sw_2 \ar@{--}[rd] \ar@{-}[ld] \ar@{--}[dd] &\\
w_2 \ar@{--}[rd] & & sw_1 \ar@{-}[dl] \\ & w_1 & }$$
The following lemma can be easily verified by using (\[dem\]).
\[lem2\] Let $\alpha\in\Pi$ be a simple root and $s=s_\alpha$. Then $${\mathcal}T_s\cdot D_w=\left\{\begin{array}{ll} (1-ve^{-\alpha})D_{sw}-v e^{-\alpha}D_w& \textrm{if }sw>w,\\
-D_w & \textrm{if } sw<w.\end{array}\right.$$
\[lem3\] Assume that the simple reflection $s=s_\alpha$ is a left ascent of $w$, i.e., $sw>w$. Then $$\begin{aligned}
{\mathcal}T_{sw}&= \sum_{x\leq w}{\mathcal T}_s(c_{w,x})D_x+\sum_{x\leq w, \;x<sx}(1-ve^{-\alpha})s(c_{w,x}) D_{sx}
\\&-\sum_{x\leq w, \;x>sx}(1-ve^{-\alpha})s(c_{w,x})D_x.\end{aligned}$$
Applying ${\mathcal}T_s$ to the equation ${\mathcal}T_w=\sum_{x\leq w}c_{w,x}D_x$ and using (\[der2\]) gives that $${\mathcal}T_{sw}=\sum_{x\leq w} s(c_{w,x}) {\mathcal}T_s \cdot D_x +(1-v)D_s(c_{w,x})D_x.$$ The lemma follows from inserting Lemma \[lem2\] into the last equation, and also from noting that $$\begin{aligned}
&(1-v)D_s-ve^{-\alpha}s=(1-ve^{-\alpha})\partial_s-1={\mathcal T}_s,\\
&(1-v)D_s-s={\mathcal T}_s-(1-ve^{-\alpha})s;\end{aligned}$$ here the second equation is an immediate consequence of the first.
By comparing the coefficients in Lemma \[lem3\] with ${\mathcal}T_{sw}=\sum_{x\leq sw} c_{sw, x}D_x$, we obtain the following inductive algorithm.
\[ind\] Assume that $w<sw$, $s=s_\alpha\in S$, and that $x\leq sw$. Then
[(i)]{} if $x\leq w$, $x<sx$, then $$c_{sw, x}={\mathcal T}_s (c_{w,x});$$
[(ii)]{} if $x\leq w$, $x>sx$, then $$c_{sw,x}=(1-ve^{-\alpha})s(c_{w,sx}-c_{w,x})+{\mathcal T}_s(c_{w,x});$$
[(iii)]{} if $x\not\leq w$, in which case $x>sx$, then $$c_{sw,x}=(1-ve^{-\alpha}) s(c_{w,sx}).$$
The three cases are illustrated in Figure \[fig-2\]. Note that in the last case we have either $x$ and $w$ incomparable, as depicted, or $x=sw>w=sx$.
$$\xymatrix{ & sw \ar@{-}[rd] \ar@{-}[ld] \ar@{-}[dd] &\\
w \ar@{-}[rd] & & sx \ar@{-}[dl] \\ & x & }\qquad
\xymatrix{ & sw \ar@{-}[ld] \\
w \ar@{-}[rd] & \\ & x \ar@{-}[dl] \\
sx & }\qquad
\xymatrix{ & sw \ar@{-}[rd] \ar@{-}[ld] &\\
w \ar@{-}[rd] & & x \ar@{-}[dl] \\ & sx & }$$
The following corollary is immediate by applying Proposition \[ind\] (i) and (iii) recursively. Throughout, we let $\Phi_w:=\Phi_+\cap w\Phi_-$ be the inversion set of $w^{-1}$.
\[cor\] We have $$c_{w,e}={\mathcal T}_w(1) \quad \text{ and } \quad
c_{w,w}=\prod_{\alpha\in \Phi_w}(1-v e^{-\alpha}).$$
Good words and shellability of Bruhat order
===========================================
Good words {#goodword}
----------
Following [@BN], we consider the notion of a [good word]{}. Assume that $x\leq w$, and introduce the sets $$\label{set}
S(x,w):=\{\alpha\in\Phi_+ \, | \, x\leq ws_\alpha<w\},\quad R(x,w)=\{s_\alpha \,|\, \alpha\in S(x,w)\}.$$ Deodhar’s inequality states that $$\label{deo}
\#S(x,w)=\#R(x,w)\geq \ell(w)-\ell(x)\,,$$ with equality holding if the Kazhdan–Lusztig polynomial $P_{w_\circ w,w_\circ x}=1$, or equivalently if the Schubert variety $X_{w_\circ x}$ is rationally smooth at the $T$-fixed point $e_{w_\circ w}$ (see [@BL]). We remark that $\#S(x,w)$ has the trivial upper bound $\ell(w)$ because of the inclusion $S(x,w)\subset \Phi_{w^{-1}}=\Phi_+\cap w^{-1}\Phi_-$, where the last set is the inversion set of $w$, of cardinality $\ell(w)$; indeed, it is well known that $\alpha\in\Phi_+$ is an inversion of $w$, i.e., $w\alpha\in\Phi_-$, if and only if $ws_\alpha<w$.
For any reduced expression $\mathfrak{w}=s_1\cdots s_n$ of $w$, let $\lambda_{x,\mathfrak{w}}$ be the set of integers $i\in [1,n]$ such that $x\leq s_1\cdots \hat{s}_i\cdots s_n$. Let $\alpha_i\in\Pi$ be such that $s_i=s_{\alpha_i}$, $i=1,\ldots, n$. Then there are bijections $$\lambda_{x,\mathfrak{w}}\to S(x,w)\to R(x,w),\quad i\mapsto \gamma_i:=s_n\cdots s_{i+1}\alpha_i\mapsto s_{\gamma_i}=s_n \cdots s_{i+1}s_i s_{i+1}\cdots s_n.$$ Moreover it is clear that $ws_{\gamma_i}=s_1\cdots \hat{s}_i\cdots s_n$. By abuse of notation, we also write $$\label{gc}
\lambda_{x,\mathfrak{w}}=( i_1,\ldots, i_d)\in{\bf N}^d$$ for the vector formed by elements of $\lambda_{x,\mathfrak{w}}$ arranged in ascending order $i_1< \cdots <i_d$. Then $\mathfrak{w}$ is called a *good word* for $x$ if $$\label{good}
x=s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_d}\cdots s_n.$$ Since $d=\#\lambda_{x,\mathfrak{w}}\geq \ell(w)-\ell(x)$, a good word exists only if (\[good\]) is a reduced expression hence $d=\ell(w)-\ell(x)$. Conversely, it is conjectured in [@BN] that if $W$ is simply-laced and $d=\ell(w)-\ell(x)$, then $w$ has a good word for $x$. This conjecture is proved in \[*loc. cit.*\] for $W=A_4$ or $D_4$ using [Sage]{}, and it is shown to be false in non-simply-laced case, e.g. for $W=B_2$.
Shellability
------------
We recall the lexicographic shellability of Bruhat order, following [@BW]. For $x, y\in W$, we say that $y$ *covers* $x$, denoted by $y\to x$, if $y>x$ and there is no $z\in W$ such that $y>z>x$. In this case $\ell(y)=\ell(x)+1$ and there is a unique $\alpha\in {\Phi}_+$ such that $s_\alpha y=x$. Moreover for any reduced expression $y=s_1\ldots s_l$, there is a unique $1\leq i\leq l$ such that $x=s_1\cdots \hat{s}_i\cdots s_l$, and one has $\alpha=s_1\cdots s_{i-1}\alpha_i$. We may also write $y\stackrel{\alpha}{\to }x$ to specify the reflection $s_\alpha$ that takes $y$ to $x$.
Consider $x\leq w$ and the Bruhat interval $[x,w]:=\{ y\in W| x\leq y\leq w\}$. Then all maximal chains ${\mathscr}{C}: w=w_0\to w_1\to \cdots \to w_d=x$ of $[x,w]$ have the same length $d=\ell(w)-\ell(x)$. Let us describe a labeling of the maximal chains of $[x,w]$. Fix once for all a reduced expression $\mathfrak{w}=s_1\cdots s_n$ of $w$. For a maximal chain ${{\mathscr}{C}}$ of $[x,w]$ as above, there is a unique sequence $i_1, \cdots, i_d$ of distinct integers in $[1,n]$ such that $w_k$ is obtained by removing $s_{i_1},\cdots, s_{i_k}$ from $\mathfrak w$, $k=1,\ldots, d$. In particular this implies that the resulting subwords representing $w_k$’s are all reduced. Then we assign the label $$\label{lab}
\lambda({{\mathscr}{C}})=(\lambda_1({{\mathscr}{C}}),\ldots,\lambda_d({{\mathscr}{C}})):= (i_1,\ldots, i_d)\in {\bf N}^d.$$
Recall that the *lexicographic order* of ${\bf N}^d$ is the linear ordering $<_L$ such that ${\bf a}=(a_1,\ldots, a_d)<_L{\bf b}=(b_1,\ldots, b_d)$ if $a_i<b_i$ in the first coordinate where they differ. The main result of [@BW] states that $[x,w]$ is lexicographically shellable. In particular this implies that
\(i) there is a unique maximal chain ${{\mathscr}{C}}^+_{x,\mathfrak{w}}$ in $[x,w]$ whose label $\lambda({{\mathscr}{C}}^+_{x,\mathfrak{w}})$ is increasing, i.e., $\lambda_1({{\mathscr}{C}}^+_{x,\mathfrak{w}})<\cdots <\lambda_d({{\mathscr}{C}}^+_{x,\mathfrak{w}})$;
\(ii) $\lambda({{\mathscr}{C}}^+_{x,\mathfrak{w}})<_L \lambda({{\mathscr}{C}})$ for any other maximal chain ${{\mathscr}{C}}$ of $[x,w]$.\
Note that the maximal chain ${{\mathscr}{C}}^+_{x,\mathfrak{w}}$ depends on the choice of the reduced word $\mathfrak{w}$ which we fix from the beginning.
Similarly, consider the reduced word $s_n\cdots s_1$ of $w^{-1}$. By applying shellability to $w^{-1}$ with this reduced word and reverting to $w$, we see that
(i$'$) there is a unique maximal chain ${{\mathscr}{C}}^-_{x,\mathfrak{w}}$ in $[x,w]$ whose label $\lambda({{\mathscr}{C}}^-_{x,\mathfrak{w}})$ is decreasing, i.e., $\lambda_1({{\mathscr}{C}}^-_{x,\mathfrak{w}})> \cdots >\lambda_d({{\mathscr}{C}}^-_{x,\mathfrak{w}})$;
(ii$'$) $\lambda({{\mathscr}{C}}^-_{x,\mathfrak{w}})>_L \lambda({{\mathscr}{C}})$ for any other maximal chain ${{\mathscr}{C}}$ in $[x,w]$.
Main Result {#sec-main}
===========
In this section we compute the coefficient $c_{w,x}$, for $x\leq w$, under either of the following two conditions for the pair $(w,x)$:
\(A) $w$ admits a reduced word $\mathfrak w$ such that $\lambda_{x, \mathfrak w}=\lambda({{\mathscr}{C}}^{-}_{x, \mathfrak w})^*=(i_1,\ldots, i_d)$;
\(B) $w$ admits a reduced word $\mathfrak w$ such that $\lambda({\mathscr}{C}_{x,\mathfrak w}^+)=
\lambda({\mathscr}{C}_{x,\mathfrak w}^-)^*=(i_1,\ldots, i_d)$.
Here we write $\lambda^*=(i_d,\ldots, i_1)\in{\bf N}^d$ for a vector $\lambda=(i_1,\ldots, i_d)\in {\bf N}^d$. Note that the reduced word $\mathfrak w$ satisfying Condition (A) is necessarily a good word for $x$.
As we will prove, both conditions guarantee that only the relations in Proposition \[ind\] (i) and (iii) are used in the recursive computation of $c_{w,x}$; these relations have the advantage of being simple, compared with the relation in part (ii).
Lemmas on good words and shellability
-------------------------------------
We first prove a few more facts regarding combinatorial properties of a reduced word.
\[6.1\] Assume that $\mathfrak{w}=s_1\cdots s_n$ is a good word of $w$ for $x$ such that $\lambda_{x,\mathfrak w}=(i_1, \ldots, i_d)$ with $i_1>1$. Then
[(i)]{} $x\not\leq s_1 w\,;$
[(ii)]{} $S(s_1x, s_1w)=S(x,w)\,;$
[(iii)]{} $s_1\mathfrak{w}:=s_2\cdots s_n$ is a good word of $s_1 w$ for $s_1 x$ and $\lambda_{s_1x, s_1\mathfrak{w}}=(i_1-1,\ldots, i_d-1)\,$.
Part (i) is obvious from the definition of good word. Part (iii) follows from (ii). To prove (ii), it suffices to show that $S(s_1x, s_1w)$ is contained in $S(x,w)$, which implies that $S(s_1x, s_1w)=S(x,w)$ because of Deodhar’s inequality $$\#S(s_1x,s_1w)\geq \ell(s_1w)-\ell(s_1x)=\ell(w)-\ell(x)=\#S(x,w).$$ Take $\alpha\in S(s_1x, s_1w)$, i.e., $s_1x\leq s_1ws_\alpha< s_1w$. We claim that $s_1ws_\alpha < ws_\alpha$. To the contrary, assume that $s_1ws_\alpha>ws_\alpha$. Then by Lemma \[lem\] we have the diamond square $$\xymatrix{ & s_1w s_\alpha \ar@{--}[rd] \ar@{-}[ld] \ar@{-}[dd] &\\
ws_\alpha \ar@{--}[rd] & & x \ar@{-}[dl] \\ & s_1x & }$$ where the two dashed lines follow from the middle vertical line. This implies that $x\leq s_1ws_\alpha <s_1 w$, a contradiction to part (i). Hence $s_1ws_\alpha < ws_\alpha$, and using Lemma \[lem\] again we obtain the diagram $$\xymatrix{ & w \ar@{-}[rd] \ar@{-}[ld] & &\\
s_1w\ar@{-}[rd] & & ws_\alpha \ar@{-}[dl] \ar@{--}[dr] \\ & s_1ws_\alpha & & x \\
& & s_1x \ar@{-}[ul] \ar@{-}[ur]& }$$ which implies that $\alpha\in S(x,w)$.
\[6.2\] Let $\mathfrak{w}=s_1\cdots s_n$ be a fixed reduced word of $w$, $\lambda({\mathscr}{C}^+_{x,\mathfrak{w}})=(i_1,\ldots, i_d)$, $\lambda({\mathscr}{C}^-_{x,\mathfrak{w}})=
(j_d,\ldots, j_1)$, where $i_1<\cdots<i_d$ and $j_1<\cdots <j_d$. Consider the reduced word $s_1\mathfrak{w}=s_2\cdots s_n$ of $s_1w$. Then
[(i)]{} if $i_1>1$, then $x\not\leq s_1w$ and $$\lambda({\mathscr}{C}^+_{s_1x, s_1\mathfrak{w}})=(i_1-1,\ldots, i_d-1);$$
[(ii)]{} if $j_1>1$, then $$\lambda({\mathscr}{C}^-_{s_1x, s_1\mathfrak{w}})=(j_d-1,\ldots, j_1-1);$$
[(iii)]{} if $i_1=1$, then $$\lambda({\mathscr}{C}^+_{x,s_1\mathfrak w})=(i_2-1,\ldots, i_d-1);$$
[(iv)]{} if $j_1=1$, then $x<s_1x$ and $$\lambda({\mathscr}{C}^-_{x, s_1\mathfrak{w}})=(j_d-1,\ldots, j_2-1).$$
Write ${\mathscr}{C}^\pm_{x,\mathfrak w}: w=w_0^\pm\to w_1^\pm\to \cdots \to w_d^\pm=x$.
\(i) The last claim is clear since we have obviously a maximal chain $${{\mathscr}{C}}: s_1 w=s_1w^+_0 \to s_1 w^+_1 \to \cdots \to s_1 w^+_d =s_1 x$$ of $[s_1x, s_1w]$ with increasing label $\lambda({{\mathscr}{C}})=(i_1-1,\ldots, i_d-1)$. We must have ${{\mathscr}{C}}={\mathscr}{C}^+_{s_1x, s_1\mathfrak{ w}}$ because of the uniqueness of increasing label. It remains to prove that $x\not\leq s_1 w$. To the contrary, assume that $x\leq s_1w=s_2\cdots s_n$. Then concatenation of $w\to s_1w$ with any maximal chain in $[x, s_1w]$ will give a maximal chain ${{\mathscr}{C}}$ in $[x, w]$ such that ${{\mathscr}{C}}<_L {\mathscr}{C}^+_{x,\mathfrak{w}}$, since $\lambda_1({{\mathscr}{C}})=1< \lambda_1({\mathscr}{C}^+_{x,\mathfrak{w}})=i_1$. This is a contradiction.
\(ii) The proof is similar.
\(iii) ${\mathscr}{C}^+_{x, s_1\mathfrak w}$ equals the following subchain of ${\mathscr}{C}^+_{x,\mathfrak w}$ $$s_1 w=w^+_1\to w^+_2\to \cdots \to w^+_d=x.$$
\(iv) $s_1x \rightarrow x$ is the last arrow in the chain ${\mathscr}{C}^-_{x,\mathfrak w}$ hence $x<s_1x$. The following subchain of ${\mathscr}{C}^-_{x,\mathfrak w}$ $$w=w^-_0\to w^-_1\to \cdots \to w^-_{d-1}=s_1 x$$ gives rise to the maximal chain of $[x, s_1 w]$ $${\mathscr}{C}: s_1w=s_1 w^-_0\to s_1 w^-_1\to \cdots\to s_1 w^-_{d-1}=x$$ with decreasing label $\lambda({\mathscr}{C})=(j_d-1,\ldots, j_2-1)$, which implies that ${\mathscr}{C}={\mathscr}{C}^-_{x,s_1\mathfrak{w}}$.
\[6.3\] Assume that $\mathfrak{w}=s_1\cdots s_n$ is a good word of $w$ for $x$ such that $\lambda_{x, \mathfrak w}=\lambda({\mathscr}{C}^-_{x,\mathfrak{w}})^*=( i_1, \dots , i_d)$ with $i_1=1$. Then
[(i)]{} $x<s_1x\,;$
[(ii)]{} $S(x, s_1w)=S(x,w)\setminus\{\gamma_1\}$, where $\gamma_1=s_n\cdots s_2\alpha_1\,;$
[(iii)]{} $s_1\mathfrak{w}=s_2\cdots s_n$ is a good word of $s_1w$ for $x\,;$
[(iv)]{} $\lambda_{x,s_1\mathfrak{w}}=(i_2-1,\ldots, i_d-1)=\lambda({\mathscr}{C}^-_{x,s_1\mathfrak{w}})^*\,$.
Part (i) and the last equality in (iv) follow from Lemma \[6.2\] (iv). Part (iii) and the first equality in (iv) are direct consequences of (ii). Finally (ii) follows from Lemma \[6.4\] below, which is of independent interest.
\[6.4\] Assume that $x<w$, $sw<w$ and $x<sx$, where $s=s_\alpha \in S$. If $\# S(x,w)=\ell(w)-\ell(x)$, then $S(x, sw)=S(x,w)\setminus \{-w^{-1}\alpha\}$.
Consider the following diamond given by Lemma \[lem\]. $$\xymatrix{ & w \ar@{--}[rd] \ar@{-}[ld] \ar@{-}[dd] &\\
sw \ar@{--}[rd] & & sx \ar@{-}[dl] \\ & x & }$$ Take $\beta\in S(x,sw)$, i.e., $sw>sws_\beta\geq x$. We claim that $\beta\in
S(x,w)$, i.e., $w>ws_\beta \geq x$. If $ws_\beta> sws_\beta$, then the claim is obvious, again by Lemma \[lem\]. If $ws_\beta< sws_\beta$, then Lemma \[lem\] gives the following diamond $$\xymatrix{ & sws_\beta \ar@{--}[rd] \ar@{-}[ld] \ar@{-}[dd] &\\
ws_\beta \ar@{--}[rd] & & sx \ar@{-}[dl] \\ & x & }$$ Hence the claim follows. Obviously $\beta\neq -w^{-1}\alpha\in S(x,w)$, because $sws_{w^{-1}\alpha}=w>sw$. Therefore we get an inclusion $S(x,sw)\subset S(x,w)\setminus\{-w^{-1}\alpha\}$. This inclusion is an equality because of Deodhar’s inequality $$\# S(x,sw)\geq \ell(sw)-\ell(x)=\ell(w)-\ell(x)-1=\# S(x,w)-1,$$ where the last equality follows from the assumption $\# S(x,w)=\ell(w)-\ell(x)$.
Main theorem
------------
We can now apply previous lemmas together with Proposition \[ind\] recursively to compute $c_{w,x}$, assuming Condition (A) or (B). As mentioned above, only cases (i) and (iii) of Proposition \[ind\] show up in the computation. In order to formulate our main result, we introduce an additional notation.
For any $\alpha\in \Phi$, let $$\label{da}
\partial_\alpha=\frac{1-e^{-\alpha}s_\alpha}{1-e^{-\alpha}},\quad {\mathcal T}_\alpha=(1-ve^{-\alpha})\partial_\alpha-1.$$ Using this notation, it is easy to see that we have $$\label{cross}
w\cdot \partial_\alpha= \partial_{w\alpha}\cdot w,\quad
w\cdot {\mathcal T}_\alpha= {\mathcal T}_{w\alpha}\cdot w.$$
\[thm-main\] Assume that either condition [(A)]{} or [(B)]{} holds. In either case, let $$\beta_i=s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots \alpha_i,\quad i=1,\ldots, n.$$ Then we have $$c_{w,x}=(1-ve^{-\beta_1})\cdots {\mathcal T}_{\beta_{i_1}}\left(\cdots
{\mathcal T}_{\beta_{i_d}}\left(\cdots (1-ve^{-\beta_n})\right)\cdots\right)\,.$$
In either case we use recursion. First assume Condition (A). If $i_1>1$, then $(s_1w, s_1 x)$ satisfies Condition (A) as well, due to Lemma \[6.1\] (iii) and Lemma \[6.2\] (ii). Moreover, we may apply Proposition \[ind\] (iii) because of Lemma \[6.1\] (i), which gives that $$c_{w,x}=(1-ve^{-\alpha_1})s_1(c_{s_1w,s_1x}).$$ If $i_1=1$, then $(s_1w, x)$ also satisfies (A) and we may apply Proposition \[ind\] (i), due to Lemma \[6.3\], which gives that $$c_{w,x}={\mathcal T}_{\alpha_1}(c_{s_1w,x}).$$ Iterating this process gives us $$\begin{aligned}
c_{w,x}&=(1-ve^{-\alpha_1})s_1\cdots {\mathcal T}_{\alpha_{i_1}}\left(\cdots {\mathcal T}_{\alpha_{i_d}}\left(\cdots
(1-ve^{-\alpha_n})s_n(1)\right)\cdots\right)\,.
$$ One may use (\[cross\]) to push the reflections $s_i$, $1\leq i \leq n$, $i\neq i_1,\ldots, i_d$ across the operators ${\mathcal T}_{\alpha_{i_1}}$, $\ldots$, ${\mathcal T}_{\alpha_{i_d}}$, noting that $x(1)=1$.
The proof assuming Condition (B) is similar. If $i_1>1$, then by Lemma \[6.2\] (i)-(ii), $(s_1w, s_1x)$ also satisfies (B) and Proposition \[ind\] (iii) applies. If $i_1=1$, then by Lemma \[6.2\] (iii)-(iv), $(s_1w, x)$ satisfies (B) and Proposition \[ind\] (i) applies.
\(i) Note that in the special cases $d=n$ and $d=0$, we recover $c_{w,e}$ and $c_{w,w}$ respectively, as given by Corollary \[cor\].
\(ii) The roots $\beta_i$ can be interpreted as follows. We have $$\{\beta_i: 1\leq i\leq n, i\neq i_1,\ldots, i_d\}=\Phi_x=\Phi_+\cap x\Phi_-.$$ Moreover, under Condition (B), the roots $\beta_{i_1},\ldots, \beta_{i_d}$ give the sequence of reflections along the maximal chain ${\mathscr}{C}^+_{x,\mathfrak w}$ of $[x,w]$, i.e., we have $${\mathscr}{C}^+_{x,\mathfrak w}: w=w_0\stackrel{\beta_{i_1}}{\to} w_1\to \cdots \stackrel{\beta_{i_d}}{\to} w_d=x.$$ Hence the calculation of $c_{w,x}$ amounts to inserting the operators $${\mathcal T}_\beta=(1-ve^{-\beta})\partial_\beta-1, \quad \beta\in \{\beta_{i_1},\ldots, \beta_{i_d}\}$$ into the product $\prod_{\alpha\in\Phi_x}(1-ve^{-\alpha})$ in a natural, combinatorial way.
Conditions (A) and (B) {#cond-a-b}
----------------------
Since these conditions are essential for our main result, we now discuss them in more detail. We start with an example.
Consider $w=s_1s_2s_1s_3s_2s_1$ and $x=s_2s_3$ in $A_3$. It is easy to see that both Conditions (A) and (B) hold in this example.
Based on thorough computer tests, we now formulate a conjecture about the equivalence of Conditions (A) and (B) in a strong sense.
\[conj\] Let $\mathfrak w$ be a reduced word for $w$ and $x\le w$. The following are equivalent:
[(i)]{} $\lambda_{x, \mathfrak w}=\lambda({{\mathscr}{C}}^{-}_{x, \mathfrak w})^*\,;$
[(ii)]{} $\lambda({\mathscr}{C}_{x,\mathfrak w}^+)=\lambda({\mathscr}{C}_{x,\mathfrak w}^-)^*\,;$
[(iii)]{} $\lambda_{x, \mathfrak w}=\lambda({\mathscr}{C}_{x,\mathfrak w}^+)\,.$
The proof of the conjecture, due to D. Muthiah and A. Puskás, is included as an Appendix. Thus, we are able use the more symmetric condition $$\lambda_{x, \mathfrak w}=\lambda({\mathscr}{C}_{x,\mathfrak w}^+)=\lambda({{\mathscr}{C}}^{-}_{x, \mathfrak w})^*\,.$$ Note that it is enough to prove ${\rm (i)}\Leftrightarrow {\rm (ii)}$, as ${\rm (i)}\Leftrightarrow {\rm (iii)}$ would easily follow; indeed, just reverse the reduced word and use the fact that inversion is an automorphism of the Bruhat order.
We now discuss some statistics related to the frequency with which Conditions (A) and (B) are satisfied. We looked at the symmetric groups $S_4$, $S_5$, and $S_6$, as well as at the hyperoctahedral groups $B_4$ and $B_5$. For each (signed) permutation $w$, we calculated (with the help of a computer) the percentage of $x\le w$ which satisfy Conditions (A) and (B). The distribution of these percentages in $S_5$ and $S_6$ is shown in Figure \[fig-3\]. It is interesting to note that this distribution is skewed right, with the mode at the right tail, while the interquartile range reaches 100% in both cases. By contrast, in type $B$, the distribution looks closer to a uniform one.
Experiments with the same Weyl groups mentioned above also showed that the formula in Theorem \[thm-main\] fails if Conditions (A) and (B) are not satisfied.
$$\begin{array}{cc}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \includegraphics[scale=0.75]{hist1.pdf} & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \includegraphics[scale=0.75]{hist2.pdf}
\end{array}$$
Casselman’s basis of Iwahori vectors {#sec-Cass}
====================================
In this section, under the shellability Condition (B) in Section 6, we compute the transition matrix between two natural bases of the Iwahori fixed vectors in a spherical representation of a semisimple $p$-adic group, considered by Casselman in [@C]. For simply-laced cases, a conjectural formula is given in [@BN], which is proved under the assumption that a good word exists; however, it seems that there is a gap in this proof, which we do not know how to fix at present. We follow the strategy of computations in [@BN], although we consider reduced words from a very different point of view. Let us first recall the basic formulations and collect a few results we need from [@BN].
Let $\chi=\chi_{\bf z}$ be an unramified character of $T(F)$, which is parametrized by an element ${\bf z}$ in the complex torus $\hat{T}$ of the L-group ${}^LG$. Let $V(\chi)=\textrm{Ind}^G_B\chi$ be the induced representation which consists of locally constant functions $f: G\to{\bf C}$ such that $f(bg)=(\delta^{1/2}\chi)(b) f(g)$, where $\delta=\det(\textrm{Ad}|_\mathfrak{n})$ is the modular character. Let $J$ be the Iwahori subgroup which is the preimage of $B({\bf F}_q)$ under the reduction $K=G({\mathcal}O_F)\to G({\bf F}_q)$. Then the space of $J$-fixed vectors $V(\chi)^J$ has dimension $|W|$, and there are two bases $\{\phi_{w, \chi}\}$ and $\{f_w\}$ of $V(\chi)^J$ parametrized by $W$.
The first natural basis $\{\phi_{w, \chi}\}$ is defined using the disjoint decomposition $G=\bigsqcup_{w\in W}BwJ$ such that $\phi_w$ is supported on $BwJ$ and $\phi_{w,\chi}|_{wJ}=1$. Let $M_w: V(\chi)\to V({}^w\chi)$ be the intertwining operator defined by $$(M_wf)(g)=\int_{N\cap wN_-w^{-1}}f(w^{-1}ng)dn.$$ Then $\{f_w\}$ is the dual basis of the linear functionals $V(\chi)\to {\bf C}$, $f\mapsto (M_wf)(1)$, $w\in W$. Casselman [@C] asks for the transition matrix between these two bases, which is in general a very difficult problem. It is better to use the basis $$\psi_{x,\chi}=\sum_{w\geq x}\phi_{w,\chi}$$ instead of $\phi_{w,\chi}$, and by Möbus inversion one has $$\phi_{x,\chi}=\sum_{w\geq x}(-1)^{\ell(w)-\ell(x)}\psi_{w,\chi}.$$ If we write $\psi_{x,\chi}=\sum_{w\in W} m(x,w)f_w$, then obviously $m(x,w)=(M_w\psi_{x,\chi})(1)$ and in [@BN] it is shown that $(m(x,w))$ is upper triangular. In \[*loc. cit*\] it is conjectured that $$m(x,w)=\prod_{\alpha\in S(x,w)}\frac{1-q^{-1}{\bf z}^\alpha}{1-{\bf z}^\alpha}$$ when the root system $\Phi$ is simply-laced and $|S(x,w)|=\ell(w) - \ell(x)$, and it is proved under the additional assumption that $w$ admits a good word for $x$.
Let $H$ be the Iwahori–Hecke algebra which consists of bi-$J$-invariant functions supported on $K$. Then $H$ has a basis $\{t_w| w\in W\}$, where $t_w$ is the characteristic function of $JwJ$, and $H$ is generated by $t_i:=t_{\sigma_i}$, $i\in I$. Let $\alpha_\chi: V(\chi)^J\to H$ be the isomorphism of left $H$-modules defined by $(\alpha_\chi f)(g)=\left. f(g^{-1})\right|_K$. Let ${\mathcal}M_w={\mathcal}M_{w, {\bf z}}: H\to H$ be the map making the following diagram commute: $$\xymatrix{
V(\chi)\ar[r]^{M_w} \ar[d]^{\alpha_\chi} & V({}^w\chi) \ar[d]^{\alpha_{{}^w\chi}} \\
H \ar[r]^{{\mathcal}M_{w}} & H
}$$ Define $\mu_{\bf z}(w)={\mathcal}M_w(1_H)\in H$. Then $$\label{mu1}
\mu_{\bf z}(\sigma_i)=q^{-1}t_i +(1-q^{-1})\frac{{\bf z}^{a_i}}{1-{\bf z}^{a_i}},$$ and for $\ell(w_1w_2)=\ell(w_1)+\ell(w_2)$ one has $$\label{mu2}
\mu_{\bf z}(w_1w_2)=\mu_{\bf z}(w_2)\mu_{w_2{\bf z}}(w_1).$$ Define $\psi(x)=\alpha_\chi(\psi_x)\in H$. Then $\psi(x)=\sum_{w\geq x}t_w$ is independent of $\chi$. For $f\in H$ let $\Lambda(f)$ be the coefficient of $1$ in the expression of $f$ in terms of the basis $t_w$. Then $$m(x,w)=\Lambda(\psi(x)\mu_{\bf z}(w)).$$ For $f, g\in H$ and $x\in W$, write $f-g\geq x$ if $f-g$ is a linear combination of $t_w$’s with $w\geq x$.
[@BN] \[bn\] Let $s=s_\alpha\in S$, $x\in W$ such that $xs>x$. Then $$\psi(x)\mu_{\bf z}(s)=\frac{1-q^{-1}{\bf z}^\alpha}{1-{\bf z}^\alpha}\psi(x),\quad \psi(xs)\mu_{\bf z}(s)- \psi(x) \geq xs.$$
Now we can give our formula for $m(x,w)$ in full root system generality, assuming that Condition (B) holds.
\[7.2\] Assume condition [(B)]{}. Let $\gamma_{i_k}= s_n\cdots s_{i_k+1}\alpha_{i_k}$, $k=1,\ldots, d$. Then $$m(x,w)=\prod^d_{k=1}\frac{1-q^{-1}{\bf z}^{\gamma_{i_k}}}{1-{\bf z}^{\gamma_{i_k}}}.$$
The proof follows the argument in [@BN], but we shall give some details for the sake of completeness. Write $\mu(s_n)=\mu_{\bf z}(s_n)$, $\mu(s_{n-1})=\mu_{s_n({\bf z})}(s_{n-1}), \ldots$, suppressing the dependence of spectral parameters. Write $\psi(x)\mu_{\bf z}(w)$ as a sum $$\begin{aligned}
[\psi(s_1\cdots\hat{s}_{i_1}\cdots \hat{s}_{i_d}\cdots s_n)\mu(s_n)-\psi(s_1\cdots\hat{s}_{i_1}\cdots \hat{s}_{i_d}\cdots s_{n-1})]\mu(s_{n-1})\cdots\mu(s_1)&+\\
[\psi(s_1\cdots\hat{s}_{i_1}\cdots \hat{s}_{i_d}\cdots s_{n-1})\mu(s_{n-1})-\psi(s_1\cdots\hat{s}_{i_1}\cdots \hat{s}_{i_d}\cdots s_{n-2})]\mu(s_{n-2})\cdots\mu(s_1)&+\\
\cdots & \\
[\psi(s_1\cdots\hat{s}_{i_1}\cdots \hat{s}_{i_d})\mu(s_{i_d})-C(d)\psi(s_1\cdots\hat{s}_{i_1}\cdots \hat{s}_{i_d})]\mu(s_{i_d-1})\cdots\mu(s_1) &+\\
C(d)[\psi(s_1\cdots\hat{s}_{i_1}\cdots s_{i_d-1})\mu(s_{i_d-1})-\psi(s_1\cdots\hat{s}_{i_1}\cdots s_{i_d-2})]\mu(s_{i_d-2})\cdots\mu(s_1)& +\\
\cdots & \\
C(d)\cdots C(1)[\psi(s_1)\mu(s_1)-\psi(1)]&+\\
C(d)\cdots C(1)\psi(1),&\end{aligned}$$ where $$C(k)=\frac{1-q^{-1}{\bf z}^{\gamma_k}}{1-{\bf z}^{\gamma_k}},\quad k=1,\ldots, d.$$ We will show that the linear functional $\Lambda$ annihilates every summand except the last, so that $m(x,w)=C(d)\cdots C(1)$.
Since we have the reduced words $w_k^+:=s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_k} s_{i_k+1}\cdots s_n$, $k=1,\ldots,n$, which form the maximal chain ${{\mathscr}{C}}^+_{x,\mathfrak{w}}$, we see that $s_1\cdots \hat{s}_{i_1}\cdots s_{i_k}> s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_k}$. Therefore by Proposition \[bn\] the summands of the form $$\prod_{j>k}C(j)[\psi(s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_k})\mu(s_{i_k})-C(k)
\psi(s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_k})]\mu(s_{i_k-1})\cdots \mu(s_1)$$ are all equal to zero. Note that the spectral parameter of $\mu(s_{i_k})$ is $s_{i_k+1}\cdots s_n{\bf z}$ and one has $(s_{i_k+1}\cdots s_n{\bf z})^{\alpha_{i_k}}={\bf z}^{s_n\cdots s_{i_k+1}\alpha_{i_k}}={\bf z}^{\gamma_{i_k}}.$
Every other summand is a constant multiple of the form $$\label{other}
[\psi(s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots s_j)\mu(s_j)-\psi(s_1\cdots \hat{s}_{i_1}\cdots\hat{s}_{i_2}\cdots s_{j-1})]\mu(s_{j-1})\cdots \mu(s_1).$$ Since $s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots s_j$ is reduced, by Proposition \[bn\] we have $$\psi(s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots s_j)\mu(s_j)-\psi(s_1\cdots \hat{s}_{i_1}\cdots\hat{s}_{i_2}\cdots s_{j-1})\geq s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots s_j.$$ Applying (\[mu1\]), (\[mu2\]) and arguing as in [@BN] one can deduce that (\[other\]) is annihilated by $\Lambda$ unless $$\label{eqn-no}
s_1\cdots \hat{s}_{i_1}\cdots \hat{s}_{i_2}\cdots s_j \leq s_1\cdots s_{j-1} =s_1 \cdots s_{j-1} \hat{s}_j .$$ Assume that is true, let $d'=\max\{1\leq k\leq d: i_k<j\}$, $x'=s_1\cdots \hat{s}_{i_1}\cdots\hat{s}_{i_{d'}}\cdots s_j$ and $\mathfrak w'=s_1\cdots s_j$. Recall that we have the reduced words $w_k^-:=s_1\cdots \hat{s}_{i_{d-k+1}}\cdots \hat{s}_{i_d}\cdots s_n$, $k=1,\ldots, d$, which make the maximal chain ${\mathscr}{C}^-_{x,\mathfrak w}$. Consider the following subchain of ${\mathscr}{C}^-_{x,\mathfrak w}$ $$w^-_{d-d'}\to w^-_{d-d'+1}\to\cdots \to w^-_d = x.$$ By taking reduced subwords, it gives rise to a maximal chain of $[x',\mathfrak w']$ $${\mathscr}{C}: \mathfrak w'=w_0' \to w_1'\to \cdots \to w_{d'}'= x'$$ where $w_i'=s_1\cdots \hat{s}_{d'-i+1}\cdots \hat{s}_{d'}\cdots s_j$, $i=1,\ldots, d'$. Then $\lambda({\mathscr}{C})=(i_{d'},\ldots, i_1)$ is decreasing, which implies that ${\mathscr}{C}={\mathscr}{C}^-_{x', \mathfrak w'}$. But similarly to the proof of Lemma \[6.2\] (i), this contradicts (\[eqn-no\]) because ${\mathscr}{C}^-_{x', \mathfrak w'}$ is lexicographically maximal. This finishes the proof of the theorem.
Given the equivalence of Conditions (A) and (B), proved in the Appendix, the Bump-Nakasuji result [@BN] in full root system generality immediately follows from Theorem \[7.2\].
Appendix: Proof of Conjecture \[conj\] .2 cm By Dinakar Muthiah and Anna Puskás {#appendix}
================================================================================
In this appendix, we prove Conjecture \[conj\]. The conjecture is that the following three conditions are equivalent.
(i) ${{\lambda}_{x,{{\mathfrak{w}}}}}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}})^{\ast}}\,;$
(ii) ${{\lambda}({{\mathscr{C}}_{x,{{\mathfrak{w}}}}^+})}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}})^{\ast}}\,;$
(iii) ${{\lambda}_{x,{{\mathfrak{w}}}}}={{\lambda}({{\mathscr{C}}_{x,{{\mathfrak{w}}}}^+})}\,.$
As mentioned right below Conjecture \[conj\], it suffices to prove that ${\rm (i)}$ and ${\rm (ii)}$ are equivalent.
We will keep the notations in the previous sections. In particular, we have $x\leq w$ two elements of $W,$ ${{\mathfrak{w}}}=s_1\cdots s_n$ a reduced word for $w;$ ${{\lambda}_{x,{{\mathfrak{w}}}}}=({\lambda}_1,\ldots ,{\lambda}_k),$ ${{\lambda}({{\mathscr{C}}_{x,{\mathfrak{w}}}^+})} = (i_1, \cdots, i_d)$ and ${{\lambda}({\mathscr{C}}^-_{x,{\mathfrak{w}}})}^{\ast} = (j_1, \cdots, j_d).$
Proof of ${\rm (i)} \implies {\rm (ii)}$
----------------------------------------
\[lem:ione-equals-one-is-equivalent-to-lambdaone-equals-one\] Let $x$, $w$, ${\mathfrak{w}}$ be as before, $\lambda_{x,{\mathfrak{w}}} = (\lambda_1, \cdots, \lambda_k)$, and ${{\lambda}({{\mathscr{C}}_{x,{\mathfrak{w}}}^+})} = (i_1, \cdots, i_d)$. Then $i_1 = 1$ if and only if $\lambda_1 = 1$.
Assume first that $i_1=1.$ Then the chain ${{{\mathscr{C}}_{x,{{\mathfrak{w}}}}^+}}$ starts with ${{\mathfrak{w}}}_{\widehat{1}},$ and hence $x\leq {{\mathfrak{w}}}_{\widehat{1}}$ and thus ${\lambda}_1=1.$ For the other direction, assume ${\lambda}_1=1.$ Omitting the first simple reflection from ${{\mathfrak{w}}}$ only decreases its length by $1,$ hence $\ell({{\mathfrak{w}}}_{\widehat{1}})=\ell(w)-1.$ Composing $w\rightarrow {{\mathfrak{w}}}_{\widehat{1}}$ with a maximal chain from ${{\mathfrak{w}}}_{\widehat{1}}$ to $x$ gives a maximal chain ${\mathscr{C}}$ from ${{\mathfrak{w}}}$ to $x$ whose label starts with $1.$ Then ${{{\mathscr{C}}_{x,{{\mathfrak{w}}}}^+}}\leq_L {\mathscr{C}}$ implies $i_1=1.$
\[rmk:wgoodif1\] If ${\rm (i)}$ holds for $x$ and ${{\mathfrak{w}}},$ i.e. ${{\lambda}_{x,{{\mathfrak{w}}}}}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}})^{\ast}},$ then ${{\mathfrak{w}}}$ is a good word of $w$ for $x.$ (Omitting all the reflections from ${{\mathfrak{w}}}$ that appear in ${{\lambda}_{x,{{\mathfrak{w}}}}}$ is the same as taking the last element of the maximal chain ${{\mathscr{C}}^-_{x,{{\mathfrak{w}}}}};$ that last element is $x.$)
[${\rm (i)} \implies {\rm (ii)}$.]{}
We proceed by induction on $\ell(w) + (\ell(w) - \ell(x))$; the base case is trivial.
Assume that ${\rm (i)}$ holds for a pair $x,{{\mathfrak{w}}}$, i.e. ${{\lambda}_{x,{{\mathfrak{w}}}}}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}})^{\ast}}$. We would like to show that ${\rm (ii)}$ holds for $x,{{\mathfrak{w}}}$ as well, i.e. ${{\lambda}({{\mathscr{C}}_{x,{{\mathfrak{w}}}}^+})}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}})^{\ast}}$.
Consider ${\lambda}_1=j_1,$ the first index in the labels ${{\lambda}_{x,{{\mathfrak{w}}}}}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}})^{\ast}}.$ We distinguish between two cases according to whether ${\lambda}_1=j_1=1$ or ${\lambda}_1=j_1>1.$
[**Case 1:**]{} ${\lambda}_1=j_1=1.$ Then by Lemma \[6.3\] (iv), we have that ${\rm (i)}$ holds for the pair $x,{{\mathfrak{w}}}'=s_1{{\mathfrak{w}}}$. Then by induction, ${\rm (ii)}$ holds for $x$ and ${{\mathfrak{w}}}'$, i.e. ${{\lambda}({{\mathscr{C}}_{x,{{\mathfrak{w}}}'}^+})}={{\lambda}({\mathscr{C}}^-_{x,{{\mathfrak{w}}}'})^{\ast}}.$
By Lemma \[6.2\] (iii) and (iv), we have: $$\label{eq:11}
(i_2-1,\ldots ,i_d-1)=(j_2-1,\ldots ,j_d-1)\,.$$ Together with $i_1=1$ (Lemma \[lem:ione-equals-one-is-equivalent-to-lambdaone-equals-one\]) we conclude that $i_r=j_r$ for every $1\le r\leq d$, hence ${\rm (ii)}$ holds for the pair $x,{{\mathfrak{w}}}$.
[**Case 2:**]{} ${\lambda}_1=j_1>1$. By Lemma \[6.1\] (iii) and Lemma \[6.2\] (ii), ${\rm (i)}$ holds for the pair $x' = s_1 x$ and ${{\mathfrak{w}}}' = s_1 {{\mathfrak{w}}}$. By induction, ${\rm (ii)}$ also holds for $x',{{\mathfrak{w}}}'$. By Lemma \[lem:ione-equals-one-is-equivalent-to-lambdaone-equals-one\], $i_1 > 1$. Thus by Lemma \[6.2\] (i) and (ii), we have $(i_1 - 1, \cdots, i_d -1) = (j_1 - 1, \cdots, j_d -1)$, which implies $(i_1, \cdots, i_d) = (j_1, \cdots, j_d )$.
Proof of ${\rm (ii)} \implies {\rm (i)}$
----------------------------------------
\[lem:remove-one-node-lambda-result\] Let $x$, $w$, ${\mathfrak{w}}$ be as before. Write ${{\lambda}({{\mathscr{C}}_{x,{\mathfrak{w}}}^+})} = (i_1, \cdots, i_d)$, ${{\lambda}({\mathscr{C}}^-_{x,{\mathfrak{w}}})}^* = (j_1, \cdots, j_d)$.
Suppose $j_1 = 1$, then:
- $x \leq s_1 w\,;$
- $\lambda_{x, s_1{\mathfrak{w}}} = (\lambda_{x,{\mathfrak{w}}} \backslash \{1\}) - 1\,.$
Suppose $i_1 > 1$, then:
- $s_1 x \leq s_1w\,;$
- $\lambda_{s_1 x, s_1{\mathfrak{w}}} \supseteq \lambda_{x,{\mathfrak{w}}} - 1\,.$
Here we write $(\lambda_{x,{\mathfrak{w}}} \backslash \{1\}) - 1$ and $\lambda_{x,{\mathfrak{w}}} - 1$ to refer to the set obtained by subtracting $1$ from all elements.
Note that $s_1{\mathfrak{w}}$ is a reduced word for $s_1w,$ and $s_1w<w.$
First suppose $j_1=1.$ By Lemma \[6.2\] (iv) we have $x<s_1x.$ By Lemma \[lem\] we may draw the diagram
(-[2]{},0) – (0,[2]{}); (0,-[2]{}) – ([2]{},0); (0,-[2]{}) – (-[2]{},0); (0,-[2]{}) – (0,[2]{}); (-0.3\*[2]{},-0.2\*[2]{}) node \[\] ; (-[2]{},0) node \[fill=white\] [$s_1w$]{}; (0,[2]{}) node \[fill=white\] [$w$]{}; (0,-[2]{}) node \[fill=white\] [$x$]{}; ([2]{},0) node \[fill=white\] [$s_1x$]{};
and conclude that $x \leq s_1 w.$ Let $1<t\leq n$ and ${{\mathfrak{w}}}_{\widehat{t}}:=s_1\cdots \widehat{s_t}\cdots s_n,$ and $s_1{{\mathfrak{w}}}_{\widehat{t}}:=s_2\cdots \widehat{s_t}\cdots s_n(=(s_1{{\mathfrak{w}}})_{\widehat{t-1}}).$ To show $\lambda_{x, s_1{\mathfrak{w}}} = (\lambda_{x,{\mathfrak{w}}} \backslash \{1\}) - 1,$ it suffices to prove $$\label{eq:toprove_1}
x\leq {{\mathfrak{w}}}_{\widehat{t}} \;\;\Longleftrightarrow\;\; x\leq s_1{{\mathfrak{w}}}_{\widehat{t}}\:.$$ (Note that we are slightly abusing notation. For example, when we write $x\leq {{\mathfrak{w}}}_{\widehat{t}}\:$, we mean $x \leq w_{\widehat{t}}$ where $w_{\widehat{t}}$ is the Weyl group element obtained by multiplying out the word ${{\mathfrak{w}}}_{\widehat{t}}\:$.)
To prove , we use Lemma \[lem\] again. We have either ${{\mathfrak{w}}}_{\widehat{t}}<s_1{{\mathfrak{w}}}_{\widehat{t}}$ or ${{\mathfrak{w}}}_{\widehat{t}}>s_1{{\mathfrak{w}}}_{\widehat{t}}\:;$ we may accordingly draw one of the following two diagrams.
; ;
These diagrams together imply that holds in both cases.
Next suppose $i_1>1.$ The argument in this case is very similar to the one above. We claim $x>s_1x.$ Assume to the contrary that $x<s_1x.$ Then again by Lemma \[lem\] we may draw the following diagram.
(-[2]{},0) – (0,[2]{}); (0,-[2]{}) – ([2]{},0); (0,-[2]{}) – (-[2]{},0); (0,-[2]{}) – (0,[2]{}); (-0.3\*[2]{},-0.2\*[2]{}) node \[\] ; (-[2]{},0) node \[fill=white\] [$s_1w$]{}; (0,[2]{}) node \[fill=white\] [$w$]{}; (0,-[2]{}) node \[fill=white\] [$x$]{}; ([2]{},0) node \[fill=white\] [$s_1x$]{};
This contradicts the statement of Lemma \[6.2\] (i) that $x\nleq s_1w.$ Hence we have $x>s_1x,$ and consequently the diagram
(-[2]{},0) – (0,[2]{}); (0,-[2]{}) – ([2]{},0); (0,-[2]{}) – (-[2]{},0); (0,0) node \[\] ; ([2]{},0) – (0,[2]{}); (-[2]{},0) node \[fill=white\] [$s_1w$]{}; (0,[2]{}) node \[fill=white\] [$w$]{}; (0,-[2]{}) node \[fill=white\] [$s_1x$]{}; ([2]{},0) node \[fill=white\] [$x$]{};
shows that $s_1 x \leq s_1w.$ Take $1<t\leq n$ and ${{\mathfrak{w}}}_{\widehat{t}}$ and $s_1{{\mathfrak{w}}}_{\widehat{t}}$ as in the case $i_1=1$ above. To prove $\lambda_{s_1 x, s_1{\mathfrak{w}}} \supseteq \lambda_{x,{\mathfrak{w}}} - 1$ we need to show $$\label{eq:toprove_not1}
x\leq {{\mathfrak{w}}}_{\widehat{t}} \;\;\Longrightarrow\;\; s_1x\leq s_1{{\mathfrak{w}}}_{\widehat{t}}\:.$$ First consider the case when ${{\mathfrak{w}}}_{\widehat{t}}<s_1{{\mathfrak{w}}}_{\widehat{t}}\:.$ Then if $x\leq {{\mathfrak{w}}}_{\widehat{t}}$ we have $$s_1x< x\leq {{\mathfrak{w}}}_{\widehat{t}}<s_1{{\mathfrak{w}}}_{\widehat{t}}\:,$$ whence $s_1x<s_1{{\mathfrak{w}}}_{\widehat{t}}\:.$ If on the other hand ${{\mathfrak{w}}}_{\widehat{t}}>s_1{{\mathfrak{w}}}_{\widehat{t}}\:,$ then again by Lemma \[lem\] we have the diagram
(-[2]{},0) – (0,[2]{}); (0,-[2]{}) – ([2]{},0); (0,-[2]{}) – (-[2]{},0); (0,0) node \[\] ; ([2]{},0) – (0,[2]{}); (-[2]{},0) node \[fill=white\] [$s_1{{\mathfrak{w}}}_{\widehat{t}}$]{}; (0,[2]{}) node \[fill=white\] [${{\mathfrak{w}}}_{\widehat{t}}$]{}; (0,-[2]{}) node \[fill=white\] [$s_1x$]{}; ([2]{},0) node \[fill=white\] [$x$]{};
which proves .
[${\rm (ii)} \implies {\rm (i)}$.]{}
Let $x,w, {\mathfrak{w}}$ be as before. We proceed by induction on $\ell(w) + (\ell(w) - \ell(x))$; the base case is trivial.
Let us assume ${{\lambda}({{\mathscr{C}}_{x,{\mathfrak{w}}}^+})} = {{\lambda}({\mathscr{C}}^-_{x,{\mathfrak{w}}})}^*$. Write $\lambda_{x,{\mathfrak{w}}} = (\lambda_1, \cdots, \lambda_k)$, ${{\lambda}({{\mathscr{C}}_{x,{\mathfrak{w}}}^+})} = (i_1, \cdots, i_d)$, and ${{\lambda}({\mathscr{C}}^-_{x,{\mathfrak{w}}})} = (j_d, \cdots, j_1)$. Our assumption means that $i_r = j_r$ for all $r$.
[**Case 1:**]{} $i_1 = j_1 = 1$. In this case $\lambda_1 =1$ by Lemma \[lem:ione-equals-one-is-equivalent-to-lambdaone-equals-one\], and Lemma \[lem:remove-one-node-lambda-result\] tells us that $x \leq s_1w$ and: $$\label{eq:01}
\lambda_{x, s_1{\mathfrak{w}}} = (\lambda_{x,{\mathfrak{w}}} \backslash \{1\}) - 1\,.$$
Then by Lemma \[6.2\] (iii) and (iv), we have that: $$\label{eq:3}
{{\lambda}({{\mathscr{C}}_{x,s_1{\mathfrak{w}}}^+})} = {{\lambda}({\mathscr{C}}^-_{x,s_1{\mathfrak{w}}})}^* = (i_2-1,\cdots, i_d -1) \,.$$ By induction, we know: $$\label{eq:10}
\lambda_{x,s_1{\mathfrak{w}}} = {{\lambda}({\mathscr{C}}^-_{x,s_1{\mathfrak{w}}})}^* \,.$$ By and , $\lambda_{x,{\mathfrak{w}}} = (i_1, \cdots, i_d)$. Therefore $\lambda_{x,{\mathfrak{w}}} = {{\lambda}({\mathscr{C}}^-_{x,{\mathfrak{w}}})}^*$.
[**Case 2:**]{} $i_1 = j_1 > 1$. The argument is very similar. In this case, $\lambda_1 > 1$ by Lemma \[lem:ione-equals-one-is-equivalent-to-lambdaone-equals-one\], and Lemma \[lem:remove-one-node-lambda-result\] tells us that $s_1x \leq s_1w$ and: $$\label{eq:5}
\lambda_{s_1x, s_1{\mathfrak{w}}} \supset \lambda_{x,{\mathfrak{w}}} - 1 \,.$$
By Lemma \[6.2\] (i) and (ii), we have that: $$\label{eq:6}
{{\lambda}({{\mathscr{C}}_{s_1x,s_1{\mathfrak{w}}}^+})} = {{\lambda}({\mathscr{C}}^-_{s_1 x,s_1{\mathfrak{w}}})}^* = (i_1-1,\cdots, i_d -1) \,.$$ By induction, we know: $$\label{eq:indconcl}
\lambda_{s_1x,s_1{\mathfrak{w}}} = {{\lambda}({\mathscr{C}}^-_{s_1x,s_1{\mathfrak{w}}})}^*\,.$$ In particular: $$\label{eq:7}
\# \lambda_{s_1x,s_1{\mathfrak{w}}} = \ell(w) - \ell(x) \,.$$ By Deodhar’s inequality: $$\label{eq:8}
\# \lambda_{x,{\mathfrak{w}}} \geq \ell(w) - \ell(x) \,.$$ So (\[eq:5\]), (\[eq:7\]), and (\[eq:8\]) together imply: $$\label{eq:9}
\lambda_{s_1x, s_1{\mathfrak{w}}} = \lambda_{x,{\mathfrak{w}}} - 1 \,.$$ By , and , $\lambda_{x,{\mathfrak{w}}} = (i_1, \cdots, i_d)$. Therefore $\lambda_{x,{\mathfrak{w}}} = {{\lambda}({\mathscr{C}}^-_{x,{\mathfrak{w}}})}^*$.
[CERP]{}
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B. Brubaker, D. Bump and A. Licata. . :41–68, 2015.
D. Bump and M. Nakasuji. . :1238–1253, 2011.
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[^1]: $^{\star}$K.-H.L. was partially supported by a grant from the Simons Foundation (\#318706).
[^2]: $^{\dagger}$ C.L. was partially supported by the NSF grant DMS–1362627.
|
---
abstract: |
Under a single-index regression assumption, we introduce a new semiparametric procedure to estimate a conditional density of a censored response. The regression model can be seen as a generalization of Cox regression model and also as a profitable tool to perform dimension reduction under censoring. This technique extends the results of Delecroix [*et al.*]{} (2003). We derive consistency and asymptotic normality of our estimator of the index parameter by proving its asymptotic equivalence with the (uncomputable) maximum likelihood estimator, using martingales results for counting processes and arguments of empirical processes theory. Furthermore, we provide a new adaptive procedure which allows us both to chose the smoothing parameter involved in our approach and to circumvent the weak performances of Kaplan-Meier estimator (1958) in the right-tail of the distribution. Through a simulation study, we study the behavior of our estimator for small samples.\
[*Keywords:*]{} asymptotic normality; empirical processes; censoring; martingales for counting processes; pseudo-maximum likelihood; single-index model
author:
- 'OLIVIER BOUAZIZ$^*$ and OLIVIER LOPEZ'
title: '**Conditional density estimation in a censored single-index regression model**'
---
Laboratoire de Statistique Théorique et Appliquée,\
Université Paris VI,\
175 rue du Chevaleret\
75013 Paris, France\
Introduction
============
A major issue of recent papers dealing with censored regression is to propose alternatives to the popular Cox regression model. This model, also known as multiplicative hazard regression model (see [@Cox72]), states some semiparametric model on the conditional hazard function. Estimation in this model is traditionally performed using pseudolikelihood techniques, and the theoretical properties of these procedures are covered by a large number of papers (see e.g. [@Flemming91]). However, in some situations, the assumptions of Cox regression model are obviously not satisfied by the data set. In this paper, our aim is to perform estimation in a semiparametric regression model which allows more flexibility than the Cox regression model. This new technique can be seen as a particularly interesting alternative, since it is valid in a larger number of situations than the multiplicative hazard model.
Alternatives to Cox regression model mostly focus on the estimation of a conditional expectation, or of a quantile regression model. Koul [*et al.*]{} (1981), [@Stute99], Delecroix [*et al.*]{} consider mean-regression models where the regression function belongs to a parametric family, but with an unknown distribution of the residuals. Parametric quantile regression was studied by Gannoun [*et al*]{}. On the other hand, [@Lu05] and [@newLopez] considered a semiparametric single-index regression model. Single-index regression models were initially introduced to circumvent the so-called “curse of dimensionality” in nonparametric regression (see. e.g. [@Ichimura93]), by assuming that the conditional expectation only depends on an unknown linear combination of the covariates. Another appealing aspect of such kind of models is that they include the Cox regression model as a particular case. The main assumption of this model is that the conditional density only depends on an unknown linear combination of the covariates, while the multiplicative hazard model states a similar assumption on the conditional hazard rate. In this paper we focus on estimation of the parameter in a regression model in which the conditional density of the response satisfies a single-index assumption. We provide asymptotic results for a new M-estimation procedure for the index parameter. This procedure can be seen as a generalization of the method of Delecroix [*et al.*]{} (2003) to the case of censored regression.
As in the uncensored case, we show that, regarding to the estimation of the parametric part of our model, there is an asymptotic equivalence between our semiparametric approach and a parametric one relying on some prior knowledge on the family of regression functions. For the nonparametric part, we use kernel estimators of conditional densities as in Delecroix [*et al.*]{} (2003). Since the performance of kernel estimators strongly relies on the choice of the smoothing parameter, we also provide a method to choose this parameter adaptively. Another technical issue in our approach concerns a truncation parameter involved in our procedure. This problem of truncation directly comes from the censored framework, where estimators of the underlying distribution functions sometimes fail to estimate correctly the tail of the distribution. This problem is traditionally circumvented for example by assuming integrability assumptions on the response and censoring distribution, see e.g. [@Stute99]. On the other hand, truncation procedure consists of removing the observations which are too large in the estimation of the regression function, see e.g. [@Heuchenne07], or condition (2.2) in [@Brunel06] which can be interpreted as such kind of truncation. Until now, the truncation bounds which were used were arbitrary fixed, and usually no method is proposed to discuss a method for choosing this truncation bound in practical situations. Therefore, in the new method we propose, we also provide a data-driven procedure to choose the truncation parameter. In our practical implementations, we used a criterion based on an asymptotic discussion which focuses on the mean-squared error associated with the estimation of the single index parameter. We also suggest some possible adaptations to other type of criterion which are covered by our theoretical results.
In section \[secreg\], we introduce our censored regression model and present our estimation procedure. It relies on the Kaplan-Meier estimator (1958) of the distribution function, and on semiparametric estimators of the conditional density. Following the procedure of Delecroix [*et al.*]{} (2003), we considered kernel based estimators. Our theoretical results are presented in section \[theorique\]. In section \[sec\_simul\] we report simulation results and analysis on real data. Section \[lemme\] contains the detailed proof of our Main Lemma which states the asymptotic equivalence of estimating the parameter in the semiparametric and parametric models. All the technicalities are postponed to the section \[technical\].
Censored regression model and estimation procedure {#secreg}
==================================================
Notations and general setting {#secnot}
-----------------------------
Let $Y_1, \ldots, Y_n$ be i.i.d. copies of a random response variable $Y
\in \mathbb{R},$ and let $X_1, \ldots,X_n$ be i.i.d. copies of a random vector of covariates $X\in \mathcal{X},$ where $\mathcal{X}$ is a compact subset of $\mathbb{R}^d$. Introducing $C_1,\ldots,C_n$ i.i.d. replications of the censoring variable $C\in \mathbb{R},$ we consider the following censored regression framework, where the observations are $$\begin{cases}
Z_i = Y_i\wedge C_i & 1\leq i\leq n,\\
\delta_i \;= \mathds{1}_{\{Y_i\leq C_i\}} & 1\leq i\leq n, \\
X_i \in {\mathcal{X} \subset \mathbb{R}^d} & 1\leq i \leq n.
\end{cases}$$ Let us introduce some notations for the distribution functions of the random variables appearing in this model, that is $H(t)= \mathbb{P}(Z\leq t),$ $F_X(t)= \mathbb{P}(X\leq t),$ $F_Y(t)= \mathbb{P}(Y\leq t),$ $F_{X,Y}(x,y)= \mathbb{P}(X\leq x,Y\leq y)$ and $G(t)= \mathbb{P}(C\leq t).$ A major difficulty arising in censored regression models stands in the unavailability of the empirical distribution function to estimate functions $F_Y,$ $F_{X,Y}$ and $G,$ which must be replaced by Kaplan-Meier estimators.
We are interested in estimating $f(y|x),$ where $f(y|x)$ denotes the conditional density of $Y$ given $X=x$ evaluated at point $y.$ If one has no insight on the function $f,$ it becomes necessary to perform nonparametric estimation of the conditional density. In absence of censoring, a classical way to proceed is to use kernel smoothing, see e.g. [@Bashtannyk01]. However, the so-called “curse of dimensionality” prevents this approach from being of practical interest when the number of covariates is important ($d>3$ in practice). Therefore it becomes relevant to consider semiparametric models which appear to be a good compromise between the parametric (which relies on strong assumptions on the function $f$ which may not hold in practice) and the nonparametric approach (which relies on fewer assumptions). In the following, we will consider the following semiparametric single-index regression model, $$\label{model} \exists \, \theta_0 \in \Theta \subset\mathbb{R}^ d \; s.a. \; f(y|x)=f_{\theta_0}(y,x'\theta_0),$$ where $f_{\theta}(y,u)$ denotes the conditional density of $Y$ given $X'\theta=u$ evaluated at $y.$ For identifiability reasons, we will impose that the first component of $\theta_0$ is one. In comparison to Cox regression model for absolute continuous variables, our model (\[model\]) is more general, since it only assumes that the law of $Y$ given $X$ depends on an unknown linear combination of the covariates, without imposing additional conditions on the conditional hazard rate.
Model (\[model\]) has been considered by Delecroix [*et al*]{}. (2003) in the uncensored case. However, their procedure can not be directly applied in the censored framework since the responses variables are not directly observed. As a consequence, the empirical distribution function is unavailable, and most of the tools used in this context are not at our disposal. A solution consists of using procedures relying on Kaplan-Meier estimators for the distribution function. An important difficulty arising in this type of techniques stands in the poor behavior of Kaplan-Meier estimators in the tail of the distribution. A practical way to prevent us from this kind of drawback is to consider truncated version of the variable $Y.$ In the following, we will consider $A_{\tau}$ a sequence of compacts included in the set $\{t:\tau_1\leq t \leq \tau\},$ for $\tau\leq \tau_0,$ where $\tau_0<\inf\{t:H(t)=1\}.$ Using only the observations in $A_{\tau}$ allow us to avoid the bad behavior of usual Kaplan-Meier estimators in the tail of the distribution. Moreover, this technique of truncation is particularly adapted to our problem of estimating $\theta_0.$ In our framework, this truncation does not lead to any asymptotic bias, since, denoting by $f^{\tau}(\cdot|x)$ the conditional density of $Y$ given $X=x$ and $Y\in A_\tau,$ for any $\tau<\infty,$ we have, under (\[model\]), $$\label{model2}
f^{\tau}(y|x)=f^{\tau}_{\theta_0}(y,x'\theta_0),$$ where $f^{\tau}_{\theta}(y,u)$ denotes the conditional density of $Y$ given $X'\theta=u$ and $Y\in A_\tau$ evaluated at $y,$ and where the parameter is the same in (\[model\]) as in (\[model2\]). In section \[tauadapt\], we will discuss a new method allowing to choose $\tau$ from the data in order to improve the performance in estimating $\theta_0$.
Estimation procedure
--------------------
We will extend the idea behind the procedure developed by Delecroix [*et al.*]{} (2003), adapting it to our censored framework. First assume that we know the family of functions $f^{\tau}_{\theta}.$ This approach is a modification of the maximum likelihood estimation procedure. Define, for any function $J\geq 0,$ $$\begin{aligned}
L^{\tau}(\theta,J) &= E\left[\log f^{\tau}_{\theta}(Y_i,\theta'X_i)J(X_i)\mathds{1}_{Y_i\in A_{\tau}}\right]=\int \log f^{\tau}_{\theta}(y,\theta'x)J(x)\mathds{1}_{y\in A_{\tau}}dF_{X,Y}(x,y).\end{aligned}$$ Here, $J$ is a positive trimming function which will be defined later in order to avoid denominators problems in the nonparametric part of the model, see section \[J\]. From (\[model2\]), $\theta_0$ maximizes $L^{\tau}(\theta,J)$ for any $\tau<\infty,$ this maximum being unique under some additional conditions on the regression model and $J.$ Since, in our framework, $F_{X,Y}$ and $f^{\tau}_{\theta}$ are unknown, it is natural to estimate them in order to produce an empirical version of $L^{\tau}(\theta,J).$
### Estimation of $F_{X,Y}$
In the case where there is no censoring (as in Delecroix [*et al.*]{} (2003)), $F_{X,Y}$ can be estimated by the empirical distribution function. In our censoring framework, the empirical distribution function of $(X,Y)$ is unavailable, since it relies on the true $Y_i'$s which are not observed. A convenient way to proceed consists of replacing it by some Kaplan-Meier estimator such as the one proposed by [@Stute93]. Let us define the Kaplan-Meier estimator ([@Kaplan58]) of $F_Y,$ $$\begin{aligned}
\hat{F}_Y(y) &= 1-\prod_{i:Z_i\leq t}\left(1-\frac{1}{\sum_{j=1}^n\mathds{1}_{Z_j\geq Z_i}}\right)^{\delta_i}
\\
&=\sum_{i=1}^n \delta_i W_{in}\mathds{1}_{Z_i\leq y},\end{aligned}$$ where $W_{in}$ denotes the “jump” of Kaplan-Meier estimator at observation $i$ (see [@Stute93]). To estimate $F_{X,Y},$ Stute proposes to use $$\hat{F}(x,y)=\sum_{i=1}^n \delta_iW_{in}\mathds{1}_{Z_i\leq y,X_i\leq x}.$$ Let us also define the following (uncomputable) estimator of the distribution function, $$\tilde{F}(x,y)=\sum_{i=1}^n \delta_iW_{i}^*\mathds{1}_{Z_i\leq y,X_i\leq x},$$ where $W_i^*=n^{-1}[1-G(Z_i-)]^{-1}.$ The link between $\hat{F}$ and $\tilde{F}$ comes from the fact that, in the case where $\mathbb{P}(Y=C)=0,$ $$\label{explicit_jump}
W_{in}=n^{-1}[1-\hat{G}(Z_i-)]^{-1},$$ where $\hat{G}$ denotes the Kaplan-Meier estimator of $G$ (see [@Satten01]). Asymptotic properties of $\hat{F}$ can be deduced from studying the difference with the simplest but uncomputable estimator $\tilde{F}.$
If we know the family of regression functions $f^{\tau}_{\theta},$ it is possible to compute the empirical version of $L^{\tau}(\theta,J)$ using $\hat{F},$ that is $$\begin{aligned}
L_n^{\tau}(\theta,f^{\tau},J) &= \int \log f^{\tau}_{\theta}(y,\theta'x)J(x)\mathds{1}_{y\in A_\tau}d\hat{F}(x,y) \\
&= \sum_{i=1}^n \delta_iW_{in}\log f^{\tau}_{\theta}(Z_i,\theta'X_i)J(X_i)\mathds{1}_{Z_i\in A_{\tau}}.\end{aligned}$$ In the case $J\equiv 1,$ the estimator of $\theta_0$ obtained by maximizing $L^{\tau}_n$ would turn out to be an extension of the maximum likelihood estimator of $\theta_0,$ used in presence of censoring.
Estimation of $f^{\tau}_{\theta}$
---------------------------------
In our regression model (\[model2\]), the family $\{f_{\theta}^{\tau},\theta\in \Theta\}$ is actually unknown. As in Delecroix [*et al.*]{} (2003), we propose to use nonparametric kernel smoothing to estimate $f_{\theta}^{\tau}.$ Introducing a kernel function $K$ and a sequence of bandwidths $h,$ define $$\begin{aligned}
\label{kernel_estimator} \hat{f}_{\theta}^{h,\tau}(z,\theta'x) &= \frac{\int
K_h(\theta'x-\theta'u)K_h(z-y)\mathds{1}_{y\in A_{\tau}}d\hat{F}(u,y)}{\int
K_h(\theta'x-\theta'u)\mathds{1}_{y\in A_{\tau}}d\hat{F}(u,y)},\end{aligned}$$ where $K_h(\cdot)=h^{-1}K(\cdot/h).$ Also define $f^{*h,\tau}$ the kernel estimator based on function $\tilde{F},$ that is $$\begin{aligned}
f^{*h,\tau}_{\theta}(z,\theta'x) &= \frac{\int
K_h(\theta'x-\theta'u)K_h(z-y)\mathds{1}_{y\in A_{\tau}}d\tilde{F}(u,y)}{\int
K_h(\theta'x-\theta'u) \mathds{1}_{y\in A_{\tau}}d\tilde{F}(u,y)}.\end{aligned}$$ $f^{*h,\tau}$ will play an important role in studying the asymptotic behavior of $\hat{f}^{h,\tau}.$ Indeed, $f^{*h,\tau}$ is theoretically more easy to handle with, since it relies on sums of i.i.d. quantities, which is not the case for $\hat{F}.$ Since $f^{*h,\tau}$ can be studied by standard kernel arguments, the most important difficulty will arise from studying the difference between $\hat{f}^{h,\tau}$ and $f^{*h,\tau}.$
In the following, we will impose the conditions below on the kernel function.
\[noyau\]Assume that
- $K$ is a twice differentiable and four order kernel with derivatives of order 0, 1 and 2 of bounded variation. Its support is contained in $[-1/2,1/2]$ and $\int_{\mathbb{R}}K(s)ds =1$,
- $\|K\|_{\infty}:=\sup_{x\in \mathbb{R}}|K(x)|<\infty$,
- $\mathcal{K}:=\{K\big((x-\cdot)/h\big):h>0, x\in \mathbb{R}^d\}$ is a pointwise measurable class of functions,
- $h\in \mathcal{H}_n\subset [an^{-\alpha},bn^{-\alpha}]$ with $a,b\in \mathbb{R},$ $1/8<\alpha<1/6$ and where $\mathcal{H}_n$ is of cardinality $k_n$ satisfying $k_nn^{-4\alpha}\rightarrow 0.$
The trimming function $J$ {#J}
-------------------------
The reason behind introducing function $J$ has to be connected with the need to prevent us from denominators close to zero in the definition (\[kernel\_estimator\]). Ideally, we would need to use the following trimming function, $$\label{J0} J_0(x,c)=\tilde{J}(f_{\theta_0'X},\theta_0'x,c),$$ where $c$ is a strictly positive constant, $f_{\theta_0'X}$ denotes the density of $\theta_0'X$ and $\tilde{J}(g,u,c)=\mathds{1}_{g(u)>c}.$ Unfortunately, this function relies on the knowledge of parameter $\theta_0$ and $f_{\theta_0'X}.$ Therefore, we will have to proceed in two steps, that is first obtain a preliminary consistent estimator of $\theta_0,$ and then use it to estimate the trimming function $J_0$ which will be needed to achieve asymptotic normality of our estimators of $\theta_0.$
We will assume that we know some set $B$ on which $\inf\{f_{\theta'X}(\theta'x): x\in B, \theta \in \Theta\}>c,$ where $c$ is a strictly positive constant. In a preliminary step, we can use this set $B$ to compute the preliminary trimming $J_B(x)=\mathds{1}_{x\in B}.$ Using this trimming function, and a deterministic sequence of bandwidth $h_0$ satisfying $(A4)$ in Assumption \[noyau\], we define a preliminary estimator $\theta_n$ of $\theta_0,$ $$\label{preliminary} \theta_n =\operatorname*{arg\,min}_{\theta\in \Theta}L^{\tau}_n(\theta,\hat{f}^{h_0,\tau},J_B).$$ Let us stress the fact that $B$ is assumed to be known by the statistician. This is a classical assumption in single-index regression (see Delecroix [*et al.*]{}, (2006)). However, in practice, the procedure does not seem very sensitive to the choice of $B.$ The bandwidth $h_0$ we consider in the preliminary step can be any sequence decreasing to zero slower than $n^{-1/2}.$ Adaptive choice of $h_0$ could be considered (using, for instance, the same choice as in the final estimation step, see below). However, since we will only need $\theta_n$ to be a preliminary consistent estimator, and the final estimator will not be very sensitive to an adaptive choice of $h_0$ while computing $\theta_n,$ we do not consider this case in the following.
With, at hand, this preliminary estimator $\theta_n,$ we can compute an estimated version of $J_0$ which will happen to be equivalent to $J_0$ (see Delecroix [*et al.*]{} (2006) page 738), that is $$\label{Jhat} \hat{J}_0(x,c)=\tilde{J}(\hat{f}^{h_0,\tau}_{\theta_n'X},\theta_n'x,c).$$
For each sequence of bandwidths satisfying (A4) in Assumption \[noyau\], and for each truncation bound $\tau,$ we can define an estimator of $\theta_0$ $$\label{thetahat} \hat{\theta}^{\tau}(h)=\operatorname*{arg\,max}_{\theta \in \Theta_n} L^{\tau}_n(\theta,\hat{f}^{h,\tau},\hat{J}_0),$$ where $\Theta_n$ is a shrinking sequence of neighborhoods accordingly to the preliminary estimation. However, as for any smoothing approach, the performance of this procedure strongly depends on the bandwidth sequence. Therefore it becomes particularly relevant to provide an approach which automatically selects the most adapted bandwidth according to the data. Then, the new question arising from the censored framework comes from the adaptive choice of the truncation parameter $\tau.$
Adaptive choice of the bandwidth {#hadapt}
--------------------------------
Our procedure consists of choosing from the data, for each $\theta,$ a bandwidth which is adapted to the computation of $f^{\tau}_{\theta}(z,u).$ For this, we use an adaptation of the cross-validation technique of [@Fan04], that is $$\hat{h}^{\tau}(\theta)=\operatorname*{arg\,min}_{h \in \mathcal{H}_n}\sum_{i=1}^n W_{in}\mathds{1}_{Z_i\in A_{\tau}}\left\{\int_{A_{\tau}} \hat{f}^{h,\tau}_{\theta}(z,\theta'X_i)^2dz-2\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i)\right\}.$$ This criterion is (up to a quantity which does not depend on $h$) an empirical version of the ISE criterion defined in (3.3) in [@Fan04] (in a censored framework), that is $\int_{A_{\tau}}\int\{\hat{f}^{h,\tau}_{\theta}(z,\theta'x)-f^{\tau}_{\theta}(z,\theta'x)\}^2f_{\theta'X}(\theta'x)dxdz.$
The estimator of $\theta_0$ with an adaptive bandwidth is now defined as $$\label{thetahat2} \hat{\theta}^{\tau}=\operatorname*{arg\,max}_{\theta \in
\Theta_n} L_n^{\tau}(\theta,\hat{f}^{\hat{h},\tau}, \hat{J}_0).$$ In the above notation, $\hat{h}$ depends on $\theta$ and $\tau,$ which was not emphasized to shorten the notation.
Adaptive choice of $\tau$ {#tauadapt}
-------------------------
As we already mentioned, the Kaplan-Meier estimator does not behave well in the tail of the distribution. For example, if some moment conditions are not satisfied, it is not even $n^{1/2}$-consistent. Moreover, even in the case where an appropriate moment condition holds, it may happen (at least for a finite sample size) that the weights corresponding to the large observations are too important and considerably influence the estimation procedure. For this reason, we introduced a truncation by a bound $\tau.$ However, a large number of existing procedure which also rely on such kind of truncation do not consider the problem of choosing $\tau$ from the data. We propose to select $\tau$ from the data in the following way. Suppose that we have a consistent estimator of the asymptotic mean squared error, $$E^2(\tau)=\limsup_n E\left[\|\hat{\theta}^{\tau}(\hat{h}^{\tau})-\theta_0\|^2\right],$$ say $\hat{E}^2(\tau)$ satisfying $$\label{consist_var} \sup_{\tau_1\leq\tau\leq \tau_0}|\hat{E}^2(\tau)-E^2(\tau)|\rightarrow 0, \text{ in probability}.$$ Such an estimator will be proposed in section \[sec\_simul\]. Using this empirical estimator, we propose to choose $\tau$ in the following way, that is $$\hat{\tau}=\operatorname*{arg\,min}_{\tau_1\leq \tau \leq \tau_0}\hat{E}^2(\tau).$$
Our final estimator of $\theta_0$ is based on an adaptive bandwidth and an adaptive choice of truncation parameter $\tau,$ that is $$\hat{\theta}=\hat{\theta}^{\hat{\tau}}.$$
As we already said, truncating the data does not introduce additional bias in the estimation of $\theta_0$. On the other hand, removing too many data points could strongly increase the variance and removing some of the largest data points will decrease it. Then, our selection procedure $\hat{\tau}$ is based on estimating the variance of $\hat{\theta}$ and consists of taking from the data the truncation parameter $\tau$ that seems to be the best compromise between these two aspects.
Asymptotic results
==================
[\[theorique\]]{}
Consistency
-----------
The assumptions needed for consistency can basically be split into three categories, that is identifiability assumptions, assumptions on the regression model (\[model2\]) itself and finally assumptions on the censoring model.
**Identifiability assumption and assumption on the regression model.**
\[identif\] Assume that for all $\tau_1\leq \tau\leq \tau_0$ and all $\theta\in \Theta-\{\theta_0\},$ $$L_{\tau}(\theta_0,J_B)-L_{\tau}(\theta,J_B)>0.$$
\[lgnu\] Assume that for $\theta_1, \theta_2 \in \Theta$, for a bounded function $\Phi(X)$ and for some $\gamma>0$, we have $$\sup_{\tau}\|f^{\tau}_{\theta_1}(y,\theta_1'x)-f^{\tau}_{\theta_2}(y,\theta_2'x)\|_{\infty} \leq \|\theta_1-\theta_2\|^{\gamma}\Phi(X).$$
**Assumptions on the censoring model.**
$\mathbb{P}(Y=C)=0.$
This classical assumption in a censored framework avoids problems caused by the lack of symmetry between $C$ and $Y$ in the case where there are ties.
\[ident\] Identifiability assumption : we assume that
- $Y$ and $C$ are independent.
- $\mathbb{P}(Y\leq C|X,Y)=\mathbb{P}(Y\leq C|Y).$
This last assumption was initially introduced by [@Stute93]. An important particular case in which assumption \[ident\] holds is when $C$ is independent from $(X,Y).$ However, assumption \[ident\] is a more general and widely accepted assumption, which allows the censoring variables to depend on the covariates.
\[consistency\] Under Assumptions \[identif\] to \[ident\], $$\label{consist1} \sup_{\theta\in \Theta,{\tau_1\leq \tau \leq \tau_0} }|L_n^{\tau}(\theta,\hat{f}^{h_0,\tau},J_B)-L^{\tau}(\theta,J_B)|=o_P(1),$$ and consequently, $$\theta_n \rightarrow_{\mathbb{P}}\theta_0.$$
To show (\[consist1\]), we will proceed in two steps. First we consider $L_n^{\tau}(\theta,f^{\tau},J_B)-L^{\tau}(\theta,J_B)$ (parametric problem), and then $L_n^{\tau}(\theta,\hat{f}^{h_0,\tau},J_B)-L_n^{\tau}(\theta,f^{\tau},J_B).$
**Step 1.** From Assumption \[lgnu\], the family $\{\log(f_{\theta}^{{\tau}}(\cdot,\theta'\cdot )),\theta \in \Theta,\,{\tau_1\leq \tau \leq \tau_0}\}$ is seen to be $P-$ Glivenko-Cantelli. Using an uniform version of [@Stute93] leads to $\sup_{\theta}|L_n^{\tau}(\theta,f^{\tau},J_B)-L^{\tau}(\theta,J_{B})|\rightarrow_{\mathbb{P}} 0.$
**Step 2.** We have, on the set $\Theta'B,$ $$|\log \hat{f}_{\theta}^{h_0,\tau}(y,u)-\log f_{\theta}^{{\tau}}(y,u)|\leq c^{-1}[\hat{f}_{\theta}^{h_0,\tau}(y,u)-f_{\theta}^{{\tau}}(y,u)].$$ Hence, $$\begin{aligned}
\sup_{\theta,{\tau}}|L_n^{\tau}(\theta,\hat{f}^{h_0,\tau},J_{B})-L_n(\theta,f^{\tau},J_{B})| & \leq c^{-1}\sup_{\theta,y,u,\tau}|\hat{f}_{\theta}^{h_0,\tau}(y,u)-f_{\theta}^{{\tau}}(y,u)|\mathds{1}_{u\in \Theta'B,y\leq\tau}\int d\hat{F}{(x,y)}
\\
& \leq c^{-1}\sup_{\theta,y,u,\tau}|\hat{f}_{\theta}^{h_0,\tau}(y,u)-f_{\theta}^{{\tau}}(y,u)|\mathds{1}_{u\in \Theta'B,y\leq\tau}.\end{aligned}$$ Using the uniform convergence of $\hat{f}_{\theta}^{h,\tau}$ (see Proposition \[convergence\_rate\] and Lemma \[leplusdur\]), deduce that\
$\sup_{\theta,\tau}|L^{\tau}_n(\theta,\hat{f}^{h_0,\tau},{J_B})-L^{\tau}_n(\theta,f^{\tau},{J_B})|\rightarrow_{\mathbb{P}} 0.$
Asymptotic normality {#sec_normal}
--------------------
To obtain the asymptotic normality of our estimator, we need to add some regularity assumptions on the regression model.
\[regularity\] Denote by $\nabla_{\theta}f_{\theta}^{\tau}(y,x)$ (resp. $\nabla^2_{\theta}f_{\theta}^{\tau}(y,x)$) the vector of partial derivatives (resp. the matrix of second derivatives with respect to $\theta$) of $f_{\theta}^{\tau}$ with respect to $\theta$ and computed at point $(\theta,x,y).$ Assume that for $\theta_1, \theta_2 \in \Theta$, for a bounded function $\Phi(X)$ and for some $\gamma>0$, we have $${\sup_\tau} \|\nabla^2_{\theta}f^{\tau}_{\theta_1}(y,x)-\nabla^2_{\theta}f^{\tau}_{\theta_2}(y,x)\|_{\infty} \leq \|\theta_1-\theta_2\|^{\gamma}\Phi(X).$$
\[donsker1\] Using the notation of [@Vaart96] in section 2.7, define $$\begin{aligned}
\mathcal{H}_1 &=
\mathcal{C}^{1+\delta}(\theta_0'\mathcal{X}\times
A_{\tau},M), \\
\mathcal{H}_2 &=
x\mathcal{C}^{1+\delta}(\theta_0'\mathcal{X}\times
A_{\tau},M)+\mathcal{C}^{1+\delta}(\theta_0'\mathcal{X}\times A_{\tau},M)\end{aligned}$$ Assume that $f^{\tau}_{\theta_0}(\cdot,\cdot)\in \mathcal{H}_1$ (as a function of $\theta_0'x$ and $y$) and $\nabla_{\theta}f^{\tau}_{\theta_0}(\cdot,\cdot)\in \mathcal{H}_2.$
If the family of functions $f^{\tau}$ [was]{} known (parametric problem), the asymptotic normality of $\hat{\theta}$ could be deduced from elementary results on Kaplan-Meier integrals (see section \[technical\] for some brief review of these results), as in [@Stute99] or in Delecroix [*et al.*]{} (2008). Using this kind of results, we can derive the following Lemma (see section \[technical\] for the proof) which is sufficient to obtain the asymptotic law of $\hat{\theta}$ in the parametric case, from Theorem 1 and 2 of [@Sherman94].
\[parametric\] Under Assumptions \[regularity\] and \[donsker1\], we have the following representations:
1. on $o_P(1)-$neighborhoods of $\theta_0,$ $$L_n^{\tau}(\theta,f^{ \tau},J_0)=L^{\tau}(\theta,J_0)+(\theta-\theta_0)'T_{1n}(\theta)+(\theta-\theta_0)'T_{2n}(\theta)(\theta-\theta_0)
+T_{3n}(\theta)+T_{4n}(\theta_0),$$ with $\sup_{\theta,{ \tau}}|T_{1n}|=O_P(n^{-1/2}),$ $\sup_{\theta,{ \tau}}|T_{2n}|=o_P(1),$ $\sup_{\theta,{ \tau}}|T_{3n}|=O_P(n^{-1})$ and $T_{4n}(\theta_0)=L^{\tau}_n(\theta_0,f^{ \tau},J_0).$
2. on $O_P(n^{-1/2})-$ neighborhoods of $\theta_0,$ $$L_n^{\tau}(\theta,f^{ \tau},J_0)=n^{-1/2}(\theta-\theta_0)'W_{n,\tau}-\frac{1}{2}(\theta-\theta_0)'V_{\tau}(\theta-\theta_0)+T_{4n}(\theta_0)
+T_{5n}(\theta),$$
with $\sup_{\theta,{ \tau}}|T_{5n}|=o_P(n^{-1}),$ and defining $f_1(x,y)=f_{\theta_0}^{{ \tau}^{-1}}(y,\theta_0'x)J_0(x,c)\nabla_{\theta}f_{\theta_0}^{{ \tau}}(y,x),$ $$\begin{aligned}
W_{n,\tau} &= \frac{1}{n^{1/2}}\sum_{i=1}^n \psi(Z_i,\delta_i,X_i;f_1\mathds{1}_{A_{\tau}}), \\
V_{\tau} &= E\left[f_{\theta_0}^{{ \tau}^{-2}}(Y,\theta_0'X)J_0(X,c)\nabla_{\theta}f_{\theta_0}^{{ \tau}}(Y,X)\nabla_{\theta}f_{\theta_0}^{{ \tau}}(Y,X)'
\mathds{1}_{Y\in A_\tau}\right],\end{aligned}$$ where $\psi$ is defined in Theorem \[km\].
In the following Theorem, we show that the semiparametric estimator proposed in section \[secreg\] has the same asymptotic law as in the fully parametric case.
\[maintheorem\] Define $\tau^*=\operatorname*{arg\,min}_{\tau} E^{2}(\tau).$ Under Assumptions \[noyau\] to \[donsker1\], we have the following asymptotic i.i.d. representation, $$\label{iid}
\hat{\theta}-\theta_0= -\frac{1}{n^{1/2}}V_{\tau^*}^{-1}W_{n,\tau^*}+o_P(n^{-1/2}),$$ where $V_{\tau}$ and $W_{n,\tau}$ are defined in Lemma \[parametric\]. As a consequence, $$n^{1/2}(\hat{\theta}-\theta_0)\Longrightarrow \mathcal{N}(0,\Sigma_{\tau^*})$$ where $\Sigma_{\tau^*}=V_{\tau^*}^{-1}\Delta_{\tau^*}(f_1) V_{\tau^*}^{-1},$ $\Delta_{\tau^*}(f_1)=Var\left(\psi(Z,\delta,X;f_1\mathds{1}_{A_{\tau^*}})\right)$ and $f_1$ is defined in Lemma \[parametric\].
This Theorem is a consequence of the Main Lemma below. This result shows that, asymptotically speaking, maximizing $L^{\tau}_n(\theta,\hat{f}^{h,\tau},J)$ is equivalent to maximizing $L^{\tau}_n(\theta,f^{ \tau},J).$
\[mainlemma\] Under Assumptions \[noyau\] to \[donsker1\], $$L^{\tau}_n(\theta,\hat{f}^{h,\tau},\hat{J}_0)=L^{\tau}_n(\theta,f^{ \tau},J_0)+(\theta-\theta_0)'R_{1n}(\theta,h,\tau)+(\theta-\theta_0)'R_{2n}(\theta,h,\tau)(\theta-\theta_0)+\tilde{L}^{\tau}_n(\theta_0),$$ where $$\begin{aligned}
\sup_{\theta \in \Theta_n,h\in \mathcal{H}_n,\tau_1\leq \tau\leq \tau_0} R_{1n}(\theta,h,\tau) &= o_P(n^{-1/2}), \\
\sup_{\theta \in \Theta_n,h\in \mathcal{H}_n,\tau_1\leq \tau\leq \tau_0} R_{2n}(\theta,h,\tau) &= o_P(1).\end{aligned}$$ and $$\tilde{L}^{\tau}_n(\theta_0)=A^{\tau}_{1n}(\theta_0,\hat{f}^{h,\tau})-B^{\tau}_{4n}(\theta_0,\hat{f}^{h,\tau})$$ where $A^{\tau}_{1n}(\theta_0,\hat{f}^{h,\tau})$ and $B^{\tau}_{4n}(\theta_0,\hat{f}^{h,\tau})$ are defined in the proof of this Lemma.
In view of Theorem 1 and 2 of [@Sherman94], this result will allow us to obtain the rate of convergence of our estimators, and then the asymptotic law is the same law as the asymptotic law in the parametric problem.
Define $$\begin{aligned}
\Gamma_{0n}(\theta,\tau,h) &= L^{\tau}_n(\theta,\hat{f}^{h,\tau},\hat{J}_0), \\
\Gamma_{1n}(\theta,\tau) &= L^{\tau}_n(\theta,\hat{f}^{\hat{h},\tau},\hat{J}_0), \\
\Gamma_{2n}(\theta) &= L^{\hat{\tau}}_n(\theta,\hat{f}^{\hat{h},\hat{\tau}},\hat{J}_0).\end{aligned}$$ We now apply Theorem 1 and 2 in Sherman (1994) to $\Gamma_{in},$ for $i=0,1,2.$ From our Main Lemma and Lemma \[parametric\], we deduce, that the representation (11) in Theorem 2 of [@Sherman94] holds for $i=0,1,2,$ on $O_P(n^{-1/2})-$ neighborhoods of $\theta_0,$ with $W_n$ and $V$ defined in Lemma \[parametric\]. The asymptotic representation (\[iid\]) is a by-product of the proof of Theorem 2 in [@Sherman94] and of the i.i.d. representations of Kaplan-Meier integrals (see Theorem \[km\]).
Simulation study and real data analysis {#sec_simul}
=======================================
Practical implementation of the adaptive choice of $\tau$ {#secsecsimul}
---------------------------------------------------------
From the proof of Theorem \[maintheorem\], we have the representation $$\hat{\theta}-\theta_0=-\frac{1}{n}\sum_{i=1}^n V_{\tau}^{-1}\psi(Z_i,\delta_i,X_i;f_1\mathds{1}_{A_{\tau}})+o_P(n^{-1/2}).$$ As in [@Stute95], the function $\psi$ of Theorem \[km\] can be estimated from the data in the following way by $$\hat{\psi}(Z,\delta,X;\hat{f_1}\mathds{1}_{A_{\tau}})=\frac{\delta\hat{f_1}(X,Z)}{1-\hat{G}(Z-)}+\int \frac{\int_{y}^{\tau_0}\int_{\mathcal{X}} \hat{f_1}(x,t)d\hat{F}(x,t) dM^{\hat{G}}(y)}{1-\hat{H}(y)},$$ where $\hat{f}_1$ is our kernel estimator of $f_1$ and $\hat{H}$ is the empirical estimator of $H$. To consistently estimate $\Delta(f_1),$ we use the general technique proposed by [@Stute96], that is $$\label{varianceestimee}
\hat{\Delta}_{\tau}(f_1) = \frac{1}{n}\sum_{i=1}^n \left[\hat{\psi}(Z_i,\delta_i,X_i;\hat{f}_1)-\frac{1}{n}\sum_{i=1}^n\hat{\psi}(Z_i,\delta_i,X_i;\hat{f}_1)\right]^{\otimes 2},$$ where $\otimes 2$ denotes the product of the matrix with its transpose. A consistent estimator of $V_{\tau}$ can then be computed as $$\hat{V}_{\tau}=\int \hat{f}^{{ h,\tau}^{-2}}_{\hat{\theta}}(y,\hat{\theta}'x)\hat{J}_0(x,c)\nabla_{\theta}\hat{f}_{\hat{\theta}}^{{ h,\tau}}(y,x)\nabla_{\theta}\hat{f}_{\hat{\theta}}^{{ h,\tau}}(y,x)'
\mathds{1}_{y\in A_\tau}d\hat{F}(x,y).$$ To estimate the asymptotic mean squared error we use $$\hat{E}^2_{\tau}=\frac{1}{n}\hat{W}_{n,\tau}'\hat{V_{\tau}}^{-1}\hat{V_{\tau}}^{-1}\hat{W}_{n,\tau}.$$
Simulation study
----------------
In order to check the finite sample behavior of our estimators of $\theta_0$, we conducted some simulations using a similar model as the one in Delecroix [*et al.*]{} (2003). We considered the following regression model, $$Y_i = \theta_0'X_i+\varepsilon_i, \quad i=1, \ldots,n$$ where $Y_i\in \mathbb{R}$, $\theta_0=(1,0.5,1.4,0.2)'$ and $X_i \sim \otimes^4 \{0.2\,\mathcal{N}(0,1)+0.8\,\mathcal{N}(0.25,2)\}.$ The errors are centered and normally distributed with conditional variance equal to $|\theta_0'X|$. We used the kernel $$K(u) = 2k(u)-k*k(u)$$ where $*$ denotes the convolution product and $$k(u)=\frac{3}{4}(1-u^2)\mathds{1}_{|u|\leq 1}$$ is the classical Epanechnikov kernel. The censoring distribution was selected to be exponential with parameter $\lambda$ which allows us to fix the proportion of censored responses ($p=25\%$ and $p=40\%$ in our simulations). $\hat{h}$ was chosen using a regular grid between 1 and 1.5.
Our estimator $\hat{\theta}^{\hat \tau}$ was compared with two other estimators, that is $\hat{\theta}^{\infty}$ which does not rely on an adaptive choice of $\tau,$ and $\hat{\theta}^{ADE}$ which is obtained using the average derivative method of [@Lu05]. In the tables below we report our results over $100$ simulations from samples of size $100$ and $200$ for two different rates of censoring. Recalling that the first component of $\theta_0$ is imposed to be one, we only have to estimate the three other components. For each estimator, the Mean Squared Error $ E(\|\hat{\theta}-\theta_0\|^2)$ is decomposed into bias and covariance.
-------------------------------------------------------------------------------------------------
$p=25\% , n=100$ Bias Variance MSE
---------------------------- ---------------------------- --------------------------- -----------
$\hat{\theta}^{ADE}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.6714181
-0.112\\ 0.14 & 0.005 & -0.022 \\
-0.551\\ 0.005 & 0.075 & 0.016 \\
-0.155 \end{array}\right)$ -0.022 & 0.016 & 0.116
\end{array}\right)$
$\hat{\theta}^{\infty}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.1841227
0.057\\ 0.033 & 0.012 & 0.001 \\
0.215\\ 0.012 & 0.073 & -0.004 \\
0.048 \end{array}\right)$ 0.001 & -0.004 & 0.027
\end{array}\right)$
$\hat{\theta}^{\hat \tau}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.1825980
0.07\\ 0.034 & 0.002 & 0.002 \\
0.221\\ 0.002 & 0.074 & 0 \\
0.028\\ 0.002 & 0 & 0.02
\end{array}\right)$ \end{array}\right)$
-------------------------------------------------------------------------------------------------
: Biases, variances and mean squared errors for $25 \%$ of censoring and sampling of size $100$.
--------------------------------------------------------------------------------------------------
$p=40\% , n=100$ Bias Variance MSE
---------------------------- ---------------------------- ---------------------------- -----------
$\hat{\theta}^{ADE}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 1.280163
-0.334\\ 0.159 & 0.009 & -0.014 \\
-0.743\\ 0.009 & 0.268 & 0.048 \\
-0.158 \end{array}\right)$ -0.014 & 0.048 & 0.165
\end{array}\right)$
$\hat{\theta}^{\infty}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.3829797
0.127\\ 0.11 & -0.034 & -0.01 \\
0.296\\ -0.034 & 0.101 & 0.021 \\
0.096 \end{array}\right)$ -0.01 & 0.021 & 0.059
\end{array}\right)$
$\hat{\theta}^{\hat \tau}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.2239023
0.074\\ 0.064 & -0.005 & -0.004 \\
0.176\\ -0.005 & 0.051 & 0.014 \\
0.061\\ -0.004 & 0.014 & 0.069
\end{array}\right)$ \end{array}\right)$
--------------------------------------------------------------------------------------------------
: Biases, variances and mean squared errors for $40 \%$ of censoring and sampling of size $100$.
-------------------------------------------------------------------------------------------------
$p=25\%,n=200$ Bias Variance MSE
---------------------------- ---------------------------- --------------------------- -----------
$\hat{\theta}^{ADE}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.7620268
-0.189\\ 0.096 & 0.003 & 0.006 \\
-0.578\\ 0.003 & 0.148 & -0.016 \\
-0.133 \end{array}\right)$ 0.006 & -0.016 & 0.131
\end{array}\right)$
$\hat{\theta}^{\infty}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.0910719
0.073\\ 0.033 & 0.004 & -0.004 \\
0.133\\ 0.004 & 0.023 & 0.002 \\
0.015 \end{array}\right)$ -0.004 & 0.002 & 0.012
\end{array}\right)$
$\hat{\theta}^{\hat \tau}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.0364064
0.034\\ 0.007 & 0.001 & 0.004 \\
0.107\\ 0.001 & 0.011 & 0 \\
0.014\\ 0 & 0 & 0.006
\end{array}\right)$ \end{array}\right)$
-------------------------------------------------------------------------------------------------
: Biases, variances and mean squared errors for $25 \%$ of censoring and sampling of size $200$.
--------------------------------------------------------------------------------------------------
$p=40\%,n=200$ Bias Variance MSE
---------------------------- ---------------------------- --------------------------- ------------
$\hat{\theta}^{ADE}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 1.078027
-0.109\\ 0.146 & -0.02 & 0.056 \\
-0.763\\ -0.02 & 0.143 & -0.014 \\
-0.053 \end{array}\right)$ 0.056 & -0.014 & 0.192
\end{array}\right)$
$\hat{\theta}^{\infty}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.2521227
0.104\\ 0.109 & 0.008 & 0.042 \\
0.151\\ 0.008 & 0.049 & 0.003 \\
0.077 \end{array}\right)$ 0.042 & 0.003 & 0.055
\end{array}\right)$
$\hat{\theta}^{\hat \tau}$ $\left(\begin{array}{c} $\left(\begin{array}{ccc} 0.07533921
0.043\\ 0.018 & -0.001 & 0.002 \\
0.14\\ -0.001 & 0.022 & 0.002 \\
0.021\\ 0.002 & 0.002 & 0.014
\end{array}\right)$ \end{array}\right)$
--------------------------------------------------------------------------------------------------
: Biases, variances and mean squared errors for $40 \%$ of censoring and sampling of size $200$.
To give a precise idea of the number of observations which are removed from the study by choosing $\tau$ adaptively, introduce $N=\sharp\{1\leq i \leq n, Z_i\leq \hat{\tau}\}.$ In the following table \[poi\], we evaluated $E[N]$ in the different cases we considered in the simulation study. We also mention the average weight allocated to the largest (uncensored) data point, first in the case where we consider the whole data set (we denote it Weight$^{\infty}$), then in the truncated data set where we removed all data points with $Z_i\geq \hat{\tau}$ (we denote it Weight$^{\hat \tau}$).
$\hat{\mathbb E}(N)$ Weight$^{\infty}$ Weight$^{\hat \tau}$
---------------- ---------------------- ------------------- ----------------------
$n=100,p=25\%$ $90$ $0.0667$ $0.0204$
$n=100,p=40\%$ 87 $0.124$ $0.0236$
$n=200,p=25\%$ $185$ $0.0402$ $0.0119$
$n=200,p=40\%$ 172 $0.0997$ $0.0122$
: Last observed data in the truncating model and weight allocated to the largest observation in each model for different sample sizes and censoring rates.[]{data-label="poi"}
\[poids\]
Clearly the MSE deteriorates when the percentage of censoring increases. According to the simulations, $\hat{\theta}^{\hat \tau}$ and $\hat{\theta}^{\infty}$ outperform $\hat{\theta}^{ADE},$ while, as expected, choosing adaptively $\tau$ improves the quality of the estimation. This is not obvious in the case where there are only $25\%$ of censoring. However, in the case where the level of censoring is high, estimation of the tail of the distribution by Kaplan-Meier estimators becomes more erratic, and the importance of choosing a proper truncation appears in the significant difference between the MSE of $\hat{\theta}^{\hat \tau}$ and $\hat{\theta}^{\infty}.$ Moreover, the importance of truncation becomes obvious if we look at table \[poi\]. We see that, in the case where there is 40 % of censoring, the weight allocated to the largest data-point if we do not use truncation can be up to 10 times (approximatively) the weight allocated to the largest observation in the truncated data set. The ratio is less important in the case where there is 25 % of censoring, but still consequent (in this case, the ratio is approximatively 3). Therefore, it seems that, considering the whole data set, the weight allocated to the largest observation can have a too strong influence on the estimation procedure, which explains the difference of performance of the estimators with or without truncation.
Example : Stanford Heart Transplant Data
----------------------------------------
We now illustrate our method using data from the Stanford Heart Transplant program. This data set was initially studied by [@Miller82]. 184 of 249 patients in this program received a heart transplantation between October 1967 and February 1980. From this data, we considered the survival time as the response variable $Z$, age as the first component of $X$ and the square of age as the second [component]{}. Patients alive beyond February 1980 were considered censored. For easier comparison to previous work on this data set, we concentrate our analysis on the 157 patients out of 184 who had complete tissue typing. Among these 157 cases, 55 were censored.
Several methods of estimation have already been applied to this data set to estimate the following regression model, $$Z=\alpha+\beta'X+\varepsilon(X), \label{modelheart}$$ where $\beta=(\beta_1,\beta_2)',$ $E[\varepsilon(X)|X]=0,$ see [@Miller82], Wei [*et al.*]{} (1990), Stute [*et al.*]{} (2000). Furthermore, nonparametric lack-of-fit tests have shown that the regression model (\[modelheart\]) seemed reasonable, see Stute [*et al.*]{} (2000) and [@Lopez08]. Therefore it seems to us appropriate to experiment our model on this data set. This strengthens the assumption on the residual, by assuming that $\varepsilon(X)=\varepsilon(\theta_0'X),$ where $\theta_0=(1,\beta_2/\beta_1)',$ but allows more flexibility on the regression function.
In the following table, we present our estimators and recall the values of the estimators of $\beta_2/\beta_1$ for the linear regression model (\[modelheart\]). We first computed $\hat{\theta}^{\infty},$ which is our estimator using the whole data set, that is with $\tau=+\infty,$ and compared it to the one obtained by choosing $\tau$ from the data as in section \[secsecsimul\]. In this last case, $\hat{\tau}=Z_{(90)}$ where $Z_{(i)}$ denotes the $i-$th order statistic, this means that it conducted us to remove the $67$ largest observations to estimate $\theta_0$ (but not to estimate Kaplan-Meier weights, which were computed using the whole data set). We computed Weight$^{\infty}=0.0397,$ and Weight$^{\hat{\tau}}= 0.0076$ for the truncated data. Adaptive bandwidth was $1.7$ for $\hat{\theta}^{\infty},$ and $1.3$ for $\hat{\theta}^{\hat \tau}.$ The estimated value of the mean-squared error was $E^2_{\infty}=0.1089375$ and $E^2_{\hat{\tau}}=0.01212701$ for $\hat{\theta}^{\infty}$ and $\hat{\theta}^{\hat \tau}$ respectively.
--------------------------------------------------------------------------------------------------------------------------
Estimator of $\theta_{0,2}=\beta_2/\beta_1$
---------------------------------------------------------------------------- ---------------------------------------------
Miller and Halpern -0.01588785
Wei *[et al.]{} & 63.75\
Stute *[et al.]{} & -0.01367034\
$\hat{\theta}^{\infty}$ (without adaptive choice of $\tau$) & -0.07351351\
$\hat{\theta}^{\hat \tau}$ (with adaptive choice of $\tau$) & -0.0421508\
**
--------------------------------------------------------------------------------------------------------------------------
: Comparison of different estimators of $\theta_{0,2}$.
Our estimators seem relatively close to the ones obtained by [@Miller82] and Stute [*et al.*]{} (2000) using respectively the Buckley-James method and the Kaplan-Meier integrals method for the linear regression model.
Proof of Main Lemma {#lemme}
===================
First, the same arguments as in Delecroix [*et al.*]{} (2006) apply to replace $\hat{J}_0$ by $J_0.$ Define $J_{\theta}(x,c)=\mathds{1}_{f_{\theta'X}(\theta'x)\geq c}.$ From Assumption \[lgnu\] on the density of $\theta'x$, deduce that, on shrinking neighbourhoods of $\theta_0,$ $J_0(x,c)$ can be replaced by $J_{\theta}(x,c/2).$ Using a Taylor expansion, write $$\begin{aligned}
L^{\tau}_n(\theta,\hat{f}^{h,\tau},J_0)-L^{\tau}_n(\theta,f^{\tau},J_0) &= \sum_{i=1}^n \delta_i W_{in}\mathds{1}_{Z_i\in A_{\tau}}\log\left(\frac{\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i)}{f_{\theta}^{{\tau}}(Z_i,\theta'X_i)}\right)J_0(X_i,c) \\
&= \sum_{i=1}^n \frac{\delta_i W_{in}\mathds{1}_{Z_i\in A_{\tau}}\left(\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i)-f_{\theta}^{{\tau}}(Z_i,\theta'X_i)\right)J_0(X_i,c)}{f_{\theta}^{{\tau}}(Z_i,\theta'X_i)} \\
& -\sum_{i=1}^n \frac{\delta_i W_{in}\mathds{1}_{Z_i\in A_{\tau}}\left[\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i)-f_{\theta}^{{\tau}}(Z_i,\theta'X_i)\right]^2J_0(X_i,c)}{\phi(f_{\theta}^{{\tau}}(Z_i,\theta'X_i),\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i))^2} \\
&= A^{\tau}_{1n}(\theta,\hat{f}^{h,\tau})-B^{\tau}_{1n}(\theta,\hat{f}^{h,\tau})\end{aligned}$$ where $\phi(f_{\theta}^{{\tau}}(Z_i,\theta'X_i),\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i))$ is between $\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i)$ and $f_{\theta}^{{\tau}}(Z_i,\theta'X_i).$
**Step 1.** We first study $A_{1n}.$ A Taylor expansion leads to the following decomposition, $$\begin{aligned}
A^{\tau}_{1n}&= (\theta-\theta_0)'\sum_{i=1}^n \frac{\delta_i W_{in}\mathds{1}_{Z_i\in A_{\tau}}\big(\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(Z_i,X_i)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i)\big)J_{\theta}(X_i,c/2)}{f_{\theta}^{{\tau}}(Z_i,\theta'X_i)} \\ & \quad+ (\theta-\theta_0)'\left[\sum_{i=1}^n \frac{\delta_i W_{in}\mathds{1}_{Z_i\in A_{\tau}}\big(\nabla^2_{\theta}\hat{f}^{h,\tau}_{\tilde{\theta}}(Z_i,X_i)-\nabla^2_{\theta}f_{\tilde{\theta}}^{{\tau}}(Z_i,X_i)\big)J_{\theta}(X_i,c/2)}{{2}f_{\theta}^{{\tau}}(Z_i,\theta'X_i)}\right](\theta-\theta_0) \\
& \quad+ \frac{1}{n}\sum_{i=1}^n \frac{\delta_iW_{in}\mathds{1}_{Z_i\in A_{\tau}}\big(\hat{f}^{h,\tau}_{\theta_0}(Z_i,\theta_0'X_i)-f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)\big)}{f_{\theta}^{{\tau}}(Z_i,\theta'X_i)f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)}\\
& \quad \times (f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)-f_{\theta}^{{\tau}}(Z_i,\theta'X_i))J_0(X_i,c)J_{\theta}(X_i,c/2)+
{A^{\tau}_{1n}(\theta_0,\hat{f}^{h,\tau})}\\
&= {A^{\tau}_{1n}(\theta_0,\hat{f}^{h,\tau})}+(\theta-\theta_0)'A^{\tau}_{2n}(\theta_0,\hat{f}^{h,\tau})+(\theta-\theta_0)'A^{\tau}_{3n}(\tilde{\theta},\hat{f}^{h,\tau})(\theta-\theta_0)+A^{\tau}_{4n}(\theta,\hat{f}^{h,\tau}),\end{aligned}$$ for some $\tilde{\theta}$ between $\theta$ and $\theta_0.$ Observe that, using the uniform consistency of $\nabla_{\theta}^2\hat{f}^{h,\tau}_{\theta}$ (deduced from Proposition \[convergence\_rate\] and Lemma \[leplusdur\]), we obtain $\sup_{\tilde{\theta}\in \Theta_n,\tau\leq \tau_0,h\in \mathcal{H}_n}A^{\tau}_{3n}(\tilde{\theta},\hat{f}^{h,\tau})=o_P(1).$ We now study $A^{\tau}_{2n}(\theta_0,\hat{f}^{h,\tau}).$ Using the expression (\[explicit\_jump\]) of the jumps of Kaplan-Meier estimator, observe that $$\begin{aligned}
& A^{\tau}_{2n}(\theta,\hat{f}^{h,\tau})
\\ &\quad= \sum_{i=1}^n \frac{W_i^*\mathds{1}_{Z_i\in A_{\tau}}\big(\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(Z_i,X_i)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i)\big)J_{\theta}(X_i,c/2)}{f_{\theta}^{{\tau}}(Z_i,\theta'X_i)}\\ &\qquad + \frac{1}{n}\sum_{i=1}^nW_i^*Z_G(Z_i-)\frac{\delta_i\mathds{1}_{Z_i\in A_{\tau}}\big(\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(Z_i,X_i)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i)\big)J_{\theta}(X_i,c/2)}{f_{\theta}^{{\tau}}(Z_i,\theta'X_i)}
\\
&\quad= A^{\tau}_{21n}(\theta,\hat{f}^{h,\tau})+A^{\tau}_{22n}(\theta,\hat{f}^{h,\tau}),\end{aligned}$$ where $$Z_G(t)=\frac{\hat{G}(t)-G(t)}{1-\hat{G}(t)}.$$ The term $A^{\tau}_{22n}$ can be bounded using (\[gill\]), (\[zhou\]) and Lemma \[leplusdur\], by $$\sup_{\tau \leq \tau_0}|A_{22n}(\theta,\hat{f}^{h,\tau})|\leq o_P(n^{-1/2})\times n^{-1}\sum_{i=1}^n \delta_i[1-G(Z_i-)]^{-1},$$ and the last term is $O_P(1)$ since it has finite expectation. Now for $A^{\tau}_{21n},$ first replace $\theta$ at the denominator by $\theta_0.$ We have $$\begin{aligned}
A^{\tau}_{21n}(\theta,\hat{f}^{h,\tau}) & =\sum_{i=1}^n \frac{W_i^*\mathds{1}_{Z_i\in A_{\tau}}(\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(Z_i,X_i)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i))J_{0}(X_i,c/4)}
{f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)}\\
& \quad +R^{\tau}_{n}(\theta,h)(\theta-\theta_0),\end{aligned}$$ with $\sup_{\theta\in \Theta_n,\tau\leq \tau_0,h\in \mathcal{H}_n}|R_n^{\tau}(\theta,h)|=o_P(1)$ from Assumption \[lgnu\] and the uniform consistency of $\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}$ deduced from Proposition \[convergence\_rate\] and Lemma \[leplusdur\]. Then use Assumption \[donsker1\] and Proposition \[classe de Donsker\]. Using the equicontinuity property of Donsker classes (see e.g. [@Vaart96] or [@Vaart98]), we obtain that $$\begin{aligned}
A^{\tau}_{2n}(\theta,\hat{f}^{h,\tau}) & =\iint \frac{\left[\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(y,x)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(y,x)\right]\mathds{1}_{y\in A_{\tau}}J_{0}({ x},c/4)d\mathbb{P}(x,y)}{f_{\theta_0}^{{\tau}}(y,u)}\\
& \quad +o_P(n^{-1/2}),\end{aligned}$$ where the $o_P-$rate does not depend on $\theta,$ $h,$ nor $\tau.$ From classical kernel arguments, $\sup_{y,x,\tau}|\int\big(\nabla_{\theta}f^{*h,\tau}_{\theta_0}(y,x)-\nabla_{\theta}
f^{{\tau}}_{\theta_0}(y,x)\big)\mathds{1}_{y\in
A_{\tau}}J_{0}({ x},c/4)d\mathbb{P}(x,y)|=O_{\mathbb{P}}(h^4)=o_\mathbb{P}(n^{-1/2}),$ since $nh^8\rightarrow 0.$ Then, Lemma \[leplusdur2\] concludes the proof for $A_{2n}^{\tau}(\theta,\hat{f}^{h,\tau})$. $A^{\tau}_{4n}(\theta,\hat{f}^{h,\tau})$ can be handled similarly.
**Step 2.** $B^{\tau}_{1n}$ can be rewritten as $$\begin{aligned}
&B^{\tau}_{1n}(\theta,\hat{f}^{h,\tau})\\
&\quad=\sum_{i=1}^n \delta_iW_{in}\mathds{1}_{Z_i\in A_{\tau}}J_{\theta}(X_i,c/2)\frac{\{(\theta-\theta_0)'[\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(Z_i,X_i)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i)]\}^2}{\phi(f_{\theta}^{{\tau}}(Z_i,\theta'X_i),\hat{f}_{\theta}^{{h,\tau}}(Z_i,\theta'X_i))^2}
\\
&\qquad+ 2\sum_{i=1}^n \delta_iW_{in}J_{\theta}(X_i,c/2)\mathds{1}_{Z_i\in A_{\tau}}[\hat{f}^{h,\tau}_{\theta_0}(Z_i,\theta_0'X_i)-f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)]\\
&\qquad \times (\theta-\theta_0)'[\nabla_{\theta}\hat{f}^{h,\tau}_{\tilde{\theta}}(Z_i,X_i)-\nabla_{\theta}f_{\tilde{\theta}}^{{\tau}}(Z_i,X_i)][\phi(f_{\theta}^{{\tau}}(Z_i,\theta'X_i),\hat{f}_{\theta}^{{h,\tau}}(Z_i,\theta'X_i))^2]^{-1}
\\
&\qquad+B^{\tau}_{4n}(\theta_0,\hat{f}^{h,\tau})+o_P(\|\theta-\theta_0\|^2)& \\ &\quad=(\theta-\theta_0)'B^{\tau}_{2n}(\theta_0,\hat{f}^{h,\tau})(\theta-\theta_0)+(\theta-\theta_0)'B^{\tau}_{3n}(\theta,\hat{f}^{h,\tau})+B^{\tau}_{4n}(\theta_0,\hat{f}^{h,\tau})+o_P(\|\theta-\theta_0\|^2)\end{aligned}$$ for some $\tilde{\theta}$ between $\theta$ and $\theta_0$. The third term does not depend on $\theta.$ For $B_{2n}^{\tau},$ use the uniform consistency of $\nabla_{\theta_0}\hat{f}^{h,\tau}_{\theta_0}$ (Proposition \[convergence\_rate\] and Lemma \[leplusdur\]) to obtain $\sup_{\tau\leq \tau_0,h\in \mathcal{H}_n}|B^{\tau}_{2n}(\theta,\hat{f}^{h,\tau})|=o_P(n^{-1/2}).$ Finally, for $B^{\tau}_{3n}(\theta,\hat{f}^{h,\tau}),$ from a Taylor expansion, $$\begin{aligned}
B^{\tau}_{3n}(\theta,\hat{f}^{h,\tau}) & =
2\sum_{i=1}^n \frac{\delta_iW_{in}\mathds{1}_{Z_i\in A_{\tau}}J_{\theta}(X_i,c/2)}{\phi(f_{\theta}^{{\tau}}(Z_i,\theta'X_i),\hat{f}^{h,\tau}_{\theta}(Z_i,\theta'X_i))^2}\\
& \quad \times[\hat{f}^{h,\tau}_{\theta_0}(Z_i,\theta_0'X_i)-f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)][\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(Z_i,X_i)-\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i)] \\
&\quad +(\theta-\theta_0)'R_n^{\tau}(\theta,\hat{f}^{h,\tau}),\end{aligned}$$ with $\sup_{\theta\in \Theta_n,\tau\leq \tau_0,h\in
\mathcal{H}_n}R_n^{\tau}(\theta,\hat{f}^{h,\tau})=o_P(1),$ from Proposition \[convergence\_rate\] and Lemma \[leplusdur\]. For the main term, the product of the uniform convergence rates of $\hat{f}^{h,\tau}_{\theta_0}$ and $\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}$ obtained from Proposition \[convergence\_rate\] and [Lemma \[leplusdur\]]{} is $o_P(n^{-1/2})$ for $h\in \mathcal{H}_n$.
Conclusion
==========
We proposed a new estimation procedure of a conditional density under a single-index assumption and random censoring. This procedure is an extension of the approach of Delecroix [*et al.*]{} (2003) in the case of a censored response. One of the advantage of this model is that it relies on fewer assumptions as a Cox regression model, in the case where the random variables of the model are absolutely continuous. By showing that estimating in this semiparametric model is asymptotically equivalent to estimating in a parametric one (unknown in practice), we obtain a $n^{-1/2}-$rate for the estimator of the index. This estimator can then be used to estimate the conditional density or the conditional distribution function by using traditional nonparametric estimator under censoring. A new feature of our procedure, is that it provides an adaptively driven choice of the bandwidth involved in the kernel estimators we used, and that it also provides an adaptive choice of a truncation parameter needed to avoid problems caused by the bad behavior of Kaplan-Meier [estimators]{} in the tail of the distribution. In this specific problem, this truncation does not introduce some additional bias in the procedure, and seems, according to our simulations, to increase the quality of the estimator, especially in the case where the proportion of censored responses is important. Our way of choosing $\tau$ was motivated by minimizing the MSE in the estimation of $\hat{\theta}.$ However, our method could be easily adapted to other kinds of criteria which, for example more focus on the error in estimating one specific direction, or on the error in the estimation of the conditional density itself.
Appendix {#technical}
========
Kaplan-Meier integrals for the parametric case
----------------------------------------------
We first recall a classical asymptotic representation of integrals with respect to $\hat{F}.$ See [@Stute95], [@Stute96] and S[á]{}nchez Sellero [*et al.*]{} (2005).
\[km\] Let $\mathcal{F}$ be a $VC-$class of functions with envelope $\Phi$ such as $$\label{nullity}
\Phi(x,y)= 0, \; for \; all \; y\geq \tau_0,$$ where $\tau_0\leq \tau_H.$ We have the following asymptotic i.i.d. representation, for all $\phi\in \mathcal{F},$ $$\int \phi(x,y)d\hat{F}(x,y) = \frac{1}{n}\sum_{i=1}^n \psi(Z_i,\delta_i,X_i;\phi)+R(\phi),$$ where $\sup_{\phi \in \mathcal{F}}|R(\phi)|=O_{a.s.}([\log n]^3n^{-1}),$ and $$\psi(Z_i,\delta_i,X_i;\phi)=\frac{\delta\phi(X_i,Z_i)}{1-G(Z_i-)}+\int \frac{\int_{y}^{\tau_0}\int_{\mathcal{X}} \phi(x,t)dF(x,t) dM_i^G(y)}{1-H(y)},$$ where $M_i^G(y)=(1-\delta_i)\mathds{1}_{Z_i\leq y}-\int_{-\infty}^y \mathds{1}_{Z_i\geq t}[1-G(t-)]^{-1}dG(t)$ is a martingale with respect to the filtration $\mathcal{G}_y=\{(Z_i,\delta_i,X_i)\mathds{1}_{Z_i\leq y}\}.$ Define $\Delta(\phi)=Var(\psi(Z,\delta,X;\phi)).$ Then it follows that $$\sqrt{n}\int \phi(x,y)d[\hat{F}-F](x,y)\Longrightarrow \mathcal{N}(0,\Delta(\phi)).$$
Initially, the result of Stute was derived for a single function $\phi.$ Furthermore, Theorem 1.1 in [@Stute96] gives a convergence rate which is only $o_P(n^{-1/2})$ for the remainder term, however an higher convergence rate is obtained in his proof of Theorem 1.1 for functions satisfying (\[nullity\]), which is the only case considered in our work. To obtain uniformity on a $VC-$class of functions, see S[á]{}nchez Sellero [*et al.*]{} (2005) who provided a more general representation that extends the one of Stute in the case where $Y$ is right-censored and left-truncated. Their result is really useful since it provides, as a corollary, uniform law of large numbers results and uniform central limit theorem. The representation we present in our Theorem \[km\] is a simple rewriting of Stute’s representation. Theorem \[km\] is then a key ingredient to prove Lemma \[parametric\].
We directly show the second part of the Lemma, since the first can be studied from similar techniques. From a Taylor expansion, $$\begin{aligned}
L^{\tau}_n(\theta,f^{{\tau}},J_0)&= (\theta-\theta_0)'\sum_{i=1}^n \delta_iW_{in}J_0(X_i,c)\mathds{1}_{Z_i\in A_{\tau}}\frac{\nabla_{\theta}f_{\theta_0}^{{\tau}}(Z_i,X_i)}{f_{\theta_0}^{{\tau}}(Z_i,\theta_0'X_i)} \nonumber \\
&\quad +\frac{1}{2}(\theta-\theta_0)'\sum_{i=1}^n \delta_iW_{in}J_0(X_i,c)\mathds{1}_{Z_i\in A_{\tau}}\nabla^2_{\theta}[\log f_{\tilde{\theta}}^{{\tau}}](Z_i,X_i)(\theta-\theta_0)\nonumber \\
&\quad +T_{4n}(\theta_0), \label{development}\end{aligned}$$ for some $\tilde{\theta}$ between $\theta_0$ and $\theta.$ Theorem \[km\] provides an i.i.d. representation for the first term (which corresponds to $W_{n,\tau}$ in Lemma \[parametric\]), while, from Assumption \[regularity\], the family of functions $\nabla^2_{\theta}[\log f_{\tilde{\theta}}^{{\tau}}](y,x)\mathds{1}_{y\in A_{\tau}}$ is a $VC-$class of functions satisfying (\[nullity\]). Hence the sum in the second term of (\[development\]) tends to $V$ almost surely using an uniform law of large numbers property.
The gradient of $f$
-------------------
In the following for any function $\varphi$ we will denote by $\varphi_h^{(n)}(\cdot)$ the expression $h^{-n}\varphi^{(n)}(\cdot /h)$ such as, for example $K'_h(\cdot) =h^{-1}K'\left(\cdot/h\right).$
\[expectation\] Let $$f'_{\tau}(y,u)=\partial_u f^{\tau}_{\theta_0}(y,u).$$ We have $$\nabla_{\theta}f^{\tau}_{\theta_0}(y',x)=xf_{1,y,\tau}(y',\theta_0'x)+f_{2,y,\tau}(y',\theta_0'x),$$ with $$\begin{aligned}
f_{1,\tau}(y,\theta_0'x) &= f'_{\tau}(y,\theta_0'x), \\
f_{2,\tau}(y,\theta_0'x) &= -f'_{\tau}(y,\theta_0'x)E\left[X|\theta_0'X\right].\end{aligned}$$ In particular, $E[\nabla_{\theta}f_{\theta_0}^{\tau}(Y,X)|\theta_0'X]=0.$
Direct adaptation of Lemma 5A in [@Dominitz05].
Convergence properties of $f^{*h,\tau}$
---------------------------------------
We first recall some classical properties on kernel estimators. Consider the class of functions $\mathcal K$ introduced in Assumption \[noyau\]. Let $N(\varepsilon, \mathcal K, d_Q)$ be the minimal number of balls $\{g:d_Q(g,g')<\varepsilon\}$ of $d_Q$-radius $\varepsilon$ needed to cover $\mathcal K$. For $\varepsilon>0$, let $N(\varepsilon,\mathcal K)=\sup_QN(\kappa\varepsilon,\mathcal{K},d_Q)$, where the supremum is taken over all probability measures $Q$ on $(\mathbb{R}^d,\mathcal B)$, $d_Q$ is the $L_2(Q)$-metric. From [@Nolan87], it can easily be seen that, using a kernel $K$ satisfying Assumption \[noyau\], for some $C>0$ and ${\nu>0, N(\varepsilon, \mathcal K)\leq C\varepsilon^{-\nu}, 0<\varepsilon<1}$.
\[convergence\_rate\] Under assumption \[noyau\] we have, for some $c>0$ $$\begin{aligned}
\label{uniff}
\sup_{x,y,h,\tau}\left|f^{*h,\tau}_{\theta_0}(y,\theta_0'x)-f^{\tau}_{\theta_0}(y,\theta_0'x)\right|\mathds{1}_{y\in A_\tau}J_0(x,c)
& = O_P\left(n^{-1/2}h^{-1}[\log n]^{1/2}\right),\\
\label{unifgrad}
\sup_{x,y,h,\tau}\left|\nabla_{\theta}f^{*h,\tau}_{\theta_0}(y,x)-\nabla_{\theta}f^{\tau}_{\theta_0}(y,x)\right|\mathds{1}_{y\in
A_\tau}J_0(x,c)
& = O_P\left(n^{-1/2}h^{-2}[\log n]^{1/2}\right),\\
\sup_{x,y,h,\tau,\theta}\left|\nabla^2_{\theta}f^{*h,\tau}_{\theta}(y,x)-\nabla^2_{\theta}f^{\tau}_{\theta}(y,x)\right|\mathds{1}_{y\in A_\tau}J_{\theta}(x,c)
& = O_P\left(n^{-1/2}h^{-3}[\log n]^{1/2}\right).\label{unif2}\end{aligned}$$
(\[uniff\]) is a direct application of Theorems 1 and 4 in [@Einmahl05]. For (\[unifgrad\]), we only show the convergence for the term $$\hat{r}^{h,\tau}_{\theta_0}(x,y):=\frac{1}{h}\sum_{i=1}^n\delta_iW_i^*\mathds{1}_{Z_i\in A_{\tau}}J_0(x,c)(X_i-x)K'_h(X_i'\theta_0-x'\theta_0)K_h(Z_i-y).$$ Define $$\bar{r}^{h,\tau}_{\theta_0}(x,y) =\frac{1}{h}\mathbb{E}\,\big[\mathds{1}_{Y\in A_{\tau}}J_0(x,c)(X-x)K'_h(X'\theta_0-x'\theta_0)K_h(Y-y)\big]$$ and $$r^{\tau}_{\theta_0}(x,y)=\left.\frac{\partial}{\partial u}\left\{\mathbb{E}\,\big[(X-x)|\theta_0'X=u,Y=y\big]\mathds{1}_{y\in A_{\tau}}J_0(x,c)f_{\theta_0'X,Y}(u,y)\right\}\right|_{u=\theta_0'x}.$$ Note that, from our assumptions $r^{\tau}_{\theta_0}$ is a finite quantity. Next, Theorem 4 in [@Einmahl05] yields : $$\sup_{x,y,h,\tau}\left|\hat{r}^{h,\tau}_{\theta_0}(x,y)-\bar{r}^{h,\tau}_{\theta_0}(x,y)\right|\mathds{1}_{y\in A_{\tau}}= O_P(n^{-1/2}h^{-2}[\log n]^{1/2}).$$ For the bias term, $\sup_{x,y,h,\tau}\left|\bar{r}^{h,\tau}_{\theta_0}(x,y)-r^{\tau}_{\theta_0}(x,y)\right|\mathds{1}_{y\in A_{\tau}}
= O(h^4)=o(n^{-1/2}),$ (see e.g. [@Bosq97]). As a consequence, $$\sup_{x,y,h,\tau}\left|\hat{r}^{h,\tau}_{\theta_0}(x,y)-r^{\tau}_{\theta_0}(x,y)\right|\mathds{1}_{y\in A_{\tau}}= O_P(n^{-1/2}h^{-2}[\log n]^{1/2}).$$ For (\[unif2\]), we also need an uniformity with respect to $\theta.$ The result can be deduced from the uniform convergence (with respect to $\theta,$ $x,$ $u$) of quantities such as $$\label{formegenerale}
S_n^{h,\tau}(\theta,x,y,\beta)=\frac{1}{h^{2}}\sum_{i=1}^n \delta_iW_i^*\phi(Z_i,X_i,\theta)\nabla^{\beta}_{\theta}K\left(\frac{\theta'X_i-\theta'x}{h}\right)K\left(\frac{Z_i-y}{h}\right),$$ where $\nabla^{\beta}_{\theta}K([\theta'X_i-\theta'x]h^{-1})$ for $\beta=1$ (resp. for $\beta=2$) denotes the gradient vector of function $K([\theta'X_i-\theta'x]/h)$ (resp. Hessian matrix) with respect to $\theta$ and evaluated at $\theta,$ and where $\phi$ is a bounded function with respect to $\theta$ and $x$. The function $\phi$ we consider is $\phi(Z,X,\theta)=f^{\tau}_{\theta'X}(\theta'X)^{-1}\mathds{1}_{{Z\in A_\tau}}J_0(x,c)$ with the convention $0/0=0$ and where $f^{\tau}_{\theta'X}(\theta'X)$ is the conditional density of $\theta'X$ given $Y\in A_\tau.$ (\[formegenerale\]) can be studied using the same method as [@Einmahl05]. For this, observe that the family of functions $\{(X,Z)\rightarrow \nabla_{\theta}^{\beta}K([\theta'X-\theta'x]h^{-1})K([Z-y]h^{-1}),\theta\in \Theta,x,y\}$ satisfies the Assumptions of Proposition 1 in [@Einmahl05] (see Lemma 22 (ii) in [@Nolan87]). Hence, apply Talagrand’s inequality ([@Talagrand94], see also [@Einmahl05]) to obtain that $$\sup_{\theta,x,y,h,\tau}|S_n^{h,\tau}(\theta,x,y,\alpha)-E[S_n^{h,\tau}(\theta,x,y,\alpha)]|\mathds{1}_{y\in A_{\tau}}=O_P(n^{-1/2}[\log n]^{1/2}h^{-1-\beta}).$$ Again, the bias term converges uniformly at rate $O(h^4).$
The difference between $f^*$ and $\hat{f}$
------------------------------------------
### Convergence rate of $\hat{f}$
In this section, we show that replacing $f^{*h,\tau}$ by $\hat{f}^{h,\tau}$ (which is the estimator used in practice) does not modify the rate of convergence. To give the intuition of this results, observe that $f^{*h,\tau}$ was obtained from $\hat{f}^{h,\tau}$ by replacing $\hat{G}$ by $G.$ Let us recall some convergence properties of $\hat{G}.$ We have $$\begin{aligned}
\sup_{t\leq \tau_0}|\hat{G}(t)-G(t)| &= O_P(n^{-1/2}), \label{gill} \\
\sup_{t\leq \tau_0}\frac{1-G(t)}{1-\hat{G}(t)} &= O_P(1). \label{zhou}\end{aligned}$$ See [@Gill83] for (\[gill\]) and [@Zhou92] for (\[zhou\]). From (\[gill\]), we see that the convergence rate of $\hat{G}$ is faster than the convergence rate of $f^{*h,\tau},$ which explains the asymptotic equivalence of $\hat{f}^{h,\tau}$ and $f^{*h,\tau}.$ Lemma \[leplusdur\] makes things more precise and also gives a representation of the difference between $\nabla_{\theta}f_{\theta_0}^{*h,\tau}$ and $\nabla_{\theta}\hat{f}_{\theta_0}^{h,\tau}$ which is needed in the proof of Main Lemma. Also required to prove our Main Lemma, Lemma \[leplusdur2\] below gives a technical result on the integral of this difference.
\[leplusdur\] Under the Assumption of Lemma \[parametric\], we have for some $c>0$ $$\begin{aligned}
\label{01} \sup_{x,y,h,\tau}\left|\hat{f}_{\theta_0}^{h,\tau}(y,\theta'x)-f^{*h,\tau}_{\theta_0}(y,\theta'x)\right|\mathds{1}_{y\in A_\tau}J_0(x,c) &= O_P(n^{-1/2}), \\
\label{02} \sup_{x,y,h,\tau}\left|\nabla_{\theta}\hat{f}_{\theta_0}^{h,\tau}(y,x)-\nabla_{\theta}f^{*h,\tau}_{\theta_0}(y,x)\right|\mathds{1}_{y\in A_\tau}J_0(x,c) &= O_P(n^{-1/2}h^{-1}), \\ \label{03}
\sup_{x,y,h,\tau,\theta}\left|\nabla^2_{\theta}\hat{f}_{\theta}^{h,\tau}(y,x)-\nabla^2_{\theta}f^{*h,\tau}_{\theta}(y,x)\right|\mathds{1}_{y\in A_\tau}J_{\theta}(x,c) &= O_P(n^{-1/2}h^{-2}).\end{aligned}$$ Furthermore, for $x$ such as $J_0(x,c)\neq 0$, $$\begin{aligned}
\label{04} \left(\nabla_{\theta}\hat{f}_{\theta_0}^{h,\tau}(y,x)-\nabla_{\theta}f^{*h,\tau}_{\theta_0}(y,x)\right) &= \int \frac{\int_{{\mathcal{X}}}\int_{t}^{\tau_0}g^{h,\tau}_{f,x,y}(x_2,y_2)d\mathbb{P}(x_2,y_2)d\bar{M}^G(t)}{1-H(t)} \nonumber\\
& \quad +R_n(\tau,h,x,y),\end{aligned}$$ where $\bar{M}^G(y)=n^{-1}\sum_{i=1}^nM_i^G(y),$ $M_i^G$ is defined in Theorem \[km\], $\sup_{x,y,\tau,h}|R_n(\tau,h,x,y)|=$\
$O_{P}((\log n)^{1/2}n^{-1}h^{-3})$ and $g_{f,x,y}^h$ is defined by $$\begin{aligned}
g^{h,\tau}_{f,x_1,y_1}(x_2,y_2) &= \frac{1}{h}
\frac{(x_1-x_2)K'_h(\theta_0'x_1-\theta_0'x_2)K_h(y_1-y_2)}{f^{\tau}_{\theta_0'X}(\theta_0'x_1)} \\
& \quad-
\frac{K_h(\theta_0'x_1-\theta_0'x_2)K_h(y_1-y_2)f'^{\tau}_{\theta_0'X}(\theta_0'x_1)}{f^{\tau}_{\theta_0'X}(\theta_0'x_1)^2},\end{aligned}$$ where $f'^{\tau}_{\theta_0'X}$ denotes the derivative of $u\rightarrow f^{\tau}_{\theta_0'X}(u),$ the conditional density of $\theta_0'X$ given $Y\in A_{\tau}.$
\[leplusdur2\] Under the Assumptions of Lemma \[parametric\] $$\sup_{h,\tau}\int\frac{[\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(y,x)-\nabla_{\theta}f^{*h,\tau}_{\theta_0}(y,x)]\mathds{1}_{y\in A_{\tau}}J_0(x,c/4)d\mathbb{P}(x,y)}{f_{\theta_0}^{{ \tau}}(y,\theta_0'x)}=o_P(n^{-1/2}).$$
To prove (\[01\]-\[03\]), we only prove (\[03\]) since the others are similar. To prove (\[03\]), we only consider the terms in which the second derivative is involved, the others being studied analogously. Consider $$\begin{aligned}
&\frac{1}{h}\sum_{i=1}^n \delta_iW_{in}(X_i-x)K_h''(\theta'X_i-\theta'x)K_h(Z_i-y)(X_i-x)'\mathds{1}_{Z_i\in A_{\tau}}f^{\tau}_{\theta'X}(\theta'x)^{-1}\\
&=\frac{1}{h}\sum_{i=1}^n \delta_iW_{i}^*J_{\theta}(X_i,c)(X_i-x)K_h''(\theta'X_i-\theta'x)K_h(Z_i-y)(X_i-x)'\mathds{1}_{Z_i\in A_{\tau}}f^{\tau}_{\theta'X}(\theta'x)^{-1}\\
&+\frac{1}{h}\sum_{i=1}^n
\delta_iW_{i}^*Z_G(Z_i-)(X_i-x)K_h''(\theta'X_i-\theta'x)K_h(Z_i-y)(X_i-x)'\mathds{1}_{Z_i\in
A_{\tau}}f^{\tau}_{\theta'X}(\theta'x)^{-1},\end{aligned}$$ where the first term is contained in $\nabla^2_{\theta}f_{\theta}^{*h,\tau},$ while the second can be bounded by $$O_P(n^{-1/2}h^{-2})\left[\frac{1}{nh^2}\sum_{i=1}^n\delta_i\mathds{1}_{Z_i\leq \tau_0}|K''|\left(\frac{\theta'X_i-\theta'x}{h}\right)|K|\left(\frac{Z_i-y}{h}\right)\right].$$ Using the results of [@Sherman94], the term inside the brackets is $O_P(1)$ uniformly in $x,$ $y$, $\theta$ and $h.$
Now, for the representation (\[04\]), observe that $$\begin{aligned}
&\nabla_{\theta}[\hat{f}_{\theta_0}^{h,\tau}-f^{*h,\tau}_{\theta_0}](y,x)\nonumber\\
&\quad =h^{-1}\sum_{i=1}^n \delta_i W_i^*Z_G(Z_i-)(x-X_i)K_h'(\theta_0'x-\theta_0'X_i)K_h(y-Z_i)f^{\tau}_{\theta_0'X}(\theta_0'x)^{-1}\mathds{1}_{Z_i\in A_{\tau}} \nonumber \\
&\qquad-\sum_{i=1}^n \delta_i W_i^*Z_G(Z_i-)J_{0}(x,c)K(\theta_0'X_i-\theta_0'x)K_h(Z_i-y){f}'^{\tau}_{\theta_0'X}(\theta_0'x)f^{\tau}_{\theta_0'X}(\theta_0'x)^{-2}\mathds{1}_{Z_i\in A_{\tau}} \nonumber \\
&\qquad +R'_n(\tau,h,x,y), \label{decompopo}\end{aligned}$$ with $\sup_{x,y,h,\tau}|R'_n(\tau,h,x,y)|=O_P\left(n^{-1}h^{-3/2}[\log n]^{1/2}\right),$ from the convergence rate of $Z_G$ (see (\[gill\]) and (\[zhou\])) and the convergence rate of the denominator in (\[kernel\_estimator\]) and its derivative, say $(\hat{f}^{\tau}_{\theta_0'X}-f^{\tau}_{\theta_0'X})$ and $(\hat{f}'^{\tau}_{\theta_0'X}-f'^{\tau}_{\theta_0'X})$ (which are of uniform rate $O_P\left(n^{-1/2}h^{-1/2}[\log n]^{1/2}\right)$ and $O_P\left(n^{-1/2}h^{-3/2}[\log n]^{1/2}\right)$ from arguments similar as for the proofs of (\[uniff\])-(\[unif2\]) and (\[01\])-(\[03\])). An i.i.d. representation of the main term in (\[decompopo\]) can be deduced from Theorem \[km\] since the class $\{h^3g^{h,\tau}_{f,x,y},x,y,h\}$ is a VC-class from [@Nolan87].
Observe that, from classical kernel arguments $$\sup_{t}\left|\iint_{x_2,t \leq y_2 \leq \tau_0}g^{h,\tau}_{f,x,y}(x_2,y_2)J_0(x,c/4)d\mathbb{P}(x_2,y_2)d\mathbb{P}(x,y)-E[\nabla_{\theta}f_{\theta_0}^{{\tau}}(Y,X)J_0(X,c/4)]\right|=O(h^4),$$ since $K$ is of order 4. From the representation (\[04\]) in Lemma \[leplusdur\], $$\begin{aligned}
&\int[\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(y,x)-\nabla_{\theta}f^{*h,\tau}_{\theta_0}(y,x)]J_0(x,c/4)d\mathbb{P}(x,y)\nonumber \\
&\quad=\int [1-H(t)]^{-1}E[\nabla_{\theta}f_{\theta_0}^{\tau}(Y,X)J_0(X,c/4)]d\bar{M}^G(t)\nonumber \\
&\qquad+\int [1-H(t)]^{-1}\left[\iint_{x_2,t \leq y_2 \leq \tau_0}g^{h,\tau}_{f,x,y}(x_2,y_2)J_0(x,c/4)d\mathbb{P}(x_2,y_2){d\mathbb{P}(x,y)}\right. \nonumber \\
&\qquad-E[\nabla_{\theta}f_{\theta_0}^{\tau}(Y,X)J_0(X,c/4)]\bigg]d\bar{M}^G(t)\nonumber \\
&\qquad+\int R_n(\tau,h,x,y)d\mathbb{P}(x,y), \label{encoreuneffort}\end{aligned}$$ where the last term is $o_P(n^{-1/2})$ uniformly in $\tau$ and $h$. The first one is zero from Proposition \[expectation\] and since $J_0$ only depends on $\theta_0'X.$ For the second, let $\phi_n(t,h,\tau)=$\
$[1-H(t)]^{-1}\{\iint_{x_2,t \leq y_2 \leq \tau_0}g^{h,\tau}_{f,x,y}(x_2,y_2)J_0(x,c/4)d\mathbb{P}(x_2,y_2)d\mathbb{P}(x,y)-E[\nabla_{\theta}f_{\theta_0}^{\tau}(Y,X)J_0(X,c/4)]\}.$ Using the fact that $\mathcal{H}_n$ is of cardinality $k_n,$ we have, for the second term in (\[encoreuneffort\]), $$\mathbb{P}\left(\sup_{h\in \mathcal{H}_n}\left|\int \phi_n(t,h,\tau)d\bar{M}^G(t)\right|\geq \varepsilon\right) \leq k_n \sup_{h\in \mathcal{H}_n}\mathbb{P}\left(\left|\int \phi_n(t,h,\tau)d\bar{M}^G(t)\right|\geq \varepsilon\right).$$ Now apply Lenglart’s inequality (see Lenglart (1977) or Theorem 3.4.1 in Fleming and Harrington, (1991)). This shows that, for all $\varepsilon>0$ and all $\eta>0,$ $$\begin{aligned}
\label{leng1}
\nonumber &\mathbb{P}\left(\sup_{\tau \leq s\leq \tau_0}\left\{\int_0^s \phi_n(t,h,\tau)d\bar{M}^G(t)\right\}^2\geq \varepsilon^2\right)\\
&\quad\leq \frac{\eta}{\varepsilon^2}+\mathbb{P}\left(n^{-1}\int_0^{\tau_0}\phi_n^2(t,h,\tau)\frac{[1-\hat{H}(t-)]dG(t)}{1-G(t-)}\geq \eta\right).\end{aligned}$$ As mentioned before, $\sup_{t}|\phi_n(t,h,\tau)|=O(h^4).$ From (\[leng1\]) and condition on $k_n$ in Assumption \[noyau\], the Lemma follows.
### Donsker classes
As stated in Assumption \[donsker1\], to obtain a $n^{-1/2}-$convergence of $\hat{\theta},$ we need the regression function (and its gradient) to be sufficiently regular. In the Lemma below, we first show that the classes of functions defined in Assumption \[donsker1\] are Donsker, and that $\hat{f}^{h,\tau}_{\theta_0}$ also belongs to the same regular class as $f^{\tau}_{\theta_0}$ with probability tending to one.
\[classe de Donsker\] Consider the classes $\mathcal{H}_1$ and $\mathcal{H}_2$ defined in Assumption \[donsker1\]. $\mathcal{H}_1$ and $\mathcal{H}_2$ are Donsker classes. Furthermore, $\hat{f}^{h,\tau}_{\theta_0}$ and $\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}$ belong respectively to $\mathcal{H}_1$ and $\mathcal{H}_2$ with probability tending to one for some constant $M$ sufficiently large.
The class $\mathcal{H}_1$ is Donsker from Corollary 2.7.4 in [@Vaart96]. The class $\mathcal{H}_2$ is Donsker from a permanence property of Donsker classes, see Examples 2.10.10 and 2.10.7 in [@Vaart96]. We only show the proof for $\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0},$ since the one for $\hat{f}^{h,\tau}_{\theta_0}$ is similar. Write $$\begin{aligned}
& \nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(z,x) \\&=
\frac{1}{nh}\sum_{i=1}^n
\frac{\delta_i\mathds{1}_{Z_i \in A_{\tau}}(X_i-x)K'_h(\theta_0'X_i-\theta_0'x)K_h(Z_i-z)}{[1-\hat{G}(Z_i-)]f^{\tau}_{\theta_0'X}(\theta_0'x)}J_0(X_i,c/2)
\\
& \quad +\frac{1}{nh}\sum_{i=1}^n
\frac{\delta_i\mathds{1}_{Z_i \in A_{\tau}}(X_i-x)K'_h(\theta_0'X_i-\theta_0'x)K_h(Z_i-z)[\hat{f}^{\tau}_{\theta_0'X}(\theta_0'x)-f^{\tau}_{\theta_0'X}(\theta_0'x)]}{[1-\hat{G}(Z_i-)]\hat{f}^{\tau}_{\theta_0'X}(\theta_0'x)f^{\tau}_{\theta_0'X}(\theta_0'x)}J_0(X_i,c/2)
\\
& \quad -\left[\frac{1}{nh}\sum_{i=1}^n
\frac{(X_i-x)K_h'(\theta_0'X_i-\theta_0'x)J_0(X_i,c/2)}{\big(f_{\theta_0'X}^{\tau}(\theta_0'x)\big)^2}\right]
\\
& \quad \times \left[\frac{1}{n}\sum_{i=1}^n
\frac{\delta_iK_h(\theta_0'X_i-\theta_0'x)K_h(Z_i-z)\mathds{1}_{Z_i \in A_{\tau}}}{[1-\hat{G}(Z_i-)]}\right]
\\
& \quad +\left[\frac{1}{nh}\sum_{i=1}^n
\frac{(X_i-x)K_h'(\theta_0'X_i-\theta_0'x)\left[\big(\hat{f}_{\theta_0'X}^{\tau}(\theta_0'x)\big)^2-\big(f^{\tau}_{\theta_0'X}(\theta_0'x)\big)^2\right]J_0(X_i,c/2)}{\big(\hat{f}^{\tau}_{\theta_0'X}(\theta_0'x)f^{\tau}_{\theta_0'X}(\theta_0'x)\big)^2}\right]
\\ & \quad \times \left[\frac{1}{n}\sum_{i=1}^n
\frac{\delta_i\mathds{1}_{Z_i \in A_{\tau}}K_h(\theta_0'X_i-\theta_0'x)K_h(Z_i-z)}{[1-G(Z_i-)]}\right].\end{aligned}$$ From this expression, we clearly see that $\nabla_{\theta}\hat{f}^{h,\tau}_{\theta_0}(y,x)=x\phi_1(x'\theta_0,y)+\phi_2(x'\theta_0,y).$ Now we must check that $\phi_1$ and $\phi_2$ are in $\mathcal{H}_1$ with probability tending to one. Since the functions are twice continuously differentiable (from the assumptions on $K$), we only have to check their boundedness. From Lemma \[leplusdur\], this can be done at first by replacing $\hat{f}^{h,\tau}$ by $f^{*h,\tau}$ (i.e. $\hat{G}$ by the true function $G$). Among the several terms in the decomposition of $\nabla_{\theta}f^{*h,\tau},$ we will only study $$\phi(u,y) = \frac{1}{nh}\sum_{i=1}^n
\frac{\delta_i\mathds{1}_{Z_i \in A_{\tau}}X_iK'_h(\theta_0'X_i-u)K_h(Z_i-z)J_0(X_i,c/2)}{[1-G(Z_i-)]f^{\tau}_{\theta_0'X}(u)},$$ since the others are similar. We will show that the derivatives of order 0, 1 and $1+\delta$ of this function are uniformly bounded by some constant $M$ with probability tending to one.
Now a centered version of $\phi$ converges to zero at rate $O_P([\log n]^{1/2}n^{-1/2}h^{-1})$ (see [@Einmahl05]), which tends to zero as long as $nh^2\rightarrow \infty.$ Furthermore, $E[\phi]$ is uniformly bounded from our Assumption \[donsker1\] on the regression function. For the derivative, $$\begin{aligned}
\partial_u\phi(u,y) &=- \frac{1}{nh}\sum_{i=1}^n
\frac{\delta_i\mathds{1}_{Z_i \in A_{\tau}}X_iK''_h(\theta_0'X_i-u)K_h(Z_i-z)J_0(X_i,c/4)}{[1-G(Z_i-)]f^{\tau}_{\theta_0'X}(u)}\\
& \quad -\frac{1}{nh}\sum_{i=1}^n
\frac{\delta_i\mathds{1}_{Z_i \in A_{\tau}}X_iK'_h(\theta_0'X_i-u)K_h(Z_i-z)J_0(X_i,c/4){f}'^{\tau}_{\theta_0'X}(u)}{[1-G(Z_i-)]\big(f^{\tau}_{\theta_0'X}(u)\big)^2}.
$$ Again, $E[\partial_u\phi]$ is uniformly bounded from our Assumption \[donsker1\]. Now using the results of [@Einmahl05], the centered version of $\partial_u\phi$ tends to zero provided that $nh^6\rightarrow \infty.$ The same arguments apply for $\partial_y\phi.$ Hence, with $f_i(u,y)=E(\phi_i(u,y))$ we proved that $\sup_{u,y} |\partial_u^{j}\partial_y^{k}\phi_{i}(u,y)-\partial_u^{j}\partial_y^{k}f_{i}(u,y)|$ tends to zero in probability for $i=1,2$, $k+j\leq 1$. Now we have to show that $\partial_u\phi_j$ and $\partial_y\phi_j$ are $\delta-$ Hölder for $j=1,2$ with an Hölderian constant bounded by some $M$ with probability tending to one. We only prove the result for $\partial_u\phi_1.$ We have $$\begin{aligned}
\sup_{u',y',x,y}\frac{\left|\partial_u\phi_{1}(u,y)-\partial_u\phi_{1}(u',y')\right|}{\|(u,y)-(u',y')\|^{\delta}}
&= \max\left(\sup_{|u-u'|\geq n^{-1}
,y,y'}\frac{\left|\partial_u\phi_{1}(u,y)-\partial_u\phi_{2}(u',y')\right|}{\|(u,y)-(u',y')\|^{\delta}},\right.
\\
& \left.\sup_{|u-u'|\leq n^{-1}
,y,y'}\frac{\left|\partial_u\phi_{1}(u,y)-\partial_u\phi_{1}(u',y')\right|}{\|(u,y)-(u',y')\|^{\delta}}\right)
\\
&= \max(S_1,S_2).\end{aligned}$$ We have $$\begin{aligned}
S_1 &\leq
\sup_{u,y,u',y'}\frac{|\partial_uf_{1}(u',y')-\partial_uf_1(u,y)|}{\|(u',y')-(u,y)\|^{\delta}}
\\
&
+2n^{\delta}\sup_{u,y,u',y'}|\partial_u\phi_{1}(u,y)-\partial_uf_1(u,y)|.\end{aligned}$$ From our Assumptions, the first supremum is bounded, while the last is\
$O_P(n^{-1/2+\delta}[\log n]^{1/2}h^{-3})$ from the convergence rate of $\partial_u\phi_2.$ It tends to zero provided that $nh^{6+\delta}\rightarrow \infty.$ For $S_2,$ since $K$ is $\mathcal{C}^3$ with bounded derivatives, for some positive constant $M,$ $$\begin{aligned}
\sup_{\|(u,y)-(u',y')\|\leq n^{-1}
,y,y'}\frac{\left|\partial_u\phi_{1}(u,y)-\partial_u\phi_{1}(u',y')\right|}{\|(u,y)-(u',y')\|^{\delta}}
\leq M\times
O_P(1)\|\sum_{i=1}^3|K^{(i)}|\|_{\infty}\\
\times \sup_{\|(u,y)-(u',y')\|\leq n^{-1}
}\|(u,y)-(u',y')\|^{1-\delta}h^{-1}\frac{1}{nh^4}\sum_{i=1}^n\frac{\delta_i}{1-G(Z_i-)}.\end{aligned}$$ The last supremum is bounded by $O_P(1)\times
n^{-1+\delta}h^{-5},$ and it tends to zero when $nh^6\rightarrow
\infty$ (and the $O_P(1)$ term does not depend on $u,y$).
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---
abstract: 'The spin evolution of isolated neutron stars (NSs) is dominatd by their magnetic fields. The measured braking indices of young NSs show that the spin-down mechanism due to magnetic dipole radiation with constant magnetic fields is inadequate. Assuming that the NS magnetic field is buried by supernova fallback matter and re-emerges after accretion stops, we carry out Monte-Carlo simulation of the evolution of young NSs, and show that most of the pulsars have the braking indices ranging from $-1$ to 3. The results are compatible with the observational data of NSs associated with supernova remnants. They also suggest that the initial spin periods of NSs might occupy a relatively wide range.'
author:
- 'Lei Fu and Xiang-Dong Li'
title: 'Population synthesis of young isolated neutron stars: the effect of fallback disk accretion and magnetic field evolution'
---
Introduction
============
The spin evolution is one of the most outstanding problems in pulsar astronomy. In the classical models the spin-down of isolated pulsars is due to the energy loss caused by magnetic dipole radiation, which is described as $$I\dot{\Omega}=-\frac{B^2R^6\Omega^3}{6c^3},$$ where $\Omega$, $\dot{\Omega}$, $I$, $B$, and $R$ are the angular velocity and its derivative, the moment of inertia, the surface magnetic field strength, and the radius of the pulsar, respectively, $c$ is the speed of light. In realistic case the pulsar’s spin-down may deviate from that due to pure dipole radiation, and a more general power-law form is adopted, $$\dot{\Omega}=-K\Omega^n,$$ where $K$ is a coefficient proportional to the spin-down torque, and the power-law index $n$ is the so-called braking index. For a constant $K$, $n\equiv\Omega \ddot{\Omega}/\dot{\Omega}^2$. The initial spin period $P_0$ of the pulsar can be obtained by using Eq. (2) once the real spin-down age and the historically averaged braking index is known, $$P_0=\left[P^{n-1}-(n-1)\dot{P}T/P^{2-n}\right]^{1/(n-1)},$$ where $P(\equiv\Omega/2\pi)$, $\dot{P}$, and $T$ are the current period, the period derivative, and the age of the pulsar, respectively.
So far, due to the uncertainties in timing measurements (e.g. timing noise and glitches) which usually dominate the relatively small values of the second derivative of $\Omega$, it is impossible to measure the stable braking indices for the majority of pulsars [@kms94]. Only for those pulsars with relatively stable long-term spin-down, the secular braking indices have been measured. By using 23 year timing data @lpg93 first obtained the braking index of the Crab pulsar (B0531$+$21) to be $n=2.51(1)$, for which the time evolution of $\dot{\Omega}$ has been monitored for more than 40 years. The braking index of the Vela pulsar B0833$-$45 was measured by @lpg96 to be $n=1.4\pm0.2$, which is significantly less than the value expected in the magnetic dipole radiation model, indicating possible changes of the magnetic moment and/or the effective moment of inertia. Currently the braking indices have been measured in 13 pulsars [see @esp13 and references therein]. Except PSRs J0537$-$6910 and B1757$-$24 which have negative $n$, other pulsars all have $0<n<3$. The smallest one is $n=0.9\pm0.2$ for PSR J1734$-$3333 [@elk11], which has a spin period of $P=1.17\,\rm s$ and a large period derivative $\dot{P}=2.3\times 10^{-12}$ ss$^{-1}$. The deduced magnetic field strength $B\simeq 5 \times 10^{13}\,\rm G$ makes it among the highest $B$ pulsars, and similar to those of magnetars [@ozv12]. The low $n$ has been attributed to an increase of the dipole component of its magnetic field [@elk11].
The possible reasons for the deviation of $n$ from 3 have been studied extensively. They include, e.g., the pulsar wind in which the high-speed particles take the angular momentum away from the pulsar [@mt77], the distortion of the magnetic field from a pure dipole field [e.g. @bt10] and a oblique rotator in which the magnetic axis is misaligned with the axis of rotation [e.g. @cs06]. In particular, to explain that most of the measured values are less than 3, there are models invoking magnetic field increase [e.g. @br88] or fallback disk assisted spin-down [e.g. @mlr01; @mph01; @aay01; @cl06] in young pulsars.
The growth of magnetic field in young neutron stars (NSs) may be caused by the re-emergence of the magnetic field [e.g. @mp96; @gpz99; @pg07; @ho11; @bpl12; @pvg12], which is buried due to the hypercritical accretion after the supernova events [@rom90]. The re-emergence timescale depends on the total amount of accreted matter, which is $\sim 10^3-10^4$ yr for total accreted masses $\sim 10^{-4}-10^{-3}\,M_{\odot}$. Relatively weak magnetic fields ($\sim 10^{10}-10^{11}$ G) have been measured in a few young NSs in supernova remnants (SNRs) named central compact objects (CCOs) [@gha13 and references therein], which might have experienced the field burial and re-emergence process [@ho13].
Spin evolution of NSs with a surrounding fallback disk originated from the SN ejecta has been investigated by many authors [e.g. @chn00; @alpar01; @mph01; @eee09; @yps12]. In these studies the NS magnetic fields were usually assumed to be constant, neglecting their possible evolution during and after the disk accretion. In this paper we use a Monte-Carlo method to calculate the evolution of young NSs. We assume that there is a fallback disk around all newborn NSs. Depending on the accreted mass the NS magnetic fields decay to some extent, and turn back to increase after accretion stops. By taking into account the disk accretion and field evolution simultaneously, we model the evolution of the spin periods and the braking indices of the NSs. We also compare the calculated distributions of various parameters with those derived from observations of young NSs.
The structure of this paper is as follows. We introduce the fallback disk model in Section 2, and describe the evolution of the magnetic field and the spin period of NSs, which is coupled with fallback disk accretion in Section 3. We present the calculated results of our population synthesis in Section 4. Discussion and conclusions are given in Section 5.
The fallback disk mass transfer
===============================
Time-evolution of a fallback disk
---------------------------------
The SN fallback material is assumed to form from the metal-rich ejecta of core collapse SNe [@mic88; @che89]. The formation of a fallback disk requires that at least part of the fallback material possesses sufficient angular momentum. Compared with the accretion disks in binaries, the lifetime of which may be comparable with the evolutionary timescale of the binary or the donor star, the fallback disk has much shorter duration. The first possible detection of the fallback disk was made by @wck06, who reported the mid-infrared counterpart of the anomalous X-ray pulsar (AXP) 4U 0142$+$61, and interpreted this as a passive dust fallback disk outside the pulsar’s magnetosphere heated by X-ray irradiation. Here we are not concerned with the formation processes of the fallback disk which is rather complicated, but focus on its time-evolution.
The time dependence of the mass transfer rate of the disk and the size of a fallback disk was studied before [e.g. @mph01; @ea03; @eee09; @clg90]. Based on the self-similar solution of the standard thin disk [@ss73; @pri81] the evolution of the mass transfer rate at the outer annulus of the disk $\dot{M}_{\rm tr}$ and the outer disk radius $R_{\rm d}$ can be written in the following form $$\dot{M}_{\rm tr}(t) = \dot{M}_0 \left(1+t/t_0\right)^{-\alpha},$$ $$R_{\rm d}(t) = R_0 \left(1+t/t_0\right)^{2\alpha-2},$$ where the power exponent $\alpha$ is $5/4$ for the opacity dominated by bound-free absorption, $R_0$ and $\dot{M}_0$ are the initial radius and mass transfer rate of the disk, respectively. $t_0$ is the timescale of disk formation, and is usually taken to be the dynamical timescale for the disk [@mph01], $$t_{\rm d}\simeq 6.6\times 10^{-5} T_{\rm c,6}^{-1} R_{0,8}^{1/2}\; {\rm yr}.$$ where $R_{\rm 0,8}$ and $T_{\rm c,6}$ are the initial outer disk radius in units of $10^8$ cm and the temperature at $R_{\rm d}$ in units of $10^6{\rm \,K}$ (taken to be $1$ in this paper), respectively. However, @eee09 showed that $t_0$ is more likely to be close to the viscous timescale in the disk, $$t_{\rm v}\simeq 3.19\times 10^{-4} (M_0/10^{-4}M_{\odot})^{-3/7} R_{\rm 0,8}^{25/14}\; {\rm yr},$$ where $M_0=\int_0^\infty \dot{M}_{\rm tr}(t){\rm d}t=\dot{M}_0 t_0/(\alpha-1)$ is the initial disk mass, or $$t_{\rm v}\simeq 2.58\times 10^{-4} (\dot{M}_0/10^{25}{\rm \,g\,s^{-1}})^{-3/10} R_{\rm 0,8}^{5/4}\; {\rm yr}.$$ In our calculation we assign the maximum value of $t_{\rm d}$ and $t_{\rm v}$ to $t_0$.
Since the temperature of the disk always decreases with time, at the later stage the outer regions of the disk may become neutral and passive [@mph01; @eee09]. However, it was argued that the irradiation by the NSs may be able to keep the disk ionized and prohibit the transition from an active to a passive disk [@aay01]. Further calculation showed that the critical temperature $T_{\rm p}$ corresponding to the lowest ionization fraction that can generate viscosity in the disk is as low as $T_{\rm p}\sim 100\,\rm K$ [@ace13]. Thus in this work we do not consider the neutralization process in the fallback disk.
Mass transfer and accretion
---------------------------
It is noted that the fallback disk accretion process is likely to be non-conservative. For a NS with a typical mass of $M=1.4M_\odot$ the accretion rate ($\dot{M}_{\rm acc}$) is generally limited by the Eddingtom accretion rate $\dot{M}_{\rm E}\simeq 10^{18} {\rm g\,s^{-1}}\simeq 10^{-8}\,M_\odot\, \rm yr^{-1}$. The initial transfer rate of the fallback material to the central object can be hypercritical and greatly exceed the Eddingtom limit. For example, in the case of SN 1987A, @che89 suggested that the transfer rate at the time of reverse shock is $\sim 2.2\times 10^{28}\rm \,g\,s^{-1}$. More recently, @zwh08 studied the supernova fallback for a wide range of progenitor masses and various metallicities and explosion energies. The transfer rate was also found to be $\sim 10^{29}\rm \,g\,s^{-1}$ at the early phase. Note that this hypercritical accretion usually lasts very short time (less than 1 yr) compared to the spin evolution time of NSs. Super-Eddinton accretion disks are likely to be advective and emit a wind. In the adiabatic inflow-outflow solutions (ADIOS) the mass transfer rate varies radially as $\dot{M}(r)\sim \dot{M}_{\rm tr}(r/R_{\rm d})^p$ with $0\le p \le 1$ [@bb99]. The case $p = 0$ corresponds to the absence of a wind, while $p = 1$ implies strong mass loss, and the mass transfer rate at the inner edge $\dot{M}_{\rm in}$ is then not equal to $\dot{M}_{\rm tr}$. A convenient form to relate the two rates is as follows, $$\dot{M}_{\rm in} =
\left\{
\begin{array}{ll}
\dot{M}_{\rm E}(\dot{M}_{\rm tr}/\dot{M}_{\rm E})^s, \; & {\rm for}\
\dot{M}_{\rm tr} > \dot{M}_{\rm E},\\
\dot{M}_{\rm tr}, \; & {\rm for}\ \dot{M}_{\rm tr} \leqslant\dot{M}_{\rm E},
\end{array}
\right.$$ with $0\le s \le1$. For example, if $s=0.75$, an initial $\dot{M}_{\rm in}$ distributed between $10^{25}$ gs$^{-1}$ and $10^{28}$ gs$^{-1}$ (as used below) corresponds to a $\dot{M}_{0}$ between $2\times 10^{27}$ gs$^{-1}$ and $2\times 10^{31}$ gs$^{-1}$, or the fallback mass between $4\times 10^{-6}\,M_{\sun}$ and $0.04\,M_{\sun}$ for typical disk formation time of about 1 s. Since the value of $s$ (or $p$) is highly uncertain and it is unclear how much of the wind matter really leaves the system or falls back again to the disk, we don’t consider the winds from the fallback disk for simplicity (i.e., $s=1$ or $p=0$), except the mass loss due to the Eddington limit accretion. In this case the accretion rate $\dot{M}_{\rm acc}$ of the NS is usually assumed to be $\dot{M}_{\rm acc}=\min[\dot{M}_{\rm E},\dot{M}_{\rm in}]$. However, if the photon’s optical depth is too high, the radiation is trapped within a radius at which the outward diffusion luminosity equals the inward convected luminosity, so that the neutrino loss plays a crucial role, and the radiative transfer probably only dominates in the later evolution [@col71; @che89]. @che89 showed that when the mass transfer rate decreases to $\dot{M}_{\rm cr}\simeq 3 \times 10^{-4}\,M_\odot\, \rm yr^{-1}$ the trapped photons can escape from the shocked envelope and the Eddington limit is enabled. Thus we assume that above this value the transferred mass is all accreted by the NS. The accretion rate of the NS can be formulated as $$\dot{M}_{\rm acc}=
\left\{
\begin{array}{ll}
\dot{M}_{\rm in}, \; & {\rm for}\
\dot{M}_{\rm in} > \dot{M}_{\rm cr} \; {\rm or} \; \dot{M}_{\rm in} < \dot{M}_{\rm E},\\
\dot{M}_{\rm E}, \; & {\rm for}\ \dot{M}_{\rm E} \leqslant \dot{M}_{\rm in} \leqslant \dot{M}_{\rm cr}.
\end{array}
\right.$$
Spin and magnetic field evolution of NSs
========================================
Spin evolution of NSs with a fallback disk
------------------------------------------
The spin evolution of NSs with a fallback disk can be divided into three phases according to the relationship of the following three critical radii.
\(1) The magnetospheric radius $R_{\rm m}$, at which the ram pressure of the accretion flow equals the magnetic pressure. Here we assume (Ghosh & Lamb 1979) $$R_{\rm m}=0.5R_{\rm A}=0.5(\frac{B^2R^6}{\dot{M}_{\rm in}\sqrt{2GM}})^{2/7}, \nonumber$$ where $R_{\rm A}$ is the Alfv[é]{}n radius, and $G$ is the gravitational constant. Note that if the mass transfer rate is sufficiently high $R_{\rm m}$ may be smaller than the NS radius $R$. This is physically impossible thus in our calculation we use $$R_{\rm m}={\rm max}(R,0.5R_{\rm A}) \nonumber$$ as the value of magnetospheric radius.
\(2) The corotation radius $$R_{\rm co}=(GM/\Omega^2)^{1/3}, \nonumber$$ at which the Keplerian angular velocity of the disk is equal to that of the NS.
\(3) The light cylinder radius $$R_{\rm LC}=c/\Omega, \nonumber$$ at which the corotating extension of the NS is equal to the speed of light $c$.
We assumed that the pulsar activity is switched off and a disk torque is exerted on the NS when the disk is able to penetrate into the light cylinder (i.e., $R_{\rm m} < R_{\rm LC}$). Furthermore, if $R_{\rm m} < R_{\rm co}$ the NS is in the accretor phase, otherwise the accretion flow is stopped and ejected by the centrifugal barrier and the NS is in the propeller phase [@is75]. The (unified) disk torque exerted on the NS in these two cases is taken to be [@mph01], $$I\dot{\Omega}=2\dot{M}_{\rm acc}R_{\rm m}^2\Omega_{\rm K}(R_{\rm m})\left[1-\Omega/\Omega_{\rm K}(R_{\rm m}) \right],$$ where $\Omega_{\rm K}$ is the Keplerian angular velocity in the disk. The NS is spun-up when $\Omega < \Omega_{\rm K}(R_{\rm m})$ and spun-down when $\Omega > \Omega_{\rm K}(R_{\rm m})$. When $\Omega = \Omega_{\rm K}(R_{\rm m})$ the NS is spinning at the so-called “equilibrium period” given by $$P_{\rm eq}=2\pi \left(\frac{R_{\rm m}^{3}}{GM}\right)^{1/2}.$$
If $R_{\rm m} \geqslant R_{\rm LC}$ the pulsar activity starts to work, and the NS is in the ejector phase. Since the kinetic energy density in the disk has a radial dependence $\propto r^{-5/2}$ (where $r$ is the distance from the center of the NS) steeper than the electromagnetic energy density outside the light cylinder ($\propto r^{-2}$), stable equilibrium of the disk outside the light cylinder is not allowed, unless it is beyond the gravitational capture radius [@lbw92]. So in this case we consider the spin-down torque only due to magnetic dipole radiation[^1].
As the pulsar ages the radio luminosity will fade away, and finally it will become unobservable as a pulsating source. This is assumed to occur when its evolutionary track on the $B-P$ plane crosses the so-called death-line given by [@rs75] $$B =0.17\times 10^{12}P^2\,{\rm G}.$$
Magnetic field evolution
------------------------
Magnetic field is the most important parameter that determines the spin evolution and the observational properties of isolated NSs. We assume that, along with accretion, the NS magnetic fields decay with the following form [@tv86; @sms89]: $$B=\frac{B_0}{1+\Delta M/10^{-5}M_{\odot}},$$ where $B_0$ is the initial magnetic field and $\Delta M$ is the accreted mass. When accretion stops the buried field will re-diffuse to the surface due to Ohmic diffusion and Hall drift [e.g., @gpz99]. The field diffusion process is governed by the MHD induction equation, and its speed depends on the initial magnetic strength and the overall accreted mass [for recent reviews, see @gep09; @ho13]. Based on the numerical calculations of field re-diffusion [e.g. @gpz99; @ho11], we fit the results with a phenomenological law for the growth rate of the magnetic field after accretion, $$\dot{B}=0.01\left( \frac{\Delta M}{M_{\odot}} \right)^{-3} \left(1-\frac{B}{B_0}\right)^2\;{\rm G\,yr^{-1}}.$$ Figure 1 illustrates the field emergence with different accreted mass.
Synthesis of NS population
==========================
In our Monte-Carlo simulation of the NS population we adopt similar input parameters as in @yps12. In previously works, the distribution of the birth spin periods of NSs has been suggested to be in a wide range from several milliseconds to hundreds of milliseconds [e.g. @acc02; @f-gk06], with various forms, e.g., a fix value [e.g. @fcm01], a flat distribution [e.g. @khb08], and a (logrithm) Gaussian distribution [e.g. @rdfp01; @lbd93]. The SNRs may present useful constraints on the initial spins of the pulsars associated with them. There is evidence supporting that the NSs are born with both fast and slow rotation. For example, the initial spin period ($\sim 19\rm \, ms$) of the Crab pulsar estimated from the age of the Crab nebula and the measured braking index [@mt77] implies that the NS was born spinning rapidly. The CCO 1E 1207.4$-$5209 in the SNR G296.5+10.0 has a spin period of 0.424 s [@zps00] and period derivative $\dot{P} < 2.5 \times 10^{-16}$ ss$^{-1}$ [@gh07]. The characteristic age of the NS $\tau_{\rm c} >27$ Myr exceeding the age of the SNR by three orders of magnitude, suggesting that 1E 1207.4$-$5209 was born with a spin period very similar to the current value. Here we adopt three different distributions of the initial spin periods, i.e., the “fast" population with, $$\begin{aligned}
\langle{\rm log}P_0({\rm s})\rangle=-2.3,\;\sigma_{{\rm log}P_0}=0.3,\end{aligned}$$ the “slow" spin population with $$\langle{\rm log}P_0({\rm s})\rangle=-0.5,\;\sigma_{{\rm log}P_0}=0.2,$$ and the “composite” population in which we assume that $\sim 40\%$ of the pulsars are born in the slow population [@vml04]. For the initial magnetic field almost all the works adopted a logarithm Gaussian distribution, and we take the following form of @acc02: $$\langle{\rm log}B_0({\rm G})\rangle=12.35,\;\sigma_{{\rm log}B_0}=0.4.$$
We follow @yps12 to assume a logarithm uniform distribution of the initial mass transfer rate $\dot{M}_0$ ranging from $10^{25}$ to $10^{28}$ $\rm g\,s^{-1}$, which is roughly consistent with previous semi-analytical and numerical results [@che89; @mwh01; @mnh97; @zwh08]. The initial radius of the fallback disk depends on the specific angular momentum of the fallback material. At the pre-supernova stage the specific angular momentum of the iron core of a rapidly rotating star with mass $8-25M_\odot$ is $\sim 10^{16}-10^{17}\rm cm^2\,s^{-1}$ [see @hlw00], corresponding a circularization radius $\sim 10^6-10^8\rm \, cm$. Thus we randomly select the logarithm of the initial radius (in units of cm) between 6 and 8.
Figures 2-4 illustrate the evolution of NSs with different initial parameters. Here we take typical values for the mass ($1.4M_\odot$), radius ($10^6\,\rm cm$) and initial magnetic field ($\sim 2\times 10^{12}$ G) of the NS. The braking index $n$ and characteristic age $\tau$ are plotted throughout the evolution although they are measurable only in the ejector phase. We consider three cases, in two of which $\dot{M}_0$ and $R_0$ are close to the maximum and minimum of the adopted values (note that $t_{\rm v}$ and $t_{\rm d}$ are positively correlated with $R_0$), and in third one we choose the medium values for $\dot{M}_0$ and $R_0$. In each figure the initial spin period is taken to be $P_0 = 300$ ms and 5 ms in the left and right panels, and the accretor, propeller and ejector phases are shown in dotted, dashed and solid lines, respectively.
In Fig. 2 we set $\dot{M}_0=10^{28}\,\rm gs^{-1}$ and $R_0=10^8\,\rm cm$ for NSs with both fast and slow initial spins. The magnetic field decays to be below $10^{9}\,\rm G$ within $\sim 0.1$ yr as $\sim 4\times 10^{-2}M_\odot$ mass is accreted, and has not recovered to its initial value at $10^6$ yr. In the left panel the NS is correspondingly accelerated from 300 ms to $\sim 4$ ms within the first $\sim 0.1$ yr. The accretor phase lasts $\sim 10^5$ yr, followed by the propeller phase until $10^6$ yr. The time dependence of $n$ and $\tau_{\rm c}$ is more complicated, which vary drastically during the evolution. At the beginning of the evolution, due to the enormously high accretion rate, the magnetosphere radius is suppressed to the NS surface. In this early stage $n<0$ since $\ddot{\Omega}<0$. When $\dot{M}_{\rm in}$ decreases to be lower than $\dot{M}_{\rm cr}$, $\dot{M}_{\rm acc}$ is limited by the Eddington limit. At this time $R_{\rm m}$ is still equal to $R$ (note that $R_{\rm m}$ depends on $\dot{M}_{\rm in}$), and $\dot{\Omega}$ is nearly constant, thus $n\sim 0$. As $\dot{M}_{\rm in}$ continues to decrease $R_{\rm m}$ becomes larger than $R$, and $n$ rapidly rises to $\sim 50$ because $\dot{R}_{\rm m}>0$ and $\ddot{M}_{\rm acc}\sim 0$ at this time. When $\dot{M}_{\rm in}=\dot{M}_{\rm E}$, $n$ has another abrupt change. At this time $\ddot{M}_{\rm acc}$ dominates over $\dot{R}_{\rm m}$ and $n<0$. When $\dot{M}_{\rm acc}$ decreases so that $R_{\rm m}= R_{\rm co}$ the NS enters the propeller phase, and the evolution is stopped at $10^6$ year.
In Fig. 3 we set moderate values for $\dot{M}_0(=10^{27}\,\rm g\,s^{-1})$ and $R_0(=10^7\,\rm cm)$. Since the accreted mass ($\sim 10^{-3}M_\odot$) is significantly lower than in Fig. 2, the magnetic field first decays to a few $10^{10}$ G, and re-grows to its initial value within $10^6$ yr. The fast-spinning NS has experienced all three evolutionary phases while the slow one is still in the propeller phase at the age of $10^{6}$ yr. In Fig. 4 we take $\dot{M}_0=10^{25}\rm\,g\,s^{-1}$ and $R_0=2 \times 10^6\rm\,cm$ (we note that if $R_0=10^6\rm\,cm$, $R_{\rm m}$ is larger than $R_0$ at the beginning of the evolution, the fallback disk does not exist and the NS enters the ejector phase directly). With such a low mass transfer rate the accretor phase is very short, about 2 yr and $10^{-3}$ yr for slow and rapid spinning NSs, respectively. The ejector phase correspondingly starts at around $10^4$ yr and 0.1 yr.
As we are only concerned with young pulsars, in the Monte-Carlo simulation we generate 100 NSs every 100 years in the first $10^4$ years, and extrapolate the results to $10^6$ years by sampling the parameters of NSs every $10^4$ years during the calculation. The distributions of $P$ and $B$ of the NS population are plotted in Fig. 5, at the time when the NS enters the pulsar phase, and of $10^4$ yr and $10^6$ yr. Note that here “pulsars" mean NSs in the ejector phase and located above the death line in the $B-P$ diagram. The calculated $P$ distribution at $10^6$ years shows double-peak structure for both the fast and slow spin populations. The reason is as follows. When the initial mass transfer rate is high, the NS magnetic field is decayed by several orders of magnitude, and the NS spends most of the time (several $10^5$ yr) in the accretor and propeller phases. Because of the weak magnetic field, the spin-down induced by the propeller torque and magnetic dipole radiation is insignificant, giving rise to a group of rapidly spinning pulsars in both the fast and slow population. The distribution of the composite population is a hybrid of the fast and slow population and has multi-peak structure at $10^6$ yr. For the fast spin population the distributions of $P$ and $B$ at $10^4$ yr provide a natural transition between the initial pulsar state to the relatively old one (at $10^6$ yr). However it is not the case in the slow spin population because, unlike the fast spin population in which most NSs have evolved into the ejector phase at $10^4$ yr, many of the slow NSs are still in the accretor and propeller phases.
In order to show the distribution of the braking indices and the characteristic ages for pulsars with different ages, we adopt the same initial distribution of the parameters as in the previous simulations but a variable birth rate, to ensure that the total number of NSs is constant for every logarithm interval of the age [see also @yps12]. For example, for ages between $10-10^{2}\,\rm yr$, $10^{2}-10^{3}\,\rm yr$ and $10^{3}-10^{4}\,\rm yr$ the birth rate is taken to be $1\,\rm yr^{-1}$, $10^{-1}\,\rm yr^{-1}$ and $10^{-2}\,\rm yr^{-1}$ respectively. Figure 6 shows the distributions of $\tau_{\rm c}$ and $n$ versus the age. The green and red crosses represent the NSs as observable pulsars and in the accretor/propeller phase, respectively. It is seen that for the slow spin population very few NSs can enter the pulsar phase within $10^6$ years.
For comparison with observations we also plot the observational data of pulsars with dots (with error bars) in Fig. 6. Here the pulsars are those associated with SNRs and with measured braking indices. The samples of SNR-PSR associations are taken from the ATNF pulsar catalogue[^2] and a census of high-energy observations of the Galactic SNRs[^3] [@ATNF; @HEcat-GSNR]. Thus we use the SNR ages as the real ages of the pulsars. The parameters of these pulsars are listed in Table 1. For all the three populations, the characteristic ages of the pulsars usually exceed their real ages by a factor up to $\sim 10^3$. The braking indices concentrate in the range $-1<n<3$ which is consistent with observations, but they can reach $\sim -10^3$. There is no pulsar with $n>3$, since in our model NSs with $n>3$ are in the accretor or propeller phase and thus unobservable, and the magnetic field growth is the only cause for the variation of $n$. Actually the braking indices for NSs in accretor and propeller phase are distributed in a much wider range of $[-10^5,10^3]$.
Discussion and conclusions
==========================
We have carried out population synthesis calculations of the NS evolution, taking into account the supernova fallback accretion, which suppresses the NS magnetic fields, and the post-emergence of the buried field. Though the input parameters in our simulation are similar to those in @yps12, some of the important assumptions are different in the two works. First, @yps12 assume that a fallback disk always surrounds the NS and exerts a torque on it; in our model the evolutionary sequence of the NS is divided into the accretor, propeller and ejector phases, the disk torque works only in the accretor and propeller phases, and the NS acts as a pulsar only in the ejector phase. Second, we consider the hypercritical accretion from the fallback matter when $\dot{M}_{\rm in}>\dot{M}_{\rm cr}$, and include the magnetic field evolution during and after the accretion, while in @yps12 the mass accretion rate is limited by $\dot{M}_{\rm E}$ and the magnetic field is assumed to be constant. Third, for the disk evolution we take the formation time $t_0 = {\rm max}(t_{\rm v},t_{\rm d})$ rather $t_0 = t_{\rm d}$ as in @yps12. This results in a lower mass transfer rate in the disk at the same age in our case. In our work due to the magnetic field re-emergence $n$ is always smaller than 3 when the NS is in the ejector phase, while in @yps12 the deviation of $n$ from 3 is caused by the disk torque: the majority of the pulsars have $n<3$, but a considerable fraction of them have $n>3$ (especially for the slow spin population). The distributions of both $\tau_{\rm c}$ and $n$ in our work are more dispersed and extended than in @yps12, since the magnetic field usually evolves on a longer timescale than the fallback disk.
Figure 6 shows that the statistical results of the fast spin population seem to be compatible with the observed distribution of the braking indices and the ages of young pulsars. NSs born with slow spins are difficult to survive the accretor and propeller phases. On the other hand, the birth period distribution for pulsars may also present possible constraints on the initial period distribution of the NSs[^4]. Recently @nsk13 derived the kinematic ages for 52 pulsars based on the measured pulsar proper motions and positions, by modelling the trajectory of the pulsars in a Galactic potential. They found that the birth periods of these pulsars show two-population structure, one with $P_0<400$ ms and the other with 700 ms $<P_0<1.1$ s. Although this result is based on the standard magnetic-dipole braking ($n = 3$), it is unlikely to deviate far from the real situation as shown by the authors. If we compare their birth period distribution with our calculated results (Fig. 5), we find that a considerable fraction of NSs may be born with relatively slow spins. This suggests that the newborn NSs might be composed by composite populations with a wide range of the spin periods.
It should be noted that currently solid theories on SN fallback and related field evolution are lacking, and hence many parameters adopted in our model are quite uncertain. This means that our results can be only regarded as illustrative rather for real situation. However, they keep the basic features for the evolution of the braking indices and the relation between the characteristic ages and the real ages. The model may be **tested** or refined by future observations.
We do not consider late evolution of the NSs. As shown by @pvg12, the effect of the magnetic field evolution on the braking index can be divided into three qualitatively different stages, depending on the age and the internal temperature of the NS: a first stage with fallback accretion and subsequent field evolution ($n < 3$); in a second stage, the evolution is governed by almost pure Ohmic field decay, and a braking index $n > 3$ is expected; in the third stage, at late times, when the interior temperature has dropped to very low values, the Hall oscillatory modes in the NS crust result in braking indices of high absolute value and both positive and negative signs. In the model proposed by @zx12 the field evolution is caused by a long-term power law decay coupled with short-term oscillations.
This work was supported by the Natural Science Foundation of China under grant numbers 11133001 and 11203009, the National Basic Research Program of China (973 Program 2009CB824800), and the Qinglan project of Jiangsu Province.
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{width="13cm"}
{width="8.1cm"} {width="8.1cm"}
{width="8.1cm"} {width="8.1cm"}
{width="8.1cm"} {width="8.1cm"}
{width="7.7cm"} {width="7.7cm"}\
{width="7.7cm"} {width="7.7cm"}\
{width="7.7cm"} {width="7.7cm"}
{width="7.0cm"} {width="7.0cm"}\
{width="7.0cm"} {width="7.0cm"}\
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[^1]: This is different from @mph01 and @yps12, who assumed that even in the ejector phase the disk can still exist outside the light cylinder, and the inner radius of the disk always coincides with the light cylinder radius irrespective of the decreasing mass transfer rate.
[^2]: http://www.atnf.csiro.au/people/pulsar/psrcat/
[^3]: http://www.physics.umanitoba.ca/snr/SNRcat/
[^4]: Note that the birth periods of pulsars are the initial periods when the NSs enter the ejector phase. They are similar to but not identical with the initial periods of newborn NSs.
|
---
author:
- |
[Sandeep Singh Sandha]{}\
University of California, Los Angeles\
Los Angeles, California 90095\
[email protected]
bibliography:
- 'main.bib'
title: '**StreetX: Spatio-Temporal Access Control Model for Data**'
---
Abstract {#abstract .unnumbered}
--------
Cities are a big source of spatio-temporal data that is shared across entities to drive potential use cases. Many of the Spatio-temporal datasets are confidential and are selectively shared. To allow selective sharing, several access control models exist, however user cannot express arbitrary space and time constraints on data attributes using them. In this paper we focus on spatio-temporal access control model. We show that location and time attributes of data may decide its confidentiality via a motivating example and thus can affect user’s access control policy. In this paper, we present StreetX which enables user to represent constraints on multiple arbitrary space regions and time windows using a simple abstract language. StreetX is scalable and is designed to handle large amount of spatio-temporal data from multiple users. Multiple space and time constraints can affect performance of the query and may also result in conflicts. StreetX automatically resolve conflicts and optimizes the query evaluation with access control to improve performance. We implemented and tested prototype of StreetX using space constraints by defining region having 1749 polygon coordinates on 10 million data records. Our testing shows that StreetX extends the current access control with spatio-temporal capabilities.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This paper contains original work. All the figures, tables and diagrams were made by me. The paper is written by me. I have done implementation, evaluation and testing of the system. The project idea originated from my discussions with my advisor Prof. Mani B. Srivastava and Dr. Bharathan Balaji. They both helped me to formulate the problem and set goals to push the limit of current access control. Prof. Mani B. Srivastava also gave me valuable feedback to include data sharing policies. I would like to acknowledge the help of Prof. Yuan Tian in understanding the current access control models. Also, I would like to thank Dr. Bharathan Balaji, Amr Alanwar and Moustafa Alzantot for proofreading.
Introduction
============
Individuals, organizations and different sectors of city generate spatio-temporal data having volume, variety, velocity and value. Few examples of such datasets include New York City taxi data with over 1.1 billion taxi trips [@nyctaxidata:2017], location tracking data of individuals, satellite data from NASA that produces 4TB of new data everyday [@nasadata:2017], Los Angeles city traffic data [@LATrafficData:2017] and water data of 1.5 million sites [@USGSData:2017]. These datasets has potential to create innovative applications which encourage cities to collect more data. For example traffic and crash datasets are used together to predict safety of roads [@krumm2017risk]. To amplify usability, datasets are shared, stored and transformed by different entities. For instance, acute respiratory syndrome disease was controlled by the efforts of World Heath Organization only within 4 months after its emergence using extensive data sharing [@destro2014perspectives]. Several open data portals give public access to city information [@LAData:2017; @USOpenData:2017]. But some datasets are confidential and are only shared with authorized entities or shared partly. For example, the health department may share confidential data with law enforcement agencies.
We focus on the space and time attributes of data to define access control policy. Space and time attributes often decide the privacy and confidentiality of data. For example, let us consider Alice who is participating in a health study. During the study she monitors her multiple vitals and location for several days while doing normal daily activities. Bob is interested in health information of Alice for his research. Alice is willing to share her data so as to understand her health problems better. But Alice wants selective data sharing with Bob according to her privacy needs. She wants to share her data when she is within Los Angeles excluding her home and only during working hours with a hourly time resolution. Alice also wants Bob to not share her data further. [In order to restrict access to data different types of access control models exist, but they lack support to specify spatio-temporal constraints on data attributes. Thus, Alice cannot express her desired access control using earlier models. Alice may have to create and share subset of her data manually.]{}
Access control based on space and time attributes restricts data at confidential locations and periods. Controlling the resolution of data limits the application usability and ability of a user to extract sensitive information. Data sharing policies affect the capabilities of user to share data further and thus can control data dissemination. [However, providing spatio-temporal access control on large datasets is not trivial. Multiple arbitrary space and time constraints may result in conflicts and can also severely penalize the performance of the system. Thus it require conflict resolution strategies and policy optimization techniques.]{}
In this paper, we present StreetX which provides simple language to specify multiple arbitrary constraints on space and time attributes and data sharing policies. StreetX policy language present constraints in human understandable semantics. [ StreetX automatically resolves conflicting access control policies and uses optimization strategies when applying policies on user queries. ]{} We show that using StreetX, Alice can easily express her access control policy for Bob. To evaluate the proposed access control model, we implemented and tested a prototype of StreetX to evaluate random user query along with space constraints specified using polygon of latitude and longitude having 1749 coordinates on 10 Million spatio-temporal data records. StreetX filters user queries using policy constraints and does query evaluation in 72 milliseconds on an average. [The remainder of this paper is organized as follows: in section 2 we discuss the related work; section 3 introduces the StreetX model; section 4 discusses access control and language. section 5 presents StreetX architecture and implementation details; section 6 gives evaluation and testing results; finally, in section 7 we conclude and point to future works.]{}
Related Work
============
[In this section, we survey access control models, policy definitions and related systems and illustrate that none of these support multiple arbitrary spatio-temporal constraints on data attributes.]{} In order to control the granularity of access control many techniques have been proposed. Traditional models such as Role-Based Access Control [@sandhu1996role] and Attribute-Based Access Control [@hu2015attribute] are used extensively. Role-Based Access Control regulates access based on the role of individual user. User’s role is associated with privileges and functions which user can perform. Attribute-Based Access Control uses attributes of user, data, context or action to restrict access. Multiple extensions to Role-Based Access Control using user’s space and time context are proposed by researchers. For example TRBAC [@bertino2001trbac] activates user roles at certain time periods and LRBAC [@ray2006lrbac] controls user privileges based on her physical location. There are similar works [@le2012strobac; @samuel2007framework; @toahchoodee2009ensuring; @ray2008spatio] where user’s role gets activated based on location or time of user requesting access to resource. However they don’t consider the space and time attributes of datasets to define spatio-temporal access control. PlexC [@le2012plexc] is a policy language designed for exposure control of data by allowing users to define access control policies based on different roles, disclosure level, time and location rules based on user context. PlexC doesn’t consider multiple space regions and time windows on data attributes. PDVLoc [@mun2014pdvloc] is designed for location based mobile services using which individual user can control location data sharing. PDVLoc is an individual data store with fine granular access control, policy recommendation and trace audit. PDVLoc supports space policies based on circular range. SensorSafe [@choi2011sensorsafe] also provides fine granular access control with major focus on privacy rules and data obfuscation techniques. It allows single space region definition in access control. Both SensorSafe and PDVLoc have no support for multiple arbitrary space boundaries, time windows and data sharing policies. Cell level access control also exists in databases like Apache Hbase [@hbase:2017] and Apache Accumulo [@accumulo:2017]. In Accumulo and Hbase each data cell can be associated with a label which can be later used to selectively control data access. Along with support for simple labels, logical *AND* and *OR* operations on labels are also supported. However these databases don’t support spatio-temporal access control. ArcGIS [@arcgis:2017] is an industry standard geographic information system, but it only supports Role-Based Access Control model. Data sharing policies are explored by attaching policies with data by Saroiu et al [@saroiu2015policy]. Every time an organization request access to the user’s data, it has to explicitly acknowledge the user policy however, policies are not enforce. Organization may later misuse the data against the sharing policy which may have legal consequences. [We extend the idea to specify data sharing policies along with spatio-temporal access control. However, StreetX doesn’t enforce the data sharing policies. ]{}
[Collectively, these systems represent an effort to develop fine granular access control. However, no system allows multiple arbitrary constraints on space and time attributes of data in a multi-user setting which require expressible language model, conflict resolution and query optimization techniques. Also handling handling large datasets require distributed architecture to be scalable.]{}
StreetX Model
=============
StreetX serves as a data hub for spatio-temporal data belonging to different entities. Streetx allows data owner to define fine granular spatio-temporal access control based on data attributes and specifying data sharing policies. Heterogeneous spatio-temporal datasets are transformed into a collection of data streams. Each data stream has an owner which specifies the access control policy and data sharing policy for other users. Both owners and users can query the data streams to which they have access to. StreetX supports rich spatio-temporal query across multiple data streams. While evaluating each query StreetX makes sure that the access control policy associated with the individual data stream is respected. Below we define the concept of data stream, owner, user and policy in our system.
**Data Stream**: Data Stream uniquely identifies the type of data. For example we may have two data streams one for the temperature sensor and other for the accelerometer sensor. A dataset with multiple measurements can be transformed into multiple data streams, one for each measurement. Each data stream consists of (latitude, longitude, time, value) and is uniquely identified by a data stream id ([$d_i$]{}). The set of data streams in the system is $D=\{ d_1,...,d_n\}$.
**Owner**: Every data stream has an owner. Owner is the entity who has created the data stream and has added data into it. Owner may own multiple data streams. Owner is the one which control the access and data sharing policies of the data stream. Owner creates policies for users with whom she wants to share her data stream. For simplicity we consider a data stream can have only one owner and ownership cannot be transfered. Every owner is uniquely identified by an owner id ([$o_i$]{}). The set of owners in the system is $O =\{o_1,...,o_m\}$.
**User**: User is the entity interested to use the data stream of owner. User can query the data stream, during which the owner policies defined on the data stream for the user are respected. A user may have access to several data streams and can query them simultaneously. In case user is the owner of the data stream, her query is answered directly. Every user is uniquely identified by a user id ([$u_i$]{}). The set of users in the system is $U =\{u_1,...,u_l\}$. [The set of owners ($O$) is a subset of the set of users ($U$).]{}
**Policy**: Policy defines the access of user to the data stream. There are two types of policies which we consider in StreetX: access control policy and data sharing policy. Access control policy is specified in terms of space and time attributes of the data stream. Access control policy can also affect the resolution of space and time attributes of data. Data sharing policy defines the sharing capabilities of the user for the data. Owner may define multiple policies on her data streams for same user. Every policy is uniquely identified by a policy id ([$p_i$]{}). The set of policies in the system is $P =\{p_1,...,p_r\}$.
Policy & Language
=================
StreetX has two different types of policies: Access control policies and data sharing policies. StreetX policy definitions are using abstract language constructs presented in Table \[table:1\].
-------------------------------------------------------------------
**Constructs** **Meaning & Examples**
---------------- --------------------------------------------------
**What** Specifies the set of data streams
on which policies are defined using
data stream ids. Eg: What(**$d_1$, $d_2$**)
**Where** Specifies the multiple semantic
space regions along with a
optional NOT keyword to
represent exclusion.
Eg: Where(**LA**, NOT **HOME**)
**When** Specifies the multiple semantic
windows of time along with a
optional NOT keyword to
represent exclusion.
Eg: When(**WorkingHours**,
“11/1/2016-11/31/2016")\
**How** & Specifies the resolution restrictions\
&on space and time. Resolution\
& keywords are reserved.\
&Eg: How(**Hour**, **ZipCode**)\
**Whom** & Specifies the set of users to whom\
&this policy applies using user ids.\
&Eg: Whom(**$u_1$, $u_2$**)\
**Who** & Specifies the data sharing policy\
&of owner using reserved keywords.\
&Eg: Who(**DenyDataSharing**)\
-------------------------------------------------------------------
: Policy Language[]{data-label="table:1"}
In Table \[table:1\], the *What* construct select the set of data streams on which the policy is being applied and the *Whom* construct selects the set of users whose access control is affected by the policy. Any policy at minimum must have *What* and *Whom* constructs. The *Where*, *When* and *How* constructs are defining the access control policy based on space and time attributes of data, which is detailed in the Section \[AccConPol\]. The *Who* construct defines the data sharing policy explained in Section \[datasharingpolicy\]. In *Where* construct, union and exclusion of different space regions can be defined and similar is allowed with time windows in *When* construct.
User Defined & Language Keywords {#keywords}
--------------------------------
Table \[table:1\] shows a subset of keywords used in our language. There are two types of keywords: user defined keywords and language reserved keywords. The user defined keywords are used in *Where* and *When* constructs to define space regions and time windows. For example the *LA, HOME* and *WorkingHours* are user defined keywords in Table \[table:1\]. [We assume that in future, keywords can be defined using graphical user interface extension to StreetX, which allows uploading GeoJSON [@GeoJSON:2017] definitions of regions and specify time constraints.]{} GeoJSON can be used to define arbitrarily complex space region. The language reserved keyword *NOT* can be used in *Where* and *When* constructs to exclude a particular space region or time window. The sample user keyword definitions are shown in Table \[table:2\].
[|c|c|]{} **Keywords** & **Definitions**\
HOME & {“Name":“HOME",“Type":\
&“Where",“Polygon":\[{$lat_1$,$lng_1$}\
&,{$lat_2$,$lng_2$},...,{$lat_1$,$lng_1$}\]}\
WorkingHours & {“Name":“WorkingHours",\
&“Type":“When",“RepeatedHour"\
&:“9AM-5PM",“ExcludeDay":\
&\[“saturday",“sunday"\]}\
The language reserved keywords to specify the resolution in **How** construct and data sharing policy specification in **Who** construct are shown in Table \[table:3\].
**Construct** **Keywords**
--------------- ----------------------------------
How:Time Second, Minute, Hour, Day,
Week, Month, Year.
How:Space ZipCodes, County, City, Country.
Who AllowDataSharing,
DenyDataSharing,
PolicyUpdateEffect.
: Language Keywords[]{data-label="table:3"}
Access Control Policy {#AccConPol}
---------------------
Access control policy can be specified using the abstract language constructs which are shown in Table \[table:1\]. The constructs are very near to the user’s natural language and uses semantic keywords. Looking back to our previous example of Alice and Bob, where Alice wanted to selectively share her historical health data with Bob. The access control policy to specify the space within Los Angeles excluding the home, during the working hours and with hourly resolution can be defined by Alice using Equation \[eq:1\], where $d_h$ identifies the health data stream of Alice, and $u_b$ identifies the user Bob. The *HOME* and *WorkingHour* keywords are defined in Table \[table:2\]. Similarly *LA* keyword can also be defined. $$\label{eq:1}
\begin{split}
\text{What($d_h$).Where(LA, NOT HOME).}\\
\text{When(WorkingHours).How(Hour).Whom($u_b$)}
\end{split}$$
Data Sharing Policy {#datasharingpolicy}
-------------------
Owner can specify data sharing policy in the *Who* construct described in Table \[table:1\] using the language reserved keywords defined in Table \[table:3\]. *Who(AllowDataSharing)* specifies that owner has allowed data sharing. The data shared by the user is at the resolution specified by the *How* construct. *Who(AllowDataSharing,PolicyUpdateEffect)* specifies that data sharing is allowed by owner, but in future owner may change the policy to deny, and all the data shared by the user to multiple other entities will not be available to them. *PolicyUpdateEffect* controls the effect of policy update by owner on multi-hop sharing. *Who(DenyDataSharing)* specifies that data sharing is not allowed. [StreetX allows owner to specify data sharing policies which are informed to users. Violating these policies may have legal consequences. However StreetX doesn’t enforces the data sharing policies.]{} [The data sharing policies affect both raw data and processed data. We didn’t separate this because user may use identity function on the raw data stream to replicate it or may encrypt the raw data stream values which might be used to regenerated raw data stream later.]{} In our previous example, the final policy of Alice for Bob along with specified data sharing is expressed in Equation \[eq:datasharing\].
$$\label{eq:datasharing}
\begin{split}
\text{What($d_h$).Where(LA, NOT HOME).}\\
\text{When(WorkingHours).How(Hour).Whom($u_b$)}\\
\text{.Who(DenyDataSharing)}
\end{split}$$
![System Architecture[]{data-label="fig:arch"}](fig/Arch){width="50.00000%"}
StreetX Workflow
----------------
In order to understand the workflow of StreetX, let us consider our earlier example of Alice and Bob. Alice defines a data stream $d_h$ and uploads her entire health data. She defines keywords mentioned in Table \[table:2\] and creates a policy shown in Equation \[eq:1\] for user Bob identified by $u_b$. Now, Bob can query Alice data, however data exposed to Bob is controlled by the policy of Alice. Our assumption is that health data of Alice is defined by only one data stream. She may define different data stream for each heath measurement type and can define access control policies on set of data stream. Suppose later Alice discovered that, her data within UCLA campus contains some confidential information, she can update her policy for Bob to hide her data while she is in UCLA campus and also keep the previous policy constrains as shown in Equation \[eq:workshop\]. StreetX allows Alice to define and update her policy easily.
$$\label{eq:workshop}
\begin{split}
\text{What($d_h$).Where(LA, NOT HOME, NOT}\\
\text{UCLA).When(WorkingHours).How(Hour)}\\
\text{.Whom($u_b$).Who(DenyDataSharing)}
\end{split}$$
StreetX Architecture & Implementation
=====================================
StreetX architecture is shown in Figure \[fig:arch\]. *Distributed Database* is used for scalability and performance reasons. *Distributed Database* stores the data streams, metadata defining the owners, users and policies. *Spatio-Temporal* layer provides the capabilities which are used to enforce the access control policies based on space and time. *Access Control and Data Management layer* have multiple components. *Data Ingestion* module is used to ingest spatio-temporal data into the *Distributed Database*. *Policy Definition* module provides abstract language constructs presented in Table \[table:1\] and keywords definition support discussed in section \[keywords\]. *Policy Metadata* module manages the information about policies, owners, data streams and users affected by policies. *Query* module is used by owners and users to query the spatio-temporal data to which they have access to. *Query* module uses information from *Policy Metadata* module to identify the set of relevant policies for a particular query. The conflicts in the policies if any are resolved by the *Policy Conflict Resolution* module. The policies are applied by *Policy Application* module which translate the user query along with extra constraints introduced due to the policies into a spatio-temporal query. This transformed query is now evaluated by the *Spatio-temporal layer*. The *API Layer* exposes the functionality of StreetX to authenticated owners and users. We implemented a prototype of StreetX in Java. We used Apache HBase [@hbase:2017] for *Distributed Database* layer. Apache HBase is an open-source NoSQL scalable distributed database. Apache HBase stores the spatio-temporal data streams, and metadata. We use GeoMesa [@geomesa:2017] as spatio-temporal layer on top of Apache HBase. GeoMesa is open source suite of tools that provide geospatial analytics support for spatio-temporal data. [GeoMesa supports Common Query Language [@cql:2017] which offers geospatial, temporal and attribute operators.]{}
Data Ingestion & Query
----------------------
*Data Ingestion* module allows owner to push data into StreetX. StreetX supports the ingestion of spatio-temporal data expressed in the form of data streams. The data ingestion module inserts data into Apache HBase. *Query* module exposes data streams to user. StreetX implementation supports spatio-temporal query on arbitrary set of data streams via JavaScript Object Notation (JSON) [@json:2017] format presented in Representation \[lst:querytrans\].
{"(*\bfseries userId *)":$u_b$,"(*\bfseries DsID *)":[$d_1$,...,$d_i$],
"(*\bfseries SpaceBox *)":[$lat_{min}$,$lat_{max}$,$lng_{min}$,$lng_{max}$],
"(*\bfseries TimeRange *)":[$TimeStamp_{min}$,$TimeStamp_{max}$]}
Policy Application
------------------
*Policy Application* module of Figure \[fig:arch\] translates the policy into constraints on space and time attributes of data. *Policy Application* module also performs the required computation for controlling resolution, which is discussed in Section \[resolution\]. For example to apply policy of Alice specified in Equation \[eq:datasharing\], we translate the policy to respective space and time constraints in JSON using keyword definitions of Table \[table:2\]. The sample output of translation of Equation \[eq:datasharing\] is shown in Representation \[lst:Sptrans\]. In order to translate *WorkingHours*, we use the meta data of data stream $d_h$ to know the start time and end time of data and create time windows of daily working hours in UNIX timestamps excluding Saturdays and Sundays. Visualizing the space region for this policy may look as shown in Figure \[fig:trans1\]. In Figure \[fig:trans1\], we have shown only 2 regions for simplicity, but policy language can express multiple arbitrary regions.
{"(*\bfseries Space *)":{"Allow":[{"Keyword":"(*\bfseries LA*)",
"Polygon":[{"lat":$lat_{i1}$,"lng":$lng_{i1}$},
..]}],"Deny":"[{"Keyword":"(*\bfseries HOME*)",
Polygon:[{"lat":$lat_{j1}$,"lng":$lng_{i1}$},
..]}]},
"(*\bfseries Time *)":[{"Keyword":"(*\bfseries WorkingHours*)",
Allow:[{"start:":$t_1$,"end":$t_2$},..]}]}
![ Space Region for Translation 1. Showing a subset of Los Angeles (*LA*) Region having 1749 coordinates in blue and definition of *HOME* Region using bounding box with 4 coordinates highlighted by red markers.[]{data-label="fig:trans1"}](fig/LA_home1){width="50.00000%"}
User query presented in Representation \[lst:querytrans\] and policy translation presented in Representation \[lst:Sptrans\] are further translated into the effective set of constraints for evaluation. The constraints can be grouped into set of space polygons and set of time windows for each data stream. For simplicity if we consider a single data stream, the final effective constraint translation is presented in Representation \[lst:finaltrans\]. Here {$R_{qb}$}$_{QuerySpaceBOX}$ and {$T_{1}$,$T_{2}$}$_{QueryTimeRange}$ are from user query (here Bob). {$R_{LA}$}$_{SpacePolicy}$, {$R_{HOME}$}$_{SpacePolicy}$ and {$T_{i-1}$,$T_{i}$}$_{QueryTimeRange}$ are from access control policy of Alice given in Equation \[eq:1\]. In prototype implementation, StreetX allows the policy constraints expressed in Representation 2. We plan to extend it with the proposed language constructs in future. Representation \[lst:finaltrans\] can be directly translated into Common Query Language supported by by GeoMesa, however it may adversely affect the performance. We use query optimization strategies discussed in Section \[queryoptimization\] to improve the query evaluation time.
{"(*\bfseries Space *)":{{$R_{qb}$}$_{QuerySpaceBOX}$ AND
{{$R_{LA}$}$_{SpacePolicy}$ AND {NOT $R_{HOME}$}$_{SpacePolicy}$}},
"(*\bfseries Time *)":[{$T_{min}$,$T_{max}$}$_{QueryTimeRange}$ AND
{{$T_{1}$,$T_{2}$}$_{QueryTimeRange}$,... AND {$T_{n-1}$,$T_{n}$}}]
Query Optimization {#queryoptimization}
------------------
[ Final access control constraints shown in Representation \[lst:finaltrans\] can affect the query evaluation performance. In our testing discussed in Section \[testing\], considering the direct user query on data stream without access control as baseline. The performance is degraded by a factor of 4 times due to access control constraints of Figure \[fig:trans1\] even though less number of data points are fetched. We reject the user query and return 0 results if it cannot be satisfied by looking at all the constraints. In order to do this, we check the satisfiability of Representation \[lst:finaltrans\] to see if user query intersects the allowed space regions and time windows specified by policy. For example, to checking the satisfiability of spatial constraints in Representation \[lst:finaltrans\], we do the spatial query to see if {$R_{qb}$}$_{QuerySpaceBOX}$ intersects {$R_{LA}$}$_{SpacePolicy}$ and is not completely within {$R_{HOME}$}$_{SpacePolicy}$. StreetX keeps policy constraints in memory and applies them to every query before translating it and evaluating using GeoMesa. If, we directly evaluate an unsatisfiable query on GeoMesa, it will still return 0 results, but takes significantly more time which affects query performance as discussed in Section \[testing\]. In future, We also plan to group together the similar constraints because complexity of contraints directly affect the performance as discussed in Section \[testing\]. For example, two space regions can be grouped together, if one of them is completely within the other. ]{}
Space & Time Resolution {#resolution}
-----------------------
StreetX handles resolution constraints on a particular data stream by creating a separate data stream with the required resolution. For example, to implement the hourly resolution, we convert all the time values within an hour to the same value, making them indistinguishable in time attribute, the similar concept is extended to space attribute. For spatial resolution, we assume the availability of required spatial boundaries. StreetX redirects all the user queries to new data stream which is not exposed to owner directly and exists only if resolution policy is defined by the Owner. Other approach is to create resolution on the fly from original data stream after doing the query. It will require processing of time and location values for every record in the result set and may also discard some of the records, because resolution can also affect the number of points fetched. We didn’t use this approach due to performance reason, however replicating data stream uses extra storage. In prototype implementation, StreetX assumes the data stream with desired resolution.
Policy Conflict Resolution
--------------------------
*Policy Conflict Resolution* module identifies conflicting policies at the time of policy definition. Allowing multiple space and time restrictions may result in conflicts. Conflict happens when multiple policies exist for the same user defined over the same data stream. Three different scenario’s in conflict are possible: Firstly, When either the space regions or time windows from different policies don’t overlap with each other. Such a scenario is explained in Figure \[fig:trans2\], where policy P1, P2, P3, P4 and P5 have space region constraints of type allow. In this case, union of all policies is followed , even though their time windows may overlap. In such conflicts, multiple policies are defined over different subsets of data within data stream and thus are not really conflicting in terms of constraints, a union of policies is followed by doing union on all the constraints. Second scenario happens when both space regions and time windows of different policies overlap, which means both are defined over same set of data within data stream. Consider for example two policies P1 and P2 where space region and time windows of both policies overlap. For simplicity, consider space regions are as shown in Figure \[fig:trans3\] and time windows are exactly same for both policies. [In this case, StreetX follows the principle of least privilege while doing union of policies. StreetX gives high priority to the constraints denying space regions and time windows.]{} A third scenario which requires special treatment is an extension to second scenario. It happens when policies have different resolutions. For example, consider the Figure \[fig:trans3\]. If policy P1 and policy P2 have different resolution, then enforcing resolution constraints is a challenge. Currently, StreetX create different data streams for policy P1 and Policy P2, if they have different resolution. This will require StreetX to subdivide the query and use resolution of P1 for R1 and resolution of P2 for R2. But for R3, decision cannot be simply made based on principle of least privilege because two different resolutions may not be comparable sometime. For example, resolution based on time cannot be compared with resolution based on space. StreetX model cannot resolve the third scenario completely without attaching priorities with policies. We have not implemented *Policy Conflict Resolution* module in prototype of StreetX. We assume that all the desired constraints are already grouped together by following union of policies and are expressed in Representation 2.
![Non-Overlapping Space Regions for Different Policies. []{data-label="fig:trans2"}](fig/PApplication2){width="3in"}
![Overlapping Space Region for Policy P1 and Policy P2.[]{data-label="fig:trans3"}](fig/PApplication3){width="3in"}
Evaluation and Testing {#testing}
======================
We deployed prototype of StreetX on a single instance of Microsoft Azure [@azure:2017] DS4\_V2 having 8 cores and 28GB RAM on Ubuntu Server 16.04 using HBase v1.2.5. We tested the performance of StreetX in applying space policy constraint using a region shown in Figure 2, having 7649 coordinates extracted from the website of Los Angeles County Department of Public Works [@LAcountyshape:2017]. The policy definition used in this testing is shown in Equation \[eq:streetxtesting\].
$$\label{eq:streetxtesting}
\begin{split}
\text{What($d$).Where(LA, NOT HOME).Whom($u$)}\\
\end{split}$$
The *LA* space region has a complex shape as shown in Figure \[fig:trans1\] which is roughly 5,000 $km^2$. Table \[table:3\] presents summary of testing. A data stream having 10 Million points was generated randomly with location variation in an area $A$ of 500 KM by 500 KM around Los Angeles and time variation $T$ of 1 year between 2014 to 2015. User query is generated randomly in the format shown in Representation \[lst:querytrans\] using a *spacebox* of 50KM by 50KM within $A$ and *timerange* of 2 weeks within $T$. Three different types of queries are performed as shown in Table \[table:3\]. The results presented are the average of 1000 user queries. ${Query_d}$ is direct user query without any policy application. It gives the baseline for number of points fetched with no space restrictions and performance time. ${Query_a}$ is user query evaluated along with policy of Equation \[eq:streetxtesting\] without optimizations discussed in Section \[queryoptimization\]. ${Query_b}$ is user query evaluated along with policy of Equation \[eq:streetxtesting\] and optimizations discussed in Section \[queryoptimization\].
[|c|c|c|c|]{} **Dataset** & **Query** & **No. of points** & **Time (ms)**\
10 Million &$Query_d$ & 5230 &540\
10 Million & $Query_a$ &202 &2325\
10 Million & $Query_b$ & 202 &72\
$Query_d$ serves as a baseline when user query is evaluated without any policy. As expected with policy restrictions the number of points fetched decreased from 5230 to 202 in $Query_a$ and $Query_b$. But $Query_a$ takes 4 times more time then even $Query_d$ though it retrieves less points. This extra time is due to the complex space policy restrictions. $Query_b$ takes only 72 milliseconds to evaluate on an average. The reason behind this is rejection of the queries which cannot be satisfied by *Policy Application* module without evaluating them using GeoMesa. In our experiments of generating 1000 random user queries, 900 queries were rejected by *Policy Application* module using optimizations discussed in Section \[queryoptimization\]. $Query_b$ also keeps the policies in memory which does the language translation faster. In order to understand the effect of number of coordinates in space constraint, we performed $Query_a$ and $Query_b$ by decreasing the number of coordinates in *LA* space region from 7649 by 3 times to 2550 by random selection. The results are shown in Table \[table:5\], which depict the effect of size of constraints on performance. The results indicate that user should express the resolution of space boundary with the least possible number of coordinates. The increase in performance of $Query_a$ is more than 6 times whereas for $Query_b$ it is less, which is because $Query_b$ is already optimized. Our results show that StreetX can express and evaluate complex spatio-temporal access control policies over large datasets.
**Dataset** **Query** **No. of points** **Time (ms)**
------------- ----------- ------------------- ---------------
10 Million $Query_a$ 258 345
10 Million $Query_b$ 258 63
: StreetX Testing With 2550 Space Constraint Coordinates[]{data-label="table:5"}
Conclusions & Future Works
==========================
In this paper, we presented StreetX to enhance access control capabilities with fine granular constraints on space and time attributes of data. StreetX serves novel applications by allowing selective data sharing based on confidential locations and time periods using simple abstract language. One such evaluation was presented by us where space policy was defined using complex space region. Arbitrary spatio-temporal constraints may result in conflicts which are handled by StreetX. StreetX also uses query optimization strategies which enhance performance significantly. Our evaluation shows StreetX is scalable and can represent and evaluate arbitrary spatio-temporal policies over large datasets in an optimized manner.
In future, we plan to implement all the proposed features and do further testing of StreetX by evaluating it using real spatio-temporal datasets. We plan to integrate StreetX with city scale data hub projects like MetroInsight [@MetroInsight:2017] and health data hub initiatives like MD2K [@MD2K:2017] to enhance data sharing of real systems. Different extensions to StreetX are possible, like allowing users to specify policies in natural language and built in support for general shape definitions of cities, counties etc. We also plan to explore use of trusted computing platforms [@pearson2002trusted] to enforce data sharing policies.
|
---
abstract: 'We study the influence of a linear nonlocal spatial coupling on the interaction of fronts connecting two equivalent stable states in the prototypical $1$-D real Ginzburg-Landau equation. While for local coupling the fronts are always monotonic and therefore the dynamical behavior leads to coarsening and the annihilation of pairs of fronts, nonlocal terms can induce spatial oscillations in the front, allowing for the creation of localized structures, emerging from pinning between two fronts. We show this for three different nonlocal influence kernels. The first two, mod-exponential and Gaussian, are positive-definite and decay exponentially or faster, while the third one, a Mexican-hat kernel, is not positive definite.'
author:
- 'Lendert Gelens$^{1,2}$, Manuel A. Matías$^2$, Damià Gomila$^{2}$, Tom Dorissen$^1$, and Pere Colet$^2$'
bibliography:
- 'Nonlocal.bib'
title: 'Formation of localized structures in bistable systems through nonlocal spatial coupling II: The nonlocal Ginzburg Landau Equation'
---
Introduction {#Sect::Introduction}
============
We have shown recently [@GelensPRL2010] that a nonlocal interaction term can induce oscillatory tails in otherwise monotonic fronts connecting two equivalent homogeneous steady states (HSSs) in the Ginzburg-Landau equation (GLE) for a 1-dimensional (1-D) real field. As a consequence the interaction between a pair of fronts has an oscillatory dependence with the distance between the fronts with an exponentially decaying envelope. The oscillatory dependence allows for a pair of fronts to be pinned at specific distances determined by the tail profile. In particular localized structures (LSs) can arise as a consequence of the pinning.
In [@PartI], which we will refer to as Part I here, we have presented a suitable framework to understand the effect of linear nonlocal spatial coupling on the shape of a class of fronts connecting two equivalent HSSs allowing to determine the parameter regions where fronts have an oscillatory profile. Here we apply this general framework to rationalize and extend the results advanced in [@GelensPRL2010] and elucidate the region in parameter space where LSs can exist for different forms of nonlocal interaction.
In particular in Part I we considered 1-D extended systems described by a real field with a nonlocal interaction term $s F(x,\sigma)$, in which $s$ is a parameter that controls the overal strength and sign of the coupling while $F(x,\sigma)$ can be written as the convolution of a spatially nonlocal kernel $K_{\sigma}(x)$, with the field $A(x)$ $$F(x,\sigma) = \int_{-\infty}^{\infty} \! K_{\sigma}(x-x')\,A(x')dx'\ ,
\label{Eq::nonlocal_term}$$ where the parameter $\sigma$ controls the spatial extension (width) of the coupling.
Here we will consider three different interaction kernels that illustrate the generality of the spatially nonlocal effects considered. Two of the kernels are positive definite, Gaussian and mod-exponential, while the third is a non-positive definite kernel, a Mexican-hat kernel. Table \[tab:kernels\] gives the expression in real and Fourier space of the three kernels used in the present work, while Fig. \[Fig::kernel\_shapes\] displays its shape. Moreover, these three kernels are relevant in different applications.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**Gaussian** **mod-exponential** **Mexican hat**
------------------------ -- --------------------------------------------- ------------------------------------------------------- --------------------------------------------------------------
$K_{\sigma}(x)$: $\frac{1}{ \sqrt{2 \pi} \sigma} e^{-x^2/ 2 $\frac{1}{ 4 \sigma} e^{- {\lvertx\rvert}/ 2 \sigma}$ $\frac{\sqrt{2}}{\sqrt{\pi} \sigma}
\sigma^2}$ \left(1- b \frac{x^2}{\sigma^2}\right) e^{-x^2/ 2 \sigma^2}$
$\hat{K}_{\sigma}(k)$: $e^{-k^2 \sigma^2 / 2 }$ $\frac{1}{1+4 $ 2 (1+ b (-1 + \sigma^2 k^2) ) e^{-\sigma^2 k^2/ 2 }$
\sigma^2 k^2}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[Fig::kernel\_shapes\] Representations of the three nonlocal interaction kernels used in this work ($\sigma=1)$: (a) Gaussian; (b) mod-exponential (Laplacian); (c) Mexican-hat ($b=1/2$). The right panels (d)-(f) show the Fourier transforms of the kernels in panels (a)-(c), respectively.](figure1-01.eps){width="8.6cm"}
Spatially nonlocal interactions in which the spatial interaction kernel is positive definite are either attractive (activatory) or repulsive (inhibitory) depending on the sign of $s$. The most usual kernels of this type are the Gaussian and the exponential, that have been studied in several contexts like competition effects in Ecology [@Emilio_2009; @*Emilio_2010; @Clerc_PRE_2005; @Clerc_PRE_2010], Nonlinear Optics [@Krolikowski_PRE_2001; @*Wyller_PRE_2002; @*Ouyang_PRE_2006], reaction-diffusion systems [@Bordyugov_PRE_2006] and Neuroscience [@Ermentrout_RPP; @Hellwig; @Coombes05]. Physically, in reaction-diffusion systems an spatially nonlocal interaction with an exponential kernel arises whenever an adiabatic elimination of a fast diffusing substance is performed [@Shima_PRE_2004]. More general exponential decaying kernels with a variable exponent, including the exponential and Gaussian as particular cases, have been also considered [@Emilio_2009; @*Emilio_2010]. In some instances it is possible to reconstruct an interaction kernel from experimental data, as it is done in Ref. [@Minovich] for a thermal nonlinear optical medium.
Non monotonic kernels were introduced in the context of neuroscience. Neurons are intrinsically discrete units, but one can make use of continuous neural field models that are coarse grained description of the spatiotemporal evolution at the tissue level. These models, like the two-layer (one activatory and the other inhibitory) network Wilson-Cowan model [@WilsonCowan1972] typically include nonlocal effects. A Mexican-hat kernel was introduced by Amari [@Amari1977] to describe in terms of an effective field model a mixed population of activatory and inhibitory neurons. Since then, Mexican-hat kernels displaying local activation and lateral inhibition characteristics, have been used extensively in Neuroscience [@Ermentrout_RPP; @Coombes05] and also in reaction-diffusion [@Hildebrand2001]. The reverse situation, local inhibition and lateral excitation has been also considered [@Ermentrout_RPP]. Here we will study a kernel resulting from the combination of two Gaussians as shown in Table I and Fig. 1(c) which also allows to recover the results for a Gaussian kernel in the limit $b\rightarrow0$.
The manuscript is organized as follows. In Section \[Sect::GLE\] we briefly describe the GLE equation with nonlocal interaction. In Section \[Sect::moments\] we discuss the moment expansion of the kernel. Sections \[Sect::gauss\], \[Sect::expon\] and \[Sect::Mexhat\] are devoted to the Gaussian, mod-exponential and Mexican-hat kernels respectively. Finally concluding remarks are given in \[Sect::conclu\].
The Ginzburg-Landau equation with nonlocal interaction {#Sect::GLE}
======================================================
The prototypical cubic Ginzburg-Landau equation (GLE) for a real field $A$ in 1 spatial dimension can be written as [@Cross_RevModPh_1993], $$\partial_t A = \mu A - A^3 + \partial_{xx} A \, .
\label{eq:gle}$$ The parameter $\mu$ is the gain coefficient. The coefficients of the diffusion and cubic terms are set to one by suitably rescaling the spatial and temporal scales without loss of generality. The GLE is symmetric under the parity transformation $x\leftrightarrow -x$.
For $\mu<0$ the origin, $A_s=0$, is the only steady state (stable) in the system. At $\mu=0$ the system exhibits a pitchfork bifurcation, and two stable, symmetry related, HSSs appear at $A_s=\pm\sqrt{\mu}$. For $\mu>0$ the system is bistable and exhibits front solutions (kinks and anti-kinks) that connect the two HSSs. The fronts always decay to the HSS in a monotonic way.
We now consider an additional nonlocal term $F(x,\sigma)$ defined as in Eq. (\[Eq::nonlocal\_term\]). We assume that the kernel is real and preserves the symmetry under the parity transformation $x\leftrightarrow -x$, namely $K_{\sigma}(x)=K_{\sigma}(-x)$. The extension of the GLE with nonlocal coupling can then be written as $$\partial_t A = (\mu - sM_0) A - A^3 +
\partial_{xx} A +
s F(x,\sigma) ,
\label{Eq::RGL}$$ where $s$ determines the strength of the nonlocal term. The term $-s M_0 A$ where $M_0 = \int_{-\infty}^{\infty} K_{\sigma}(x) dx$ compensates for the local contribution of $F(x,\sigma)$. Through this compensation the nonlocal system (\[Eq::RGL\]) has the same HSSs as the GLE with local coupling (\[eq:gle\]).
To analyze the linear stability of a HSS we consider perturbations of the from $A = A_s + \epsilon \exp{(\Gamma t + i k x)}$. Linearizing for small $\epsilon$ one obtains for Eq. (\[Eq::RGL\]) the dispersion relation: $$\Gamma (k) = \mu' -k^2 + s (\hat{K}_{\sigma}(k)-M_0) \, ,
\label{Eq::HSS_stab_k}$$ where $$\hat{K}_{\sigma}(k) = \int_{-\infty}^{\infty} \! K_{\sigma}(x)\,e^{-ikx} dx ,
\label{eq:Kernel_Fourier}$$ is the Fourier transform of the kernel. Owing to the kernel symmetries $\hat{K}_{\sigma}(k)=\hat{K}_{\sigma}(-k)$ and the dispersion relation $\Gamma(k)$ depends on $k$ only through $k^2=u$ and we can write $$\tilde \Gamma (u) = \mu' - u + s (\tilde{\hat{K}}_{\sigma}(u)-M_0) \, ,
\label{Eq::HSS_stab_u}$$ where $\mu^{\prime}=\mu-3 A_s^2$. For $\mu\le 0$, $A_s=0$ and $\mu^{\prime}=\mu$ while for $\mu> 0$, $A_s=\pm\sqrt{\mu}$ and $\mu^{\prime}=-2\mu<0$. As a consequence all stable steady states of the GLE are associated to a negative value for $\mu^{\prime}$.
A given HSS becomes unstable if the maximum of $\Gamma(k)$ becomes positive at some $k_c$. If $k_c=0$ the instability is associated to a homogeneous perturbation while if $k_c \neq 0$ the system undergoes a modulational instability (MI). For the local GLE, $s=0$, the dispersion relation has a parabolic shape with a single maximum at $k=0$ where $\Gamma(0)=\mu'$. Changing the parameter $\mu$ the parabola moves rigidly in the vertical direction, so the maximum of the dispersion relation is always located at zero and therefore none of the HSSs can undergo a MI. As for homogeneous instabilities, the zero HSS becomes unstable at $\mu=0$, where the two steady states at $A_s=\pm\sqrt{\mu}$ are born (pitchfork bifurcation). The two non-zero symmetric HSSs are always stable in the parameter region where they exist. As we will see later, nonlocality induces a MI if, for some $k_c$, the last term in eq. (\[Eq::HSS\_stab\_k\]) overcomes the stabilizing $\mu^{\prime}-k^2$ terms, making $\Gamma(k_c)=0$ for a finite $k_c$.
We focus now on stationary spatial structures. Setting the time derivative to zero in (\[Eq::RGL\]) one has, $$\partial_{xx} A = (- \mu + s M_0) A + A^3 - s F(x,\sigma) \, .$$ Defining the intermediate variable $V$, one obtains the following 2-dimensional [*spatial*]{} dynamical system $$\begin{aligned}
A' &=V \nonumber\\
V' &=(-\mu +sM_0)A+A^3 -s F(x,\sigma) \ ,
\label{eq:spatdynsys} \end{aligned}$$ where the prime stands for derivatives with respect to the spatial variable $x$. The fixed points of (\[eq:spatdynsys\]) corresponds to solutions for $A$ which do not depend on $x$, thus to HSSs. Close to a HSS the shape of the fronts starting or ending at it can be obtained by considering a perturbation of the form $A(x) = A_s + \epsilon \exp(\lambda x)$ (where, in general, $\lambda$ is complex) and linearizing for small $\epsilon$. The spatial eigenvalues fulfill $$\Gamma_s(\lambda)=0$$ where $\Gamma_s(\lambda)$ is the dispersion relation (\[Eq::HSS\_stab\_k\]) replacing $k$ by a complex $-i\lambda$, namely $$\Gamma_s(\lambda) = \Gamma (-i\lambda) = \mu' + \lambda^2 + s (\hat{K}_{\sigma}(-i\lambda)-M_0) \, .$$ $\Gamma_s(\lambda)$ depends on $\lambda$ only through $\lambda^2=-u$, thus spatial eigenvalues can also be obtained from $\tilde \Gamma(u_0)=0$. If $u_0$ is real then there is a doublet of spatial eigenvalues $\lambda_0=\pm\sqrt{-u_0}$ with $\lambda_0$ real for $u_0<0$ or purely imaginary for $u_0>0$. If $u_0$ is complex then $u_0^*$ is also a zero and therefore complex spatial eigenvalues come one in quartets $\lambda_0=\pm q_0 \pm i k_0$.
As discussed in Part I [@PartI], if the eigenvalues are well separated the leading eigenvalues, i.e. those with the smallest real part, determine the asymptotic approach to the HSS. If the leading eigenvalues are a real doublet, fronts approach monotonically to the HSS. If the leading eigenvalues are an imaginary doublet, the HSS is modulationally unstable. The most interesting case is when the leading eigenvalues are a complex quartet, since fronts starting or ending at the HSS have oscillatory tails and thus LSs may arise as a consequence of the tail interaction.
Varying parameters two doublets can collide and lead to a complex quartet and viceversa. As explained in Part I the collision is signaled by a real double zero (RDZ) of $\tilde \Gamma (u)$, namely $\tilde \Gamma(u_c)=\tilde \Gamma'(u_c)=0$ for $u_c \in R$. If $u_c>0$ two imaginary doublets become a complex quartet signaling a Hamiltonian-Hopf (HH) bifurcation. If the HH occurs at the maximum of $\Gamma(k)$ it corresponds to a MI. If $u_c<0$ two real doublets become a complex quartet which corresponds to the so-called Belyakov-Devaney (BD) transition [@Devaney76]. Moving on top of the RDZ manifold $u_c$ changes value and eventually can change sign, so that a MI becomes a BD and viceversa. This happens when $\tilde \Gamma (0)=\tilde \Gamma' (0)=0$ and corresponds to a quadruple zero (QZ) of the dispersion relation when written as function of $k$, $\Gamma(k)$ [@PartI].
As shown in Part I, besides the QZ, there are two other codim-2 points that play a relevant role in organizing the phase space dynamics, namely the cusp point where two BD or two HH manifolds start (or end) and the 3DZ$(i\omega)$ in which the HSS becomes simultaneously unstable to homogeneous and finite wavelength perturbations.
Moment expansion {#Sect::moments}
================
A qualitative understanding of the effects of a nonlocal coupling by means of a moment expansion is possible for kernels that in Fourier space have no singularities at finite distances, as is the case of the Gaussian kernel to be considered in Sec. \[Sect::gauss\]. Proceeding as indicated in Part I, the nonlocal interaction can be written as a series of spatial derivatives of $A$ (see also [@Murray_Springer_2002]) $$F(x,\sigma)= \sum_{j=0}^{\infty} \frac{M_{2j}}{(2j)!} \frac{\partial^{2j} A}{\partial x^{2j}} \, ,
\label{eq::taylorexp}$$ where $M_j = \int_{-\infty}^{\infty} x^j K_{\sigma}(x) dx$. To describe the BD and MI transitions which involve 4 spatial eigenvalues we need to keep the expansion terms at least up to fourth order derivatives.
For the GLE (\[Eq::RGL\]) truncating the expansion of the kernel at fourth order one has $$\partial_t A = \mu A - A^3 +
\left(1+ \frac{1}{2!}s M_2\right)\nabla^2 A +
\frac{1}{4!} s M_4 \nabla^4 A \, .
\label{Eq::momexpan}$$ Eq. (\[Eq::momexpan\]) is related to the widely studied Swift-Hohenberg equation [@SH1977; @*MalomedNT90; @*Bestehorn90; @BurkeK07]. An analogous truncation for a spatially nonlocal interaction was considered in Ref. [@Gelens_PRA_2007; @*Gelens_PRA_2008]. Notice Eq. (\[Eq::momexpan\]) only makes sense if $sM_4<0$ since otherwise large wavenumber perturbations will always be amplified leading to divergences. Table \[tab:moments\] gives the values of the moments for the non-singular kernels considered in this article.
**Moment ** **Gaussian ** **Mexican hat**
-------------- ---------------- ---------------------------- --
$M_0$ 1 $2 (1 - b)$
$M_2$ $\sigma^2$ $2 (1 - 3b) \sigma^2$
$M_4$ $3 \sigma^4$ $6 (1 - 5 b) \sigma^4$
$M_6$ $15 \sigma^6$ $30 (1 - 7 b ) \sigma^6$
$M_8$ $105 \sigma^8$ $210 (1 - 9 b ) \sigma^8$
: First moments $M_i$ for the Gaussian and the Mexican-hat kernels[]{data-label="tab:moments"}
The dispersion relation for the $4$th order truncated moment expansion is, $$\tilde{\Gamma}(u) = \mu^{\prime} - \frac{2+s M_2}{2} u+ \frac{s M_4}{24} u^2\ .
\label{eq:momexpdispeg}$$ The spatial eigenvalues are, $$u_0=-\lambda_0^2=6 \frac{2+s M_2\pm \sqrt{(2+sM_2)^2-2\mu' s M_4/3}}{s M_4} .
\label{eq:momexeig}$$ The RDZ manifold of $\tilde \Gamma (u)$, which signals HH and BD transitions, is given by $\tilde \Gamma(u_c)=\tilde \Gamma'(u_c)=0$: $$\begin{aligned}
4 \mu^{\prime}-(2+s M_2) u_c=0 \label{eq:momexpdz1}\\
s M_4 u_c= 6(2+s M_2)\ .
\label{eq:momexpdz2}\end{aligned}$$ Combining these two equations in order to eliminate $u_c$ yields the RDZ manifold, $$\mu^{\prime}_{\rm RDZ} =\frac{3(2+s M_2)^2}{2s M_4} \ ,
\label{eq:momexpdzm}$$ which is of codim-$1$ in the 3-D $(\mu^{\prime},sM_2,s M_4)$ parameter space. Since we are considering $sM_4<0$, $\mu'_{\rm RDZ}$ is always negative. Setting $u_c=0$ in (\[eq:momexpdz1\]–\[eq:momexpdz2\]) one obtains the QZ manifold $$\mu^{\prime}_{\rm QZ}=0\, , \quad s_{\rm QZ} =-2/M_2 \ .
\label{eq:QZmomexp}$$
For a fixed $sM_4$, considering the $(s,\mu^{\prime})$ parameter space the RDZ manifold has the shape of a parabola with vertex at the QZ point and unfolding towards negative $\mu^{\prime}$. In the part of the RDZ with $s<s_{\rm QZ}$, $u_c>0$, and corresponds to a MI while the other part corresponds to a BD. For parameter values in the region between the BD and MI lines, where the leading eigenvalues are a complex quartet, fronts connecting two equivalent homogeneous solutions has oscillatory tails and localized structures can be formed.
In some cases one is interested in the effect of the nonlocality for fixed values of the parameters of the local GLE, namely for a given $\mu'$, then it is convenient to rewrite eq. (\[eq:momexpdzm\]) so that $s$ is isolated $$s_{\rm RDZ}=\frac{-(M_2+4\mu M_4)\pm 2\sqrt{4\mu^2 M_4^2+2\mu M_2 M_4}} {M_2^2}\ ,
\label{eq:scmomexp}$$ where the $+$ solution corresponds to the BD transition and the $-$ solution to the MI.
For kernels whose moments can be written as $M_j=\sigma^j {\cal M}_j$ (cf. subsection IIIA of Part I), Eq. (\[eq:scmomexp\]) becomes, $$s_{\rm RDZ}=\frac{1}{{\cal M}_2} \left[-\frac{1}{\sigma^2}-4\mu \frac{{\cal M}_4}{{\cal M}_2}\pm 2 \sqrt{4\mu^2 \frac{{\cal M}_4^2}{{\cal M}_2^2}+2\mu \frac{{\cal M}_4}{{\cal M}_2 \sigma^2} } \right] \, .
\label{eq:scmomexpsig}$$ In the limit of nonlocal interaction range going to zero, $\sigma\rightarrow 0$, one has $s_{\rm RDZ}\rightarrow -1/({\cal M}_2\sigma^2)$ for both BD and MI transitions. For infinite range nonlocality, $\sigma\rightarrow\infty$, $s_{\rm RDZ}\rightarrow 0$ for the BD transition while $s_{\rm RDZ}\rightarrow -8\mu{\cal M}_4/{\cal M}_2^2$ for the MI. These predictions will be compared with the results for a Gaussian kernel in the next section.
The Gaussian kernel {#Sect::gauss}
===================
In this section we analyze the influence of a nonlocal Gaussian kernel in the shape of the front starting (or ending) at an HSS of the GLE. Without loss of generality this kernel can be normalized so that $M_0=1$. In terms of $u=-\lambda^2$ the dispersion relation obtained linearizing around the HSS can be written as (cf. Table \[tab:kernels\]), $$\tilde{\Gamma}(u)=\mu^{\prime}-s-u+s \exp(-\sigma^2 u/2)\ .
\label{eq:gammagauss}$$ The spatial eigenvalues are the zeros of (\[eq:gammagauss\]), a transcendental equation that can be solved analytically in terms of the Lambert’s W function (see Appendix \[sec:Lambertap\]). The result is, $$u_0=-\lambda_0^2=\mu^{\prime}-s+\frac{2}{\sigma^2} W_l\left[\frac{s\sigma^2}{2}
\exp\left(\frac{\sigma^2}{2} (-\mu^{\prime}+s) \right)\right] \, ,
\label{eq:lambertsol}$$ where $l\in \mathbb{Z}$ and $W_l(x)$ is the $l$th branch of Lambert’s W function, $W(x)\in\mathbb{C}$, and, thus, the spectrum of spatial eigenvalues is infinite (numerable).
In order to determine the location of the MI and BD instabilities of the HSSs, we look for the RDZ of (\[eq:gammagauss\]) which is given by $\tilde \Gamma(u_c)=\tilde \Gamma'(u_c)=0$: $$\begin{aligned}
\exp(-\sigma^2 u_c/2)=-\frac{2}{s \sigma^2} \label{eq:k2temdgau}\\
\mu^{\prime}=s+\frac{2}{\sigma^2}+u_c\ .
\label{eq:muMI}\end{aligned}$$ One consequence of (\[eq:k2temdgau\]) is that BD and MI transitions require that $s<0$, as $u_c$ has to be real. Combining (\[eq:k2temdgau\]) and (\[eq:muMI\]) to eliminate $u_c$ leads to the condition defining the RDZ manifold of $\tilde{\Gamma}(u)$. This manifold has one dimension less than the dimensionality of the parameter space. Since we have three parameters, $\mu'$, $s$ and $\sigma$, the RDZ manifold is a 2-dimensional manifold given by $$\frac{s\sigma^2}{2} \exp\left(\frac{\sigma^2}{2} (-\mu^{\prime}+s)\right)=-\frac{1}{e}\ .
\label{eq:sdefeqngk}$$ The RDZ manifold given by (\[eq:sdefeqngk\]) corresponds in Eq. (\[eq:lambertsol\]) to the branching point of the two real branches of Lambert’s W function (see Appendix \[sec:Lambertap\]), in which these two branches both merge and finish. Solving Eq. (\[eq:sdefeqngk\]) for $\mu^{\prime}=\mu^{\prime}_{\rm RDZ}(s,\sigma)$ one has $$\mu^{\prime}_{\rm RDZ}=s+\frac{2}{\sigma^2}\left[1+\ln \left(-\frac{s\sigma^2}{2}\right)\right]\ .
\label{Eq::Gauss_muRDZ}$$ A cut of the $\mu'_{\rm RDZ}$ manifold for $\sigma=2$ is shown in solid lines in Fig. \[Fig::Gauss\_muRDZ\].
The codim-$2$ QZ manifold (which is a line in our $3$-D parameter space), can be obtained by setting $u_c=0$ in (\[eq:k2temdgau\]) and (\[eq:muMI\]) or, alternatively, locating the sub-manifold of RDZ (\[eq:sdefeqngk\]) in which $u_c=0$. The result is that the QZ manifold is defined by, $$\mu^{\prime}_{\rm QZ}=0\, , \quad s_{\rm QZ} =-2/\sigma^2 \ .
\label{eq:QZgau}$$ For the parameters of the QZ, $\tilde \Gamma''(0) <0$, therefore, in the notation of Part I, this is a QZ$^-$ point. In the $(s,\mu^{\prime})$ plane shown in Fig. \[Fig::Gauss\_muRDZ\] the QZ point is located at (-1/2,0). The character of the two pieces of the RDZ manifold, BD or MI, can be elucidated by calculating $u_c$ on top of the manifold, such that, respectively, $u_c<0$ and $u_c>0$. Substituting (\[Eq::Gauss\_muRDZ\]) in (\[eq:muMI\]) one obtains $$u_c=\frac{2}{\sigma^2} \ln \left(-\frac{s\sigma^2}{2}\right)\ .
\label{Eq::Gauss_uc}$$ Therefore the part of the RDZ manifold in which $s\sigma^2<-2$ has a positive $u_c$ and corresponds to a MI while the part in which $0>s\sigma^2>-2$ corresponds to a BD. In Fig. \[Fig::Gauss\_muRDZ\] the MI is located at the left of the QZ point and the BD at the right. Fronts starting (or ending) at the HSS have oscillatory tails for parameter values in the region between the MI and BD lines. This region is labeled as 3 in Fig. \[Fig::Gauss\_muRDZ\] in agreement with the notation used in Part I. The other parameter regions of the figure are also labeled as in Part I. For $s<0$ we refer to Part I for a detailed description of the regions and the transitions between them. The $s=0$ line corresponds to the GLE with local coupling for which there are only 2 spatial eigenvalues which are a real doublet for $\mu'<0$ and an imaginary doublet for $\mu'>0$. At $\mu=0$ the two components of the doublet collide at the origin (Hamiltonian-pitchfork bifurcation). For $s>0$, despite the presence of the Gaussian nonlocal kernel, the spatial dynamics shows a qualitative behavior is similar to that for $s=0$.
![(Color online) Boundaries in the $(s,\mu')$ plane at which the leading spatial eigenvalues of the GLE with a Gaussian kernel exhibit different transitions for $\sigma=2$. Sketches indicate the location of the leading eigenvalues in the (${\rm Re}(\lambda),{\rm Im}(\lambda)$) plane ($\times$ signal simple eigenvalues, $\bullet$ double eigenvalues, and $\square$ quadruple eigenvalues). Dotted lines labeled as MIm4 and BDm4 show the MI and BD transitions given by the 4th moment expansion (\[eq:momexpdzm\]).[]{data-label="Fig::Gauss_muRDZ"}](Gauss_muRDZ.eps){width="8.cm"}
The second derivative of $\tilde \Gamma(u)$, given by $$\tilde \Gamma''(u)= \frac{s\sigma^4}{4} \exp(-u\sigma^2/2) \,
\label{Eq::Gauss_Gamma_second}$$ does not vanish for any value of $u$ provided $s \neq 0$. This indicates that there is no cusp point for the GLE with a Gaussian nonlocal kernel. For $s=0$ the second derivative vanishes but this corresponds to the GLE with only local interaction whose dispersion relation is linear in $u$, thus it has nothing to do with a cusp point.
We now look for 3DZ and 3DZ$(i\omega)$ codim-2 points which, as discussed in Part I, correspond to the coincidence of a simple zero at the origin $\tilde \Gamma(0)=0$ and a RDZ at finite distance, $\tilde \Gamma(u_c)=\tilde \Gamma'(u_c)=0$. The first condition, $\tilde \Gamma(0)=0$ implies $\mu=0$, thus 3DZ and 3DZ$(i\omega)$ can be obtained setting $\mu_{\rm RDZ}=0$ in Eq. (\[Eq::Gauss\_muRDZ\]) and looking for solutions with non-zero $u_c$. There is no such a solution and therefore the GLE with a Gaussian nonlocal kernel does not have any 3DZ$(i\omega)$ or 3DZ points.
The absence of cusp and 3DZ points, indicates that the GLE with a Gaussian nonlocal kernel does not have any crossover manifold. This has strong implications on the location of the complex quartets in the $({\rm Re}(\lambda), {\rm Im}(\lambda))$ plane. In particular if a real doublet is leading the spatial dynamics, changing parameters complex quartets can not overcome the real doublet. In this case, the only way oscillatory tails can appear is after a BD transition in which two real doublets collide to become a complex quartet.
We now consider the effect of the nonlocal Gaussian kernel for given values of the parameters of the local dynamics, namely for a given $\mu^{\prime}$. Solving (\[eq:sdefeqngk\]), e.g. for $s_{\rm RDZ}(\mu^{\prime},\sigma)$, one obtains, $$s_{\rm RDZ}=\frac{2}{\sigma^2} W_l\left[-\exp\left(\frac{\sigma^2}{2} \mu^{\prime}-1\right)\right]\ ,
\label{eq:ssigmagauss}$$ that leads to two real branches since the argument of $W$ is in the interval $[-1/e,0]$. These two pieces of the RDZ manifold are organized by the codim-$2$ QZ manifold (\[eq:QZgau\]). Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\] shows three cuts of the $s_{\rm RDZ}(\mu^{\prime},\sigma)$ for different values of $\mu^{\prime}$. At $\mu^{\prime}=0$ one has the QZ manifold (\[eq:QZgau\]) \[Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\] (a)\], from which the BD and MI branches emerge as $\mu^{\prime}$ is decreased \[see panel (b)\]. The upper branch has $u_c<0$ and therefore it corresponds to a BD while the lower branch corresponds to the MI. LSs exist for parameter values in the region between the BD and MI curves. The BD and MI branches separate as $\mu^{\prime}$ is further decreased \[see panel (c)\], thus the region where LSs exists becomes larger. The asymptotic limit of both transitions as $\sigma\rightarrow\infty$ is $s_{\rm BD}(\sigma\rightarrow\infty)\rightarrow 0^-$ and $s_{\rm MI}(\sigma\rightarrow\infty)\rightarrow \mu^{\prime}$ [^1].
![(Color online) Boundaries in the $(\sigma,s)$ plane at which the leading spatial eigenvalues of the GLE with a Gaussian kernel exhibit different transitions. Panel (a) shows the QZ manifold at $\mu'=0$. Panels (b) and (c) show the BD and MI manifolds at $\mu'=-1$ and $\mu'=-6$ respectively. The short-dashed curves show the BD transition as predicted by the 4th moment expansion, Eq. (\[eq:scmomexpsig\]). Long-dashed horizontal lines show the asymptotic values for $\sigma \rightarrow \infty$. Sketches represent the location of the leading eigenvalue.[]{data-label="Fig::SpatialEigBoundary_Sigma_S_realGLE"}](figure3-01.eps){width="8.cm"}
![Location of the first spatial eigenvalues for the GLE with a Gaussian nonlocal kernel in the complex $\lambda$ plane (shown as black dots) for $\mu'=-6$. For comparison the two grey dots show the location of the eigenvalues for the local GLE ($s=0$). The hyperbola, given by Eq. (\[Eq::spatialDyn\_realGLE4\_amp\]), in greyscale represents an approximation for the location of the spatial eigenvalues. For panels (a-c) on the top row, $s=1$, for (d-f) on the middle row $s=-1$ and for (g-i) on the bottom row $s=-7.5$. For panels (a),(d) and (g) on the left column $\sigma=0.3$, for (b), (e) and (h) on the middle column $\sigma=2$ and for (c), (f) and (i) on the right column $\sigma=3$. For all values of $\sigma$ the number of spatial eigenvalues is infinite: the plot just presents the region around the origin in the complex plane.[]{data-label="Fig::spatEig_hyperbola"}](spatEig_hyperbola_9panels.eps){width="8cm"}
![(Color online) The solid line shows front profile for the GLE with a Gaussian nonlocal kernel for $\mu'=-6$, $s=-1$ and (a) $\sigma=0.5$ and (b) $\sigma=1$. The red dashed lines show the approximations given by Eqs. (\[Eq::Front\_fit\_monotonous\]) and (\[Eq::Front\_fit\_BD\]) (see text). \[Fig::Gauss\_front\_profile\] ](figure5_fit-01.eps){width="8.6cm"}
To illustrate the main behaviors exhibited by the system we plot in Fig. \[Fig::spatEig\_hyperbola\] the location in the complex $\lambda$ plane of the first few spatial eigenvalues of (\[eq:lambertsol\]) for $\mu^{\prime}=-6$ and three values of $s$ (at different rows) and three values $\sigma$ (at different columns). For attractive nonlocal interaction, $s>0$, only the principal branch, $W_0$, is real thus the spectrum contains only one real doublet \[panels (a-c)\]. For small $\sigma$ it is located very close to the real doublet of the local dynamics, as shown in panel (a). There is also an infinite number of complex eigenvalues but they are located outside the region shown in panel (a). As $\sigma$ increases the location of the spatial eigenvalues approaches the imaginary axis. Since, as discussed before, the GLE with Gaussian kernel has no crossover manifolds, the real doublet is always the eigenvalue located closer to the imaginary axis (see panels (b) and (c)). Neither a BD transition can exist for $s>0$ because there is no other real doublet with which the leading real doublet can collide. As a consequence for $s>0$ the spatial dynamics is always lead by a real doublet and fronts decay monotonically.
For repulsive nonlocal interaction, $s<0$, and $\sigma$ small, the argument of $W_l(x)$ in (\[eq:lambertsol\]) is in the range $x\in [-1/e,0]$, and both $W_0(x)$ and $W_{-1}(x)$ are real (see Appendix \[sec:Lambertap\]) and, as a consequence, there are two pairs of real eigenvalues. The pair located closer to the origin is shown in panel (d) for $s=-1$ and in panel (g) for $s=-7.5$. Increasing $\sigma$ the two real doublets approach each other and collide at the BD transition, which corresponds to the branching point of $W$ beyond which there is no real solution. Panels (e) and (h) correspond to parameters at the right of the BD curve in Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\]), and one finds a leading complex quartet. Fig. \[Fig::Gauss\_front\_profile\] illustrates the change of the front profile when crossing the BD line. From Eq. (\[eq:sdefeqngk\]) one obtains that for $\mu'=-6$ and $s=-1$ the BD line is located at $\sigma=\sqrt{(2/5)W_0(5/e)} \approx 0.5708076$. For $\sigma=0.5$, at the left of the BD line, the fronts are monotonic. Fig. \[Fig::Gauss\_front\_profile\] (a) shows the detailed shape of the front close to the HSS corresponding to $A_s=\sqrt{\mu}$. The overall profile of the front connecting the two HSS is shown in the inset. Close to the HSS the front is well described by an exponential of the form $$A(x)-A_{st} \approx c_1 e^{q_1 x}\ ,
\label{Eq::Front_fit_monotonous}$$ where $q_1=-2.753$ is the leading spatial eigenvalue and the coefficient $c_1$ has been fitted to $c_1=-4.507$. When crossing the BD line oscillations in the front profile appear initially with an infinite wavelength. The front profile for $\sigma=1$ is shown in Fig. \[Fig::Gauss\_front\_profile\] (b). Again close to the HSS the front profile is very well described by an exponential of the form $$A(x)-A_{st} \approx c_1 e^{q_1 x} \cos(k_1 x + \phi_1)
\label{Eq::Front_fit_BD}$$ where $q_1=-2.01$ and $k_1=1.01$ are the real and imaginary part of the spatial eigenvalue and the coefficients $c_1=2.02$ and $\phi_1=2.91$ have been fitted.
Once crossed the BD line, for $s=-1$ increasing $\sigma$ the spatial eigenvalues get closer to the imaginary axis. Nevertheless for $s>\mu'$ the situation remains qualitatively the same no matter how large is $\sigma$, as shown in Fig. \[Fig::spatEig\_hyperbola\] (f). For $s<\mu'$, as the range of interaction $\sigma$ increases one crosses the MI line, so that the HSS becomes modulationally unstable. Beyond the MI line the spatial dynamics is lead by two imaginary doublets as shown in Fig. \[Fig::spatEig\_hyperbola\] (i).
The spatial eigenvalues lie on a hyperbola-like curve for $\sigma$ high enough (cf. Fig. \[Fig::spatEig\_hyperbola\]). Although the analytical solution is available, (\[eq:lambertsol\]), it does not yield a geometrically transparent picture of the locus of the curve on which the spatial eigenvalues lie. This behavior can be easily understood by neglecting the linear term versus the exponential one in (\[eq:gammagauss\]), $$\exp (-\sigma^2 u_0/2) = 1-\mu'/s\ .
\label{Eq::spatialDyn_realGLE2old}$$ Using $u_0=-(q_0+ik_0)^2$ one gets $$\exp \left[\sigma(q_0^2-k_0^2)/2 + i \sigma q_0 k_0 \right] = 1-\mu'/s\ .
\label{Eq::spatialDyn_realGLE2B}$$ From the modulus of (\[Eq::spatialDyn\_realGLE2B\]) one has, $$q_0^2-k_0^2 = \frac{2}{\sigma^2} \log\left|1-\frac{\mu'}{s}\right| , \label{Eq::spatialDyn_realGLE4_amp}$$ which represents a hyperbola with eccentricity $\sqrt{2}$ in the complex plane. The RHS of (\[Eq::spatialDyn\_realGLE2B\]) is real and for $\mu'/s<1$ is positive, thus the phase of the exponent must be 0 or multiple of $2\pi$: $$q_0 k_0 = \frac{2 n \pi}{\sigma^2} \, , \quad n \in {\cal Z} \, .
\label{Eq::spatialDyn_realGLE4}$$ Conversely for $\mu'/s>1$, the RHS of (\[Eq::spatialDyn\_realGLE2B\]) is negative and $$q_0 k_0 = \frac{(2 n + 1) \pi}{\sigma^2} \, , \quad n \in {\cal Z} \, .
\label{Eq::spatialDyn_realGLE5}$$ Eqs. (\[Eq::spatialDyn\_realGLE4\]) and (\[Eq::spatialDyn\_realGLE5\]) can be seen as a selection criterion, which has to be satisfied by a point on the hyperbola to be a spatial eigenvalue of the system. These hyperbolas are shown in Fig. \[Fig::spatEig\_hyperbola\]. The approximation is meaningless for $\sigma \ll 1$ (cf. Fig. \[Fig::spatEig\_hyperbola\](a), (d) and (g)). When $\sigma \geq 2$ it is clear that the above equation of the hyperbola provides a good approximation of the location of the spatial eigenvalues and the ’selection criterion’ in fact gives eigenvalues that lie increasingly close to the real ones. Notice that for panels in top and middle rows $\mu'/s>1$, thus the selection criterion is given by (\[Eq::spatialDyn\_realGLE5\]). Instead for the panels in the bottom row $\mu'/s<1$, thus the criterion is given by (\[Eq::spatialDyn\_realGLE4\]). Therefore the hyperbola in the panels of the bottom row has a conjugated shape as compared to the one in the panels of the top and middle rows.
![Bifurcation diagram of the HSS of the nonlocal GLE with a Gaussian kernel and for $\sigma = 2$: (a) $s = 0$ (local case); (b) $s = -1$. Stable solutions are shown in a solid line, while the unstable ones in a dashed line.[]{data-label="Fig::realGLE_hombif"}](realGLE_hombif.eps){width="8.6cm"}
![Location of the MI in parameter space for the GLE with Gaussian kernel as a function of the nonlocal strength $s$ for different values of the interaction range: The solid, long-dashed and short-dashed lines correspond, respectively, to $\sigma = 2, 3, 5$, while the dotted line represents the $\sigma\rightarrow\infty$ limit. []{data-label="Fig::MI_GLE"}](figure7-01.eps){width="8.6cm"}
Regarding temporal instabilities associated to the MI, there is a finite range of values of $\mu\in [\mu_{M1},\mu_{M2}]$ around the pitchfork bifurcation, $\mu^{\prime}=0$, where HSSs are modulationally unstable. For the parameters of Fig. \[Fig::realGLE\_hombif\], $\sigma=2$ and $s=-1<-1/(2\sigma^2)$, $\mu^{\prime}_{\rm MI}=-1+(1+\ln 2)/2=-0.153426$, and, thus, $\mu_{\rm M1}=\mu^{\prime}_{\rm MI}$ and $\mu_{\rm M2}=-\mu^{\prime}_{\rm MI}/2=-0.076713$. In turn, in Fig. \[Fig::MI\_GLE\] the dependence of $\mu'_{\rm M1}$ on $s$ for three different values of $\sigma$ is plotted. In the limit of $\sigma\rightarrow\infty$ this dependence is given by the line $\mu^{\prime}=s$. The effect of a nonlocal nonlinear response in a MI is also discussed in Ref. [@Krolikowski_PRE_2001; @*Wyller_PRE_2002].
Finally, we compare the results obtained here with an expansion up to the 4th moment as discussed in Sect. \[Sect::moments\]. Fig. \[Fig::Gauss\_muRDZ\] shows in dashed lines the location of the BD and MI manifolds given by (\[Fig::Gauss\_muRDZ\]). The prediction given by the 4th moment expansion is quite good for the BD transition. The prediction for the MI manifold, while following the correct trend, becomes quite off as soon as one moves away from the QZ point where $u_c=0$. As for the dependence of $s_{\rm RDZ}$ on the kernel width for a fixed $\mu'$, one can use Eq. (\[eq:scmomexpsig\]) with ${\cal M}_2=1$ and ${\cal M}_4=3$ (see Table \[tab:kernels\]). As shown in Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\] for the BD transition, the result given by 4th moment approximation (dashed line) is in good agreement with the exact one (solid line). In the case of the MI, the 4th order expansion does work as well. In the limit of zero interaction range, it correctly predicts an asymptotic behavior $s_{\rm RDZ}\rightarrow -1/\sigma^2$, however in the limit of infinite interaction range, $\sigma\rightarrow\infty$, the prediction is $s_{\rm RDZ}\rightarrow -24\mu$ which is out of the figure. Therefore the results for the MI given by the 4th order expansion are outside the parameter region plotted in Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\].
The mod-exponential kernel {#Sect::expon}
==========================
We consider here the effect of a kernel whose profile decays exponentially in space on the tails of fronts starting (or ending) in an HSS of the GLE. In Fourier space the mod-exponential kernel is a Lorentzian (cf. Table \[tab:kernels\]) $$\tilde{\hat{K}}(u)= \frac{1}{1+4\sigma^2 u}\ ,
\label{Eq::Lorentzian_u}$$ which in the complex plain has a singularity at $u=-1/(4 \sigma^2)$. The dispersion relation obtained linearizing around the HSS is given by, $$\tilde \Gamma(u)=\mu^{\prime}-u-s+ \frac{s}{1+4\sigma^2 u}\ .
\label{eq:dispersion_exp_kernel_u}$$ The spatial eigenvalues are the zeros of $\tilde \Gamma(u)$. In this case there are only 4 spatial eigenvalues $\lambda_0$ given by $$\begin{aligned}
u_0 & =-\lambda_0^2= \nonumber \\
& =\frac{1}{2} \left[\mu'-s-\frac{1}{4\sigma^2}\pm\sqrt{\left(s-\mu'+\frac{1}{4\sigma^2}\right)^2+\frac{\mu'}{\sigma^2}}
\right].
\label{eq:speigenexp}\end{aligned}$$
MI and BD instabilities are located on the RDZ manifold of $\Gamma(u)$ which is given by $\tilde \Gamma(u_c)=\tilde\Gamma'(u_c)=0$: $$\begin{aligned}
u_c&=&\frac{-1\pm 2\sigma\sqrt{-s}}{4\sigma^2} \label{eq:k2temdexp}\\
\mu^{\prime}&=&s+\frac{1}{4\sigma^2}+2 u_c\ .
\label{eq:muMIexp}\end{aligned}$$ MI and BD transitions require $u_c$ real, thus MI and BD transitions can only exist for $s<0$, namely, for repulsive nonlocal interaction. Combining (\[eq:k2temdexp\]) and (\[eq:muMIexp\]) to eliminate $u_c$ one obtains the RDZ manifold which in the $(\mu',s,\sigma)$ parameter space is the surface given by $$\mu^{\prime}_{\rm RDZ}=s-\frac{1}{4\sigma^2}\pm\frac{\sqrt{-s}}{\sigma}\ .
\label{Eq::Exponential_muRDZ}$$
Setting $u_c=0$ in (\[eq:k2temdexp\]) and (\[eq:muMIexp\]) one obtains the QZ codim-2 bifurcations (lines in the 3D parameter space). It turns out that there are two QZ lines. The first one takes place for finite $\sigma$ and is given by $$\mu'_{\rm QZ1}=0, \, \, s_{\rm QZ1}=-\frac{1}{4 \sigma^2} \ .
\label{eq:quadzero_exponential}$$ For the parameters of QZ1, $\tilde \Gamma''(0)=32 s_{\rm QZ1} \sigma^4=-8 \sigma^2<0$, thus, in the notation of Part I, this is a QZ$^-$ point. The second QZ line is located at $$\sigma_{\rm QZ2}=\infty, \quad \mu'_{\rm QZ2}=s \, .$$ For the parameters of QZ2, $\tilde \Gamma''(0)= 32 s \sigma_{\rm QZ2}^4 =-\infty$, thus this is also a QZ$^-$ point, albeit a particular one since the second derivative is infinite. As a consequence the parabola described by the RDZ close to QZ2 is infinitely narrow, and the BD and MI lines unfold from QZ2 practically tangentially.
In the part of the RDZ manifold that starts from the side $s \sigma^2 < -1/4$ of QZ1 and the side $s<\mu'$ of QZ2, $u_c>0$, it thus corresponds to a MI bifurcation. On the other part $u_c<0$, thus corresponds to a BD.
Cusp or 3DZ codim-2 points can not exist for the GLE with a mod-exponential kernel since they require at least 6 spatial eigenvalues. The crossover manifold does not exist either in this case.
We now address the effect of the mod-exponential kernel for a given value of $\mu'$. It is convenient to rewrite (\[Eq::Exponential\_muRDZ\]) as, $$s_{\rm RDZ}=\mu^{\prime}-\frac{1}{4\sigma^2}\pm \frac{1}{\sigma} \sqrt{-\mu^{\prime}}\ .
\label{eq:ssigmaexp}$$ The $+$ and $-$ signs correspond to the BD and MI manifolds respectively. The MI and BD transitions are shown in Fig. \[Fig::s\_sigma\_nlRGLE\_Laplacian\_mu\_3\] for the non-zero HSSs for $\mu'=-6$ corresponding to $\mu=3$. LSs are found in the parameter region bounded by the BD and MI curves. The existence of QZ2 leads to a significant difference with the Gaussian kernel (cf. Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\]). Increasing $\sigma$ the BD and MI lines tend asymptotically one to the other and meet at the QZ2. As a consequence the region of LSs narrows as $\sigma$ increases. The curve $s_{\rm BD}$ has a maximum at $\sigma^{\dag}=1/(2\sqrt{2\mu})$ (i.e., at $\sigma=1/\sqrt{24}=0.204124$ in Fig. \[Fig::s\_sigma\_nlRGLE\_Laplacian\_mu\_3\], where it reaches $s=0$, the maximum value of $s$ for which the RDZ manifold exists. For $\mu\rightarrow 0$ the BD and MI curves approach each other and meet at QZ1.
![\[Fig::s\_sigma\_nlRGLE\_Laplacian\_mu\_3\] (Color online) Boundaries in the $(\sigma, s)$-plane separating the regions of monotonic and oscillatory tails in the GLE with a mod-exponential kernel for $\mu'=-6$ obtained from (\[eq:ssigmaexp\]). The horizontal dashed line at $s=-6$ shows the asymptotic limit of MI and BD lines for $\sigma \rightarrow \infty$. ](figure8-01.eps){width="8cm"}
Considering the MI and comparing with the local case shown in Fig. \[Fig::realGLE\_hombif\](a), one also finds that there is a finite range of values of $\mu\in[\mu_{M1},\mu_{M2}]$ around the pitchfork bifurcation, $\mu=0$, that are modulationally unstable. For the parameters of Fig. \[Fig::realGLE\_hombif\](b), $\sigma=2$ and $s=-1<-1/(4\sigma^2)$, $\mu_{M1}=-9/16$ and $\mu_{M2}=-\mu_{M1}/2=9/32$.
Written in terms of $u$ the mod-exponential kernel in Fourier space (\[Eq::Lorentzian\_u\]) has a pole of order 1 at $u=-1/(4 \sigma^2)$. As a consequence a moment expansion around $u=0$ such as the one discussed in section 3.1 of Part I will converge only for $|u|<1/(4 \sigma^2)$, and therefore it will be of limited use. A truncation up to $M_4$ identifies the spatial behavior around the first QZ, in particular the existence of spatial tails, and thus of LSs, for $s<0$ but as $\sigma$ is increased the predictions from moment expansion are quite off. This kernel can be considered as the simplest with singularities and no further approximations can be obtained from a Laurent expansion (Sect. 3.2 of Part I) since $1/(1+4\sigma^2u)$ is already the first and only term of that expansion. Still, proceeding as in Sect. 3.2 of Part I, one can obtain an exact transformation for the nonlocal interaction term. Using (\[Eq::Lorentzian\_u\]) the nonlocal interaction can be written as $$\hat{F}(k,\sigma) = \frac{1}{1+4\sigma k^2} \hat{A}(k)\ ,$$ or equivalently $$(1+4\sigma k^2) \hat{F}(k,\sigma) = \hat{A}(k)\ .$$ In real space this leads to $$(1-4\sigma \partial_{xx})F(x,\sigma) = A(x)\ .$$ which is an ordinary differential equation. Therefore the GLE with a mod-exponential nonlocal kernel can be written as a partial differential equation coupled to an ordinary differential equation, $$\begin{aligned}
&&\partial_t A =(\mu - s) A - A^3 + \partial_{xx} A + s F(x,\sigma) \nonumber \\
&&\partial_{xx} F(x,\sigma) = \frac{1}{4\sigma} (F(x,\sigma)-A(x)) \, ,
\label{eq:GLE_exponential_kernel_transform}\end{aligned}$$ where we have used that $M_0=1$. This treatment of the mod-exponential kernel was introduced in Ref. [@Ermentrout93], and used also by other authors [@Coombes05; @Clerc_PRE_2010].
The Mexican-hat kernel {#Sect::Mexhat}
======================
In this Section, we discuss in detail the effects of a spatially nonlocal kernel that is not everywhere positively defined. More precisely the kernel we consider consists of two Gaussians and has an extra parameter $b>0$ (cf. Table \[tab:kernels\]) that regulates the spatial extension of the negative sector of the kernel. The total area of this kernel is given by $M_0=2(1-b)$ (cf. Table \[tab:kernels\]), and for $b>1$, $M_0<0$. For $s>0$ one has short-range attraction (activation) and medium to long-range repulsion (inhibition). In this case, activation dominates globally for $b<1$ while otherwise overall inhibition is stronger than activation. For $s>0$ one has inhibition in the short-range and activation in the medium to long-range and globally inhibition dominates for $b<1$ while activation does otherwise.
The dispersion relation for this kernel is (cf. Eq. (\[Eq::HSS\_stab\_u\]) and Table \[tab:kernels\]), $$\tilde{\Gamma}(u)=\mu^{\prime}- 2s (1 - b) - u + 2 s ( 1 - b + b \sigma^2 u ) e^{-\sigma^2 u/2},
\label{Eq::Mexhat_Gamma_u}$$ and setting $\tilde{\Gamma}(u)=0$ does not lead to a closed expression for the spatial eigenvalues, that have to be obtained numerically for this kernel. The RDZ manifold is given by $\tilde \Gamma(u_c)=\tilde \Gamma'(u_c)=0$: $$\begin{aligned}
\frac{1}{s\sigma^2}&=(-1+3b-b\sigma^2 u_c) e^{-\sigma^2 u_c/2}
\label{Eq::Mexhat_RDZ1}\\
\mu^{\prime}&=2s-2sb + \frac{2}{\sigma^2} + u_c - 4 s b e^{-\sigma^2 u_c/2} .
\label{Eq::Mexhat_RDZ2}\end{aligned}$$ Since now the parameter space $(\mu,s,\sigma,b)$ is $4$-D, the RDZ manifold is a 3-D hyper-surface. Eq. (\[Eq::Mexhat\_RDZ1\]) can be solved analytically for $u_c$, $$u_{c,l}= \frac{1}{\sigma^2} \left[ 3 - \frac{1}{b} -2 W_l(\chi)\right]\ .
\label{Eq::Mexhat_uc}$$ where $l=0,-1$ are the indices of the two real branches of the Lambert $W$ function (see Appendix \[sec:Lambertap\]) and $$\chi=\chi(b,s,\sigma)=\frac{1}{2bs\sigma^2}\exp\left(\frac{3b-1}{2b}\right)\ .$$ Substituting (\[Eq::Mexhat\_uc\]) into (\[Eq::Mexhat\_RDZ2\]) one gets $$\mu^{\prime}_{{\rm RDZ,}l}=2s(1-b) + \frac{5}{\sigma^2} -\frac{1}{b\sigma^2} -\frac{2W_l(\chi)}{\sigma^2} -\frac{2}{\sigma^2 W_l(\chi)} \ .
\label{eq:mexhat_mu_rdz}$$ For $-1/e < \chi < 0$ the RDZ manifold has two branches which we label $l=0$ and $l=-1$ as the indices of the $W$ function (see Appendix \[sec:Lambertap\]). For $\chi > 0$ the RDZ manifold has a single branch given by $l=0$. For $\chi < -1/e$, $W$ does not take real values thus there is no RDZ manifold. The asymptotic behavior of the RDZ branches for large $s$ is given by $$\begin{aligned}
\mu'_{{\rm RDZ,}0}=&2s\left[1-b-2b \exp{\left(\frac{1-3b}{2b}\right)}\right]+\frac{3b-1}{b\sigma^2} \nonumber \\ & +\mathcal{O}(s^{-1}) \label{Eq::Mexhat_muRDZ0_asymptotic} \\
\mu'_{{\rm RDZ},-1} =& 2s(1-b)+ \mathcal{O}\left(\log(s)\right) \label{Eq::Mexhat_muRDZ-1_asymptotic} \, .\end{aligned}$$
The part of the RDZ manifold with $u_c<0$ corresponds to a BD transition while the part with $u_c>0$ corresponds to a HH bifurcation. Note that the BD and HH parts of the RDZ manifold are not directly related to the index $l$ of $\mu^{\prime}_{{\rm RDZ,}l}$. Instead, as it will be discussed below, $\mu^{\prime}_{{\rm RDZ,}0}$ have both BD and HH parts and the same applies to $\mu^{\prime}_{{\rm RDZ,}-1}$. For the moment, we will distinguish the part of the BD and MI manifolds in which the second derivative of the dispersion relation $\Gamma(u)$, given by $$\tilde \Gamma''(u)=\frac{s\sigma^4}{2}\left(1-5b+b\sigma^2 u\right) e^{-\sigma^2 u/2},
\label{Eq::Mexhat_Gamma_second}$$ is positive from that where is negative. The part of the RDZ with $\tilde \Gamma''(u_c)>0$ corresponds to a local minimum of the dispersion relation crossing zero and in this section it will be referred to as HH$^+$ or BD$^+$, while the part with $\tilde \Gamma''(u_c)<0$ corresponds to a local maximum of the dispersion relation crossing zero and will be referred to as as HH$^-$ or BD$^-$. If the local maximum signaled by the HH$^-$ turns out to be the global maximum then it corresponds to a modulational instability of the HSS and, as in previous sections, it will be referred to as MI.
Setting $u_c=0$ in Eqs. (\[Eq::Mexhat\_RDZ1\]–\[Eq::Mexhat\_RDZ2\]) one finds the QZ manifold, which in the $(\mu,s,\sigma,b)$ parameter space is a 2-D surface given by, $$\mu_{\rm QZ}^{\prime}=0 \, ; \quad s_{\rm QZ}=\frac{1}{\sigma^2 (3b-1)}\ .
\label{Eq::Mexhat_QZ}$$ $s_{\rm QZ}$ has a divergence at $b=1/3$, and, as a consequence, the QZ manifold splits in two parts. For $b<1/3$, $s_{\rm QZ}<0$ as shown in Fig. \[Fig::Mexhat\_muRDZ\] (a), while for $b>1/3$, $s_{\rm QZ}>0$ as shown in Fig. \[Fig::Mexhat\_muRDZ\] (d). This is a clear distinction with the previous two kernels for which $s_{\rm QZ}$ was always negative.
As discussed in Part I there are two kinds of QZ points depending on the sign of $\Gamma''(u)$ at the QZ point. From (\[Eq::Mexhat\_Gamma\_second\]) we have $$\tilde \Gamma_{\rm QZ}'' (0) =\frac {s_{\rm QZ} \sigma^4}{2}(1-5b)=-\frac{1}{\sigma^2}\frac{1-5b}{1-3b}
\label{Eq::Mexhat_Gamma_second_QZ}$$ For $\tilde \Gamma_{\rm QZ}''(0)<0$ the QZ is a QZ$^-$ unfolding a BD$^-$ and a MI manifolds towards $\mu'<0$ \[see Fig. \[Fig::Mexhat\_muRDZ\] (a) or (d)\], while for $\tilde \Gamma_{\rm QZ}''(0)>0$ one has a QZ$^+$ unfolding a BD$^+$ and a HH$^+$ manifolds towards $\mu'>0$ \[see Fig. \[Fig::Mexhat\_muRDZ\] (c)\]
At difference with the previous kernels, the GLE with a Mexican-hat nonlocal kernel exhibits a codim-$2$ cusp manifold. Setting $\tilde \Gamma''(u_{\rm cusp})=0$ one has, $$u_{\rm cusp}=\frac{1}{\sigma^2}\left(5-\frac{1}{b}\right)\, .
\label{Eq::Mexhat_u_cusp}$$ From $\tilde \Gamma'(u_{\rm cusp})=0$ one obtains, $$s_{\rm cusp}=-\frac{1}{2b\sigma^2}\exp\left(\frac{5}{2}-\frac{1}{2b}\right)\, .
\label{Eq::Mexhat_s_cusp}$$ Finally setting $\tilde \Gamma(u_{\rm cusp})=0$ and using (\[Eq::Mexhat\_s\_cusp\]) one arrives to, $$\mu^{\prime}_{\rm cusp}=\frac{1}{\sigma^2} \left[9-\frac{1}{b}+\left(1- \frac{1}{b}\right) \exp\left(\frac{5}{2}-\frac{1}{2b}\right)\right]\, .
\label{Eq::Mexhat_mu_cusp}$$
   
From Eq. (\[Eq::Mexhat\_u\_cusp\]) and Eq. (\[Eq::Mexhat\_uc\]), one finds that at the cusp $W_l(\chi_{\rm cusp})=-1$, which is the branching point of the Lambert $W$ function where the two real branches $W_0$ and $W_{-1}$ originate (cf. Appendix \[sec:Lambertap\]). These two $W$ branches associated to the two branches of the $\mu^{\prime}_{\rm RDZ}$ manifold which in parameter space emerge from the cusp one tangent to the other \[see for example Fig. \[Fig::Mexhat\_muRDZ\] (a)\]. Since $W_{-1}(\chi)<-1$ for any value of $\chi$ (cf. Appendix \[sec:Lambertap\]), on the branch $\mu^{\prime}_{{\rm RDZ},-1}$, $u_{c,1}>u_{\rm cusp}$ and, as a consequence, the sign of the second derivative $\tilde \Gamma''(u_{c,1})$ is that of $s$. On the contrary on the branch $\mu^{\prime}_{{\rm RDZ},0}$, $u_{c,0}<u_{\rm cusp}$ and $\tilde \Gamma''(u_{c,0})>0$ for $s<0$ while $\Gamma''(u_{c,0})<0$ for $s>0$.
For $u_{\rm cusp}<0$ the cusp is, in the notation of Part I, a C$^-$ unfolding a BD$^+$ and a BD$^-$ manifolds. If the cusp is located at $s<0$ as is the case of Fig. \[Fig::Mexhat\_muRDZ\] (a), then the BD$^+$ corresponds to $\mu^{\prime}_{{\rm RDZ},0}$ and the BD$^-$ to $\mu^{\prime}_{{\rm RDZ},-1}$. Moving away from the cusp along the BD$^-$ line, $u_{c,-1}$ increases and eventually it reaches zero at the QZ$^-$ where the BD$^-$ becomes a MI. The C$^-$ cusp also unfolds a crossover manifold at which a real doublet and a complex quartet are located at the same distance from the imaginary axis which we label as XR. In parameter space XR can be seen as the natural continuation of the two BD manifolds that end one tangent to the other at the cusp \[see Fig. \[Fig::Mexhat\_muRDZ\] (a)\].
For $u_{\rm cusp}>0$ the cusp is a C$^+$ unfolding a HH$^+$ and a HH$^-$ which for a cusp located at $s<0$ correspond to $\mu^{\prime}_{{\rm RDZ},0}$ and to $\mu^{\prime}_{{\rm RDZ},-1}$, respectively \[see Fig. \[Fig::Mexhat\_muRDZ\] (c)\]. Moving along the HH$^+$ manifold away from the cusp, $u_{c,0}$ decreases and eventually it reaches zero at the QZ$^+$ point where the HH$^+$ becomes a BD$^+$. Also unfolding from C$^+$ there is a crossover manifold (labeled as XI) at which a imaginary doublet and a complex quartet are located at the same distance from the real axis. Similarly as before, in parameter space XI can be seen as the continuation of the two HH manifolds ending at the cusp.
The GLE with a Mexican-hat kernel also has a 3DZ and a 3DZ$(i\omega)$ codim-2 points. As discussed in part I, at these points a simple zero at the origin $\tilde \Gamma(0)=0$ and a RDZ at finite distance, $\tilde \Gamma(u_c)=\tilde \Gamma'(u_c)=0$, take place simultaneously. Since $\tilde \Gamma(0)=0$ implies $\mu=0$, the 3DZ and 3DZ$(i\omega)$ can be obtained setting $\mu_{\rm RDZ}=0$ in Eq. (\[eq:mexhat\_mu\_rdz\]) and looking for solutions with non-zero $u_c$. Thus, within the $\mu=0$ hyper-plane the location of the 3DZ and 3DZ$(i\omega)$ is given by the implicit equation $$0=2s\sigma^2(1-b) + 5 -\frac{1}{b} -2W_l(\chi) -\frac{2}{W_l(\chi)}\ .
\label{Eq::Mexhat_3DZ}$$ The 3DZ$(i\omega)$ point is located on the HH$-$ manifold that unfolds from the C$^+$ cusp \[see Figs. \[Fig::Mexhat\_muRDZ\] (c) and (d)\] and have a significant effect on it. At this point HH$-$, which at the cusp is a local maximum of the dispersion relation crossing zero, becomes a global maximum. Thus the HH bifurcation becomes a MI \[see Fig. 8 of Part I\]. Similarly the 3DZ point is located on the BD$^+$ manifold that unfolds from the C$^-$ cusp \[see Fig. \[Fig::Mexhat\_muRDZ\] (a)\].
The sextuple zero point, which organizes the overall scenario, takes place when $u_{\rm cusp}=0$, which in the $(\mu,s,\sigma,b)$ parameter space is the line $$b_{\rm SZ}=\frac{1}{5}\, ; \quad s_{\rm SZ}=-\frac{5}{2\sigma^2}\, ; \quad \mu_{\rm SZ}=0\ .
\label{eq::Mexhat_SZ}$$
The parameter space portrait for the GLE with a Mexican-hat kernel is as follows. For $b=b_{\rm SZ}$ \[Fig. \[Fig::Mexhat\_muRDZ\] (b)\] the SZ unfolds a BD$^+$ ($\mu^{\prime}_{{\rm RDZ},0}$), a MI ($\mu^{\prime}_{{\rm RDZ},-1}$), a XR and a XI manifolds in a similar way as described in Part I for the six-order dispersion relation in $\lambda$. For $s$ negative and large the asymptotic behavior of BD$^+$ is given by Eqs. (\[Eq::Mexhat\_muRDZ0\_asymptotic\]) and that of MI by (\[Eq::Mexhat\_muRDZ-1\_asymptotic\]). The part for $s<0$ of Fig. \[Fig::Mexhat\_muRDZ\] (b) can be directly compared with Fig. 1(b) of Part I. We refer to Part I for a detailed explanation of all the regions surrounding the SZ as well as the transitions between them. The regions relevant for the existence of stable LSs are region 3 where the spatial dynamics is lead by a complex quartet and the part of region 4 close to the crossover XR where the spatial dynamics results from the combination of a real doublet and a complex quartet. The $s=0$ line corresponds to the GLE with local coupling with two spatial eigenvalues which are real $\mu'<0$ and imaginary for $\mu'>0$. For $s>0$ and $\mu<0$ there is a BD$^-$ line given by $\mu^{\prime}_{{\rm RDZ},0}$ which separates region 3 led by a complex quartet from region 7 led by two real doublets, in which fronts are monotonic. For large positive $s$ follows the asymptotic behavior (\[Eq::Mexhat\_muRDZ0\_asymptotic\]). Thus the BD$^+$ line for $s<0$ and the BD$^-$ line for $s<0$ have the same oblique asymptote. For $s \rightarrow 0$ the BD$^-$ line goes to $-\infty$. When crossing $\mu=0$ from region 7 into 1, the components of the doublet closer to the origin collide leading to a imaginary doublet. This is a Hamiltonian-pitchfork bifurcation as described in Part I.
For $b<b_{\rm SZ}$ the SZ unfolds a QZ$^-$ within the $\mu'=0$ hyperplane and a cusp located at $\mu'<0$ as shown in Fig. \[Fig::Mexhat\_muRDZ\] (a) for $b=0.1$. From Eq. (\[Eq::Mexhat\_s\_cusp\]), for $b<b_{\rm SZ}$, $u_{\rm cusp} <0$, thus this is a C$^-$ cusp unfolding two BD manifolds and a crossover XR. The BD$-$ manifold (given by $\mu^{\prime}_{{\rm RDZ},-1}$) connects C$^-$ with QZ$^-$ where it becomes an MI. The BD$+$ manifold (given by $\mu^{\prime}_{{\rm RDZ},0}$) connects C$^-$ with the 3DZ located at the $\mu'=0$. For large negative $s$ the asymptotic behavior of both manifolds is given by Eqs. (\[Eq::Mexhat\_muRDZ0\_asymptotic\]) and (\[Eq::Mexhat\_muRDZ-1\_asymptotic\]) respectively. The overall picture shown in the part part of Fig. \[Fig::Mexhat\_muRDZ\] (a) corresponding to $s<0$ has the same structure as Fig. 1(a) of Part I, to which we refer for a discussion. As for $s>0$ the regions are the same as in Fig. \[Fig::Mexhat\_muRDZ\] (b) but the BD$^-$ line given by $\mu^{\prime}_{{\rm RDZ},0}$ has moved down to more negative values of $\mu'$ \[Notice the different vertical scale in Figs. \[Fig::Mexhat\_muRDZ\] (a) and (b)\] and region 3 has narrowed. As a consequence one has a large parameter region for $s<0$ where localized structures may be formed which includes region 3 unfolding from QZ$^-$ and the part of region 4 close to XR while for $s>0$ there is a narrow region 3 located at small $s$ and reachable only for strongly negative values of $\mu'$.
If $b$ is further decreased the $C^-$ cusp moves towards smaller values of $s$ and more negative values for $\mu'$. At the same time the BD$^+$ line born at the right of the cusp becomes more vertical. The BD$^-$ line at $s>0$ is located further down and region 3 keeps narrowing. In the limit $b \rightarrow 0^+$ the cusp goes to $s_{\rm cusp} \rightarrow 0^+$ and $\mu'_{\rm cusp}\rightarrow -\infty$ while the BD$^-$ line located in the $s>0$ semi-plane goes also to $-\infty$. The result is that one recovers the parameter space diagram obtained for the Gaussian kernel (Fig. \[Fig::Gauss\_muRDZ\]).
For $b>b_{\rm SZ}$ the SZ unfolds a QZ$^+$ and a 3DZ$(i\omega)$ manifolds located within the $\mu'=0$ hyperplane and a cusp located at $\mu'>0$ as shown in Fig. \[Fig::Mexhat\_muRDZ\] (c) for $b=0.3$. From Eq. (\[Eq::Mexhat\_s\_cusp\]), for $b_{\rm SZ}$, $u_{\rm cusp} >0$, thus this is a C$^+$ cusp unfolding two HH manifolds and a crossover XI. The HH$^+$ manifold (given by $\mu^{\prime}_{{\rm RDZ},0}$) connects C$^+$ with QZ$^+$ where it becomes a BD$^+$. The HH$^-$ manifold (given by $\mu^{\prime}_{{\rm RDZ},-1}$) connects C$^+$ with the 3DZ$(i\omega)$. Globally the part for $s<0$ of Fig. \[Fig::Mexhat\_muRDZ\] (c) has the same regions and transitions as those obtained for $b>0$ for the six-order dispersion relation considered in Part I (Fig. 1 (c)). For $s<0$ region 3 now unfolds from the 3DZ$(i\omega)$ point and has a sharp-pointed shape and thus is narrower than in Figs. \[Fig::Mexhat\_muRDZ\] (a) and (b). For $s>0$ the regions are the same as in Fig. \[Fig::Mexhat\_muRDZ\] (b) but the BD$^-$ line given by $\mu^{\prime}_{{\rm RDZ},0}$ has moved up and region 3 has widened significantly. Still in order to reach region 3 for $s>0$ it is necessary that $\mu'$ is not too close to zero.
If $b$ is further increased the QZ$^+$ point moves towards more negative values for $s$. For $b \rightarrow 1/3^-$, $s_{\rm QZ} \rightarrow -\infty$ \[cf. Eq. (\[Eq::Mexhat\_QZ\])\]. Also the slope and the ordinate at the origin of the oblique asymptote of $\mu^{\prime}_{{\rm RDZ},0}$ tend to zero as $b \rightarrow 1/3$ as can be seen from Eq. (\[Eq::Mexhat\_muRDZ0\_asymptotic\]). For $b>1/3$, the QZ point is located at $s_{\rm QZ}>0$ starting from $s_{\rm QZ}=\infty$ at $b=1/3^+$ and monotonically approaching $s_{\rm QZ}=0$ as $b$ increases. The QZ is in fact a QZ$^-$ point since for $b>1/3$, $\tilde \Gamma''_{\rm QZ}(0)<0$ \[see Fig. \[Fig::Mexhat\_muRDZ\] (d) for $b=1/2$\]. The QZ$^-$ unfolds a BD$^-$ and a MI manifolds given by $\mu^{\prime}_{{\rm RDZ},0}$. The cusp C$^+$ is still present in the $\mu'>'$, $s<0$ quadrant, unfolding a HH$+$ line which asymptotically connects with the MI line for $s>0$, since both are part of $\mu^{\prime}_{{\rm RDZ},0}$. As before the HH$-$ unfolding from C$^+$ goes to the 3DZ$(i\omega)$ point where it becomes an MI. Now one has region 3 reaching all the way up to $\mu^{\prime}=0$ for both $s<0$ and $s>0$, and, in fact, region $3$ is much larger for $s>0$ than for $s<0$. From a physical point of view this can be understood by noticing that for $b>1/3$ all the moments of the Mexican-hat kernel, except $M_0$ are negative. Thus while the overall area of the kernel is positive and for $s>0$ the nonlocal interaction can be considered as globally attractive, at medium and long distances the nonlocal interaction is repulsive. This compensates the local attractive interaction leading to oscillations in the front profile.
Increasing $b$ in the half-plane $s>0$ as $b$ increases the QZ$^-$ approaches zero but there are no qualitative changes. In the half-plane $s<0$ the slope of the HH$^-$ that unfolds from C$^+$ given by by $\mu^{\prime}_{{\rm RDZ},-1}$ decreases \[e.g. Eq. (\[Eq::Mexhat\_muRDZ-1\_asymptotic\])\] and the 3DZ$(i\omega)$ point moves towards more negative values of $s$. For $b=1$ the 3DZ$(i\omega)$ point is located at $s->-\infty$. For $b>1$ the HH$^-$ does not cross $\mu'=0$ and there is no 3DZ$(i\omega)$ point and, thus, no crossover XR nor region 3 for $s<0$. The result is that for $b>1$ only region 3 unfolded by QZ$^-$ located at $s>0$ remains as parameter regions where stable LSs can be formed.
We now focus on the effect of the Mexican-hat nonlocal kernel for given parameter values of the local GLE, that is for a given $\mu^{\prime}$. In what follows we take $b = 1/2$ \[Fig. \[Fig::Mexhat\_muRDZ\] (d)\] so that the total area of the kernel is $M_0=1$, as in the kernels considered in the previous Sections.
![\[Fig::mexHat\_bound\] (Color online) Boundaries in the $(\sigma, s)$-plane separating the regions of monotonic and oscillatory tails when using a Mexican-hat shaped kernel in the nonlocal GLE. $\mu^{\prime} = -6$.](figure10-01.eps){width="8cm"}
For this kernel a closed formula for $s_{\rm RDZ}(\mu^{\prime},\sigma)$ is not available, but, nevertheless, it can be found semi-analytically by replacing (\[Eq::Mexhat\_uc\]) into (\[Eq::Mexhat\_RDZ2\]) with $b = 1/2$. The result for $\mu'=-6$ is shown in Fig. \[Fig::mexHat\_bound\]. As discussed above, the result is that one finds two sections of the RDZ manifold for $s>0$, BD$^-$ and MI, unfolding from QZ$^-$ while for $s<0$ one has a MI and a crossover XR unfolding from 3DZ$(i\omega)$ \[see also Fig. \[Fig::Mexhat\_muRDZ\] (d)\].
Fig. \[Fig::mexHatspatEig\] shows the location in the complex $\lambda$ plane of the first few spatial eigenvalues for different values of $s$ and $\sigma$. The first row corresponds to $s=30$. For $\sigma$ small the spatial dynamics is lead by a real doublet located close to the real doublet of the GLE with local coupling \[Fig. \[Fig::mexHatspatEig\] (a)\]. There is another real doublet located outside the region plotted in Fig. \[Fig::mexHatspatEig\] (a). Increasing $\sigma$ the second real doublet gets closer to the origin and collides with the first pair in a BD transition leading to a complex quartet \[Fig. \[Fig::mexHatspatEig\] (b)\] within parameter region 3, where front tails have a oscillatory profile. Further increasing $\sigma$ leads to a collision of the components of the complex quartet on the imaginary axis (MI transition) which results in two imaginary doublets Fig. \[Fig::mexHatspatEig\] (c)\] within parameter region 2.
For $b=1/2$ and $s<0$ the one finds only one real doublet in the spatial spectrum. For $s$ not too negative this real doublet leads the spatial dynamics for any value of $\sigma$ as shown in the second row of Fig. \[Fig::mexHatspatEig\] which corresponds to $s=-1$. The three panels of this row are within parameter region 4.
For a large negative $s$ the real doublet leads the dynamics only for small $\sigma$ \[see Fig. \[Fig::mexHatspatEig\] (g)\]. Increasing (g) one encounters the crossover XR after which there is a complex quartet located closer to the imaginary axis than the real doublet \[Fig. \[Fig::mexHatspatEig\] (h)\] and therefore one enters in the parameter region 3 unfolding from the 3DZ$(i\omega)$ point. For larger values of $\sigma$ there is a MI transition at which the components of the complex quartet collide on the imaginary axis. After this the spatial dynamics is lead by two imaginary pairs as shown in Fig. \[Fig::mexHatspatEig\] (i) for parameters within region 2.
![\[Fig::mexHatspatEig\] Location of the first spatial eigenvalues for the GLE with a Mexican-hat nonlocal kernel in the complex $\lambda$ plane (shown as black dots) for $b=1/2$ and $\mu^{\prime}=-6$. The top row corresponds to $s=30$ with (a) $\sigma=0.1$, (b) $\sigma=0.5$, and (c) $\sigma=2$. The middle row corresponds to $s=-1$ with (d) $\sigma=1$, (e) $\sigma=2$, and (f) $\sigma=3$. The bottom row corresponds to $s=-8$ with (g) $\sigma=1$, (h) $\sigma=2$, and (i) $\sigma=3$. For comparison the two grey dots show the location of the eigenvalues for the local GLE ($s=0$). In all the cases the number of spatial eigenvalues is infinite: the plot just presents the region around the origin in the complex plane.](figure11-01.eps){width="8cm"}
![\[Fig::Mexhat\_frontvelocity\] Velocity at which two fronts connecting the two equivalent HSSs approach each other for the GLE equation with a Mexican-hat nonlocal kernel with $b=1/2$, $\mu'=6$, $s=-1$ and different values for the interaction range $\sigma$ within parameter region 4. ](figure12-01.eps){width="8cm"}
As discussed in Part I in the part of region 4 located close to the crossover XR, the real part of the complex quartet is close to that of the real doublet and the spatial dynamics results, in fact, from the combination of the real doublet and the complex quartet. Thus, although asymptotically the front tail is monotonic, closer to the front core the complex quartet manifest introducing oscillations, whose role in the fronts interaction can lead to the existence of stable LSs. At difference with the case where the leading eigenvalues are complex, and the front tail has oscillations asymptotically, here the front shows oscillations only close to the core, and thus the locking of two fronts can only occur at short distances.
To determine the part of region 4 where this occurs we consider the full evolution equation (\[Eq::RGL\]) with the Mexican-hat nonlocal kernel. We set the initial condition such that there are two fronts connecting the two equivalent HSSs and look at the velocity at which the two fronts approach each other. In Ref. [@GelensPRL2010] (cf. Fig. 1) it was shown that the interaction of two monotonic fronts (in systems with two equivalent states) decays exponentially (both for local and spatially nonlocal interactions where the kernel decays faster than exponentially). In the part of region 4 where the fronts are no longer monotonic, the envelope of the interaction of two oscillatory fronts still decays exponentially, but at some particular distances the fronts pin and the relative velocity drops to zero. Fig. \[Fig::Mexhat\_frontvelocity\] shows the dependence of the relative velocity $v(d)$ on the distance between the fronts $d$ for a interaction for different values of the interaction range $\sigma$.
One can see that when the range of nonlocal interaction vanishes ($\sigma=0$) the logarithm of the relative velocity grows linearly when decreasing $d$. Switching on the nonlocal interaction with a small interaction range (e.g. $\sigma=1$) a similar linear growth is encountered, albeit with a smaller slope, for $d > 3$. For smaller values of $d$ the two fronts have a stronger interaction and the logarithm of the velocity increases linearly when decreasing $d$ but at a much slower rate. For $\sigma=1$ although the real part of the complex quartet is not far away from the real doublet, as shown in Fig. \[Fig::mexHatspatEig\] (d), the separation is still sufficient to warrant a quasi-monotonic front shape as shown in Fig. \[Fig::Mexhat\_profile\] (a). Close to the HSS the front profile is well described by an exponential of the form (\[Eq::Front\_fit\_monotonous\]) with $q_1=-1.433$, which is the value for leading spatial eigenvalue, and fitting the amplitude to $c_1=-1.299$. As a consequence, the velocity at which two connected fronts approach each other is a monotonic function of the front separation.
As the range of the nonlocal interaction $\sigma$ increases, the real part of the complex quartet keeps approaching the real doublet as shown in Fig. \[Fig::mexHatspatEig\] (e) for $\sigma=2$, and a plateau appears in the velocity when the distance is around 3 (see line for $\sigma=2$ in Fig. \[Fig::Mexhat\_frontvelocity\]). For $\sigma=3$ clearly the velocity goes to zero at $d=d_0\approx 2.3$. Fronts starting at a initial distance larger than this one will approach each other until reaching $d_0$ while fronts starting at a distance slightly smaller than $d_0$ will separate until reaching $d_0$. Thus $d_0$ is a stable distance at which the fronts lock forming a LS.
Proceeding in this way one can determine numerically the boundary within region 4 where the oscillations induced by the complex quartet close to the front core enable the formation of LSs. This boundary is labeled as LO in Fig. \[Fig::mexHat\_bound\]. For small nonlocal interaction ranges the line LO approaches to the crossover XR. This comes from the fact that for small values of $\sigma$ the real parts of the doublet and the quartet separate faster than for large values of $\sigma$, as illustrated in Fig. \[Fig::mexHatspatEig\] (compare panels (d) and (g) for instance). Thus the parameter region where one must account for both the real doublet and the complex quartet is smaller for small $\sigma$.
Fig. \[Fig::Mexhat\_profile\](b) shows an oscillatory front profile for $\sigma=3$ within the parameter region between XR and LO. At difference from the BD transition here the oscillations appear at a finite spatial wavenumber and close to the HSS the front profile is very well described by a combination of two exponentials of the form $$A(x)-A_{st} \approx c_1 e^{q_1 x}+ c_2 e^{q_2 x} \cos(k_2 x + \phi_2)
\label{Eq::Front_fit_crossover}$$ where $q_1=-0.518$ is the real spatial eigenvalue while $q_2=-0.6495$ and $k_2=0.734$ are the real and imaginary part of the complex spatial eigenvalue. The coefficients $c_1=-0.7014$, $c_2=0.9892$ and $\phi_2=-1.584$ have been fitted.
![(Color online) Spatial front profile for the GLE with a Mexican-hat nonlocal kernel for $b=1/2$, $\mu'=-6$, $s=-1$ and (a) $\sigma=1$ and (b) $\sigma=3$. The red dashed lines show the approximations given by Eqs. (\[Eq::Front\_fit\_monotonous\]) and (\[Eq::Front\_fit\_crossover\]) (see text). \[Fig::Mexhat\_profile\] ](figure13_fit-01.eps){width="8cm"}
Conclusions {#Sect::conclu}
===========
In this manuscript we have applied the general framework developed in Part I (see Ref. [@PartI]) to illustrate the effect of nonlocal interactions using the GLE as a prototypical example. In particular the work presented here allows for a detailed explanation of some of the findings advanced in [@GelensPRL2010]. One of the main results of [@GelensPRL2010] was that in spatially extended nonlinear systems exhibiting fronts connecting two equivalent homogeneous steady states, the addition of a spatially nonlocal linear interaction term can induce the creation of localized structures in systems with monotonic fronts. This interesting effect is induced by a repulsive nonlocal interaction, that is able to induce spatial tails, and, thus, lead to stable LSs. This was shown in Ref. [@GelensPRL2010] for the case of a Gaussian nonlocal influence kernel. Leveraging the general framework developed in Part I [@PartI] we rationalize these results and to show its generality by considering two other choices of the nonlocal influence kernel, a mod-exponential and a Mexican-hat. Remarkably, in the case of the two first kernels we have been able to find analytical conditions for the existence of the LSs. In the case of the Mexican-hat kernel, with coexisting attractive and repulsive interactions, LSs are obtained through two different mechanisms for both the cases with short-range excitation and long-range inhibition and the other way around. One mechanism is the Belyakov-Devaney transition [@Devaney76; @Homburg10] in which oscillations appear initially at infinite wavelength, and the other a crossover in the location of the spatial eigenvalues on the complex plain in which finite wavelength oscillations develop.
There are a number of problems that exhibit bistable dynamics and domain walls connecting them and also, presumably, spatially nonlocal effects, e.g., in chemical reactions [@Boissonade2006] and in nonlinear optics [@Esteban_06; @*Taranenko_98]. The present work shows that spatial nonlocal effects can have a big influence on these phenomena. Comparisons with experimental results can be made more quantitative by reconstructing the experimental kernel [@Minovich; @Hellwig].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Belgian Science Policy Office (BeISPO) under grant No.IAP 7-35, and by the Spanish MINECO and FEDER under Grants FISICOS (FIS2007-60327), DeCoDicA (TEC2009-14101), INTENSE@COSYP (FIS2012-30634), and TRIPHOP (TEC2012-36335), and from Comunitat Autònoma de les Illes Balears. LG acknowledges support by the Research Foundation - Flanders (FWO). We thank Prof. E. Knobloch and Dr. G. Van der Sande for interesting discussions.
Lambert’s W Function {#sec:Lambertap}
====================
The so-called Lambert’s W function [^2] is the inverse function of $x=f(y)=y\exp(y)$, i.e., $y=W(x)$ (see, e.g., [@Corless96; @MapleW] for further details). It can be seen as a generalized algorithm, a useful analogy, because as the (complex) logarithm function, Lambert’s W function is multivalued. So, we will define it as, $$x=W_l(x) \exp(W_l(x))\ ,\quad l\in \mathbb{Z}
\label{eq:lambertdef}$$ where, in principle, $x\in \mathbb{C}$ and $l\in \mathbb{Z}$ is the branch index. The principal branch, $W_0(x)$ or simply $W(x)$, has a branch point at $x=-1/e$ and a branch cut along the negative real axis $x\in [-\infty,-1/e]$ ($W_0(-1/e)=-1$), and is real valued in the interval $x\in [-1/e,\infty]$. Moreover, it is analytic at $x=0$, $W_0(0)=0$, while all the other branches have a branch point at $0$. Moreover, $W_{-1}(x)$ is real in the interval $x\in [-1/e,0]$.
The Lambert function can also be used to find the exact solution of transcendental equations of the type $x+\exp(x)=a$. Thus, a solution to the equation, $$cx+\exp(ax)=b$$ can be found with the change, $$\frac{y}{a}=b-c x$$ as $$\frac{y}{c}=W\left[\frac{a}{c}\exp(ab/c)\right]\\$$ and undoing the change of variables one gets, $$x=\frac{b}{c}-\frac{1}{a} W\left[\frac{a}{c}\exp(ab/c)\right]$$
The real branches of Lambert’s W function admit the following series expansions, valid close to $x=0$ ($x<0$ for $W_{-1}(x))$, $$\begin{aligned}
W_0(x)=\sum_{n=1}^{\infty} x^n=x-x^2+\frac{3}{2} x^3+\ldots \label{eq:powexpw0}\\
W_{-1}(x)=\ln(-x)-\ln(-\ln(-x))+\ldots
\label{eq:powexpwm1}\end{aligned}$$
![\[Fig::Lambertfunct\] A plot of the real branches of Lambert’s W function, two-valued in the range $[-1/e,0]$.](appendix-01.eps){width="8cm"}
[^1]: This can be shown taking into account that for the argument of $W$ in (\[eq:ssigmagauss\]), $\lim_{\sigma\rightarrow\infty} -\exp\left(\frac{\sigma^2}{2} \mu^{\prime}-1\right)\rightarrow 0^-$, $z(\sigma\rightarrow \infty)\rightarrow 0^-$, and remembering (see Appendix \[sec:Lambertap\]) that $W_0(0^-)\rightarrow 0$, and so $s_{\rm BD}\rightarrow 0^-$ while as $W_{-1}(0^-)\rightarrow -\infty$ it is necessary to use its series expansion. Keeping the first term, $W_{-1}(z)\sim \ln(-z)$ one gets $s_{MI}\rightarrow 2/\sigma^2 ((\sigma^2\mu^{\prime}/2-1))\sim \mu^{\prime}$, as shown in Fig. \[Fig::SpatialEigBoundary\_Sigma\_S\_realGLE\].
[^2]: This special function is quite accessible nowadays as it can be found in modern scientific programs like Mathematica (where it is called [*ProductLog*]{}), Matlab (where it is called [*lambertw*]{}, and Maple (where it is called [*LambertW*]{}).
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abstract: 'We propose a model of sub-diffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model. Constant force leads to the transition from non-ergodic sub-diffusion to seemingly ergodic diffusive behavior. However, we show it remains anomalous in a sense that the diffusion coefficient depends on the force and the anomalous exponent. For the quadratic potential we find that the anomalous exponent defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.'
author:
- 'Sergei Fedotov, Nickolay Korabel'
title: 'Sub-diffusion in External Potential: Anomalous hiding behind Normal'
---
Recently it has become clear that anomalous diffusion measured by a non-linear growth of the ensemble averaged mean squared displacement $%
\left\langle x^{2}\right\rangle \sim t^{\mu}$ with the anomalous exponent $%
\mu \neq 1$ is as widespread and important as normal diffusion with $\mu =1$ [@Kla08]. Sub-diffusion with $\mu <1$ was observed in many physical and biological systems such as porous media [@Dra99], glass-forming systems [@Weeks02], motion of single viruses in the cell [@Sei01], cell membranes [@Sax97; @Rit05], and inside living cells [Wei03,Gol06,Tolic04]{}. Many examples of sub-diffusive processes in biological systems can be found in recent reviews [@Fra13; @Bar12]. Nowadays new tools are available including super-resolution light optical microscopy techniques to deal with biological in vivo data which allows to monitor a large number of trajectories at the single-molecule level and at nanometer resolution [@Ser08; @Jaq08; @Nor14]. Using these techniques it is possible to discriminate between anomalous ergodic processes where the ensemble and time averages coincide and non-ergodic processes where ensemble and time averages have different behavior [@He08; @Lub08; @Mer13]. Two important observations have been made about anomalous transport in living cells: (1) anomalous transport is usually a transient phenomenon before transition to normal diffusion or saturation due to confined space [Bro09,Neu08,Sax01]{} (2) ergodic and non-ergodic processes may coexist as it was observed in plasma membrane [@Wei11].
Several models are proposed to describe ergodic and non-ergodic anomalous processes such as non-ergodic continuous time random walk (CTRW) with power-law tail waiting times, ergodic anomalous process generated by fractal structures, fractional Brownian-Langevin motion characterized by long correlations and time dependent diffusion coefficient [@Kla08; @Sok12; @Bre13]. The standard CTRW model for sub-diffusion of a particle in an external field $F(x)$ randomly moving along discrete one-dimensional lattice can be described by the generalized master equation for the probability density $p(x,t)$ to find the particle at position $x$ at time $t$ $$\frac{\partial p}{\partial t}=-i(x,t)+w^{+}(x-a)i(x-a,t)+w^{-}(x+a)i(x+a,t),
\label{MM}$$where $a$ is the lattice spacing and $i(x,t)$ is the total escape rate from $%
x$ $$i(x,t)=\frac{1}{\Gamma(1-\mu)\tau _{0}^{\mu }}\mathcal{D}_{t}^{1-\mu }p(x,t). \label{ieq}$$Here $\tau _{0}$ is a constant timescale and $\mathcal{D}_{t}^{1-\mu }$ is the Riemann-Liouville fractional derivative defined by $$\mathcal{D}_{t}^{1-\mu }p(x,t)=\frac{1}{\Gamma (\mu )}\frac{\partial }{%
\partial t}\int_{0}^{t}\frac{p(x,\tau )}{(t-\tau )^{1-\mu }}d\tau .$$The probabilities of jumping to the right $w^{+}(x)$ and to the left $w^{-}(x)$ are $$w^{+}(x)=\frac{1}{2}+\beta a F(x),\quad w^{-}(x)=\frac{1}{2}-\beta a F(x).
\label{ww}$$Series expansion of Eq. (\[MM\]) together with Eq. (\[ieq\]) and Eq. (\[ww\]) leads to the fractional Fokker-Planck equation (FFPE) [@Met99; @Met00] $$\frac{\partial p}{\partial t}=D_{\mu }\left[ \frac{\partial ^{2}}{\partial
x^{2}}-\beta \frac{\partial }{\partial x}F(x)\right] \mathcal{D}_{t}^{1-\mu
}p, \label{eq:FFPE}$$where the generalized diffusion $D_{\mu }=a^{2}/(2\;\Gamma (1-\mu )\tau
_{0}^{\mu }).$ The stationary solution of Eq. (\[eq:FFPE\]) is the Boltzmann distribution. There exist a huge literature on this equation [Met99,Met00]{} and its generalization for time dependent forces [Hei07,Mag08,Henry10,Eule09,Sok06,Shu08,Shkilev12]{}.
One of the main assumptions in this literature, which is not always clearly stated is that, as long as a random walker is trapped at a particular point $%
x$, the external force $F(x)$ does not influence the particle. It is clear from Eq. (\[ieq\]) that the escape rate $i(x,t)$ does not depend on the external force $F(x).$ The force only acts at the moment of escape inducing a bias. The question is how to take into account the dependence of the escape rate on $%
F(x)?$ To the author’s knowledge this is still an open question. One of the main aims of this Letter is to propose a model which deals with this problem. We find that the dependence of escape rate on force drastically changes the form of the master equation (\[MM\]) and FFPE (\[eq:FFPE\]). We observe transient anomalous diffusion and transition from non-ergodic to normal ergodic behavior. However, we show that this seemingly normal process could be still anomalous masked by normal behavior. Our findings suggest that a closer inspection of experimental results could be necessary in order to discriminate between normal and anomalous processes.
*Model.—* We consider a random particle moving on a one dimensional lattice under assumption that an external force acts on a particle at all times not only at the moment of jump as in Eq. (\[MM\]). The implication of this assumption is the dependence of the random trapping time on the external force (not just jumping probabilities as in (\[ww\])). Some discussion of situation when the external force influence the rates and jumps can be found in [@Hei07]. The main physical idea behind our model is that there exists two independent mechanism of escaping from the point $x$ with two different random residence times. The first mechanism is due to external force with the escape rate proportional to $F(x).$ The second one is the sub-diffusive mechanism involving the rate inversely proportional to the residence time. The latter generates the power law waiting time distribution with the infinite first moment.
Regarding the first mechanism, we define the jump process from the point $x$ as follows. We assume that the rate of jump to the right $\mathbb{T}_{x}^{+}$ from $x$ to $x+a$ is $\nu aF(x)$ when $F(x)\geq 0$ and the rate of jump to the left $\mathbb{T}_{x}^{-}$ from $x$ to $x-a$ is $-\nu
aF(x)$ when $F(x)\leq 0.$ For this jump model the random waiting time $T_{F}$ at the point $x$ is defined by the exponential survival probability $\Psi
_{F}(x,\tau )$ involving the external force $F(x)$ $$\Psi _{F}(x,\tau )=\Pr \left\{ T_{F}>\tau \right\} =\exp \left( -\nu
a|F(x)|\tau \right).$$where $\nu $ is the intensity of jumps due to force field. For example, one can think of the escape rate $\mathbb{T}_{x}^{+}$ that is defined in terms of the potential field $U(x)$ that is $\mathbb{T}_{x}^{+}=-\nu \left[
U(x+a)-U\left( x\right) \right] >0,$ there $F(x)=-U^{\prime }(x)+o(a^{2})$ for $U^{\prime }(x)\leq 0.$ The second mechanism involves the sub-diffusive random walk with the escape rate $\lambda (x,\tau )$ from the point $x$, which is inversely proportional to the residence time $\tau.$ In this case the random waiting time $T_{\lambda }$ at the point $x$ is defined by the survival probability $$\Psi _{\lambda }(x,\tau )=\Pr \left\{ T_{\lambda }>\tau \right\} =\exp
\left( -\int_{0}^{\tau }\lambda (x,s)ds\right) . \label{FFFF}$$The question now is how to implement the jumping process due to external force into the sub-diffusive random walk scheme? When the random walker makes a jump to the point $x$, it spends some random time (residence time) before making another jump to $x+a$ or $x-a$. Let us denote this residence time $T_{x}$. The key point of our model is that we define this residence time as the minimum of two: $T_{\lambda }$ and $T_{F}$ $$T_{x}=\min \left( T_{\lambda },T_{F}\right) . \label{min}$$For the anomalous sub-diffusive case this model could lead to the drastic change in the form of the fractional master equation. The main reason for this is that the external force $F(x)$ plays the role of tempering factor preventing the random walker to be anomalously trapped at point $x$.
Because of the independence of two mechanisms, in our model the rate of jump $\mathbb{T}_{x}^{+}$ to the right from $x$ to $x+a$ and the rate of jump $%
\mathbb{T}_{x}^{-}$ to the left from $x$ to $x-a$ can be written as the sum$$\mathbb{T}_{x}^{+}=%
\begin{cases}
\omega ^{+}(x)\lambda (x,\tau )+\nu aF(x), & F(x)\geq 0, \\
\omega ^{-}(x)\lambda (x,\tau ),\quad & F(x)<0%
\end{cases}
\label{T+}$$and$$\mathbb{T}_{x}^{-}=%
\begin{cases}
\omega ^{+}(x)\lambda (x,\tau ), & F(x)\geq 0, \\
\omega ^{-}(x)\lambda (x,\tau )-\nu aF(x),\quad & F(x)<0.%
\end{cases}
\label{T-}$$Although it is straightforward to consider general $\omega ^{+}(x)$, $\omega
^{-}(x)$, for simplicity in what follows we consider $\omega ^{+}(x)=\omega
^{-}(x)=1/2$. In our model the asymmetry of random walk occur only from the force dependent rate. Let us explain the main idea of Eqs. (\[T+\]) and (\[T-\]). The external force $F(x)\geq 0$ increases the sub-diffusive rate of jumps to the right $\lambda (x,\tau )/2$ and does not change the sub-diffusive rate of jumps to the left. The essential property of Eqs. (\[T+\]) and (\[T-\]) is that the rate $\lambda (x,\tau )$ depends on the residence time variable $\tau $. This dependence makes any model involving the probability density $p(x,t)$ non-Markovian. For the Markov case with $%
F(x)=0,$ $\lambda ^{-1}(x)$ has a meaning of the mean residence time at the point $x$. When the parameter $\nu =0$ and the rates are $\mathbb{T}%
_{x}^{+}=w^{+}(x)\lambda (x,\tau ),$ $\mathbb{T}_{x}^{-}=w^{-}(x)\lambda
(x,\tau )$, we obtain the standard fractional Fokker-Planck equation ([eq:FFPE]{}). Notice that Eq. (\[min\]) is consistent with the expression for the effective escape rate $\mathbb{T}_{x}^{+}+\mathbb{T}_{x}^{-}$ as a sum of two rates $\lambda (x,\tau )+\nu a|F(x)|$. Similar situation has been considered in [@Fed13].
After incorporation of the force dependent escape rates we can obtain generalized master equation (see Supplementary Materials for the derivation). By expanding the RHS of the master equation to the second order in jump size $a$, we get a fractional diffusion equation $$\frac{\partial p}{\partial t}=\frac{\partial ^{2}}{\partial x^{2}}\left[
D_{\mu }e^{-\nu a|F(x)|t}\mathcal{D}_{t}^{1-\mu }\left[ p(x,t)e^{\nu
a|F(x)|t}\right] \right] - \label{main_eq}$$$$-a^{2} \nu \frac{\partial }{\partial x}\left[ F(x)p(x,t)\right] .$$ This equation is fundamentally different from the classical FFPE ([eq:FFPE]{}) because it involves the external force in both terms on the right hand side. One can see that the force $F(x)$ not only determines the advection term as in Eq. (\[eq:FFPE\]), but also plays the role of tempering parameter through the factor $e^{\nu a|F(x)|t}$. Similar factor occurs in sub-diffusive equation with the death or evanescent process [@Abad10; @Fal13]. However, here we consider the system with constant total number of particle.
The stationary solution $p_{st}(x)$ of Eq. (\[main\_eq\]) obeys the standard equation$$-a^{2}\nu F(x)p_{st}(x)+\frac{d}{dx}\left[ D_{F}(x)p_{st}(x)\right] =0.
\label{stat_eq}$$(see a supplement material for details). Interesting property of this equation is that the effective diffusion constant $D_{F}(x)$ depends on the external force and anomalous exponent $$D_{F}(x)=D_{\mu }\left( \nu a|F(x)|\right) ^{1-\mu }. \label{D_F}$$This fact implies that the Boltzmann distribution is no longer stationary solution of (\[stat\_eq\]). For the quadratic potential $U(x)=\kappa x^{2}/2
$ with $F(x)=kx,$ we find that for large $x$ the stationary density $%
p_{st}(x)$ has the form $$p_{st}(x)\sim \exp (-A|x|^{1+\mu }), \label{stat}$$where $A>0$ is a constant. One can see that the form of stationary density is determined by the anomalous exponent $\mu $. In this case the particles spread further compared to the Boltzmann case. The reason is the dependence of the effective diffusion constant $D_{F}(x)$ on force $F(x).$ Note that for the sub-diffusive fractional Fokker-Planck equation (\[eq:FFPE\]) the anomalous exponent only determines the slow power law relaxation rate, while the stationary density converges to Boltzmann equilibrium which does not depend on $\mu .$
*Numerical simulations.—* We consider two particular cases: (1) constant force $F$ corresponding to the linear potential and (2) the quadratic potential $U(x)=\kappa x^{2}/2$ both in the infinite domain. We concentrate on the behavior of the density function $p(x,t)$, the mean $%
\left\langle x(t)\right\rangle $ and the variance $\sigma (t)=\left\langle
x^{2}\right\rangle -\left\langle x\right\rangle ^{2}$ calculated using an ensemble of trajectories from the initial distribution $p(x,0)=\delta (x)$. We also calculate the time averaged variance of a single trajectory of length $T$, $\sigma _{T}(\Delta ,T)=\delta ^{2}(\Delta ,T)-(\delta (\Delta
,T))^{2}$, where $\delta ^{n}(\Delta ,T)=\int_{0}^{T-\Delta }(x(t+\Delta
)-x(t))^{n}dt/(T-\Delta )$, $n=1,2$. This quantity become a standard tool to assess ergodic properties of a system been equivalent to its ensemble averaged counterpart only for ergodic case.
When the external force $F$ is constant, we observe the transition from sub-diffusion at short times to seemingly normal diffusion at long times. The density function changes from the distinct sub-diffusive shape for short times to the Gaussian shape propagator at longer times (see the inset of figure \[FIG1\]). The average position of the ensemble behaves as $\left<x(t)\right>=F t$. The ensemble averaged variance $\sigma(t)$ grows as a power law for short times, $\sigma(t) \sim t^{\eta}$, and transition to a normal diffusive linear growth $\sigma (t)\sim 2D_{F}t$ for longer times. However, in this case the diffusion coefficient $D_{F}$ depends on the force $F$ and anomalous exponent $\mu $. We conclude that although the variance $%
\sigma (t)$ is linearly proportional to time, this dependence reveals the anomalous nature of the process even in the limit $t\rightarrow \infty $. Numerical calculations confirms the analytical result for the diffusion coefficient Eq. (\[D\_F\]) (see figure \[FIG1\]). Second observation is that the power law behavior at short times involves the exponent $\eta
(F)>\mu $ which depends on force $F$. For $\mu =0.3$ they are estimated to be $\eta
\approx 0.39$ for $F=0.0001$, $\eta \approx 0.47$ for $F=0.001$ and $\eta
\approx 0.6$ for $F=0.01$. This can be interpreted as an enhancement of sub-diffusion coursed by the constant force. Such enhancement should be taken into account in the analysis of biological experiments where sub-diffusion usually appears as transient before the transition to the normal diffusion [@Fra13]. For the large value of $F$ the exponent $\eta
$ tends to one while in the small force limit $\eta \rightarrow \mu $. The time averaged variance calculated for constant force grows linearly $\sigma
_{T}(\Delta ,T)\sim \Delta $ and shows minor scatter between single trajectories (figure \[FIG2\]). After averaging over different trajectories, it grows with the coefficient $2 D_F$ which is equal to the ensemble average value. This shows that the non-ergodic sub-diffusive system becomes an ergodic one.
Now we consider the quadratic potential $U(x)=\kappa x^{2}/2.$ The system becomes again non-ergodic despite the tempering affect of the force. To confirm this we calculate the time averaged variance (inset of figure [FIG2]{}). As expected it shows large fluctuations among different trajectories typical for non-ergodic systems. Note that even with this typical behavior, it can be easily distinguished in experiments since in our case the mean of the time averaged variance converges to a constant, while for standard CTRW in a bounded region it grows as a power of the anomalous exponent, $\left\langle \sigma_{T}(\Delta )\right\rangle \sim \Delta ^{1-\mu
}$. Regarding the shape of the stationary density, numerical simulations are in good agreement with analytical results Eq. (\[stat\]) (see figure [FIG3]{}).
*Summary.—* In this Letter we have presented a model of anomalous sub-diffusive transport in which the force acts on the particle at all times not only at the moment of jump. This leads to the dependence of jumping rate on the force with the dramatic change of particles behavior compared to the standard CTRW model. We have derived a new type of fractional diffusion equation which is fundamentally different from the classical fractional Fokker-Planck equation. In our model the force $F(x)$ not only appears in the drift term as in Eq. (\[eq:FFPE\]), but also determines the structure of the diffusion term controlling the spread of particles. The constant external force leads to the natural tempering of the broad waiting time distribution and, as a result, to the transition to a seemingly normal diffusion (linear growth of the mean squared displacement) and equivalence of the time and ensemble averages. However, this may lead to a wrong conclusion in analyzes of experimental results on transient sub-diffusion [@Fra13] that the process is normal for large times. We have found that contrary to normal diffusion process in the external force field, the diffusion coefficient depends on the force and anomalous exponent. This fact implies that the Boltzmann distribution is no longer stationary solution. External perturbations and noise fluctuations are not separable which reflects the non-Markovian nature of the process even for large times.
Our results would be possible to test in experiments, for example, by considering a bead which is moving sub-diffusively in an actin network. The motion of such a beat can be described by a random walk type of dynamics [@Wong04]. Force-measurements could be realized by using optical trap and tweezers which are the nano-tools capable of performing such measurements on individual molecules and organelles within the living cell [@Nor14]. When the force is constant the dependence of the measures diffusion coefficient on the strength of the force would reveal the predicted power law behavior $F^{1-\mu }$. For quadratic potential it could be possible to retrieve the form of the stationary profile (\[stat\]) with the slow decay compared to Boltzmann distribution for large $x$.
Acknowledgements {#acknowledgements .unnumbered}
================
SF and NK acknowledge the support of the EPSRC Grant EP/J019526/1.
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SUPPLEMENTARY MATERIALS {#supplementary-materials .unnumbered}
=======================
To derive Eq. (\[main\_eq\]) we use the structured probability density function $\xi (x,t,\tau )$ with the residence time $\tau $ as auxiliary variable. This density gives the probability that the particle position $%
X(t) $ at time $t$ is at the point $x$ and its random residence time $T_{x}$ at point $x$ is in the interval $(\tau ,\tau +d\tau ).$ The density $\xi
(x,t,\tau )$ obeys the balance equation $$\frac{\partial \xi }{\partial t}+\frac{\partial \xi }{\partial \tau }%
=-\left( \mathbb{T}_{x}^{+}(x,\tau )+\mathbb{T}_{x}^{-}(x,\tau )\right) \xi .
\label{basic}$$We consider only the case when the residence time of random walker at $t=0$ is equal to zero, so the initial condition is $$\xi (x,0,\tau )=p_{0}(x)\delta (\tau ), \label{initial}$$where $p_{0}(x)$ is the initial density. The boundary condition in terms of residence time variable ($\tau =0)$ can be written as [@Cox] $$\begin{aligned}
\xi (x,t,0) &=&\int_{0}^{t}\mathbb{T}_{x}^{+}(x-a,\tau )\xi (x-a,t,\tau
)d\tau + \notag \\
&&\int_{0}^{t}\mathbb{T}_{x}^{-}(x+a,\tau )\xi (x+a,t,\tau )d\tau .
\label{arr1}\end{aligned}$$We solve (\[basic\]) by the method of characteristics for $\tau <t$ $$\xi (x,t,\tau )=j\left( x,t-\tau \right) \Psi_{\lambda} (x,\tau )e^{-\Phi(x) \tau
},\quad \tau <t, \label{je}$$where $$\Phi(x) = \nu a |F(x)|.$$The solution Eq. (\[je\]) is written in terms of the integral arrival rate $%
j(x,t)=\xi (x,t,0)$ and in terms of the survival function Eq. (\[FFFF\]) $$\Psi_{\lambda} (x,\tau ) = e^{-\int_{0}^{\tau } \lambda (x,s) ds}.$$Our purpose now is to derive the master equation for the probability density$$p(x,t)=\int_{0}^{t^{+}}\xi (x,t,\tau )d\tau . \label{denG}$$Let us introduce the integral escape rate to the right $i^{+}(x,t)$ and the integral escape rate to the left $i^{-}(x,t)$ as$$i^{\pm }(x,t)=\int_{0}^{t^{+}} \omega^{\pm}(x) \lambda(x,\tau )\xi (x,t,\tau )d\tau .
\label{i1}$$Note that the integration with respect to the residence time $\tau $ in ([denG]{}) and (\[i1\]) involves the upper limit $\tau =t,$ where we have a singularity due to the initial condition (\[initial\]). Then the boundary conditions (\[arr1\]) can be written in a simple form: $$\begin{aligned}
j(x,t) &=&i^{+}(x-a,t)+i^{-}(x+a,t) \notag \\
&&+%
\begin{cases}
\nu aF(x-a)p(x-a,t), & F\geq 0, \\
-\nu aF(x+a)p(x+a,t),\quad & F<0.%
\end{cases}
\label{jj}\end{aligned}$$It follows from (\[initial\]), (\[je\]) and (\[i1\]) that $$\begin{aligned}
i^{\pm }(x,t) &=&\int_{0}^{t}\psi ^{\pm }(x,\tau )j(x,t-\tau )e^{-\Phi(x)
\tau }d\tau \notag \\
&&+\psi ^{\pm }(x,t)p_{0}(x)e^{-\Phi(x) t}, \label{i55}\end{aligned}$$where $\psi ^{+}(x,\tau )=\omega^{+}(x) \lambda(x,\tau ) \Psi_{\lambda} (x,\tau )$ and $\psi^{-}(x,\tau )=\omega^{-}(x) \lambda(x,\tau )\Psi_{\lambda} (x,\tau ).$ Substitution of ([initial]{}) and (\[je\]) to (\[denG\]), gives $$\begin{aligned}
p(x,t) &=&\int_{0}^{t}\Psi_{\lambda} (x,\tau )j(x,t-\tau )e^{-\Phi(x) \tau }d\tau
\notag \\
&&+\Psi_{\lambda} (x,t)p_{0}(x)e^{-\Phi(x) t}. \label{p11}\end{aligned}$$The balance equation for probability density $p(x,t)$ can be written as$$\frac{\partial p}{\partial t}=-i^{+}(x,t)-i^{-}(x,t)+j(x,t)-\Phi(x) p(x,t).
\label{balan}$$Let us find a closed equation for $p(x,t)$ by expressing integral rates $%
i^{\pm }(x,t)$ and $j(x,t)$ in terms of the density $p(x,t).$ We apply the Laplace transform $\hat{f}(s)=\int_{0}^{\infty }f (\tau )e^{-s\tau
}d\tau $ to (\[i55\]), and (\[p11\]), and obtain $$\hat{i}^{\pm }(x,s)=\frac{\hat{\psi}^{\pm }(x,s+\Phi(x))}{\hat{\Psi}%
(x,s+\Phi(x))}\hat{p}(x,s), \label{new55}$$which after the inversion of the Laplace transform and using the shift theorem gives $$i^{\pm }(x,t)=\int_{0}^{t}K^{\pm }(x,t-\tau )e^{-\Phi(x) (t-\tau)} p(x,\tau)
d\tau.$$The memory kernels $K^{+}(x,t)$ and $K^{-}(x,t)$ are defined by Laplace transforms $$\hat{K}^{\pm }\left( x,s\right) =\hat{\psi}^{\pm }(x,s)/\hat{\Psi}_{\lambda}\left(
x,s\right). \label{new5}$$Now we consider the sub-diffusive case where $\lambda (\tau)$ is inversely proportional to the residence time $\tau:$ $$\lambda (\tau )= \mu/(\tau _{0}+\tau),\qquad 0<\mu <1.
\label{s33}$$For simplicity we consider $$\omega^{-}=\omega^{+}=1\slash2.$$ It is straightforward to generalize to non-homogeneous systems by considering space dependent $\lambda(x)$ and space dependent anomalous exponent $\mu(x)$, this case we consider elsewhere [@Che05; @Fed13]. From Eqs. (\[FFFF\]) and ([s33]{}) it follows that the survival function has a power-law dependence $$\Psi_{\lambda} (\tau)= \tau _{0}^{\mu} \left( \tau _{0}+\tau \right)^{-\mu}.$$ The waiting time density functions $\psi^{\pm} (\tau )$ are $$\psi^{+}(\tau)=\psi^{-}(\tau)= \mu \tau _{0}^{\mu} (\tau _{0}+\tau )^{-1-\mu}\slash2.
\label{Pareto}$$Using the Tauberian theorem their Laplace transforms are $\hat{\psi}%
^{\pm}\left( s\right) \simeq (1- g s^{\mu})/2$ as $s\rightarrow 0$, where $g =
\Gamma (1-\mu )\tau _{0}^{\mu}$. From (\[new5\]) we obtain the Laplace transforms $$\hat{K}^{+}(s) = \hat{K}^{-} (s) \simeq s^{1-\mu}\slash (2 g),\qquad s\rightarrow 0.$$Therefore, the integral escape rates to the right $i^{+}$ and to the left $%
i^{-}$ in the sub-diffusive case are $$i^{+}(x,t)=i^{-}(x,t) = e^{-\Phi(x) t}\mathcal{D}_{t}^{1-\mu}\left[ p(x,t)e^{\Phi(x)
t}\right] \slash (2 g). \label{gen_i}$$By introducing the total integral escape rate $$i(x,t)=i^{+}(x,t)+i^{-}(x,t),$$and expanding the right-hand side of Eq. (\[balan\]) to second order in jump size $a$ we obtain the following fractional equation $$\frac{\partial p}{\partial t} = - a^{2}\nu \frac{\partial }{\partial x}\left[ F(x)p(x,t)\right] + \frac{a^{2}}{2}\frac{\partial ^{2}i}{\partial x^{2}},
\label{sup_main}$$which using Eq. (\[gen\_i\]) leads to the main equation of the paper Eq. (\[main\_eq\]).
Now we derive the equation for the stationary solution Eq. (\[stat\_eq\]). Writing the escape rate $i(x,t)$ in Laplace form $$\hat{i}(x,s) = \frac{(s+\Phi(x))^{1-\mu}}{g} \hat{p}(x,s)$$and taking the limit $s \rightarrow 0$ corresponding to $t \rightarrow \infty$, we obtain the stationary escape rate $$i_{st}(x) = \frac{\Phi(x)^{1-\mu}}{g} p_{st}(x).
\label{i_st}$$where the stationary density is defined in a standard way $p_{st}(x) = \lim_{s \rightarrow 0} s \hat{p}(x,s)$. Taking the time derivative in Eq. (\[sup\_main\]) to zero and substituting Eq. (\[i\_st\]) we obtain the stationary advection-diffusion equation $$-a^{2}\nu \frac{d}{dx}\left[ F(x) p_{st}(x) \right] + \frac{d^2}{dx^2}\left[ D_{F}(x)p_{st}(x)\right] =0.
\label{stat_eq_2}$$Integrating Eq. (\[stat\_eq\_2\]) and taking into account that the flux of the particles is zero we obtain Eq. (\[stat\_eq\]).
|
---
abstract: 'We have identified a set of optical emission-line features 700(12 kpc) to the southwest of the nucleus of Centaurus A, roughly opposite to the radio jet and well-known optical emission filaments associated with the northern radio structure (along the axis of the southwestern radio lobes, although there is no coherent counterjet at this radius). We use integral-field optical spectroscopy to trace the ratios of strong emission lines, showing changes in excitation across the region, and significant local reddening. The emission regions are spatially associated with far-infrared emission peaks in one of two cold dust clouds identified using [*Herschel*]{} and [*Spitzer*]{} data, and there may be a mismatch between the low temperature of the dust and the expected heating effect of young stars. The strong emission lines have ratios consistent with photoionization in normal H II regions, requiring only modest numbers of OB stars; these stars and their cooler accompanying populations must be obscured along our line of sight. These data fit with a picture of fairly ordinary formation of clusters in a large giant molecular cloud, or network of such clouds. The location, projected near the radio-source axis and within the radius where a starburst wind has been inferred on the other side of the galaxy, raises the question of whether this star-forming episode was enhanced or indeed triggered by an outflow from the central parts of Centaurus A. However, optical emission-line ratios and line widths limit the role of shocks on the gas, so any interaction with an outflow, associated either with the radio source or star formation in the gas-rich disk of Centaurus A, can at most have compressed the gas weakly. We speculate that the presence of similar star-forming regions on both sides of the galaxy, contrasted with the difference in the character of the emission-line clouds, reflects the presence of a collimated radio jet to the northeast and perhaps anisotropic escape of ionizing radiation from the AGN as well. In this view, the star formation on the southwestern side of Cen A could be enhanced by a broad outflow (whether originated by a starburst or AGN), distinct from the radio jet and lobes.'
author:
- 'William C. Keel'
- 'Julie K. Banfield and Anne Medling'
- 'Susan G. Neff'
title: Optical Detection of Star Formation in a Cold Dust Cloud in the Counterjet Direction of Centaurus A
---
Introduction
============
Centaurus A (NGC 5128), as the nearest galaxy hosting a large double radio source, has long played a special role in our understanding of similar objects. It displays a host of characteristic features observable in unique detail - a large-scale double source spanning 600 kpc, radio and X-ray jets on scales up to 3 kpc, merger signatures in stars, gas, and dust, and optical emission-line features often attributed to interaction between the propagating jet and ambient interstellar medium, generating both shock ionization and star formation.
Optical emission regions and possible young stars near the northeastern radio jet were identified by [@Blanco75]; spectroscopy by [@Osmer78] confirmed that both normal H II region and bright supergiants were present. [@GrahamPrice] used higher-resolution spectra to show that large “turbulent" velocities in the range 400 km s$^{-1}$ appear across few-arcsecond scales, well beyond what could be produced in star-forming regions. Data presented by [@PDC75] show that multiple ionizing mechanisms may be at work, with line ratios in some parts of the first-discovered emission regions have \[O I\] too strong for ordinary H II regions ionized by stars. The ionization of the gas in these filaments is complex; [@Morganti91] suggest a dominant role for photoionization by the beamed continuum of a small scale jet. In contrast, [@Sutherland] present models showing that the small-scale velocity structure matches shock excitation of the spectral lines; [*Chandra*]{} X-ray data analyzed by [@EvansKoratkar] support this through showing very hot gas along one side of the most prominent filament. [@Rejkuba] used color-magnitude diagrams to identify local young supergiant stars; comparison of their locations to the ionized gas structure provides additional evidence for multiple ionizing mechanisms, while a deeper [*Hubble Space Telescope*]{} analysis by [@Crockett] makes this case more stringently in the brightest emission region.
[@Santoro2015] show that the dynamics of ionized gas in the prominent filaments are consistent with the ambient H I clouds, including components in regular rotation about the galaxy as well as being entrained by the radio jet. Additional MUSE observations by [@Santoro2016] show that even on small scales in these filaments, there is a changing mix of ionization mechanisms, with embedded star formation and photoionization by the distant AGN both indicated by emission-line ratios. [@McKinley2017] suggest that one of the outermost emission-line filaments may result from interaction with a wind rather that a collimated jet, and speculate that the southwestern jet may not intersect suitable cold gas to produce similar effects in the “counterjet" direction.
As part of a study of these emission features, we have found an emission-line feature along the “counterjet" direction, which seems to have been previously unremarked[^1]. We present here morphological and spectroscopic observations of this object, which appears to consist of (possibly multiple) H II regions powered by young stars. The H$\alpha$ emission is spatially coincident with the peaks of the cold dust feature found by [@Auld] from [*Herschel*]{} observations, furnishing an interesting puzzle as to how a star-forming region coexists with the dust without generating a higher temperature than observed.
The radio structure does not have a distinct counterpart to the northern jet on the southern side at this distance. The regions we observe fall in a minimum in radio flux between inner and outer lobes [@Junkes], making direct interaction with a jet unlikely.
In computing sizes and luminosities, we adopt a distance 3.7 Mpc (scale 17.9 pc arcsecond$^{-1}$), following the Cepheid results of [@Tully2013] and the red-giant studies from, e.g., [@Crnojevic] and [@Tully2015].
Observations
============
Identification and optical imaging
----------------------------------
The emission region was identified in May 2014 using an H$\alpha$ filter of FWHM 75 Å on the remotely-operated SARA 0.6m telescope at Cerro Tololo, Chile [@SARA]. A CCD system from ARC of San Diego operated at -110 C; the pixel scale was 0.38. We coadded images totaling 7 hours’ integration in this filter, and 70 minutes of continuum imaging in the R band, obtained between May and August 2014. Flux calibration used [@Landolt] standard stars, carried to the narrow filter using the ratio of filter widths. The narrowband image (Fig. \[fig-halphamontage\]) shows a set of diffuse H$\alpha$ emission regions, and two starlike objects with strong residual H$\alpha$ flux after continuum subtraction using the $R$ image. Color terms in this subtraction will be modest, because H$\alpha$ is near the center of the $R$ band. The image gives a total H$\alpha$+\[N II\] flux within the brightest $15 \times 15$ region of $6.1 \times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$, about 50% lower than implied by the flux calibration of the integral-field spectroscopic data.
For comparison, we also make use of a similar H$\alpha$/$R$ image pair centered on the core of Centaurus A, summing total exposures of 4 hours in H$\alpha$ and 50 minutes in $R$, and H$\alpha$ and $I$ images of a location in the northeast emission filaments obtained in February 1987 using the ESO/MPI 2.2m telescope at La Silla, as described by [@Keel89]
{width="130.mm"}
Coordinates were derived using the astrometry.net Web service [@Lang], automatically matching field stars to coordinate catalogs. The emission-line features span an extreme length of 26 (460 pc) in projection; J2000 coordinates are given in Table \[tbl-coords\]. This region lies 12 kpc in projection from the core of Centaurus A.
[lcc]{} Brightest & 13 24 35.224 & -43 09 09.5\
North & 13 24 35.186 & -43 09 06.5\
East & 13 24 35.636 & -43 09 06.9\
Northeast & 13 24 36.192 & -43 08 52.2\
& 13 24 41.371 & -43 07 29.83\
& 13 24 44.032 & -43 07 03.7\
Optical spectroscopy
--------------------
Data cubes in both blue and red grating settings were obtained using the WiFeS integral-field spectrograph [@Dopita2007] at the 2.3m ANU Advanced Technology Telescope at Siding Spring. The field of view spans $25 \times 38$, with 1 sampling. Simultaneous exposures in the blue (3500-5700 Å) and red (5400-7000 Å) ranges were obtained on 4 March 2016, for 900 seconds. The field covered the three southern discrete components, as well as more diffuse emission to their northeast, and the southern edge of the NE component.
Strong, narrow emission lines appear; the mean heliocentric radial velocity is $cz=773 \pm 6$ km s$^{-1}$ (internal error) for the brightest region, and $759 \pm 21$ km s$^{-1}$ for the fainter one just to its north. These compare to the consensus systemic velocity 547 km s$^{-1}$ from NED. All lines are narrow, close to the instrumental resolution; measured FWHM values range from 1.3-1.8 Å. These observations used the B3000/R7000 grating combination, with nominal resolutions 1.7 Å in the blue and 0.9 Å at H$\alpha$.
The associated continuum is quite faint; even summed over all spatial pixels in the brightest knot, the continuum S/N is only 0.9 per 0.77-Å pixel at 5000 Å and 1.2 per 0.44-Å pixel near H$\alpha$. This limits what we can learn about associated starlight. In measuring the Balmer emission lines, we therefore consider the full range of plausible corrections for absorption in young stellar populations. While H$\alpha$ absorption in old populations is weak, with equivalent widths near 2 Å it can be as strong as 12 Å in type A stars [@JHC]. Corresponding values for H$\beta$ are 4–16 Å . Since the line emission has large equivalent widths, we include the ranges in these corrections in our uncertainties on Balmer decrement, other ratios involving these Balmer lines, and reddening. Error contributions from noise were evaluated from empty continuum regions near various emission lines. We follow [@Santoro2016] in converting from Balmer decrements to reddening values and H$\alpha$ attenuation, which include both foreground Milky Way and internal contributions. Although the Milky Way absorption is significant in the direction, with $A(\rm H \alpha ) = 0.25$ magnitude from the results of [@SchlaflyFinkbeiner], internal attenuation is dominant in each of the three regions where we can measure the Balmer decrement.
Discussion
==========
Relation to cold dust cloud
---------------------------
Fig. \[fig-IRoverlay\] overlays the H$\alpha$ on-band image with contours of a [*Spitzer*]{} $24\mu$m MIPS observation described by [@Brookes]; [@Auld] showed that the cloud was clearly detected at this wavelength, where angular resolution is better than the longer-wavelength data from either [*Spitzer*]{} or [*Herschel*]{}. The two emission-line peaks are closely associated with the 24$\mu$m peak locations. There are no similar H$\alpha$ or 24$\mu$m features within $\approx 300$, leaving little doubt that these are associated physically rather than only along the line of sight.
![Narrowband H$\alpha$ image as in Fig. \[fig-halphamontage\], overlaid with contours from the 24$\mu$m [*Spitzer*]{} MIPS observation. Contours are spaced at intervals 0.10 MJy sr$^{-1}$, spaced sparsely to show the H$\alpha$ peaks. The area shown is $120 \times 150$ with north at the top.[]{data-label="fig-IRoverlay"}](Halpha24mv2crop.eps){width="60.mm"}
Ionizing sources and star formation
-----------------------------------
To evaluate likely ionization mechanisms, we measured emission-line ratios in four spatial regions (Fig. \[fig-regions\]), fitting Gaussian profiles and linear baselines, with results given in Table \[tbl-lineratios\]. These regions, selected by position and surface brightness, differ significantly in Balmer decrement H$\alpha$/H$\beta$.
![Monochromatic H$\alpha$ image summed across the line’s emission profile from the WiFeS data cube, spanning $24 \times 36$ with north at the top. The regions summed in Table \[tbl-lineratios\] are indicated.[]{data-label="fig-regions"}](halpharegions.eps){width="80.mm"}
All the \[S II\] line ratio values cluster near the low-density limit of I($\lambda 6717$)/I($\lambda 6731$)=1.43 (mean of all values $1.49 \pm 0.06$).
[lcccc]{} F(H$\alpha$) (erg cm$^{-2}$ s$^{-1}$) & $9.3 \times 10^{-15}$ & $1.3 \times 10^{-15}$ & $1.1 \times 10^{-15}$ & $1.0 \times 10^{-15}$\
H$\alpha$ EW (Å ) & $156 \pm 6$ & $319 \pm 10$ & $58 \pm 9$ & $780 \pm 300$\
\[N II\] $\lambda 6583$/H$\alpha$ & $0.28 \pm 0.04$ & $0.36 \pm 0.06$ & $0.40 \pm 0.13$ & $0.44 \pm 0.18$\
\[S II\] $\lambda 6717/ \lambda 6731$ & $1.53 \pm 0.13$ & $1.52 \pm 0.28$ & $1.52 \pm 0.18$ & $2.0 \pm 0.3$\
\[S II\] $\lambda 6717+6731$/H$\alpha$ & $0.18 \pm 0.02$ & $0.27 \pm 0.07$ & $0.49 \pm 0.10$ & $0.25 \pm 0.07$\
\[O I\] $\lambda 6300$/H$\alpha$ & $0.025 \pm 0.006$ & $0.018 \pm 0.002$ & ... & ...\
\[O III\] $\lambda 5007$/H$\beta$ & $0.69 \pm 0.10$ & $0.60 \pm 0.30$ & $0.47 \pm 0.14 $ & ...\
\[O II\] $\lambda 3726+3729$/\[O III\] $\lambda 5007$ & $1.6 \pm 0.5$ & $< 1.0$ & $4.3 \pm 1.3$ & ...\
H$\alpha$/H$\beta$ & $7.0 \pm 0.8$ & $10.8 \pm 2.7$ & $5.1 \pm 0.9$ & ...\
E$_{B-V}$ & $0.84 \pm 0.11$ & $1.25 \pm 0.23$ & $0.54 \pm 0.1$ & ...\
A(H$\alpha$) & $2.10 \pm 0.18$ & $3.14 \pm 0.51$ & $1.35 \pm 0.40$ & ...\
L(H$\alpha$) (erg s$^{-1}$) & $1.1\times 10^{38}$ & $3.8 \times 10^{37}$ & $5.7 \times 10^{36}$ & $ > 2.0 \times 10^{36}$\
The location of the regions in the BPT line-ratio diagrams [@BPT], using the revised dividing curves from [@Kewley], classifies all of them as photoionized by hot stars (Fig. \[fig-BPTplots\]). This fits with the narrow line widths measured from the WiFeS data, $\leq 50$ km s$^{-1}$, indicating that shocks fast enough to add significantly to the ionization levels do not add significantly to the overall ionization level.
These emission regions coincide spatially with the cold dust cloud seen in [*Herschel*]{} data by [@Auld], who consider limits on the star formation set from the dust temperature and (lack of) associated UV sources. Their highest allowed SFR is set from the 24$\mu$m flux, 0.00012 M$_\odot$ year$^{-1}$. The GALEX-based near- and far-UV limits are much lower ($<2 \times 10^{-4}$ and $< 5 \times 10^{-5}$ M$\odot$ year$^{-1}$ respectively), indicating that any associated population of young stars must be obscured from our point of view.
As a star-forming region, the line emission we detect suggests that this object is modest in scale, with an H$\alpha$ luminosity close to $10^{38}$ ergs s$^{-1}$. This is a few times greater than that of the Orion Nebula M42, requiring only a few ionizing stars ($< 10$). To compare with star-formation rates in other environments, we follow [@Kennicutt2007], using estimating SFR from a linear combination of H$\alpha$ and 24$\mu$m luminosities, as calibrated for disk H II regions in M51. For the entire southwest complex in Centaurus A, this conversion gives a modest increase in effective H$\alpha$ luminosity, and a total SFR $1.3 \times 10^{-3}$ M$_\odot$ year$^{-1}$(for a Salpeter initial-mass function). This SFR is an order of magnitude greater than the 24$\mu$m limit from [@Auld], and correspondingly larger than their UV limits. This difference makes sense if the ionizing stars (presumably in clusters) are largely obscured along our line of sight, but the 24$\mu$m flux is still interestingly low to be associated with even a few ionizing stars. One can, for example, picture a geometry in which the dust blocks optical and UV light over a small solid angle about the stars, but still hides them from our direction.
For individual star-forming regions, the relation between long-term star-formation rate and tracers of massive stars, such as H$\alpha$, has a strong stochastic element. For example, using the stellar-atmosphere results from [@Vacca] and [@Sternberg] as in [@GildePaz] shows that the expected H$\alpha$ luminosities from nebulae completely encompassing the star range from $2 \times 10^{36}$ erg s$^{-1}$ at spectral type B0 to $10^{38}$ erg s$^{-1}$ at O3. The entire ionizing flux in this complex could be provided by the equivalent of 9 O7 stars, so small-number statistics would change the emission-line output strongly as individual stars are formed and evolve. Even so, there may be more to learn; the limits on IR emission from [@Auld] are quite low in comparison to the SFR rate inferred from H$\alpha$ emission.
The associated dust cloud matches one of the two regions in the H I “shells" of Centaurus A where [@Charmandaris] detected CO emission, implying a typical H$_2$/H I ratio near unity and consistent with conditions for star formation in the inner regions of spirals. With an estimated H$_2$ mass of $2 \times 10^7$ M$_\odot$ and linear scale $\approx 0.5$ kpc, this would be either an exceptionally large giant molecular cloud or a collection of more usual clouds (noting that geometrically it may not be easy to distinguish these cases). Under these conditions, it would be common for parts of a “blister" H II region to be highly obscured in the optical unless viewed face-on, which fits with the low UV limits on radiation from young stars.
In H$\alpha$ luminosity and Balmer decrement, these star forming regions are similar to those found embedded in the northeastern filaments (in what is sometimes known as the “necklace" structure) by [@Santoro2016]. In comparison with the [@Santoro2016] regions, these are comparable in scale and observed luminosity, although perhaps more luminous when dereddened (Fig. \[fig-Northeast-Southwest\]). We might speculate that in both cases, interaction with a central outflow [@Neff2015] has compressed ambient H I to trigger star formation, but that the southwestern structure lacks the additional ingredients of a direct view of the AGN and a collimated radio jet which enhance both effects in the gas kinematics and ionization on the “jet" side.
![Comparison of the newly observed star-forming regions (right panel) with the H II regions A and B found in the northeastern emission-line jet by [@Santoro2016] (left). The additional filamentary features on the northwest side are ionized by the AGN, either photoionized or via interaction with the radio jet. Both panels show H$\alpha$ emission and are at the same angular and intensity scales. The jet region was imaged using the ESO/MPI 2.2m telescope as described by [@Keel89]. []{data-label="fig-Northeast-Southwest"}](NortheastSouthwest.eps){width="60.mm"}
Summary
=======
We have described a set of optical emission-line regions found 12 kpc to the southwest of the nucleus of Centaurus A, closely coincident with dust and gas structures previously reported. The emission-line ratios as well as UV and FIR properties are well accommodated as a set of normal H II regions, photoionized by only a few OB stars.
Even in a system as disturbed as Centaurus A, this is a distant place to find apparently normal H II regions. Fig. \[fig-halphanucblob\] shows the location superimposed on an H$\alpha$ image of the inner regions, where the rich distribution of star-forming regions is strongly confined to the warped remnant disk, within a radius of 212 (3.8 kpc). [@Charmandaris] suggest that an outflow from the central regions of the system has played a role in compressing gas (possibly concentrated dynamically in a way similar to the stellar shells) so as to trigger such distant star formation. On this basis, we might speculate that star formation can be triggered by a broad outflow (driven either by the AGN or strong star formation in the inner disk), but the southwest region lacks the additional factors producing the rich optical emission along the north jet. This difference could mean that there is no collimated radio jet on this side, that the southwestern cloud of gas and dust is not in the right place to be ionized by AGN radiation, or that the AGN radiation does not escape effectively on this side.
{width="130.mm"}
The IFU spectroscopy was facilitated by a Twitter interaction between two of the authors. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration This research has made use of NASA’s Astrophysics Data System Bibliographic Services. Support for AMM is provided by NASA through Hubble Fellowship grant HST-HF2-51377 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofisica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatorio Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autonoma de Mexico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
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[^1]: As this study was in progress, we found that the emission feature appears in the long-exposure, very deep composite image presented by Rolf Olsen at http://www.rolfolsenastrophotography.com/Astrophotography/Centaurus-A-Extreme-Deep-Field/ and is described in his text as possibly related to the jet: “A corresponding faint trace of nebulosity, likely related to the otherwise invisible Southern jet, is also noticeable as a small red smudge on the opposite side of the galaxy core".
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abstract: 'We suggest that orbifold field theories which are obtained from non-commutative [${\cal N}=4\ $]{}SYM are UV finite. In particular, non-supersymmetric orbifold truncations might be finite even at finite values of $N_c$.'
---
[**A Note on Non-Commutative Orbifold Field Theories** ]{}
0.5cm
Adi Armoni
0.5cm
Centre de Physique Th[é]{}orique de l’[É]{}cole Polytechnique
91128 Palaiseau Cedex, France
0.5cm
[email protected]
0.5cm
Introduction
============
Recently non-commutative Yang-Mills theory [@con] attracted a lot of attention, mainly due to discoveries of new connections to string theory[@CDS; @DH; @SW]. In a recent paper[@BS], it was suggested that the divergences of the non-commutative Yang-Mills theory are dictated by the large $N_c$ limit of the theory. Namely, that divergences occur only in planar diagrams (the observation that the planar commutative and non-commutative theories are the same up to phases in the Green functions, was already made in[@filk]. A careful analysis of the divergences was carried out in ref.[@KW]).
Another direction of research is the study of orbifold field theories - motivated by the AdS/CFT correspondence [@Mal]. It was conjectured by Kachru and Silverstein [@KS] that orbifolds of $AdS_5 \times S^5$ which act on the $S^5$ part define a large $N_c$ finite theory, even when the R-symmetry is completely broken and the theory is not supersymmetric. This conjecture was later proved using both field theory [@BJ] and string theory [@BKV] techniques.
In this note we would like to consider non-commutative Yang-Mills theories which are obtained by an orbifold truncation of [${\cal N}=4\ $]{}SYM. We suggest that these theories are UV finite, namely that there are no divergent Feynman diagrams, even when the theory under consideration is non-supersymmetric and the number of colors is finite. Note, however, that our conjecture relies on recent suggestions [@BS; @MSV] about UV finiteness of non-planar graphs in the non-commutative theory. The later were not fully proved yet, but seems to be necessary if the non-commutative theory is renormalizable.
Orbifold field theories
=======================
Orbifold field theories are obtained by a certain truncation of a supersymmetric Yang-Mills theory. Let us consider the special case of [${\cal N}=4\ $]{}. The truncation procedure is as follows: consider a discrete subgroup $\Gamma$ of the [${\cal N}=4\ $]{}R-symmetry group $SU(4)$. For each element of the orbifold group, a representation $\gamma$ inside $SU(|\Gamma |N_c)$ should be specified. Each field $\Phi$ transform as $\Phi \rightarrow r \gamma ^\dagger \Phi \gamma$, where $r$ is a representation matrix inside the R-symmetry group. The truncation is achieved by keeping invariant fields. The resulting theory has a reduced amount of supersymmetry, or no supersymmetry at all. It was conjectured[@KS], based on the AdS/CFT conjecture, that the truncated large $N_c$ theories are finite as the parent [${\cal N}=4\ $]{}theory. Later it was proved [@BJ] that the planar diagrams of the truncated theory and parent theories are identical. In particular it means that the perturbative beta function of the large $N_c$ daughter theory is zero and that the theory is finite.
In the cases of [${\cal N}=2\ $]{}truncations, there is only one-loop (perturbative) contribution to the beta function. Its vanishing indicates the perturbative finiteness of the daughter theory at finite $N_c$ as well.
In [${\cal N}=1\ $]{}truncations the situation is more subtle. Indeed, the theory is finite at finite values of $N_c$, but the finiteness is due to Leigh-Strassler type of arguments[@LS]. In that case one should consider the $SU(N_c)$ version of the theory (and not the $U(N_c)$ theory which is obtained from the string theory orbifold), since the $U(1)$ beta function is always positive at the origin. In addition ${1\over
N_c^2}$ shifts of the Yukawa couplings are needed[@OT].
In the non-supersymmetric case there are no known examples of finite theories at finite $N_c$, though attempts in this direction using orbifolds were recently made[@FS]. As we shall see, due to non-commutativity such examples can be found.
Perturbative behavior of non-commutative Yang-Mills theories
============================================================
Recently, several authors[@KW; @BS] analyzed the renormalization behavior of non-commutative Yang-Mills theories (for an earlier discussion see[@filk]. Related works are [@MS; @SJ]). We briefly review their analysis. The Feynman rules of the commutative and non-commutative theories in momentum space are very similar. In fact the only difference is that each vertex of the non-commutative theory acquires a phase, $\exp{i \Sigma _{i<j} k_i
\wedge k_j}$ (the Moyal phase), with respect to the vertex of ordinary commutative theory[@filk]. For planar diagrams, this phase cancels at internal loops and the only remnant is an overall phase. Therefore the planar commutative and non-commutative theories are similar, in accordance with recent findings[@HI; @MR; @LW]. In particular, thermodynamical quantities such as the free energy and the entropy are not affected by non-commutativity at the planar limit [@MR].
Another claim[@BS; @KW] is that the oscillations of the Moyal phases at high momentum would regulate non-planar diagrams, namely that UV divergences of non-planar diagrams would disappear. There are two exceptional cases in which non-planar diagrams will still diverge[@KW]: (i). When the non-planar diagrams consist of planar sub-diagrams which might diverge. (ii). For specific values of momentum (a zero measure set) the Moyal phase can be zero. Indeed, it was shown lately [@MSV] that in theories which contain scalars - there are non-planar divergences. The theories that we will consider contain scalars and therefore contain infinities, however [@MSV] interpret these divergences as [*Infra-Red*]{} divergences. The reason is as follows: The integrals which are associated with non-planar graphs converge unless the Moyal phase is set to zero by a vanishing incoming momentum. In this case new type of divergences appear, and it seems that infinite number of counterterms are needed. Therefore the non-commutative theory seems to be non-renormalizable. However, the authors of ref.[@MSV] suggest that the re-summation of these divergent non-planar graphs would yield a finite result. Their observation is based on the similarity between the present case and the standard (commutative) IR divergences. Note that Infra-Red divergences are, anyways, expected in theories with massless particles. Thus, ’truly’ divergences in the non-commutative theory occur only in planar graphs.
Finiteness of non-commutative orbifold field theories
=====================================================
Let us now consider an orbifold truncation of non-commutative [${\cal N}=4\ $]{}SYM. These theories can be defined perturbatively by a set of Feynman rules. The natural definition would be to attach to each vertex the corresponding Moyal phase[@filk]. According to ref.[@BS], the only potential divergences are the ones which arise in planar diagrams. Non-planar diagrams are expected to be finite, except for some “accidental” divergences [@BS]. Note that the infinities which do occur in these graphs are associated with the Infra-Red [@MSV] and therefore do not contradict our claim that orbifold field theories are UV finite.
According to the analysis of [@BJ], the planar diagrams of an orbifold theory can be evaluated by using the corresponding diagrams of the parent non-commutative [${\cal N}=4\ $]{}. These diagrams are finite since they differ from the commutative [${\cal N}=4\ $]{}only by an overall phase. In this way sub-divergences of non-planar diagrams will also be canceled. We therefore conclude that any orbifold truncation of non-commutative non-compact [${\cal N}=4\ $]{}SYM is finite. In particular, it means that we might have a rich class of non-supersymmetric gauge theories which are finite, even at finite $N_c$ (in contrast to ordinary non-supersymmetric orbifold field theories, where the two loop beta function is generically non-zero). Note that though these theories might be finite, they are certainly not conformal.
Let us consider a specific example[@KT]. The example is an $SU(N) \times SU(N)$ gauge theory with 6 scalars in the adjoint of each of the gauge groups and 4 Weyl fermions in the $(N,\bar N)$ and 4 Weyl fermions in the $(\bar N,N)$ bifundamental representations. It is the theory which lives on dyonic D3 branes of type 0 string theory and can be also understood, from the field theory point of view, as a $Z_2$ orbifold projection of [${\cal N}=4\ $]{}SYM[@NS]. The large $N_c$ commutative theory contains a line of fixed points. We suggest that its analogous non-commutative (finite $N_c$) theory is finite at that line. Namely, at $g_1=g_2=h_1=h_2$, where $g_1,g_2$ are the gauge couplings and $h_1,h_2$ are the Yukawa couplings. Note that in contrast to the commutative case the position of the line of finite theories is [*not*]{} corrected by ${1\over N_c}$ contributions.
Finally, it might be interesting to understand how the finiteness of these theories arise from string theory orbifolds.
I would like to thank O. Aharony, N. Itzhaki and B. Kol for useful discussions and comments. This work was supported in part by EEC TMR contract ERBFMRX-CT96-0090.
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abstract: 'Here, we present our on-going work on the determination of stellar parameters of giants in the Galactic Disks and Bulge observed with UVES on the VLT. We present some preliminarily results.'
date: '?? and in revised form ??'
---
Our project aims at discerning whether the Galactic bulge and the thick disk are evolutionarily connected, by differentially studying their respective chemical evolution. Here, we present our initial step dealing with the detailed determination of the fundamental parameters of our target stars: metal-rich ($-0.5<\mathrm{[Fe/H]}<-0.1$), late-type ($4000\, \mathrm K <
\mathrm{T} < 5000\, \mathrm K$) giants. This step is important for the subsequent step of accurately determining the abundances from these rich spectra.
One advantage of our project is that it is a strictly differential analysis of spectra, with homogeneously determined stellar parameters. Thus, the stars will be analysed in an identical way in order to firmly establish the picture of the abundance evolution of the two populations in relation to each other. We will analyse standard stars, thick disk and bulge stars.\
Here we present the analysis of a sub-sample of our stars: 8 thick-disk and 8 standard stars. We are currently working on more thick disk and bulge giants. The spectra were recorded with UVES at S/N$=75$ and $R=60\,000$, and were reduced with REDUCE (Piskunov & Valenti, 2002). The Fe line measurements were done in IRAF and the line synthesis and analysis was done in the MARCS model-atmosphere environment. Our analysis is based on the discussions in Fulbright et al. (2006), who analysed bulge giants.
Equivalent widths for more than 120 Fe [i]{} absorptions lines in the middle and red parts of the spectra were measured. To be able to determine the surface gravities also eight Fe [ii]{} lines were analysed. By requiring that the iron abundance should be independent of the excitation potential, we modified the temperature of the models (increments of 25 K). The $\log g$ was estimated by studying the difference in the abundance for Fe [i]{} and Fe [ii]{}. Finally, diagrams with abundance vs $\log W/\lambda$ for Fe [i]{} were constructed to determine the microturbulence velocity. When these parameters are defined it is possible to assign to the star its metallicity. For the standard stars, it is possible to get accurate starting values from photometry for $T_{\mathrm{eff}}$ and $\log g$ and then iterate and find the spectroscopically based parameters. For the thick disk stars more iterations were required.\
The photometrically and spectroscopically determined parameters for the standard stars agree well, which is reassuring for our analysis of the thick disk sample. Finally, we compared the spectroscopically derived iron abundances for the thick disk stars with iron abundances derived from DDO photometry using the calibration by Holmberg & Flynn (2004). The agreement is in general good. Table 1 summarizes our results.
Our full project includes three sets of stars, local standards, thick disk stars, and stars in the Galactic bulge. The aim with the current study is to investigate how well we can trust different methods for derivation of the stellar parameters. For the Bulge stars we will have significantly less good information as compared to our data for the local standard stars. Differential reddening will be one of the greatest problems there. With the current results it does appear that $\log g$ can be derived using excitation balance in iron (see Table 1 for the standard stars). This is in agreement with what was found by Kraft & Ivans (2003).
We will also further investigate possible departures from the assumption of LTE in the derivation of the iron abundances in such cool giant stars. For this we plan to use e.g. the methods and model by Collet et al. (2005)
------------------- -------------------- -------- -------------------- ---------- ----------------------- ------------------------ ---------- -------------------
Star $T_{\mathrm{eff}}$ \# of $T_{\mathrm{eff}}$ $\log g$ $\log g$ $\xi_{\mathrm{micro}}$ \[Fe/H\] \[Fe/H\]
$\mathrm{(phot)}$ colors (spec) (Hipp) ($\mathrm{FeII/FeI}$) \[kms$^{-1}$\] (spec) $\mathrm{(phot)}$
Standard stars:
HD113226 4974 (22) 3 4961 2.80 2.80 1.5 0.07
HD123139 4753 (51) 1 4740 2.76 2.50 1.5 -0.15
HD124897 4240 (37) 7 4270 1.93 1.35 1.7 -0.62
HD138716 4743 (43) 4 4683 3.02 3.02 1.1 -0.14
HD139663 4233 (15) 2 4377 1.83 1.83 1.7 0.03
HD161096 4503 (15) 2 4534 2.48 2.46 1.5 0.16
HD171443 4223 (51) 1 4276 1.85 1.55 1.5 -0.12
HD175190 4176 (21) 6 4259 1.00 1.50 1.6 -0.22
Thick Disk stars:
HD 1378 4007 (51) 1 4148 1.00 1.5 -0.10 -0.13
HD 2763 4169 (78) 3 4246 1.50 1.5 -0.23 -0.22
HD 3356 4142 (16) 4 4272 1.25 1.5 -0.50 -0.35
HD 3524 4016 (51) 1 4029 1.00 1.3 -0.34 -0.47
HD 3709 4254 (9) 4 4327 1.75 1.6 -0.16 -0.05
HD 4303 4575 (22) 6 4625 1.75 1.5 -0.20 -0.14
HD 4955 4043 (51) 1 4135 1.00 1.5 -0.38 -0.40
HD 8349 4029 (51) 1 4154 1.25 1.4 -0.20 -0.27
------------------- -------------------- -------- -------------------- ---------- ----------------------- ------------------------ ---------- -------------------
: Preliminary fundamental parameters of a sub-sample of our programme stars[]{data-label="tab:kd"}
We would like to thank Kjell Eriksson for useful discussions.
2005, *A&A* 442, 643
2006, *ApJ* 636, 821
2004, *MNRAS* 352, 440 2003, *PASP* 115, 143
2002, *A&A* 385, 1095
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abstract: 'Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between $P_3$, the probability that all three vertices in the triangle connect, and $P_0$, the probability that none connect. A linear approximation for $P_3(P_0)$ is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for $P_3(P_0)$ gives thresholds for the kagomé, site-bond honeycomb, (3-12$^2$) lattice, and “stack-of-triangle" lattices that compare favorably with numerical results.'
author:
- 'Robert M. Ziff'
- Hang Gu
bibliography:
- 'SiriusZiffRapid.bib'
title: 'Universal condition for critical percolation thresholds of kagomé-like lattices'
---
Percolation is the study of long-range connectivity in random systems. The value of the site or bond occupation probability where that connectivity first appears is percolation threshold $p_c$ [@StaufferAharony94]. Finding exact and approximate $p_c$’s for percolating systems on various lattices is a long-standing problem that continues to receive much attention today (e.g., [@Kondrat08; @RiordanWalters07; @ScullardZiff06; @ZiffScullard06; @Ziff06; @Scullard06; @ScullardZiff08; @Parviainen07; @QuintanillaZiff07; @NeherMeckeWagner08; @JohnerGrimaldiBalbergRyser08; @Ambrozic08; @FengDengBlote08; @Wu06; @MajewskiMalarz07; @WagnerBalbergKlein06; @TarasevichCherkasova07; @MayWierman05; @HajiakbariZiff08]).
![$\triangle$-$\triangle$ duality for lattices in simple triangular array. (left) Shaded triangles represent any collection of internal bonds. (right) Result of $\triangle$-$\triangle$ transformation where blue (dark) triangles are the dual triangles, and form the same arrangement as on the left but rotated $180^\circ$.[]{data-label="tritri"}](tritriFig1.eps){width="0.6\hsize"}
All known exact $p_c$’s are for two-dimensional lattices that can be represented as arrays of triangular units self-dual in the triangle-triangle ($\triangle$-$\triangle$) transformation, as illustrated in Fig. \[tritri\] for the case of a simple triangular array. When this duality is satisfied, $p_c$ is determined by the simple condition [@Ziff06; @ChayesLei06] $$P_3' = P_0' \ ,
\label{P30}$$ where $P_3'$ is the probability that all three vertices of the triangular unit connect, $P_0'$ is the probability that none connect, and the prime indicates a $\triangle$-$\triangle$-dual system. The shaded triangular units can contain any collection of bonds, including correlated bonds which can mimic site percolation, connecting the three vertices.
If, for example, the triangular unit is simply a triangle of three bonds, each occupied with probability $p$, then $P_0' = q^3$ and $P_3' = p^3 + 3 p^2 q$, where $q = 1 - p$, and (\[P30\]) yields the bond criticality condition for the triangular lattice as $q^3 = p^3 + 3 p^2 q$ which has the solution $p_c = 2 \sin \pi/18 = 0.34729636$ [@SykesEssam64]. Likewise, taking a star of three bonds as the basic unit gives $P_0' = q^3 + 3 q^2 p$ and $P_3' = p^3$, and (\[P30\]) yields $ q^3 + 3 q^2 p = p^3$ or $p_c = 1 - 2 \sin \pi/18 = 0.65270365$ for the honeycomb lattice [@SykesEssam64]. Eq. (\[P30\]) has been applied to many other lattices that satisfy $\triangle$-$\triangle$ duality, including “martini" [@Scullard06; @Ziff06; @Wu06], bowtie [@Wierman84; @ZiffScullard06], and “stack-of-triangle" [@HajiakbariZiff08] lattices, to find exact $p_c$’s.
However, when $\triangle$-$\triangle$ duality is not satisfied, then Eq. (\[P30\]) cannot be used to find $p_c$. For example, the $\triangle$-$\triangle$ transformation for the kagomé lattice is shown in Fig. \[kagdual\], and it can be seen that, while the lattice can be broken up into non-touching shaded triangular units, the $\triangle$-$\triangle$ transformation gives a different lattice altogether, and so the self-duality condition is not satisfied. Likewise, site percolation on the honeycomb lattice, which can be represented as bond percolation on the kagomé lattice with all three bonds correlated (see Fig. \[honeycombfig\]), is also non-self-dual.
![(left) Shaded triangles in the generalized kagomé configuration. (right) Result of $\triangle$-$\triangle$ transformation, showing that this system is not self dual.[]{data-label="kagdual"}](kagdualFig2.eps){width="0.5\hsize"}
Nevertheless, for any system that can be broken up into identical disjoint isotropic triangular units, $p_c$ must be determined by a unique condition that depends only upon the connections probabilities $P_0$ and $P_3$ of the triangular units. In this paper we consider lattices of the kagomé form, as shown in \[fourfigs\](d), and investigate the corresponding relation between $P_3$ and $P_0$. The kagomé form includes several unsolved lattices of interest as discussed below. While we can’t find exact thresholds for these lattices (indeed, they are likely insolvable), we can make very precise predictions on their values and unify their study.
{width="0.35\hsize"} {width="0.35\hsize"}
First we consider the “double honeycomb" lattice, shown in Fig. \[fourfigs\](b), which is of the kagomé form and is the one exactly soluble lattice of this form. It can be constructed by replacing each bond of a honeycomb lattice (Fig. \[fourfigs\](a)) by two bonds in series, which implies that its $p_c$ is the square root of the $p_c$ for the honeycomb lattice: $$p^\star = \sqrt{1 - 2 \sin \pi /18} = 0.80790076$$ For this lattice, which we indicate by a star, we have $$\begin{aligned}
P_0^* &=& {q^\star}^3 + 3 {q^\star}^2 p^\star = 0.09652861 \label{P0star} \\
P_3^* &=& {p^\star}^3 = 0.52731977
\label{P3star}\end{aligned}$$ where $q^\star = 1 - p^\star$. Note, Eq. (\[P30\]) is far from being satisfied.
![Steps in the derivation of the linear relation Eq. (\[linear\]): (a) the honeycomb lattice, (b) double-honeycomb forms a kagomé class of lattice, (c) all up-stars replaced by triangular units, forming martini configuration satisfying $\triangle-\triangle$ duality, (d) remaining stars replaced by triangular units, forming the kagomé configuration.[]{data-label="fourfigs"}](fourfigsFig4.eps){width="0.8\hsize"}
Next, generalizing the considerations in [@ScullardZiff06], we develop an approximate linear relation between $P_3$ and $P_0$ for all lattices of the kagomé form, that is exact at the point $(P_0^*$, $P_3^*)$. Consider the systems shown in Fig. \[fourfigs\]. In (c) we replace all the up-stars of (b) with general shaded triangular units with a given net connectivity $P_0$ and $P_3$. This produces a generalized “martini" configuration, which falls under the general triangular class of Fig. \[tritri\], with connectivities (as follows from the diagram in (c)): $$\begin{aligned}
P_0' &=& P_0 +3 P_2 ({q^\star}^2 + 2 q^\star p^\star) + P_3 ({q^\star}^3 + 3 {q^\star}^2 p^\star) \nonumber \\
P_3' &=& P_3 {p^\star}^3 \end{aligned}$$ Eq. (\[P30\]) then yields the exact criticality condition for system (c): $$P_3 = P_3^* + b (P_0 - P_0^*)
\label{linear}$$ where $b = 1/(2 - p^\star) = 0.83885634$. As a final step, we hypothesize that Eq. (\[linear\]) represents an approximation to $p_c$ of the “full" kagomé system with both up and down triangles shown in Fig. \[fourfigs\](d). The justification is that in going from (a) to (b), we replaced one set of stars by shaded triangles satisfying (\[linear\]), and the system remained at criticality. Now we replace the second identical set of stars by the same shaded triangles, and we expect that the system remains close to criticality.
![Plot of $P_3 - [P_3^* + b (P_0 - P_0^\star)]$ vs. $P_0 - P_0^*$ using data of Table \[table:P0P2P3\], showing deviations from Eq. (\[linear\]). Points are numerical data, and the curve is a plot of Eq. (\[cubic\]). The locations of some specific systems are also shown.[]{data-label="siriusdifference"}](siriusdifferenceFig5.eps){width="1\hsize"}
system $p_c$(linear)$^a$ $p_c$(cubic)$^b$ $p_c$(numerical) $P_0$ $P_2$ $P_3$
------------------ -------------------- ------------------ -------------------- ------------ ------------ ------------
double honeycomb 0.80790076 — — 0.09652861 0.12538387 0.52731977
$(3,12^2)$ 0.74042118$^c$ 0.74042081 0.74042195(80)$^e$ 0.10045606 0.12297685 0.53061341
kagomé 0.52440877$^{c,d}$ 0.52440516 0.52440499(2)$^f$ 0.10757501 0.11861544 0.53657867
honeycomb (site) 0.69891402 0.69702981 0.69704024(4)$^f$ 0.30297019 0 0.69702981
$\infty$ subnet — — 0.628961(2)$^g$ 0.09652861 0.12538387 0.52731977
subnet 4 0.62536437 0.62536431 0.625365(3)$^g$ 0.09823481 0.12433811 0.52875085
subnet 3 0.61933204 0.61933180 0.6193296(10)$^g$ 0.10016607 0.12315455 0.53037028
subnet 2 0.60086322 0.60086202 0.6008624(10)$^g$ 0.10402522 0.12078995 0.53360494
In Table \[table:2\] we compare the predictions of the linear relation (\[linear\]) with the numerical results for several systems. The $p_c$(linear) estimates are found by putting the corresponding expressions for $P_0$ and $P_3$ into Eq. (\[linear\]) and solving numerically for $p$. For the kagomé lattice, we use $$P_0 = q^3, \qquad
P_3 = p^3+3p^2 q \ .
\label{kagP}$$ For the $(3,12^2)$-lattice (shown for example in Ref. [@ScullardZiff06]) we use$$\begin{aligned}
P_0 &=&1 - 3 p^2 - 3 p^3 +6 p^{7/2} +3 p^4 -4 p^{9/2} \cr
P_3 &=& 3 p^{7/2} - 2 p^{9/2} \ .\end{aligned}$$ For site percolation on the honeycomb lattice, $p_c = P_3 = 1 - P_0$, and Eq. (\[linear\]) yields explicitly $p_c = 1/[{p^\star}^2(3 - p^\star)] = 0.69891402$. The agreement between $p_c$(linear) and numerical results is especially good for systems where $P_0$ is near $P_0^\star$.
To test the behavior of $P_3(P_0)$ over a more complete range of values, we carried out new simulations using the gradient percolation method [@RossoGouyetSapoval85; @ZiffSapoval86] on a general kagomé systems. We fixed $P_0 = 0, 0.5, 0.1, 0.15,$ and $0.25$ and allowed $P_3$ to vary linearly in the vertical direction, with the estimate of the critical value found as the fraction of $P_3$-triangles in the frontier. We considered systems of different gradients and extrapolated the estimates to infinity to find the values of $P_3$ given in Table \[table:P0P2P3\].
In Fig. \[siriusdifference\] we plot the difference between the measured $P_3$ and the predictions of Eq. (\[linear\]) as a function of $P_0$ for these systems. The first derivative at $P_0 = P_0^*$ appears to be zero, which would imply that Eq. (\[linear\]) represents the *exact* linear term in the behavior of $P_3$ vs. $P_0 - P_0^*$. The numerical data also suggests that (\[linear\]) gives an upper bound for $P_3(P_0)$ for all $P_0$. Fitting the data to a cubic equation, assuming that $P_3'(P_0^\star) = b$ exactly, we find $$P_3 = P_3^* + b (P_0 - P_0^*) + c(P_0 - P_0^*)^2 + d(P_0 - P_0^*)^3
\label{cubic}$$ with $c = -0.05987$ and $d = -0.1038$. This curve fits all the data points $P_3$ within $\pm 10^{-5}$. The results of using this equation to predict $p_c$ are shown in Table \[table:2\] under the heading “cubic", and all are within the expected error of about $\pm10^{-5}$, and more accurate as $P_0$ approaches $P_0^\star$. For the kagomé case, our prediction $p_c = 0.52440516$ compares favorably to the recent precise result 0.52440499(2) of Ref. [@FengDengBlote08] (which appeared after our analysis was complete) and the previous value 0.5244053(3) [@ZiffSuding97].
![Lattices with subnets 2, 3 and 4 (left to right)[]{data-label="figr18_2"}](subnetFig6.eps){width="\hsize"}
We next apply our general relation for $P_3$ vs. $P_0$ to get very accurate $p_c$’s for a class of lattices in which each triangle of the kagomé arrangement contains a “stack-of-triangles" as shown in Fig. \[figr18\_2\]. In Ref. [@HajiakbariZiff08] the similar stack-of-triangles were studied in a regular triangular arrangement, and explicit expressions for $P_0$ and $P_3$ were found by exact enumeration for these three subnets. We can use those same expressions to analyze the subnets on the kagomé lattice as well. For subnet 2, we have [@HajiakbariZiff08] $$\begin{aligned}
P_0 &=&q^9+9 p q^8+33 p^2 q^7+54 p^3 q^6+21 p^4 q^5+3 p^5 q^4 \nonumber \\
%P_2 &=& p^2q^7+10p^3q^6+32p^4q^5+22p^5q^4+7p^6q^3+p^7q^2 \nonumber \\
P_3 &=& 9p^4q^5+57p^5q^4+63p^6q^3+33p^7q^2+9p^8q+p^9 \nonumber \\\end{aligned}$$ with $q = 1 - p$. For subnets 3 and 4, see Ref. [@HajiakbariZiff08].
We insert these expressions for $P_0$ and $P_3$ into Eqs. (\[linear\]) and (\[cubic\]) to find the linear and cubic estimates for $p_c$. The resulting values are shown in Table \[table:2\], along with results of numerical simulations. For subnets 3 and 4, the predictions of (\[linear\]) and especially (\[cubic\]) are expected to be very accurate, because $P_0$ is so close to $P_0^*$, and indeed the precision of the numerical simulations is not high enough to see the difference between these predictions and the actual values.
As seen in Table \[table:2\], the quantities $P_0$, $P_2$ and $P_3$ evidently approach the double-honeycomb values $P_0^*$, $P_2^*$ and $P_3^*$ as the mesh of the subnet gets finer. This is because the triangular units in the fine-mesh limit can be effectively represented by a star of three bonds, with the central site in this star representing the supercritical “infinite cluster" in the central region of the triangular units [@HajiakbariZiff08]. The set of these stars creates the double-honeycomb lattice, so the $P_i$ are the same as the double-honeycomb values. Furthermore, the probability $P_{\infty,\mathrm{corner}}$ of connecting from a corner to the central infinite cluster at criticality must be identical to the double-honeycomb bond threshold, $p^\star$. Thus, we can find $p_c$ for the infinite net by running simulations of growing clusters from the corner of a single large triangular system, and adjusting $p$ until $P_{\infty,\mathrm {corner}}(p)=p^\star$. This yields $p_c(\infty) = 0.628961(2)$.
$P_0$ $P_2$ $P_3$ $p_b$ $p_s$
----------- ----------- ----------- ----------- -----------
0 0.1846972 0.4459084
0.05 0.1539432 0.4881704
0.0965286 0.1253839 0.5273198 0.6527036 1
0.1 0.1232560 0.5302320 0.6583497 0.9926153
0.15 0.0926739 0.5719784 0.7405771 0.8974788
0.2 0.0622208 0.6133375 0.8242773 0.8195766
0.25 0.0319205 0.6542385 0.9091230 0.7547482
0.3029598 0 0.6970402 1 0.6970402
: \[table:P0P2P3\] Results of simulations for $P_3$ and $P_2 = (1 - P_0 - P_3)/3$ for general kagomé systems as a function of $P_0$; values are accurate to about $10^{-6}$. These data are plotted in Fig. \[siriusdifference\]. Also shown are the equivalent site-bond probabilities $p_s$ and $p_b$ calculated from Eqs. (\[pands\]). The third row is the double-honeycomb system and the final row represents site percolation on the honeycomb lattice [@FengDengBlote08].
Finally, we note that a realization of the general kagomé system for $P_0 \ge P_0^*$ is given by site-bond percolation on the honeycomb lattice, as represented in Fig. \[honeycombfig\]. For the site-bond basic unit of Fig. \[sitebondfig\], we have $$\begin{aligned}
P_0 &=& 1-p_s + p_s [(1-\sqrt{p_b})^3 + 3 (1-\sqrt{p_b})^2 \sqrt{p_b}] \nonumber \\
P_3 &=& p_s p_b^{3/2}
\label{sitebondeq}\end{aligned}$$ which can be inverted to yield: $$p_b = \left(\frac{3 P_3}{2 P_3 - P_0 + 1} \right)^2 \ , \qquad p_s = P_3/p_b^{3/2}
\label{pands}$$ In Table \[table:P0P2P3\], we list the values of $p_b$ and $p_s$ that correspond to the measured values of $P_3(P_0)$. We can also put Eq. (\[sitebondeq\]) into Eq. (\[linear\]) and simplify using Eqs. (\[P0star\]) and (\[P3star\]) to find an approximate expression for the critical line on the $p_s$–$p_b$ plane: $$p_s = \frac{{p^\star}^2}{p_b( 1 - B (\sqrt{p_b} - p^\star))}$$ where $B = p^\star/(3 - {p^\star}^2)$. We can improve upon this relation by using the cubic function of $P_3(P_0)$ given in Eq. (\[cubic\]); this adds the additional terms $C (\sqrt{p_b} - p^\star)^2 + D (\sqrt{p_b} - p^\star)^3$ to the above formula, where $C = 9 {p^\star}^2(2 - p^\star)^3/(3 - {p^\star}^2)^3 c = -0.0460682$ and $D = -0.01681$.
In conclusion, we have shown how the notion of a unique relation between $P_3$ and $P_0$, first studied in the context of self-dual systems [@Ziff06; @ChayesLei06], extends to the non-self-dual kagomé configuration. The approximate linear expression we found, Eq. (\[linear\]), appears to be exact to first order, and the simulation results shown in Fig. \[siriusdifference\] suggest that that expression provides upper bounds to $p_c$ for these systems. We conjecture that this is indeed the case. The numerically refined cubic relation of Eq. \[cubic\] allows very accurate thresholds to be predicted for a wide variety of systems, and an explicit expression for the criticality condition of site-bond percolation on the honeycomb lattice to be written.
This work was supported in part by the U. S. National Science Foundation Grant No. DMS-0553487.
|
---
abstract: 'We show that both the baryon asymmetry of the universe and dark matter (DM) can be accounted for by the dynamics of a single axion-like field. In this scenario, the observed baryon asymmetry is produced through spontaneous baryogenesis—driven by the early evolution of the axion—while its late-time coherent oscillations explain the observed DM abundance. Typically, spontaneous baryogenesis via axions is only successful in regions of parameter space where the axion is relatively heavy, rendering it highly unstable and unfit as a dark matter candidate. However, we show that a field-dependent wavefunction renormalization can arise which effectively “deforms” the axion potential, allowing for efficient generation of baryon asymmetry while maintaining a light and stable axion. Meanwhile, such deformations of the potential induce non-trivial axion dynamics, including a tracking behavior during its intermediate phase of evolution. This attractor-like dynamics dramatically reduces the sensitivity of the axion relic abundance to initial conditions and naturally suppresses DM isocurvature perturbations. Finally, we construct an explicit model realization, using a continuum-clockwork axion, and survey the details of its phenomenological viability.'
author:
- Kyu Jung Bae
- Jeff Kost
- Chang Sub Shin
bibliography:
- 'references.bib'
title: |
Deformation of Axion Potentials: Implications for Spontaneous Baryogenesis,\
Dark Matter, and Isocurvature Perturbations
---
=1
Introduction\[sec:Introduction\]
================================
A wide array of cosmological observations indicate that the universe has a significant matter-antimatter asymmetry, as quantified by the baryon-to-photon ratio [@Iocco:2008va; @Ade:2015xua] \[eq:etaobs\] \_B = (6.100.14)10\^[-10]{} , in which is the baryon-number density and $n_{\gamma}$ is the photon number density. An essential task of fundamental physics is to explain this figure in terms of microphysical processes in the early universe. Along these lines, a set of necessary conditions for the production of baryon asymmetry were obtained in a seminal paper by Sakharov [@Sakharov:1967dj]: (i) violation of baryon number ($B$) symmetry[^1], (ii) violation of the discrete $C$ and $CP$ symmetries, and (iii) departure from thermal equilibrium. Typically, satisfying these conditions is a starting point for building any model of baryogenesis. However, it is important to note that the last condition includes an implicit assumption of $CPT$ invariance. Indeed, at thermal equilibrium, $CPT$ symmetry guarantees that the energy spectra and thermal distributions of baryons and antibaryons are equal, thereby enforcing .
By contrast, dynamical scenarios can arise in which $CPT$ is violated *spontaneously*, effectively lifting this degeneracy [@Cohen:1987vi; @Cohen:1988kt]. In this way, baryon asymmetry could be generated at equilibrium, provided processes occur at a sufficient rate in the plasma \[, that condition (i) is satisfied\]. A model of such “spontaneous baryogenesis” is typically realized by coupling the baryon current $J^{\mu}_B$ to a tensor field which attains some non-zero vacuum expectation value (VEV). A straightforward example comes in the form of a scalar field $\phi$ coupled derivatively to the baryon current: \[eq:spont\_term\] \_ \_J\_B\^ , where $M$ is a cutoff scale. It is often reasonable to assume negligible spatial variation in $\phi$, such that the interaction reduces to . In the absence of any scalar field motion, this term has no effect. However, as soon as an energy gap is induced between baryon-antibaryon pairs. In other words, the “velocity” of the scalar field acts as an *effective chemical potential* for baryon number. The production of $n_B$ through this mechanism proceeds as long as processes are coupled to the thermal bath. However, as the universe expands the bath cools, and these eventually decouple, fixing the baryon asymmetry of the universe (BAU).
A candidate for the scalar $\phi$ can emerge in a variety of contexts: , inflatons [@Brandenberger:2003kc; @Takahashi:2015ula], flat directions [@Chiba:2003vp; @Takahashi:2003db], radions [@Alberghi:2003ws], quintessence fields [@Li:2001st; @DeFelice:2002ir; @Li:2002wd], scalar curvature [@Davoudiasl:2004gf], Higgs fields [@Cohen:1991iu; @Kusenko:2014lra], However, scalars with an approximate shift symmetry such as pseudo–Nambu Goldstone bosons (pNGB) are particularly well-motivated in this context [@Carroll:2005dj; @Kusenko:2014uta; @DeSimone:2016ofp; @DeSimone:2016bok]. These fields may couple linearly to total derivatives, such as topological Chern-Simons interactions with SM gauge fields . An axion coupling to the weak gauge bosons in this way is equivalent to Eq. due to the electroweak anomaly, and thus naturally leads to spontaneous baryogenesis.
In this respect, an axion-like particle [@Peccei:1977hh; @Peccei:1977ur; @Weinberg:1977ma; @Wilczek:1977pj; @Svrcek:2006yi; @Jaeckel:2010ni] — which we shall refer to simply as an “axion” — is an attractive candidate for spontaneous baryogenesis models. Moreover, it is interesting to consider whether the late-time coherent oscillations of the axion field could also play the role of dark matter (DM). Recent studies [@Kusenko:2014uta; @DeSimone:2016ofp] suggest that axion masses exceeding are necessary to generate the observed BAU, which would ruin such a prospect. In particular, a smaller curvature is associated with the potential of a lighter axion. This property generally yields a weaker chemical potential $M^{-1}{\partial}_0\phi$ and dynamics triggered at lower temperatures, both of which impair spontaneous baryogenesis. Other proposals have attempted to revive the idea, such as driving early axion dynamics with the Gauss-Bonnet term [@DeSimone:2016bok], effectively adding a linear term to the axion potential at early times. While such a scheme is interesting in that it can be implemented with the QCD axion, it requires a fine-tuning of the misalignment angle, or hierarchical mass scales, to obtain the observed DM abundance and prevent significant baryonic backreaction or isocurvature perturbations.
In this paper, we describe a novel approach to accommodate both the observed baryon asymmetry and DM abundance. Namely, we consider scenarios in which “deformations” to the sinusoidal axion potential arise from a *field-dependent wavefunction renormalization* $Z(\phi)$. A variation in $Z(\phi)$ between different regions of the potential establishes a mismatch in curvature between those respective regions. This can have a dramatic effect on axion dynamics and the overall evolution of the baryon asymmetry and DM abundance. In particular, we motivate scenarios in which toward the minimum of the potential, but elsewhere. Indeed, this implies that in its early stages of evolution the axion rolls through a region with relatively large curvature, generating a large chemical potential, and the appropriate baryon asymmetry is easily produced. However, as the field falls toward the minimum of the potential, the enhancement effectively “flattens” it, suppressing the axion mass. This enhancement also has the effect of suppressing the rate of axion decays to SM particles. These two considerations taken together imply a sufficiently stable DM candidate that can simultaneously generate the observed BAU.
Notably, the intermediate region of such potentials can give rise to highly non-trivial dynamics. In particular, we find a period of *tracking* behavior, similar to that found in quintessence models of dark energy. In this phase the axion follows an attractor-like trajectory, with its equation-of-state parameter converging rapidly to a value that depends on the details of $Z(\phi)$ and the background cosmology. Consequently, the axion relic abundance is rendered insensitive to the initial misalignment of the field, in contrast to traditional expectations. Furthermore, the axionic isocurvature perturbations also evolve in a non-trivial manner, experiencing a suppression in amplitude for as long as tracking continues, which can be a considerable duration. The generic suppression of this isocurvature mode is one of several features which leads to a different analysis of the cosmic microwave background (CMB) constraints for such models.
The paper is organized as follows. In Sec. \[sec:GeneralDescription\], we introduce a general model which forms the basis for our analysis in the remainder of the paper. We first discuss some of its important properties, illustrating the non-trivial axion dynamics that arise and producing estimates for the lifetime and relic abundance. We then incorporate the spontaneous baryogenesis mechanism into the model, outlining the different avenues by which baryon asymmetry may be produced and underscoring the significance of its interplay with the axion dynamics. We also discuss the form of isocurvature perturbations that appear in the model. The penultimate Sec. \[sec:AnExplicitModel\] is devoted to an explicit realization, in which we demonstrate how the above scheme could be furnished from a complete model construction. To this end, we consider an extra-dimensional “continuum-clockwork” model, where our axion corresponds to the lightest state in a Kaluza-Klein (KK) tower of axion modes. We determine the phenomenological viability of this model and thereby a “proof of concept” showing how the ideas in this paper may be applied within a specific setting. Finally, in Sec. \[sec:Conclusions\] we provide a summary of our main results and possible directions for future work.
This paper also contains two Appendices. In Appendix \[sec:TrackingDynamics\] we provide a brief review of the classification of tracking potentials relevant for our analysis. Meanwhile, a derivation of the Boltzmann evolution for $(B-L)$ is detailed in Appendix \[sec:BoltzmannEquations\], which carefully accounts for the various subtleties of sphaleron equilibrium.
General Model Description\[sec:GeneralDescription\]
===================================================
In this section, we delineate a general model for an axion-like field which shall serve as the basis for our analysis in this paper. We begin by defining the model and examining its dynamical evolution, and then we shift focus to incorporating a mechanism for spontaneous baryogenesis. Finally, we close the section with an analysis of the isocurvature perturbation spectrum.
Axion dynamics and relic abundance\[subsec:AxionDynamics\]
----------------------------------------------------------
Let us consider a model for an angular axion-like field $\theta(x)$ with periodic potential $U(\theta)$ and non-trivial wavefunction renormalization $Z(\theta)$, such that the Lagrangian contains a non-canonical kinetic term: \[eq:generaleffectiveL\] \_ = Z()(\_)\^2 - \^4 U() + . We have refrained from writing any topological interactions since these will not affect our discussion of the dynamics that follow. The two mass parameters that characterize the model are determined by UV physics. Namely, $f$ is the spontaneous symmetry-breaking scale associated with our axion, and $\Lambda$ is the scale of some non-perturbative physics, , the confinement scale of a non-Abelian gauge theory.[^2] In this paper, we shall not further address the origin of these parameters.
While a sinusoidal form may be associated with a potential generated through instanton effects, the wavefunction renormalization $Z(\theta)$ is less restricted. A field-dependent wavefunction renormalization may arise from a variety of mechanisms, , integrating-out heavy degrees of freedom [@Rubin:2001in; @Tolley:2009fg; @Dong:2010in] or non-minimal couplings to gravity [@Bezrukov:2007ep; @Kallosh:2013tua]. The most activity in this area has been with inflationary model building [@ArmendarizPicon:1999rj; @Alishahiha:2004eh; @Domcke:2017fix] or kinetically driven quintessence [@Chiba:1999ka; @ArmendarizPicon:2000dh; @ArmendarizPicon:2000ah]. However, apart from some exceptions [@Alvarez:2017kar], the implications of these effects have not been extensively explored in the context of axion DM models.
Any non-trivial field dependence in $Z(\theta)$ can significantly influence how the axion evolution unfolds, as it follows the equation of motion \[eq:EofM\_theta\] + \^2 + 3 H + = 0 , where dots indicate time derivatives ${\partial}/{\partial}t$.
We can continue along these lines, analyzing the dynamics according to Eq. . However, it is also instructive to introduce the canonically normalized field \[eq:can\_phi\] (x) f\_0\^[(x)]{} -0.2cm dand study its corresponding dynamics. In particular, the equation of motion takes the more familiar form \[eq:canonicaleqnofmotion\] + 3H + = 0 , where , and $\theta(\phi)$ is obtained by inverting Eq. . In this picture, the influence of $Z(\theta)$ is captured solely through the *deformations it induces on the canonical potential* ${V_{\text{eff}}}(\phi)$. Naturally, in regions of field space where , the deformation is insignificant, and ${V_{\text{eff}}}(\phi)$ is similar to the potential in the non-canonical representation. However, in regions with an enhancement , the effect is to “flatten” the canonical potential, as seen explicitly through = and the curvature = ( - ) .
In this paper, we examine the possibility that such an axion can simultaneously (i) generate the observed BAU through spontaneous baryogenesis and (ii) serve as a DM candidate with the appropriate relic abundance. At first glance, the ingredients necessary to realize this appear incompatible. Indeed, in spontaneous baryogenesis the production of baryon asymmetry is driven by the *velocity* of the axion field ${\partial}_0\theta$, and thus ultimately depends on the shape of the potential traversed by the axion during its early evolution. In other words, a sufficiently steep region within ${V_{\text{eff}}}(\phi)$ is required for baryogenesis by these means. With the usual sinusoidal axion potential, this is to equivalent requiring a sufficiently large axion mass. However, previous studies suggest this mass must be so large that the axion is rendered highly unstable and thus an unsuitable DM candidate [@Kusenko:2014uta; @DeSimone:2016ofp].
By contrast, we argue that deformations to ${V_{\text{eff}}}(\phi)$ can repair this incompatibility. For the moment, we interpret the wavefunction renormalization $Z(\theta)$ simply as a vehicle for supplying the necessary deformations. Then, an enhancement around the minimum of the potential — but elsewhere — can furnish a model with both an adequate baryon asymmetry, as well as a suppressed axion mass and decay rates.
To explore the implications of such a model more explicitly, let us consider the wavefunction \[eq:Z(theta)\] Z() {
[lcl]{} 1 & & =(1)\
1/\^[2n]{} & & || < (1)\
1/\^[2n]{} & & ||
. , where is an integer. Along with $n$, the small parameter determines the strength of the deformation. Namely, the effective axion mass is suppressed as [m\_]{}\^2 .|\_[=0]{} = \^[2n]{} .
The dynamics that arises in response is generally non-trivial and reveals trajectories qualitatively different from the traditional axion dynamics. In the following, we outline the various periods of field evolution. A schematic of the canonical potential ${V_{\text{eff}}}(\phi)$ is shown in Fig. \[fig:potential\_annotation\], with regions labeled by their associated dynamics. We shall discuss the timeline of axion field evolution, moving sequentially from right to left in the figure.
### Slow-roll and fast-roll periods\[subsubsec:SlowRollPeriod\]
Let us assume the global symmetry associated with the axion is broken either before or during inflation, and that the field is initially misaligned at some angle . Then, according to Eq. , our initial conditions have . Furthermore, we shall only consider scenarios in which the axion is a light field during inflation and the subsequent reheating epoch, such that H , where is the Hubble parameter during that era. The damping imposed by the Hubble term holds the field to a slow-roll trajectory [@Kawasaki:2011pd] \[eq:slowrolltrajectory\] - as the radiation-dominated epoch is approached. The slow-roll evolution continues for as long as the following condition is satisfied: \[eq:slowrollcondition\] [| |]{} 1 .
The Hubble damping eventually falls sufficiently to violate Eq. . The field then enters a transient “fast-roll” period in which , and the velocity of the field reaches its maximum value over the evolution .
The temperature of the thermal bath at this point is approximately \[eq:TFR\] T\_ ()\^ , in which $g_*$ is the effective number of relativistic degrees of freedom and is the reduced Planck scale. The inverse dependence on the small parameter is particularly noteworthy, as it implies the fast-roll period is driven to increasingly higher temperatures as the potential is more acutely deformed.
![ A schematic of the effective potential ${V_{\text{eff}}}(\phi)$ for the canonically normalized field $\phi$, associated with the wavefunction renormalization $Z(\theta)$ in Eq. . The non-trivial field dependence in $Z(\theta)$ induces deformations in the potential and alters the axion dynamics. The periods of evolution — slow roll, fast roll, tracking, and coherent oscillations — are labeled in their respective regions. The initial misalignment of the axion ${\phi_{\text{in}}}$ is assumed to be toward the edge of field space. []{data-label="fig:potential_annotation"}](potential_annotation){width="49.00000%"}
### Tracking period\[subsubsec:TrackingPeriod\]
In the conventional axion dynamics \[, a model with for all $\theta$\] the field would now transit into a harmonic region of the potential and undergo coherent oscillations. However, in our model, as the field enters , the wavefunction changes form to $Z(\theta)\simeq 1/\theta^{2n}$. This change, and its associated deformation in ${V_{\text{eff}}}(\phi)$, dramatically alters the field trajectory and introduces a starkly different segment of evolution. In explicit terms, the canonical potential in this region is [V\_]{}() \^2\^4 {
[lcl]{} e\^[2||/f -2]{} & & n=1\
\^[-]{} & & n>1
. . \[eq:tr\_pot\]
The sort of dynamics induced by such a potential is well-known in the literature of quintessence models [@Wetterich:1987fm; @Ratra:1987rm]. In particular, ${V_{\text{eff}}}(\phi)$ yields so-called “tracker” solutions: attractor-like field trajectories which have an identical late-time evolution for a wide range of initial conditions [@Zlatev:1998tr; @Steinhardt:1999nw]. These are characterized by an equation-of-state parameter (for axion pressure $P_{\phi}$ and energy density ${\rho_{\phi}}$) [w\_]{} = which converges to some fixed value, depending on parameters in the potential and the background cosmology. In Sec. \[subsec:Classification\] we have provided a brief overview of the identification and classification of tracker solutions. Using that technology, we deduce that tracker solutions exist with ${V_{\text{eff}}}(\phi)$ for , which drive the axion equation-of-state parameter to \[eq:wphitrack\] w\_ = for background parameter $w$.
In other words, $n$ determines whether the axion energy density dissipates less rapidly than the dominant component in the universe, which we assume is the radiation component (). The case of an exponential potential () is unique, since it implies ${w_{\phi}}$ simply traces the background $w$ and the axion abundance remains fixed. For potentials with larger $n$, the axion component behaves increasingly like vacuum energy, causing ${\Omega_{\phi}}$ to grow and eventually dominate if tracking lasts sufficiently long. Note that since we assumed $n$ is a positive integer, there is a bound and the axion energy density never dissipates more rapidly than the dominant component.
### Coherently oscillating period\[sec:OscillationPeriod\]
The tracking dynamics continues for as long as the angular field is confined to the region, , for as long as the potential has the form in Eq. . However, as the field falls to it exits the tracking regime, and the wavefunction is frozen at a constant value . The potential is then approximately quadratic: [V\_]{}()\^2 || and does not support a tracker solution.
To determine the field evolution during this final phase, we should compare the Hubble damping $H$ to the axion mass at the time tracking completes. If we find that the field will promptly begin to undergo coherent oscillations. On the other hand, if we find , then it will sit in an overdamped phase until $H$ has dropped sufficiently for oscillations to commence. Assuming tracking lasts sufficiently long for the field to converge to the tracker trajectory, we can use Eq. , , and to show that \[eq:oscillationcond\] at the time the field exits the tracking region. The ratio above is contained within , so the field commences coherent oscillations relatively soon after, regardless of $n$. Neglecting the minor $n$-dependence and any “overshooting” effect, the axion energy density at the time of oscillations is given approximately by \_[V\_]{}() \^[2]{}\^4 . \[eq:osc\_den\]
Once coherent oscillations begin, the axion equation-of-state parameter averages to and thus the axion behaves as matter. Ideally, the matter it constitutes would be abundant enough today to comprise the entirety of the DM. To calculate the relic abundance, we use that the temperature at which oscillations first occur is given approximately by : T\_\^4 . Then, employing conservation of entropy density, we find a relic abundance[^3] \[eq:general\_abundance\] [\_]{}h\^2 ()\^2 for a sufficiently long-lived axion.
It is important to note that for a deformation in the potential enhances the relic abundance, while for it is *independent of $\epsilon$*. Moreover, ${\Omega_{\phi}}$ is *independent of the initial misalignment angle ${\theta_{\text{in}}}$*, as a result of the tracking dynamics encountered in the field evolution. This insensitivity reveals a significant departure from the traditional axion cosmology and also leads to a natural suppression of axionic isocurvature perturbations. We shall discuss the perturbations in more detail below and also provide a model-specific analysis in Sec. \[sec:AnExplicitModel\].
Meanwhile, the lifetime of the axion is also enhanced if it decays primarily through an anomalous coupling to the electroweak sector. That is, the enhancement in $Z(\theta)$ near the origin generically implies a decay width and thus an enhanced lifetime \_ ()\^2 ()\^3 , alleviating constraints from axion decays as the potential is deformed with .
Incorporating the baryogenesis mechanism\[subsec:IncorporatingBaryogenesis\]
----------------------------------------------------------------------------
Let us now shift focus to embedding the mechanism for spontaneous baryogenesis in our model. We shall begin by describing the interactions necessary and discussing how they may appear in the UV theory. We then construct the Boltzmann equations for the matter-antimatter asymmetry and catalog the different ways in which production can occur.
### Spontaneous CPT violation
We must include interactions that spontaneously violate $CPT$ in the axion background, such as the effective coupling between the axion $\theta$ and the baryon current $J_B^{\mu}$: \[eq:requiredterm\] \_ \_J\_[B ]{}\^ where $\mathcal{N}$ is a constant that we clarify in what follows. The baryon current is given by J\^\_[B ]{}=\_k (q\^\_k\^q\_k +u\^\*\_k\^u\_k + d\^[\*]{}\_k\^d\_k ) , where $q_k$, $u_k$, and $d_k$ are two-component Weyl spinors for the left-handed quark doublets, right-handed up-type quarks, and right-handed down-type quarks, respectively. Note also that we have suppressed ${\text{SU}(3)}_c$ and ${\text{SU}(2)}_L$ indices and the $\sigma^{\mu}$ are the Pauli matrices.
The homogeneity of the axion field in space implies its gradient is negligible and that Eq. effectively reduces to an interaction \[eq:effectiverequiredterm\] \_ \_0 J\_B\^[0]{} . Indeed, such a term spontaneously breaks $CPT$ symmetry once the field is set in motion, inducing an energy gap between baryons and antibaryons. If interactions are occurring in the thermal bath, then a baryon asymmetry is generated.
There are several ways to motivate the appearance of an interaction such as Eq. in the effective Lagrangian. For instance, we may consider a spontaneous breaking of the baryon number symmetry ${\text{U}(1)}_B$ at high scales, in which $\theta$ is the corresponding Nambu-Goldstone boson (NGB), which in general would appear as the phase of some complex scalar field. In such a scenario, it is also necessary to specify the relationship between the axion potential and ${\text{U}(1)}_B$, as well as the effect of operators after integrating-out the radial scalar field.
However, other avenues exist through which we may generate such an interaction, even if $\theta$ is neutral under ${\text{U}(1)}_B$. By rotating the quark phases according to $$\begin{aligned}
q_k &\longrightarrow e^{i \theta/\mathcal{N}}q_k {\nonumber}\\
u_k &\longrightarrow e^{i \theta/\mathcal{N}}u_k {\nonumber}\\
d_k &\longrightarrow e^{i \theta/\mathcal{N}}d_k \ ,\end{aligned}$$ our term in Eq. can be eliminated from the action, and we obtain an equivalent operator \[eq:CSterm\] \_ , with $W_{\mu\nu}$ the $\text{SU}(2)_L$ field strength, $B_{\mu\nu}$ the $\text{U}(1)_Y$ field strength, and ${\widetilde{W}}_{\mu\nu}$ and ${\widetilde{B}}_{\mu\nu}$ their respective duals. The factors and are the weak and hypercharge gauge coupling constants, respectively. Such an axion-like coupling of $\theta$ to the gauge bosons can naturally arise in the UV theory, independent of baryon number [@Witten:1984dg; @Green:1984sg; @Choi:1985je; @Witten:1985fp; @Choi:1985bz; @Svrcek:2006yi], if the normalization of the interaction is given by = 3/n n=1,2, . In what follows, we shall take for simplicity.
### Source for B-L violation
In addition to spontaneous $CPT$-violation, to satisfy the conditions for baryogenesis some interactions must also exist. The early-universe plasma is naturally equipped with such processes through weak sphaleron transitions. However, since the weak sphalerons preserve $(B-L)$ any baryon number generated through the spontaneous baryogenesis mechanism will still be annihilated once the axion settles to its minimum vacuum value. It is therefore essential that our theory also include interactions which break $(B-L)$. A well-motivated way to invoke such terms is through physics in the neutrino sector, where heavy right-handed neutrinos offer a natural explanation of neutrino masses and other associated phenomena. In the low-energy theory these appear in the form of a Weinberg operator \_[[L-0.15cm/]{}]{} = , \[eq:wein\_op\] where $\ell_i$ is an SM lepton doublet, ${\mathcal{H}}$ is the Higgs doublet, and neutrino observables determine the mass scale . Of course, the Weinberg operator breaks the lepton number $L$ in addition to $(B-L)$, so we shall denote associated processes by ${L\hskip -0.15cm\slash}$. The existence of heavy right-handed neutrinos in the early universe may also lead to successful thermal leptogenesis through out-of-equilibrium decays [@Fukugita:1986hr]. However, our study is dissociated from these models since $M_*$ is sufficiently heavy that the right-handed neutrinos are never produced in the thermal bath. The operator in Eq. thus remains valid throughout our analysis.
### Boltzmann evolution of asymmetry
Under the spontaneous breaking of $CPT$ by the term in Eq. , one may readily show that for sufficiently rapid processes, we obtain an equilibrium number-density asymmetry \[eq:nBmLeq\] n\^\_[[B-L]{}]{}= [\_[[B-L]{}]{}]{}T\^2 , where ${\mu_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}}$ is the effective chemical potential associated with ${(B-L)}$. However, the asymmetry is not necessarily generated at equilibrium; rather, its evolution is more generally described by the Boltzmann equation \[eq:lep\_evol\] n\_[[B-L]{}]{}+ 3 H n\_[[B-L]{}]{}= -[\_[L-0.15cm/]{}]{}(n\_[[B-L]{}]{}- n\_[[B-L]{}]{}\^) . A detailed derivation of the Boltzmann equation, in which we account for the role of sphaleron transitions in the plasma, is provided in Appendix \[sec:BoltzmannEquations\]. Implementing these we can extract the interaction rate [\_[L-0.15cm/]{}]{}= , in which \[eq:gammaLVdef\] [\_[L-0.15cm/]{}]{}= (0.01) is the thermally averaged scattering rate density for processes sourced by the Weinberg operator in Eq. , and $N_f$ denotes the number of generations with Yukawa interactions in equilibrium during baryogenesis. The effective chemical potential is likewise given by \[eq:muBmL\] [\_[[B-L]{}]{}]{}= -g(N\_f)\_0 , for which the coefficient is derived in Eq. : g(N\_f) = . The weak sphaleron processes eventually decouple, and the final baryon number is set according to \[eq:sphaleronfactor\] n\_B = n\_[[B-L]{}]{} .
The characteristic scale which determines the final $n_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$, however, is the temperature ${T_{\text{D}}}$ at which the processes derived from $\mathcal{L}_{L\hskip -0.15cm\slash}$ decouple. In explicit terms, we define ${T_{\text{D}}}$ as the temperature at which the Hubble rate becomes dominant . It is expressed as [T\_]{}() , which in our study will be given by . The conclusion we draw is that, given the assumptions above, a relatively high reheating temperature is necessary for processes sourced by Eq. to ever achieve equilibrium. Indeed, depending on the temperature regime of the early radiation-dominated epoch, there are several different ways in which baryogenesis may unfold. To explore these in more detail, let us work instead with comoving quantities, such as the abundance , in which \[eq:entropydensity\] s = g\_[\*S]{}T\^3 is the entropy density. It is then straightforward to determine the corresponding Boltzmann evolution \[eq:YBmLboltzmann\] = , where the equilibrium value is defined analogously to Eq. . An integral solution follows as $$\begin{aligned}
\label{eq:lep_yield}
Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}(T) = \int_T^{{T_{\text{RH}}}} dT' \left[\frac{e^{-(T'-T)/{T_{\text{D}}}} }{{T_{\text{D}}}}\right] Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}^{\text{eq}}(T') \ .\end{aligned}$$ There are different limiting behaviors, depending on the relative size of the temperatures ${T_{\text{D}}}$ and ${T_{\text{RH}}}$, corresponding to equilibrium and out-of-equilibrium production. Below, we discuss each of these cases.
### Equilibrium production\[subsubsec:EquilibriumProduction\]
In the regime, the function enclosed within square brackets of Eq. approaches a Dirac delta function , which holds for temperatures . The asymmetry closely follows its equilibrium value in that regime and reproduces the result in Eq. . However, once the plasma cools below , equilibrium productions ceases and the asymmetry “freezes out.” Therefore, we obtain a late-time asymmetry \[eq:FOlepasy\] Y\_[[B-L]{}]{}Y\_[[B-L]{}]{}\^([T\_]{}) .
The interplay between the equilibrium production of $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ and the axion dynamics is significant in determining the final baryon asymmetry. In particular, the late-time yield can be ruined if decoupling does not occur until after the axion undergoes oscillations — in this event $\mu_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ oscillates as well, and the asymmetry is washed out. Almost as severely, if decoupling occurs within the tracking period, the field velocity and thus $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ is considerably weakened. In the context of equilibrium production we shall then assume that \[eq:washoutcondition\] H([T\_]{}) 3 [T\_]{}\^2/M\_P , which implies the field is slowly rolling at decoupling. The yield is then determined by the slow-roll trajectory in Eq. and we find an approximate expression Y\_[[B-L]{}]{}\~10\^[-3]{} . Note that taking $\epsilon$ to smaller values, and thereby deforming the potential more acutely, corresponds to an *enhancement* in the matter-antimatter asymmetry.
### Out-of-equilibrium production\[subsubsec:OutofEquilibriumProduction\]
Let us now consider the opposite temperature regime, in which reheating occurs below the scale of decoupling . In such a scenario, processes which violate ${(B-L)}$ are always out-of-equilibrium, , for all temperatures. Consequently, the asymmetry production occurs in a different fashion, weakly but persistently driven by the term in Eq. . The production mechanism here is analogous to the “freeze-in” production in the DM literature [@Hall:2009bx]. It is straightforward to obtain the yield \[eq:yield\_fi\] Y\_[[B-L]{}]{} ([\_]{}) , in which the integral $\mathcal{I}(\theta)$ is defined by \[eq:Iintegral\] () \_0\^ du , for a dimensionless temporal parameter . It can be shown using Eq. that gives only an $\mathcal{O}(1)$ contribution. Finally, assuming baryogenesis occurs at temperatures , we conclude that Y\_[[B-L]{}]{}\~10\^[-3]{} = 10\^[-3]{} \[eq:FIlepasy\] within order-of-magnitude accuracy.
We observe for both the equilibrium production in Eq. and the out-of-equilibrium production above, for a fixed axion mass *the asymmetry is enhanced by deformations in the potential*. Thus, possibilities for model-building can exist in either of these two regimes.
On another note, by comparing the scaling behavior for the axion relic abundance \[see Eq. \] to the estimates for the late-time asymmetry $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ above, we find that has some intriguing properties. In particular, while the baryon asymmetry is always enhanced by , the relic abundance is unaffected for , providing us with some modularity between these two cosmological quantities. We shall investigate these details further in the context of the explicit model constructed in Sec. \[sec:AnExplicitModel\], which is specific to the case.
Isocurvature perturbations\[subsec:IsocurvaturePerturbations\]
--------------------------------------------------------------
As alluded to above, for successful baryogenesis the axion must be relatively light during inflation, for an inflationary Hubble scale $H_I$. Therefore, it is subject to quantum fluctuations with amplitude \[eq:axionfluctuations\] \_ = , in the pure-de Sitter limit. As a result, we find corresponding fluctuations in the angular field [\_]{}= .
The sources of primordial scalar perturbations are decomposed into linearly independent adiabatic and isocurvature modes, , perturbations to the total energy density and the local equation of state, respectively. The fluctuations $\delta{\theta_{\text{in}}}$ source only the isocurvature mode, which is subdominant and tightly constrained by observations of the CMB [@Akrami:2018odb]. Furthermore, since the baryon asymmetry is generated via the effective chemical potential , baryonic isocurvature perturbations $\delta Y_B$ also exist and play an important role.
As illustrated in Sec. \[subsec:AxionDynamics\], the presence of a tracking region in the canonical potential ${V_{\text{eff}}}(\phi)$ renders the late-time axion dynamics insensitive to the initial field displacement. Consequently, the axionic isocurvature perturbations are generically suppressed with a magnitude corresponding to the duration of the tracking period.[^4] In the remainder of this section, we shall assume tracking lasts for a sufficiently long period that we may focus exclusively on the baryonic component.
As discussed above, $Y_B$ may be populated while driving the system either at equilibrium (freeze-out) or out-of-equilibrium (freeze-in). In the former case, the baryon asymmetry produced is simply evaluated at the decoupling temperature ${T_{\text{D}}}$. Therefore, the baryonic perturbation is $$\begin{aligned}
\frac{\delta Y_B}{Y_B} \simeq \frac{\delta {\mu_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}}}{{\mu_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}}}
&= \left.\frac{d \log\mu_{B-L}^{\rm eq}}{d\theta} \frac{1}{\sqrt{Z(\theta)}}\frac{H_I}{2\pi f}\right|_{{\theta_{\text{in}}}} {\nonumber}\\
&= \left.\frac{d\log\dot\theta}{d\theta}\frac{1}{\sqrt{Z(\theta)}}\frac{H_I}{2\pi f}\right|_{{\theta_{\text{in}}}} \ ,\end{aligned}$$ also evaluated at ${T_{\text{D}}}$. As before, we assume decoupling occurs during the slow-roll period, so the trajectory is given by Eq. : $$\begin{aligned}
\dot\theta\simeq -\frac{1}{5H}\frac{\Lambda^4}{f^2}\frac{1}{Z(\theta)}\frac{dU}{d\theta} \ ,\end{aligned}$$ which we have written in the non-canonical basis. Assuming the field moves negligibly from its initial misalignment , the approximate isocurvature is \[eq:PSSFO\] \_ ()\^[2]{} .\^[2]{}|\_[[\_]{}]{} where a prime denotes a derivative with respect to the field $\theta$. Interestingly, the two terms in Eq. may have opposite signs, allowing for cancellations and a vanishing perturbation. Additionally, the first term can clearly vanish if ${\theta_{\text{in}}}$ sits at any of its inflection points.
On the other hand, in the case of out-of-equilibrium production, we can derive the baryonic perturbation directly from Eq. : . |\_[[\_]{}]{} , which results in the expression for the power \[eq:PSSFI\] P\_ .()\^[2]{}()\^[2]{}|\_[[\_]{}]{} .\
The integral $\mathcal{I}({\theta_{\text{in}}})$ is over time \[see Eq. \] and therefore it is not a simple function of the potential or its derivatives. Instead, these results must be obtained numerically from the equation of motion.
The power spectra in Eq. and Eq. act as constraints on the parameter space given an explicit model realization. In the remainder of this paper, we shall consider such a concrete model, and show a more detailed study of its phenomenology and cosmological constraints, using numerical simulations where necessary.
An explicit model: The continuum-clockwork axion\[sec:AnExplicitModel\]
=======================================================================
In Sec. \[sec:GeneralDescription\], we described a general construction by which an axion-like field $\theta$ may dynamically generate matter-antimatter asymmetry in the early universe, while also serving as a plausible DM candidate. The crux of this approach is the appearance of a field-dependent wavefunction renormalization $Z(\theta)$ that meets some basic requirements. In particular, if the wavefunction is enhanced near the minimum of the axion potential, but remains elsewhere, then the potential ${V_{\text{eff}}}(\phi)$ for the canonically normalized field $\phi$ is “deformed” in a way that suppresses its mass, generically suppresses its couplings, and can dramatically alter its dynamics.
In this section, we demonstrate that models exist with the ingredients necessary to furnish such a wavefunction renormalization. As an explicit example, we focus on “continuum-clockwork” (CCW) [@Giudice:2016yja; @Craig:2017cda; @Giudice:2017suc; @Choi:2017ncj] axion models[^5], incorporating the interactions necessary for spontaneous baryogenesis along the lines of Sec. \[subsec:IncorporatingBaryogenesis\]. We show that regions of parameter space exist in which both the observed baryon asymmetry and dark matter abundance are produced. Furthermore, we show that other phenomenological constraints, such as those from decays and isocurvature perturbations, are adequately contained.
Overall features and construction\[subsec:OverallFeaturesandConstruction\]
--------------------------------------------------------------------------
The hallmark of the “clockwork mechanism” [@Kim:2004rp; @Choi:2014rja; @Choi:2015fiu; @Kaplan:2015fuy] is the generation of an exponential hierarchy of couplings in theories with exclusively $\mathcal{O}(1)$ input parameters. There have been many studies and implementations, including on the QCD axion [@Higaki:2015jag; @Higaki:2016yqk; @Higaki:2016jjh; @Farina:2016tgd; @Coy:2017yex; @Long:2018nsl], dark matter [@Hambye:2016qkf; @Kim:2017mtc; @Kim:2018xsp; @Goudelis:2018xqi], cosmological topics [@Fonseca:2016eoo; @Saraswat:2016eaz; @Kehagias:2016kzt; @Park:2018kst; @Agrawal:2018mkd], flavor physics [@Hong:2017tel; @Park:2017yrn; @Ibarra:2017tju; @Alonso:2018bcg], and generalizations or more formal aspects [@Giudice:2016yja; @Craig:2017cda; @Giudice:2017suc; @Choi:2017ncj; @Ahmed:2016viu; @Ben-Dayan:2017rvr; @Lee:2017fin; @Giudice:2017fmj; @Choi:2017ncj; @Teresi:2018eai].
To introduce this idea more explicitly, let us consider a model with scalars $\chi_i$. The clockwork mechanism typically arises through “nearest-neighbor” interactions between adjacent scalars, such as through terms proportional to , where is a dimensionless parameter. Then, the lightest mass eigenstate $\phi$ in the system exhibits most of the interesting phenomenology. In particular, if the $\chi_i$ have couplings $Q_i$ to some other sector, then these contribute to the coupling for $\phi$ as \[eq:Q\_phi\] Q\_ \_[i=0]{}\^N . That is, the coupling for the lightest state is determined through a non-uniform distribution over the $Q_i$. As their contributions are weighted by powers , the distribution is effectively “localized” toward $Q_N$, where the parameter $q$ sets the strength of the localization.
A natural extension of this idea is to construct the clockwork mechanism in the *continuum* limit , where the theory is reinterpreted as that of a discretized compact extra dimension. In the continuum, the nearest-neighbor interactions composed of are mapped onto , where , the extra spatial coordinate is $y$, and $\chi(x,y)$ is now a five-dimensional scalar field. This type of combination can be realized by bulk and boundary mass terms. Moreover, several of the phenomena found in the discrete clockwork theory are mapped onto the extra-dimensional theory in some way. In particular, the profile for the zero-mode is exponentially localized toward a boundary in the extra dimension, analogous to the localization of the coupling in Eq. . Similar phenomenological implications arise from this as well, such as the suppression of couplings to other boundary operators.
Furthermore, interesting observations can be made if the CCW theory is constructed from a five-dimensional angular field $\theta(x,y)$. Due to the periodicity , the clockwork interactions should have the general form , for periodic . As discussed in Ref. [@Choi:2017ncj], this results in a more non-trivial localization of the lightest mode along the extra dimension, which has subtle implications for the axion couplings and its dynamics. While the specific details are beyond the scope of this paper, it provides us the essential features by which we shall realize a field-dependent wavefunction renormalization of the form proposed in Sec. \[sec:GeneralDescription\].
Let us therefore begin by considering the action for this particular five-dimensional realization: \[eq:Stheta\] \_ = d\^4x dy , where we compactify over an orbifold of radius $R$, and $m$ sets the scale for bulk and boundary terms. The SM fields are assumed to be confined to the brane and flat space is assumed for tractability. Note that a massless four-dimensional mode $\phi(x)$ is found in the spectrum: \[eq:tantheta\] = e\^[my]{}u , where the function $u[\phi]$ enforces canonical normalization for $\phi$ over its domain. Namely, we define ue\^[-mR]{} in which and is a Jacobi elliptic function. Integrating out the higher KK modes and the compact dimension, we construct the low-energy effective action \[eq:ccw\_wave\] S\_ d\^4x , where it is understood that is evaluated at the brane. The wavefunction renormalization as defined in Eq. is easily extracted: Z() . In the regime that clockwork has a substantial effect, this reduces to \[eq:Zclockwork\] Z() , where is a small parameter. It is manifest that near the boundaries of field space and near the origin. Furthermore, expanding about the origin we find , which is the necessary scaling to ensure the desired “tracking” dynamics. It is then evident that the CCW axion reproduces the form of Eq. and we can conclude: *CCW axions satisfy our minimal requirements for spontaneous baryogenesis driven by a stable axion DM candidate.*
![The effective 4D canonical potential ${V_{\text{eff}}}(\phi)$ that arises for a continuum-clockwork axion, where the different curves show various choices for the clockwork parameter $mR$, and the horizontal axis is normalized such that curves are plotted over the full field range . []{data-label="fig:potential"}](potential.pdf){width="49.00000%"}
Having satisfied the minimal set of conditions from Sec. \[sec:GeneralDescription\], let us further investigate the details of this model. While the symmetry of the action in Eq. yields a massless zero-mode $\phi$, any small deviation in boundary masses will generate an effective four-dimensional potential , which in the canonical basis reads \[eq:Veff\] [V\_]{}() = . While ${V_{\text{eff}}}(\phi)$ is periodic in $\phi$, it is important to note the period is *not* given by $2\pi f$, but rather by the expression $$\begin{aligned}
\label{eq:feff}
2\pi f_{\text{eff}} &\equiv 4f K\left(1 - \epsilon^2\right) {\nonumber}\\
&\xrightarrow[~\epsilon^2\ll 1~]{} 2f\log\left(\frac{16}{\epsilon^2}\right) \ ,\end{aligned}$$ where $K(\cdot)$ is the complete elliptic integral of the first kind. As a result, the field range of the canonical four-dimensional axion is effectively extended for finite $mR$.
In Fig. \[fig:potential\] the potential is shown for several values of $mR$, normalized so that the curves all span the same domain. As soon as we exceed the potential quickly shows substantial deformations, with the minimum flattened along most of the field range. The resulting axion mass \[eq:mphi\] [m\_]{}\^2 .|\_[=0]{} = e\^[-2mR]{} , is *exponentially suppressed* relative to the standard sinusoidal potential. Indeed, the suppression of this mass scale confirms the CCW axion model is equipped with one of the imperative features.
The other feature necessary to avoid phenomenological complications is the suppression of axion decays to SM states. In the discrete clockwork theory described above Eq. , this suppression would arise for the light state $\phi$ if the SM were coupled to the endpoint $\chi_N$ opposite to where $\phi$ is localized. Analogously, in the continuum limit this corresponds to SM fields confined to the brane. In other words, if we have an interaction between the axion and some generic SM operator $\mathcal{O}(x)$ \_ d\^4x (x,0)(x) , we can write it in the canonical basis using \[eq:fieldmapping\] = fd , where $F(\cdot\,\vert \,\cdot)$ is the incomplete elliptic integral of the first kind. Let us examine how the coupling is affected for larger $mR$. Away from the minimum of the potential is quickly approached and thus the coupling is not significantly affected beyond the minor enhancement of ${f_{\text{eff}}}$. However, around the minimum Eq. reduces to , so that couplings to $\phi$ are exponentially suppressed. For example, if we take the operator , for some SM field strength $F_{\mu\nu}$ and its dual ${\widetilde{F}}_{\mu\nu}$, then the axion decay rate suffers a suppression \[eq:decaysuppression\] \_ = \^2 . Note that this suppression acts in addition to that implicitly included in the mass \[see Eq. \].
{width="\textwidth"}
Early dynamics and baryogenesis \[subsec:EarlyDynamics\]
--------------------------------------------------------
Above we have constructed the axion sector of the theory, however, we must also incorporate the necessary ingredients for baryogenesis. We shall proceed in a manner parallel with Sec. \[subsec:IncorporatingBaryogenesis\], specializing our analysis to the continuum-clockwork model. As we argued previously, the motion of the axion spontaneously breaks $CPT$ symmetry if it couples derivatively to a baryon current, as in Eq. . In addition, the Weinberg operator in Eq. provides a source for processes that violate $(B-L)$. Assuming a similar interaction in this model, the effective chemical potential from Eq. in the canonical basis takes the form [\_[[B-L]{}]{}]{}(,) = -g(N\_f) , where ${V_{\text{eff}}}(\phi)$ was used for a more succinct expression.
In order to numerically simulate the early dynamics we make several assumptions. Let us suppose a period of inflation with Hubble parameter during which the axion is misaligned from the minimum of its potential by angle . Then, the reheating epoch is modeled by assuming the energy density in the inflaton decays into radiation $\rho_R$ at some rate $\Gamma_{I}$. It follows that these quantities evolve as $$\begin{aligned}
\dot{\rho}_{I} + 3H\rho_{I} &= - \Gamma_{I}\rho_{I} {\nonumber}\\
\dot{\rho}_{R} + 4H\rho_R &= + \Gamma_{I}\rho_{I} + \Gamma_{\phi}{\rho_{\phi}}\ ,\end{aligned}$$ where the Hubble parameter is given by H\^2 = , and [\_]{}= \^2 + [V\_]{}() is the energy density in the axion field. Meanwhile, the axion equation of motion reads + (3H+\_) + = [\_[L-0.15cm/]{}]{}(n\_[[B-L]{}]{}- n\_[[B-L]{}]{}\^) . The term on the right-hand side is due to backreaction from $(B-L)$ generation and is usually negligible. As the axion is set into motion it drives the production of $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$, which follows the Boltzmann evolution in Eq. .
In Fig. \[fig:field\_dynamics\] the evolution of two types of quantities — the cosmological abundances and abundance $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ — are shown in the two rows of panels, as functions of the number of e-folds since reheating . Almost all parameters are held fixed: the mass , effective scale , and misalignment angle . However, we have varied the strength of the clockwork mechanism in each column. Therefore, the left-hand column shows dynamics for the traditional sinusoidal axion potential, the right-hand column shows a substantially deformed potential, and the center column shows an intermediate case between these two regimes.
Note that results are sensitive to inflationary scales $\Gamma_{I}$ or $H_I$ only if they are exceeded by the initial scale of curvature . The curvature exceeding $H_I$ implies the axion is a heavy field during inflation and we shall exclude this region. On the other hand, if the curvature exceeds $\Gamma_{I}$ it implies the axion is set in motion prior to reheating. Then, washout effects that suppress the asymmetry can become sizeable. In this section, we look to identify phenomenologically viable regions of parameter space, and thus as a simplifying assumption we take these scales to be comparable .
The influence from the variation of $mR$ in Fig. \[fig:field\_dynamics\] is seen most immediately in the plots of $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$. The deformation of the potential sets the axion in motion at higher temperatures. Nevertheless, we always find such that $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ *never exactly tracks the equilibrium value $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}^{\rm eq}$*, , this model exhibits the “freeze-in” spontaneous baryogenesis discussed in Sec. \[sec:GeneralDescription\], in contrast to more common examples in the literature. As a result, the asymmetry is mostly set during the fast-roll period and remains frozen at that value. Using the estimate in Eq. and evaluating at the fast-roll temperature $T_{\text{FR}}$ from Eq. we find e\^[mR/2]{} , when production ceases. The above expression demonstrates how the deformation of the potential through the clockwork mechanism (, the exponential factor $e^{\pi mR/2}$) enables sufficient baryogenesis while maintaining a relatively light axion.
{width="\textwidth"}
Another interesting dynamical feature is found in the column of Fig. \[fig:field\_dynamics\]. Although the deformation of ${V_{\text{eff}}}(\phi)$ sets the axion in motion earlier, it does not undergo coherent oscillations until much later when . Instead, during this period the trajectory is such that ${\Omega_{\phi}}$ is temporarily fixed, with a radiation-like equation of state . Indeed, we have identified the “tracking” phenomenon, which we have discussed in more generality in Sec. \[subsubsec:TrackingPeriod\]. We expect this dynamics to occur over the field range where , and taking is sufficient to generate such a region. We find that \[eq:trackingpotential\] [V\_]{}() [m\_]{}\^2f\^2e\^[[| |]{}/f]{} is a good approximation over . The enhancement of the axion field range by makes such a displacement easy to achieve. The trajectory of the tracker is found using that \[eq:trackingcondition\] = is approximately constant. The resulting solution \[eq:trackingsolution\] [| (t) |]{} -2f , naturally depends weakly on the initial field amplitude as it enters the tracking epoch. Considering that tracking ends once , the dependence on $\phi_{\rm tr}$ is ultimately washed away if . Therefore, for sufficiently deformed potentials *the late-time axion field is insensitive to the initial misalignment angle ${\theta_{\text{in}}}$*, in contrast to the standard case.
Survey of viable regions\[subsec:SurveyofViableRegions\]
--------------------------------------------------------
We are now in position to discuss the phenomenological viability of this explicit model. As a first requirement we must verify the existence of regions in parameter space which have both the observed dark matter abundance and baryon abundance . Furthermore, regions in which the axion is not sufficiently stable, , lifetimes longer than [@Chen:2003gz; @Zhang:2007zzh], must be excluded. Indeed, regions with substantial deformations, , at least moderately large $mR$, are where we expect to find viability in these respects.
As we have found above, such a regime is also associated with a tracking period for the axion. While tracking has little direct effect on the development of baryon asymmetry, it does have an marked influence on the relic abundance ${\Omega_{\phi}}$. Namely, employing the tracking-field solution in Eq. we find at the onset of coherent oscillations , so that at present day \[eq:abundance\] [\_]{}h\^2 0.12 ()\^2 , In the region of interest we find agreement with numerical computations to [10%]{}. Note that the insensitivity of ${\Omega_{\phi}}$ to the initial misalignment angle ${\theta_{\text{in}}}$ is a result of the attractor-like dynamics and distinguishes our result from the standard axion cosmology.
An analytical approximation for the baryon asymmetry can also be constructed using the general result in Eq. , taking the integrated $\mathcal{O}(1)$ factor to be unity. We find an approximate expression Y\_B \~10\^[-10]{}() e\^[(mR-12)]{} , which holds to at least order-of-magnitude accuracy throughout the parameter space.
The results of our numerical simulations span the space and are shown in Fig. \[fig:Y\_B\]. In each panel, contours show both the normalized baryon abundance (black) and axion abundance (yellow). The sole distinction between each panel is the choice for the scale ${f_{\text{eff}}}$. The green regions show exclusions due to axion decays. As expected from Eq. , for even moderately large $mR$ these regions are substantially reduced in size. The points of intersection between the two thickest curves correspond to viable configurations that match observations, and these plots show that *viable regions exist over all the panels shown* in Fig. \[fig:Y\_B\]. The only constraints not yet applied are bounds on isocurvature perturbations, which is the focus for the remainder of the section.
Isocurvature perturbations\[sec:IsocurvaturePerturbations\]
-----------------------------------------------------------
In the more general analysis provided in Sec. \[subsec:IsocurvaturePerturbations\], several significant observations were made regarding the axionic and baryonic isocurvature perturbations. Moreover, power spectra for these perturbations were found for both in-equilibrium and out-of-equilibrium production of the baryon asymmetry. We conclude this section with a more thorough treatment, in which the perturbation equations are solved numerically, within the context of the continuum-clockwork axion model.
### System of perturbations\[subsubsec:SystemofPerturbations\]
In our analysis, for perturbations of the FRW background, we use the conformal Newtonian gauge, defined by the line element [@Bardeen:1980kt; @Mukhanov:1990me]: ds\^2 = (1+2)dt\^2 - a(t)\^2(1 - 2)d\^2 , where the scalar potentials are functions of space and time. The anisotropic stress is vanishing in our model, which implies an equivalence .
The gravitational potential develops according to the Einstein equations as [@rubakovinflation] + 3H(H+ ) = - for a comoving Fourier mode $k$, where is the sum of energy density perturbations. Meanwhile, covariant stress-energy conservation gives the evolution for matter degrees of freedom. Namely, it gives axion perturbations + 3H+ = 4 - 2[V\_]{}’() , and radiation perturbations \_ - k\^2 v\_[B]{} = 4 , as well as the velocity potential $v_{B\gamma}$ of the baryon-photon fluid. However, for large-scale perturbations in our scenario $v_{B\gamma}$ has a negligible influence. Finally, the perturbations in lepton or baryon density are coupled to the axionic degrees of freedom through : \[eq:dnLeqn\] \_B - v\_[B]{} = -[\_[L-0.15cm/]{}]{}(\_B - ) + 3 , where the distinction between $Y_B$ and $Y_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ as they appear in these equations is inconsequential.
### Initial conditions
Before discussing the features of this system in some detail, let us first make our initial conditions and other ancillary assumptions clear. The isocurvature mode is formally defined by a vanishing initial condition for the gauge-invariant curvature perturbation [@Mukhanov:1990me] \[eq:Rperturbation\] + , and it follows that $\Phi$, $\dot{\Phi}$, and $\delta\rho_{\text{tot}}$ (after enforcing the Einstein equations) all have vanishing initial conditions as well [@Perrotta:1998vf].
The stress-energy fluctuations are functions both of perturbations in the field and the gravitational potential: $$\begin{aligned}
\delta{\rho_{\phi}}&= {\dot{\phi}}\delta\dot{\phi} - {\dot{\phi}}^2\Phi + {V_{\text{eff}}}'(\phi){\delta\phi}{\nonumber}\\
\delta P_{\phi} &= {\dot{\phi}}\delta\dot{\phi} - {\dot{\phi}}^2\Phi - {V_{\text{eff}}}'(\phi){\delta\phi}\ ,\end{aligned}$$ so there is a non-zero initial axion perturbation \[eq:initialaxionperturbation\] \_ .
Note that in a more traditional scenario — , the QCD axion — the potential is flat until the confining phase transition is approached, when it is finally generated by instanton effects. Although the same non-zero field fluctuation $\delta\phi$ exists, the perturbation $\delta_{\phi}$ is vanishing until the potential for the axion is generated. By contrast, in our case ${V_{\text{eff}}}(\phi)$ is established already during (or prior to) inflation. As a result of this contrast and the deformation of the potential in our model, we shall find several interesting features in the evolution of axionic perturbations, even for large-scale modes.
![The evolution of both the axionic $\mathcal{S}_{\phi\gamma}$ (blue curve) and baryonic $\mathcal{S}_{B\gamma}$ (green curve) contributions to the isocurvature mode, with a choice of parameters — , , , , — similar to previous figures. The dashed-green curve shows the result in Eq. , if spontaneous baryogenesis had occurred at equilibrium, and the shaded-blue area indicates the tracking phase, suppressing as over its duration. []{data-label="fig:perturbation_dynamics"}](perturbation_dynamics.pdf){width="49.00000%"}
### Axionic contribution
There are several significant observations to make that are unique to the axionic contribution. To simplify the discussion, we momentarily ignore the baryonic component and define two gauge-invariant entropy perturbations. One is intrinsic [@Kodama:1985bj]: \[eq:Gammaperturbation\] , where is the adiabatic sound speed of the axion fluid. The other is expressed relative to photons: \[eq:SphiRperturbation\] \_ - \_ . The relevant modes for our discussion are outside the Hubble sphere during the early evolution. At these scales, the basis of gauge-invariant perturbations and is convenient, since these two sets decouple. In particular, writing the perturbation equations in this basis and simplifying the system for the tracking regime (, taking ), we find [@Bartolo:2003ad] $$\begin{aligned}
\label{eq:trackingperturbationeqns}
\frac{1}{2}\frac{d\left[(1+{w_{\phi}})\mathcal{S}_{\phi\gamma}\right]}{d\log a} &= -\Gamma {\nonumber}\\
2\left[(1+{w_{\phi}})\mathcal{S}_{\phi\gamma}\right] - \Gamma &= \frac{d\Gamma}{d\log a} \ .\end{aligned}$$ The solutions for $\mathcal{S}_{\phi\gamma}$ undergo damped harmonic oscillations every few e-folds, rapidly suppressing the axionic isocurvature amplitude as[^6] \_ . It is instructive to view a numerical solution of the full system of perturbations in this regime, focusing on the tracking period. In Fig. \[fig:perturbation\_dynamics\] this is shown by the blue curve, where we have highlighted the tracking phase. Indeed, a few e-folds after reheating the axion begins to converge to the tracker solution, and the amplitude of the isocurvature perturbation falls as . The field eventually enters a region of the potential which is approximately harmonic, ending the tracking dynamics and thus concluding the suppression. The isocurvature $\mathcal{S}_{\phi\gamma}$ then undergoes some short-lived transient oscillations before finally settling on its asymptotic late-time value.
### Baryonic contribution
The dynamical suppression of $\mathcal{S}_{\phi\gamma}$ is significant, since it could ensure that the baryonic contribution is dominant if tracking lasts for a sufficient number of e-folds. Indeed, this dominance was an assumption we made when deriving the general power spectra in Sec. \[subsec:IsocurvaturePerturbations\]. The baryonic component is defined in an analogous way: \_[B]{} = \_B - \_ .
An example of numerical solutions for the evolution of $\mathcal{S}_{B\gamma}$ is shown in the solid-green curve of Fig. \[fig:perturbation\_dynamics\]. Apart from some early dynamical behavior as the baryon asymmetry is being established, the $\mathcal{S}_{B\gamma}$ perturbations are essentially fixed after the fast-roll period. However, there are other subtleties that we should outline.
Although the baryon asymmetry is typically produced out-of-equilibrium in this model, a useful benchmark comparison with regard to the perturbation spectrum is the case of in-equilibrium production, which we have shown as a dashed-green line in Fig. \[fig:perturbation\_dynamics\]. We find that the baryonic perturbations for different types of production typically do not differ by more than an order of magnitude throughout most of parameter space. However, there are some particularly important exceptions. To this end, it is instructive to derive an analytical approximation for the equilibrium case in terms of the canonical field $\phi$. Using the proportionality we find the perturbation by evaluating \[eq:equilibriumSBgamma\] \_B = (+ ) at the decoupling temperature ${T_{\text{D}}}$. In previous investigations (, Ref. [@DeSimone:2016ofp]), a slow-roll approximation is used and neither $\delta\phi$ or $\phi$ are assumed to significantly evolve. Under these conditions, we find an analytical expression: \[eq:equilibriumSBgamma2\] S\_[B]{} .{ -}|\_[[\_]{}]{} , where the form of the expression is influenced by the chemical potential being a function of both the velocity of the canonical field and the field itself . Additionally, note the appearance in Eq. of several critical points for the initial field displacement ${\phi_{\text{in}}}$: the two terms may have opposite signs, allowing for cancellations and a vanishing $\mathcal{S}_{B\gamma}$, and any inflection points in the canonical potential cause the first term to vanish.
The behavior of the perturbations near these points can dramatically change the isocurvature power spectrum, making the baryonic contribution subdominant. While the perturbations for out-of-equilibrium asymmetry production cannot be found analytically in this way, it is important to investigate how these effects are manifested in that case, which is a question we shall continue to address below.
![In the top two panels, numerical results for the baryonic and axionic contributions to the isocurvature power $\mathcal{P}_{\mathcal{S}\mathcal{S}}(k_*)$ are shown, respectively, as a function of the misalignment angle ${\theta_{\text{in}}}$, with the various curves showing different values of $mR$. The values and were chosen, and ${m_{\phi}}$ is set to satisfy . In the top panel, dashed curves show the approximate equilibrium result of Eq. . In the bottom panel, the isocurvature fraction $\beta_{\text{iso}}$ is computed for each curve, comprising the total effect. []{data-label="fig:iso_contours"}](iso_contours.pdf){width="49.00000%"}
{width="\textwidth"}
### Isocurvature bounds
The constraint on isocurvature from the CMB comes in the form of an upper-bound on the uncorrelated “isocurvature fraction,” from *Planck* collaboration data [@Akrami:2018odb]: \[eq:betaiso\] \_(k\_\*) < 0.038 , in which is the adiabatic power, $\mathcal{P}_{\mathcal{SS}}(k_*)$ is the isocurvature power, and each is evaluated at the pivot scale . The baryonic and axionic contributions are exactly correlated due to their common source, and appear as a weighted sum \[eq:PSS\] \_(k) \^2 , where and are the cosmological abundances for baryons and cold dark matter (CDM), respectively.
Note that the total power spectrum $\mathcal{P}_{\mathcal{S}\mathcal{S}}$ has the possibility for cancellations *between* axionic and baryonic components. This type of behavior is made more clear in the context of our model by examining the perturbations as a function of the misalignment angle ${\theta_{\text{in}}}$. In the bottom panel of Fig. \[fig:iso\_contours\], we plot the dependence of $\beta_{\text{iso}}(k_*)$ on ${\theta_{\text{in}}}$ by numerically solving the perturbation equations. The upper panels show explicitly how the weighted isocurvature sources in Eq. contribute to the bottom panel. The different curves show various choices for $mR$, while ${m_{\phi}}$ is taken to ensure the axions have the observed DM abundance at (the value used in all previous figures). We have also included the equilibrium result from Eq. with dashed curves in the top panel. Our interest is mostly in the behavior for at least moderate values of ${\theta_{\text{in}}}$, to ensure the field is misaligned sufficiently from the minimum to enable adequate production of baryon asymmetry.
In the case of Fig. \[fig:iso\_contours\], the axionic isocurvature is monotonic with ${\theta_{\text{in}}}$ and does not experience any sign changes. However, as $mR$ is increased both isocurvature contributions show vanishing points that generally do not coincide. We also confirm that as $mR$ is increased tracking effects suppress the axionic component, as seen through the overall reduction in the $\mathcal{S}_{\phi\gamma}$ amplitude. The effect is more subtle toward the edge of field space, however, as both contributions are enhanced with ${\theta_{\text{in}}}$. The accumulation of all the effects is that as we deform the potential, the baryon asymmetry is amplified exponentially as $e^{\pi mR}$, while the isocurvature is *increasingly suppressed* at moderately large misalignment angles, focused roughly around the region.
While these discussions are instructive in forming a qualitative picture of the perturbations, our interest ultimately is in producing exclusion regions over the plots in Fig. \[fig:Y\_B\]. Therefore, we solve the perturbation equations over the full parameter space and mark regions that violate Eq. . These are indicated by dark-red in Fig. \[fig:Y\_B\_iso\], while all other features and parameter choices are identical to Fig. \[fig:Y\_B\], as discussed previously. We immediately observe that the viable regions in which the baryon abundance $Y_B$ (black contours) and axion abundance ${\Omega_{\phi}}$ (yellow contours) are produced in the observed amounts *remain safely outside the exclusion region*.
Discussion and Conclusions\[sec:Conclusions\]
=============================================
In this paper, we have investigated the possibility that both the baryon asymmetry of the universe and dark matter may be accounted for by a single axion-like field. In this scenario, the early-universe dynamics of the axion drive a period of spontaneous baryogenesis, during which the observed baryon asymmetry is produced. As the axion field settles to the minimum of its potential, it undergoes coherent oscillations, which behave cosmologically as dark matter at late times. Typically, to generate the observed baryon asymmetry, a relatively “steep” axion potential is required in the region where the axion initially rolls. The corresponding axion mass is large and highly unstable against decays, making it inadequate as a dark matter candidate. However, we have shown that a field-dependent wavefunction renormalization can arise which effectively “deforms” the axion potential, inducing in a mismatch in curvature between different regions. In this way, novel possibilities have emerged, as we can not only generate the observed baryon and dark matter abundance jointly, but the axion dynamics can also exhibit dramatic modifications.
In Sec. \[sec:GeneralDescription\], we have given a general description of the type of wavefunction renormalization necessary to realize such a scenario. Namely, with an enhancement near the minimum of the axion potential, and near the edges, the necessary deformations in the canonical potential are generated. Specifically, for this has the effect of flattening the potential near its minimum while leaving its shape toward the edges of field space unaltered. The late-time mass is then suppressed by a factor of , while the effective chemical potential which efficiently drives spontaneous baryogenesis is retained. Moreover, the wavefunction enhancement also has the effect of suppressing the axion decay width by a factor of . As we have discussed, the culmination of these features is that the general arrangement in Sec. \[sec:GeneralDescription\] can yield the observed baryon asymmetry, while maintaining a sufficiently light and stable axion dark matter candidate. We have investigated the production of baryon asymmetry — both in-equilibrium and out-of-equilibrium — and found that both cases present compelling possibilities.
Meanwhile, to interpolate between the two regions of the wavefunction, we implemented a simple power-law form . As a result, we have shown that the axion exhibits a “tracking” behavior as it transits through this region. The field follows an attractor-like trajectory in which its late-time evolution is made increasingly insensitive to initial conditions. This phenomenon implies not only an axion relic abundance which is insensitive to the initial misalignment angle, but also a suppression of its isocurvature perturbations. We also have described how the axion equation-of-state parameter during this period converges to a non-trivial value , which reflects the shape of the potential through its dependence on the parameter $n$.
In Sec. \[sec:AnExplicitModel\], we have supplied a “proof of concept” by constructing an explicit model using the five-dimensional continuum-clockwork axion, which serves as a realization of the more general scenario described in Sec. \[sec:GeneralDescription\]. In particular, by integrating out the heavy KK modes and examining the theory for the lightest four-dimensional axion, we have shown that such a model furnishes a wavefunction renormalization $Z(\theta)$ with similar properties to the case of Sec. \[sec:GeneralDescription\]. The small parameter $\epsilon$ that determines the deformation of the axion potential is mapped onto a factor $e^{\pi mR}$ in the clockwork theory, such that the scale of bulk and boundary masses $m$ and the size of the extra dimension $R$ together set the strength of the deformation. We have shown (see, for example, Fig. \[fig:field\_dynamics\]) that spontaneous baryogenesis in this model is typically accomplished via out-of-equilibrium production, in contrast to many of the conventional spontaneous baryogenesis models in the literature. Moreover, we have also recovered the anticipated tracking dynamics in this model, as the clockwork parameters exceed . We have determined regions of phenomenological viability by producing a set of numerical simulations over the parameter space. Namely, in Fig. \[fig:Y\_B\] we have shown the produced baryon asymmetry and axion relic abundance over a range of parameters and found several viable regions.
We have also, at the close of Sec. \[sec:AnExplicitModel\], given a more thorough treatment of the large-scale isocurvature perturbations produced in this model, which include both axionic and baryonic components. The evolution of these components in the early universe is made non-trivial by the deformations to the potential. As anticipated in Sec. \[sec:GeneralDescription\], the axionic component is suppressed by tracking dynamics, and we have determined that in most regions the baryonic isocurvature component is dominant. Furthermore, we have demonstrated an interesting dependence of the perturbations on the initial misalignment angle ${\theta_{\text{in}}}$. There are certain critical points for ${\theta_{\text{in}}}$ where sign-changes can occur in the amplitude of either of the perturbation components, which can result in a suppression in that region. These points generally shift throughout the model parameter space. The culmination of these effects is a non-trivial bound imposed by the CMB isocurvature constraints. Although these bounds can be quite severe, we have found that the viable regions for the CCW model all remain below the isocurvature constraints (see Fig. \[fig:Y\_B\_iso\]).
To conclude, we have shown in this paper that an axion with a field-dependent wavefunction renormalization, which is enhanced near the minimum of the axion potential, can generate both the observed baryon asymmetry and dark matter relic abundance. Using the continuum-clockwork axion, we have constructed an explicit model realization of this idea. Our results also suggest directions for further research, including approaches with multiple scalar fields, where non-trivial dynamics can arise that significantly alters the effective chemical potential, , effects from temperature-dependent masses [@Dienes:2015bka; @Dienes:2016zfr]. Moreover, the CCW axion model constitutes only a single realization of the more general idea in this paper. A natural extension is to explore other models which yield similar non-canonical kinetic terms, but an altogether different set of phenomenological possibilities.
Acknowledgments\[sec:Acknowledgments\] {#acknowledgmentssecacknowledgments .unnumbered}
======================================
The authors thank Natsumi Nagata and Lauren Pearce for useful discussions. This work was supported by IBS under the project code IBS-R018-D1.
Tracking Dynamics\[sec:TrackingDynamics\]
=========================================
In this appendix, we provide a short review on cosmological tracking solutions and their general classification in terms of the scalar field potential. Throughout much of this review we closely follow the methodology and results of Ref. [@Steinhardt:1999nw]. We first give the necessary background for Sec. \[sec:GeneralDescription\] and the formalism used to deduce the class of potentials and regions in field space which exhibit tracking solutions \[see Eq. \]. Then, we tailor our analysis specifically to the continuum-clockwork axion example of Sec \[sec:AnExplicitModel\], showing that tracking solutions are a generic property of these potentials which drive the axion equation of state to that of the background .
Classification of Tracking Potentials\[subsec:Classification\]
--------------------------------------------------------------
A tracking field, by definition, is a field that converges to a given evolution in phase space, even under a variation in initial conditions. Typically, such attractor-like solutions are also associated with convergence of the equation-of-state parameter ${w_{\phi}}$ to some fixed value, but this ultimately depends on the background cosmology.
Let us consider a canonically normalized scalar field $\phi$ with potential $V(\phi)$ that evolves in an FRW spacetime: \[eq:appendixeqnofmotion\] + 3H + V’() = 0 . A useful parameter to define is the ratio of kinetic energy to potential energy of the scalar field: \[eq:xdef\] x = . After some rearrangement, Eq. can be recast as \[eq:appendixeqnofmotion2\] = M\_P[| |]{} - 1 . A tracking solution with a convergent equation of state requires that $x$ is approximately constant, such that the last term is small. The expression \[eq:trackingcond\] [| |]{} then dictates the tracking trajectory, where in the last approximation we implicitly assumed .
Naturally, for the tracking condition in Eq. to remain satisfied as the system evolves, both sides of the relation must change in the same way. Therefore, differentiating the equation of motion with respect to $\phi$, and demanding still that $x$ varies negligibly with time, we arrive at the relation \[eq:Gammadef\] 1 + . The necessary (but not sufficient) condition is that a region of the potential may yield tracking solutions if the dimensionless quantity $\Gamma$ does not vary appreciably over that field range. It acts as a determinant for classifying the features of different tracking regions and it does this *only through properties of the potential*, without reference to any dynamical information. In particular, the equation-of-state parameter to which the tracker converges is found by rearranging the above expression: \[eq:trackingw\] [w\_]{} , which is determines the evolution of ${\Omega_{\phi}}$ during tracking.
The condition in Eq. is not always sufficient because it does not guarantee that the tracking solutions are stable under small perturbations to the equation of state. An analysis shows that > 1 - is required for stable tracking solutions.[^7] Moreover, within the above range there are two distinctive behaviors. In the case that , the equation of state for the scalar field is less kinetic than the background , so the abundance ${\Omega_{\phi}}$ grows during tracking. On the other hand, in the case we find instead, and the abundance falls during that epoch. The “borderline” scenario of is also an interesting critical case for which ${w_{\phi}}$ is driven to match the background, and ${\Omega_{\phi}}$ does not evolve at all. This borderline case is found with potentials that have an exponential region. Incidentally, this is approximately the scenario we find in the continuum-clockwork axion example of Sec. \[sec:AnExplicitModel\], for which we now briefly specialize our discussion.
Tracking with Continuum-Clockwork Axion\[subsec:CCWTracking\]
-------------------------------------------------------------
Let us now examine the continuum-clockwork example of Sec. \[sec:AnExplicitModel\] and use the analysis above the identify any tracking regions for that potential. It is perhaps instructive to first consider , , the standard sinusoidal axion potential . Using the definition in Eq. we find that = 1 - . Regardless of how slowly this function varies throughout field space, it is bounded from above by and thus never can admit stable tracking solutions.
On the other hand, allowing for sufficiently large such that , we can approximate \[eq:clockworkGamma\] 1 - \^2() < 1 .
Although $\Gamma$ is always less than unity, for field values larger than we can achieve and thus find stable tracking solutions. This is easy to accomplish for if $mR$ is moderately large. Additionally, we must check that $\Gamma$ is slowly varying over a Hubble time: [| |]{} [| () |]{}\^2() 1 , where $N_e$ is the number of efolds and we used Eq. . In the field range where Eq. is viable, the above condition is easily satisfied as well, and we can therefore *always identify a tracking region of the CCW axion potential* for .
Indeed, the above analysis confirms the findings of our numerical simulations in Sec. \[sec:AnExplicitModel\], including the fact that the axion equation of state always appears radiation-like during the tracking period. Using Eq. , we find [w\_]{}w + (1+w)\^2() , which in the proper field range matches the background to an excellent approximation.
Boltzmann Equations for B-L at High Temperature\[sec:BoltzmannEquations\]
=========================================================================
In this appendix, we derive the effective chemical potential ${\mu_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}}$ used in Eq. , taking into account the details of sphaleron transitions in the Boltzmann evolution. To begin, let us consider a species $X$ which is in kinetic equilibrium at temperature $T$. Assuming some chemical potential $\mu_X$, the asymmetry in number density between particles and antiparticles is described by either Fermi-Dirac $(+)$ or Bose-Einstein $(-)$ statistics as $$\begin{aligned}
n_X = g_X \int \frac{d^3 \vec p}{(2\pi)^3} &\left[\frac{1}{\exp\left[\left(E_X - \mu_X\right)/T\right] \pm 1}\right. {\nonumber}\\
&\hspace{2mm}-\left.\frac{1}{\exp\left[\left(E_X + \mu_X\right)/T\right] \pm 1}\right] \ , \end{aligned}$$ where $g_X$ is the number of degrees of freedom for the species and is the energy. At high temperature , this is well-approximated by n\_X {
[ll]{} g\_X \_X T\^2/6 & \
g\_X \_X T\^2/3 &
. , such that a proportionality exists between the chemical potential and the number density for the species.
Let us now consider that this species is involved in some chemical process $A$, according to A: X + i + j + . Naturally, if the reaction is sufficiently rapid and it reaches chemical equilibrium, then the associated chemical potentials satisfy algebraic relations \[che\_eq\] d\_[A,X]{} \_X + d\_[A, i]{} \_i + d\_[A, j]{} \_j+= 0 , where $d_{A,X}$ ($d_{A,i}$) denotes the multiplicity of $X$ ($i$) and the signs determine the direction of the reaction. In the case of a spatially homogeneous and spontaneous violation of $CPT$ symmetry, as studied in this paper, these relations are *sourced* by an effective chemical potential $\mu_A$. That is, we instead have the relations d\_[A,X]{} \_X + d\_[A,i]{} \_i +d\_[A,j]{} \_j+ + \_A = 0 . The corresponding out-of-equilibrium evolution for the number density $n_X$ is given by the Boltzmann equation $$\begin{aligned}
\label{che_boltzmann}
\dot n_X &+ 3 H n_X \\
&=-\sum_A d_{A,X}\gamma_A\left(d_{A,X}\,\frac{\mu_X}{T}+d_{A,i}\,\frac{\mu_i}{T}+\cdots+\frac{\mu_A}{T}\right) \ . {\nonumber}\end{aligned}$$ where $\gamma_A$ is the thermally averaged interaction rate density for the process $A$ normalized by $T^3$, and the sum is over all the chemical processes involving $X$. We can solve the coupled Boltzmann equations with some set of sources $\{\mu_A\}$ and obtain any of the number densities or chemical potentials in the process, , the lepton- and baryon-number density $n_L$ and $n_B$.
Considering that all processes preserve the gauge symmetry during baryogenesis, the chemical potentials for the gauge bosons all vanish, and we can impose other additional constraints. In particular, the expectation value for the hypercharge over the chemical potentials should vanish: \[eq:U1Y\] \_i (\_[q\_i]{} + 2\_[u\_i]{} - \_[d\_i]{} - \_[\_i]{} -\_[e\_i]{})+ 2[\_]{}= 0 , where, respectively, $i$ is a flavor index, $q$ and $\ell$ are left-handed quark and lepton doublets, $u$ and $d$ are right-handed up and down quarks, $e$ is a right-handed electron, and ${\mathcal{H}}$ is the Higgs boson.
Under the above constraint, we can show that the quark number densities evolve according to $$\begin{aligned}
\label{eq:full_boltz1}
\dot n_{q_i}+ 3 H n_{q_i} =
&-\frac{\gamma_{\lambda_{u_i}}}{T}\left(\mu_{q_i} -\mu_{u_i} +{\mu_{\mathcal{H}}}\right) {\nonumber}\\
&-\frac{\gamma_{\lambda_{d_i}}}{T}\left(\mu_{q_i} -\mu_{d_i} -{\mu_{\mathcal{H}}}\right) {\nonumber}\\
&-2\frac{{\gamma_{\text{ss}}}}{T} \sum_j \left(2\mu_{q_j}-\mu_{u_j} -\mu_{d_j}\right) {\nonumber}\\
&-3\frac{{\gamma_{\text{ws}}}}{T}\Big[\sum_j \left(3\mu_{q_j} + \mu_{\ell_j}\right) +{\mu_{\text{ws}}}\Big] \end{aligned}$$ and $$\begin{aligned}
\dot n_{u_i}+ 3 H n_{u_i} =
&\phantom{+}\frac{\gamma_{\lambda_{u_i}}}{T}\left(\mu_{q_i} -\mu_{u_i} +{\mu_{\mathcal{H}}}\right) {\nonumber}\\
&+\frac{{\gamma_{\text{ss}}}}{T} \sum_j \left(2\mu_{q_j}-\mu_{u_j} -\mu_{d_j}\right) {\nonumber}\\
\dot n_{d_i}+ 3 H n_{d_i} =
&\phantom{+}\frac{\gamma_{\lambda_{d_i}}}{T}\left(\mu_{q_i} -\mu_{d_i} -{\mu_{\mathcal{H}}}\right) {\nonumber}\\
&+\frac{{\gamma_{\text{ss}}}}{T} \sum_j \left(2\mu_{q_j}-\mu_{u_j} -\mu_{d_j}\right) \ ,\end{aligned}$$ while the lepton number densities evolve as $$\begin{aligned}
\label{eq:full_boltz2}
\dot n_{\ell_i}+ 3 H n_{\ell_i} =
&-\frac{\gamma_{\lambda_{e_i}}}{T}\left(\mu_{\ell_i} -\mu_{e_i} -{\mu_{\mathcal{H}}}\right) {\nonumber}\\
&-\frac{{\gamma_{\text{ws}}}}{T} \Big[ \sum_j \left(3\mu_{q_j} + \mu_{\ell_j}\right) +{\mu_{\text{ws}}}\Big] {\nonumber}\\
&-\sum_j\frac{{{\gamma_{L\hskip -0.15cm\slash}}}_{ij}}{T}\left(\mu_{\ell_i} +\mu_{\ell_j} + 2{\mu_{\mathcal{H}}}\right) {\nonumber}\\
\dot n_{e_i}+ 3 H n_{e_i} = &\phantom{+}\frac{\gamma_{\lambda_{e_i}}}{T}\left(\mu_{\ell_i} -\mu_{e_i} -{\mu_{\mathcal{H}}}\right) \ . \end{aligned}$$ In the above, the rate densities $\gamma_{\lambda_{u_i}}$, $\gamma_{\lambda_{d_i}}$, and $\gamma_{\lambda_{e_i}}$ correspond to Yukawa interactions in the SM, while the other rate densities ${\gamma_{\text{ss}}}$ and ${\gamma_{\text{ws}}}$ correspond to strong and weak sphalerons. The source of ${(B-L)}$ violation in this paper is the Weinberg operator in Eq. , for which we denote the rate density as ${{\gamma_{L\hskip -0.15cm\slash}}}_{ij}$. The one remaining unspecified quantity ${\mu_{\text{ws}}}$ is related to the spontaneous breaking of the $CPT$ symmetry through Eq. . As the axion field rolls down its potential, it induces this effective chemical potential for the weak sphalerons: [\_]{}= \_0 .
Adding the various contributions from the Boltzmann equations above, we can determine the number-density evolution for baryons $n_B$ and leptons $n_L$ as $$\begin{aligned}
\label{eq:boltz_nb_nl}
\dot n_B + 3 H n_B =
&-3\frac{{\gamma_{\text{ws}}}}{T}\Big[\sum_i (3\mu_{q_i}+\mu_{\ell_i})+{\mu_{\text{ws}}}\Big] {\nonumber}\\
\dot n_L + 3 H n_L =
&-3\frac{{\gamma_{\text{ws}}}}{T}\Big[\sum_i (3\mu_{q_i}+\mu_{\ell_i})+{\mu_{\text{ws}}}\Big] {\nonumber}\\
&-\sum_{ij}\frac{{{\gamma_{L\hskip -0.15cm\slash}}}_{ij}}{T}(\mu_{\ell_i}+\mu_{\ell_j} + 2{\mu_{\mathcal{H}}}) \ .\end{aligned}$$ It is instructive to comment on the limit where the weak sphaleron rate is negligibly small. Taking in these equations, we find that the evolution of $n_B$ becomes trivial and that if the initial baryon number is zero. In this limit, the equation for lepton number also loses source terms, implying $n_L$ is also vanishing [@Shi:2015zwa].
With the hypercharge constraint from Eq. , and vanishing initial conditions , we can in principle solve the coupled Boltzmann equations numerically. However, we can also simplify them through some physical considerations. Let us assume that the Yukawa interactions for $N_f$ generations of fermions are in equilibrium, in addition to all gauge interactions and the strong and weak sphalerons. However, we shall ignore the Yukawa interactions of the remaining generations during baryogenesis. In such a case, baryon and lepton number are mostly generated by sphaleron processes in conjunction with axion dynamics, which leads approximately to the flavor-universal contributions n\_[B\_i]{} n\_B n\_[L\_i]{} n\_L . Furthermore, the interactions that violate ${(B-L)}$ are not flavor-diagonal. Instead, they are flavor-democratic, such that the off-diagonal components are determined by the PMNS neutrino mixing matrix. We can therefore simplify the ${(B-L)}$ rate density to \_[ij]{} [\_[L-0.15cm/]{}]{} , for all lepton flavors, where we defined ${\gamma_{L\hskip -0.15cm\slash}}$ in Eq. .
Taking these simplifications into account, we can compute the necessary chemical potentials. In particular, for the Higgs we find [\_]{}= , while for the $N_f$ generations of quarks and leptons with Yukawa interactions in equilibrium we have $$\begin{aligned}
\label{eq:chem_matter_Nf}
\mu_{u_i} &= \frac{(9+N_f)n_L-(6-5N_f)n_B}{2(3+ 5N_f) T^2} {\nonumber}\\
\mu_{d_i} &= \frac{(12 + 5N_f)n_B - (9+N_f)n_L}{2(3+ 5N_f) T^2} {\nonumber}\\
\mu_{\ell_i} &= \frac{7(1+N_f)n_L-3n_B}{2(3+ 5N_f) T^2} {\nonumber}\\
\mu_{e_i} &= \frac{3n_B- (1-3N_f)n_L}{(3+ 5N_f) T^2} \ .\end{aligned}$$ and for the remaining generations: $$\begin{aligned}
\mu_{u_i} = \mu_{d_i} &= \frac{n_B}{2T^2} {\nonumber}\\
\mu_{\ell_i} &= \frac{n_L}{T^2} {\nonumber}\\
\mu_{e_i} &= 0 \ .\end{aligned}$$ Meanwhile, the chemical potential for the left-handed quark doublets is independent of $N_f$: \_[q\_i]{} = .
The $n_B$ and $n_L$ number densities are related to each other by weak sphaleron processes: $$\begin{aligned}
n_B &= \frac{(18+31N_f-3N_f^2)n_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}- 2(3+ 5N_f){\mu_{\text{ws}}}T^2}{45 + 73N_f - 3N_f^2} {\nonumber}\\
n_L &= \frac{-3(9+14 N_f)n_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}- 2(3+ 5N_f){\mu_{\text{ws}}}T^2}{45 +73N_f - 3N_f^2} \ .\end{aligned}$$ The evolution of $n_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ is determined by the difference between the equations in Eq. and the chemical potentials above: \_[[B-L]{}]{}+ 3Hn\_[[B-L]{}]{}= -[\_[L-0.15cm/]{}]{}(n\_[[B-L]{}]{}- n\_[[B-L]{}]{}\^) , where the rate is given by [\_[L-0.15cm/]{}]{}= and the equilibrium number density is given by \[eq:nBmLeqapp\] n\_[[B-L]{}]{}\^ = - [\_]{}T\^2 . The above expression provides us with the coefficient that appears in Eq. . We are now equipped to compute the final number density $n_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}$ and therefore the final baryon asymmetry. In particular, after the weak sphalerons decouple at : $$\begin{aligned}
n_B =\frac{28}{79} n_{{B\hspace{-0.03cm}-\hspace{-0.03cm}L}}\ .\end{aligned}$$
[^1]: In circumstances where sphaleron processes are in equilibrium, $B$ violation is replaced by ${(B-L)}$-violation (where $L$ is lepton number), so that baryogenesis can also occur via leptogenesis.
[^2]: For example, in the case of the QCD axion, $f$ is associated with the scale at which the Peccei-Quinn ${\text{U}(1)}_{\text{PQ}}$ symmetry is spontaneously broken, and $\Lambda$ with the confinement scale of QCD.
[^3]: We shall often assume a temperature regime sufficiently high for this dynamics that the effective relativistic $g_*$ and entropy degrees of freedom $g_{*S}$ are approximately equal.
[^4]: In the explicit model realization covered in Sec. \[subsec:IsocurvaturePerturbations\], we give a more rigorous illustration of this phenomenon, supplemented by a full numerical simulation of the system of perturbations.
[^5]: In the formalism below we rely heavily on Ref. [@Choi:2017ncj].
[^6]: Similar effects have appeared in the literature on investigations of quintessence field perturbations [@Abramo:2001mv; @Kawasaki:2001nx; @Copeland:2006wr].
[^7]: More specifically, for a matter-dominated epoch this implies and for a radiation-dominated epoch .
|
---
abstract: 'The [NEMO]{}High Performance Computing Cluster at the University of Freiburg has been made available to researchers of the [ATLAS]{}and [CMS]{}experiments. Users access the cluster from external machines connected to the World-wide LHC Computing Grid (WLCG). This paper describes how the full software environment of the WLCG is provided in a virtual machine image. The interplay between the schedulers for [NEMO]{}and for the external clusters is coordinated through the [[ROCED]{}]{}service. A cloud computing infrastructure is deployed at [NEMO]{}to orchestrate the simultaneous usage by bare metal and virtualized jobs. Through the setup, resources are provided to users in a transparent, automatized, and on-demand way. The performance of the virtualized environment has been evaluated for particle physics applications.'
author:
- Felix Bührer
- Frank Fischer
- Georg Fleig
- Anton Gamel
- Manuel Giffels
- Thomas Hauth
- Michael Janczyk
- Konrad Meier
- Günter Quast
- Benoît Roland
- Ulrike Schnoor
- Markus Schumacher
- Dirk von Suchodoletz
- Bernd Wiebelt
date: 20 December 2018
title: Dynamic Virtualized Deployment of Particle Physics Environments on a High Performance Computing Cluster
---
Introduction {#intro}
============
Particle physics experiments at the Large Hadron Collider (LHC) need a great quantity of computing resources for data processing, simulation, and analysis. This demand will be growing with the upcoming High-Luminosity upgrade of the LHC [@HLLHCcompneeds]. To help fulfill this requirement, High Performance Computing (HPC) resources provided by research institutions can be useful supplements to the existing World-wide LHC Computing Grid (WLCG) [@wlcg] resources allocated by the collaborations.
This paper presents the concepts and implementation of providing a HPC resource, the shared research cluster [NEMO]{} [@nemo] at the University of Freiburg, to ATLAS and CMS users accessing external clusters connected to the WLCG with the purpose of accommodating data production as well as data analysis on the HPC host system. The HPC cluster [NEMO]{}at the University of Freiburg is deploying an [`OpenStack`]{} [@Openstack] instance to handle the virtual machines. The challenge is in provisioning, setup, scheduling, and decommissioning the virtual research environments (VRE) dynamically and according to demand. For this purpose, the schedulers on [NEMO]{}and on the external resources are connected through the [[ROCED]{}]{}service [@ROCED].
A VRE in the context of this paper is a complete software stack as it would be installed on a compute cluster fitted to the demands of ATLAS or CMS workloads.
Virtualization infrastructure {#sec:openstack}
=============================
Hardware virtualization has become mainstream technology over the last decade as it allows to host more than one operating system on a single server and to strictly separate users of software environments. Hardware and software stacks are decoupled, such that complete software environments can be migrated across hardware boundaries. While widespread in computer center operation this technique is rarely applied in HPC.
Computing at the University of Freiburg
---------------------------------------
The computer center at the University of Freiburg provides medium scaled research infrastructures like cloud, storage, and especially HPC services adapted to the needs of various scientific communities. Significant standardization in hardware and software is necessary for the operation of compute systems comprised of more than 1000 individual nodes combined with a small group of administrators.
The level of granularity of the software stack provided is not fine enough to directly support the requirements of world-wide efforts like the ATLAS or CMS experiments. Therefore, novel approaches are necessary to ensure optimal use of the system and to open the cluster to as many different use-cases as possible without increasing the operational effort. Transferring expertise from the operation of the established local private cloud, the use of [`OpenStack`]{}as a cloud platform has been identified as a suitable solution for [NEMO]{}. This approach provides a user defined software deployment in addition to the existing software module system. The resulting challenges range from the automated creation of suitable virtual machines to their on-demand deployment and scheduling.
Research Cluster [NEMO]{}
-------------------------
The research cluster [NEMO]{}is a cluster for research in the federal state of Baden-Württemberg in the scientific fields of Elementary Particle Physics, Neuroscience and Microsystems Engineering. Operation started on August 1, 2016. It currently consists of 900 nodes with 20 physical cores and 128GiB of RAM each. Omni-Path [@Omnipath] spans a high-speed, low-latency network of 100Gbit/s between nodes. The parallel storage has 768TB of usable capacity and is based on [<span style="font-variant:small-caps;">BeeGFS</span>]{} [@BeeGFS].
A pre-requirement to execute a VRE is the efficient provisioning of data which has to cross institutional boundaries in the CMS use-case. A signficant bandwidth is needed to transfer the input data into the VRE from the storage system at the Karlsruhe Institute of Technology (KIT) and to store back the results. The NEMO cluster is connected with two 40Gbit/s links to the main router of the University of Freiburg which itself is linked to the network of scientific institutions in Baden-Württemberg, BelWü, at 100Gbit/s.
Separation of software environments
-----------------------------------
The file system of a VRE is a disk image presented as a single file. From the computer center’s perspective this image is a “black box” requiring no involvement or efforts like updates of the operating system or the provisioning of software packages of a certain version. From the researcher’s perspective the VRE is an individual virtual node whose operating system, applications and configurations as well as certain hardware-level parameters, e.g. CPU and RAM, can be configured fully autonomously by the researcher within agreed upon limits.
To increase the flexibility in hosted software environments, the standard bare metal operation of [NEMO]{}is extended with an installation of [`OpenStack`]{}components [@hpc-symp:2016]. The [NEMO]{}cluster uses Adaptive’s Workload Manager [`Moab`]{} [@Moab] as a scheduler of compute jobs. [`OpenStack`]{}as well can schedule virtual machines on the same nodes and resources. To avoid conflicts, it is necessary to define the master scheduler which decides the job assignment to the worker nodes. Both [`Moab`]{}and [`OpenStack`]{}are unaware that another scheduler exists within the cluster and there is no API which enables them to communicate with each other. Since the majority of users still use the bare metal HPC cluster, [`Moab`]{}is deployed as the primary scheduler. It allows for detailed job description and offers sophisticated scheduling features like fair-share, priority-based scheduling, detailed time limits, etc. [`OpenStack`]{}’s task is to deploy the virtual machines, but [`Moab`]{}will initially start the VRE jobs and the VRE job will instruct [`OpenStack`]{}to start the virtual machine on the reserved resources with the required flavor, i.e. the resource definition in [`OpenStack`]{}.
When a VRE job is submitted to the [NEMO]{}cluster, [`Moab`]{}first calculates the priority and the needed resources of the job and then inserts it into its queue. When the job is in line for execution and the requested resources are available, the job runs a script which then starts the VRE on the selected node within the resource boundaries. During the run-time of the VRE a monitoring script regularly checks if the VRE is still running and terminates the job when the VRE has ended. When the job ends, [`OpenStack`]{}gets a signal to terminate the virtual machine and the VRE job ends as well. Neither [`Moab`]{}nor [`OpenStack`]{}have access inside the VRE, so they cannot assess if the VRE is actually busy or idle. The software package [[ROCED]{}]{}(described in further detail in Section \[section:roced\]) has been introduced to solve this issue. It is used as a broker between different HPC schedulers, translating resources and monitoring usage inside the virtual machine, as well as starting and stopping VRE images on demand.
Generation of the VRE image
===========================
The VREs for ATLAS and CMS software environments consist in [`OpenStack`]{}containers in the format of compatible VM images. These images are provided in an automatized way allowing versioning and archiving of the environments captured in the images.
[`Packer`]{}combined with [`Puppet`]{}
--------------------------------------
One approach to generate the image is the open-source tool [`Packer`]{} [@packer], interfaced to the system configuration framework [`Puppet`]{} [@puppet]. [`Packer`]{}allows to configure an image based on an ISO image file using a [`kickstart`]{} [@kickstart] file and flexible script-based configuration. It also provides an interface to [`Puppet`]{}making it particularly convenient if an existing [`Puppet`]{}role is to be used for the images. If the roles are defined according to the hostname of the machine as is conventional in [`Puppet`]{}with [`Hieradata`]{}, the hostname needs to be set in the scripts supplied to [`Packer`]{}. Propagation of certificates requires an initial manual start of a machine with the same hostname to allow handshake signing of the certificate from the [`Puppet`]{}server.
[`Packer`]{}’s interface to [`Puppet`]{}allows a fully automated image generation with up-to-date and version-controlled configuration. At the end of the generation run, the image is transferred to the [`OpenStack`]{}image server.
Image generation using the [`Oz`]{}toolkit
------------------------------------------
Another option to employ a fully-automated procedure is to use the [`Oz`]{}toolkit [@OZ]. All requirements and configuration options of an image can be specified through a XML template file. The partitioning and installation process of the operating system is fully automated, as [`Oz`]{}will use the remote-control capabilities of the local hypervisor. After the installation of the operating system, additional libraries and configuration files can be installed. Once the image has been created, it is automatically compressed and uploaded to a remote cloud site. This technique allows to build images in a reproducible fashion, as all templated files are version controlled using `git`. Furthermore, existing template files are easy to adapt to new sites and experiment configurations.
Interfacing batch systems and virtual resources using [[ROCED]{}]{} {#section:roced}
===================================================================
While HPC systems with support for virtualized research environments and commercial cloud providers offer the necessities to acquire computing and storage capacity by dynamic resource booking, the computing needs of high energy physics research groups additionally require workflow management systems capable of maintaining thousands of batch jobs. Some cloud providers, for example Amazon with AWS Batch [@awsbatch], provide a service for workflow management, however these offers are often limited to one specific cloud instance. To dynamically distribute batch jobs to multiple sites and manage machine life-time on specific sites, a combination of a highly-scalable batch system and a virtual machine scheduler is desirable.
[[ROCED]{}]{}
-------------
{width="0.9\linewidth"}
Many capable batch systems exist today and they can be interfaced to virtualization providers using the cloud meta-scheduler [[ROCED]{}]{}(Responsive On-demand Cloud Enabled Deployment) which has been developed at the KIT since 2010 [@ROCED]. [[ROCED]{}]{}is written in a modular fashion in python and the interfaces to batch systems and cloud sites are implemented as so-called *Adapters*. This makes [[ROCED]{}]{}independent of specific user groups or workflows. It provides a scheduling core which collects the current requirement of computing resources and decides if virtual machines need to be started or can be stopped. One or more Requirement Adapters report the current queue status of batch systems to the central scheduling core. Currently, Requirement Adapters are implemented for the Slurm, Torque/Moab, HTCondor and GridEngine batch systems. The Site Adapters allow [[ROCED]{}]{}to start, stop, and monitor virtual machines on multiple cloud sites. Implementations exist for Amazon EC2, OpenStack, OpenNebula and Moab-based virtualization at HPC centers. Special care has been put into the resilience of [[ROCED]{}]{}: it can automatically terminate non-responsive machines and restart virtual machines in case some machines have dropped out. This allows VM setups orchestrated by [[ROCED]{}]{}with thousands of virtual machines and many tens of thousands of jobs to run in production environments. The modular design of [[ROCED]{}]{}is shown in Fig. \[fig-roced\].
Using [`HTCondor`]{}as front-end scheduler {#sec:ROCED:HTCondor}
------------------------------------------
The open-source project [`HTCondor`]{}provides a workload management system which is highly configurable and modular [@HTCondor]. Batch processing workflows can be submitted and are then forwarded by [`HTCondor`]{}to idle resources. [`HTCondor`]{}maintains a resource pool, which worker nodes in a local or remote cluster can join. Once [`HTCondor`]{}has verified the authenticity and features of the newly joined machines, computing jobs are automatically transferred. Special features are available to connect from within isolated network zones, e.g. via a Network Address Translation Portal, to the central [`HTCondor`]{}pool. The Connection Brokering (CCB) service [@HTCondorCCB] is especially valuable to connect virtual machines to the central pool. These features and the well-known ability of [`HTCondor`]{}to scale to O(100k) of parallel batch jobs makes [`HTCondor`]{}well suited as a workload management system for the use cases described in this paper.
The CMS group at the KIT is using [`HTCondor`]{}for scheduling the jobs to be submitted to [NEMO]{}. The VRE for CMS contains the [`HTCondor`]{}client `startd`. This client is started after the machine has fully booted and connects to the central [`HTCondor`]{}pool at KIT via a shared secret. Due to [`HTCondor`]{}’s dynamic design, new machines in the pool will automatically receive jobs and the transfer of the job configuration and meta-data files is handled via [`HTCondor`]{}’s internal file transfer systems.
Using [`Slurm`]{}as front-end scheduler
---------------------------------------
Alternatively to the approach described above, the open-source workload managing system [`Slurm`]{} [@Slurm] has been interfaced into [[ROCED]{}]{}by the ATLAS group at University of Freiburg. [`Slurm`]{}provides a built-in functionality for the dynamic startup of resources in the *Slurm Elastic Computing* module [@SlurmElastic]. However, this module is based on the assumption of a fixed maximum startup time of the machines. In the considered case, due to the queue in the host system, the start of a resource can be delayed by a significant, varying time period. In addition the transfer of information, such as error states, from one scheduler to the other, and therefore to the user, is limited. Therefore, [[ROCED]{}]{}has been chosen as the interface between the [`Moab`]{}scheduler on the host system and the [`Slurm`]{}scheduler on the submission side.
![Implementation of [[ROCED]{}]{}with [`Slurm`]{}on the Tier-3 cluster of the WLCG used by ATLAS researchers in Freiburg.[]{data-label="fig:slurmRocedBFG"}](virtualisierung_ROCED.png){width="0.95\linewidth"}
The scheduling system is illustrated in Fig. \[fig:slurmRocedBFG\]. For [`Slurm`]{}, it is necessary that each potential virtual machine is registered in the configuration at the time of start of the [`Slurm`]{}server as well as the client. [`Slurm`]{}configurations also need to be in agreement between server and client. Therefore, a range of hostnames is registered in the configuration in a way that is mapped to potential IP addresses of virtual machines. These virtual machines have a fixed number of CPUs and memory assigned and are registered under a certain [`Slurm`]{}partition. When a job is submitted to this partition and no other resource is available, information from the [`Slurm`]{}`squeue` and `sinfo` commands is requested and parsed for the required information.
Since the ATLAS Freiburg group comprises three sub-groups, each mapped to a different production account on [NEMO]{}, special care is taken to avoid interference of resources used by another account to ensure fair share on [NEMO]{}, while nevertheless allowing jobs from one group to occupy otherwise idle resources of another group.
[[ROCED]{}]{}determines the amount of virtual machines to be started and sends the corresponding VRE job submission commands to [`Moab`]{}. After the virtual machine has booted, the hostname is set to the IP dependent name which is known to the [`Slurm`]{}configuration. A `cron` job executes several sanity checks on the system. Upon successful execution of these tests, the [`Slurm`]{}client running in the VM starts accepting the queued jobs. After completion of the jobs and a certain period of receiving no new jobs from the queue, the [`Slurm`]{}client in the machine drains itself and the machine shuts itself down. The IP address as well as the corresponding hostname in [`Slurm`]{}are released and can be reused by future VREs.
Analysis of performance and usage
=================================
The ROCED-based solution described above has been implemented and put into production by the research groups at the University of Freiburg (Institute of Physics) and the KIT (Institute of Experimental Particle Physics). To prove the usefulness of this approach statistical analyses of the performance of the virtualized setup both in terms of CPU benchmarks and usage statistics have been conducted.
Benchmarks
----------
Benchmark tests are performed with the primary goal to measure the performance of the CPU for High Energy Physics applications. Alongside the legacy HEP-SPEC06 (HS06) benchmark [@Hepspec], the performance of the compute resources is furthermore evaluated with the ATLAS Kit Validation KV [@DeSalvo:2010zza], a fast benchmark developed to provide real-time information of the WLCG performance and available in the CERN benchmark suite [@Alef:2017jyx]. The primary target is to measure the performance of the CPU for High Energy Physics applications. The KV benchmark is making use of the simulation toolkit GEANT4 [@Agostinelli:2002hh] to simulate the interactions of single muon events in the detector of the ATLAS experiment and provides as ouput the number of events produced per second. It constitutes a realistic workload for High Energy Physics jobs.\
To assess the impact of the virtualization, the performance of the identical hardware configuration (20 cores Intel Xeon E5-2630 CPUs) has been determined either deployed via the standard bare metal operation on the [NEMO]{}cluster ([NEMO]{}bare metal) and on the ATLAS Tier-3 center in Freiburg (ATLAS Tier-3 bare metal), or as virtual machines on the [NEMO]{}cluster ([NEMO]{}VM). On the ATLAS Tier-3 bare metal and on the virtual machines running on the [NEMO]{}cluster, hyper-threading (HT) technology is activated. Both are using Scientific Linux 6 [@SL6] as the operating system. On the cluster [NEMO]{}bare metal jobs are restricted to 20 cores by cgroups, since the application mix is broader than on HEP clusters. The operating system is CentOS7 [@CentOS7]. The scores of the HEP-SPEC06 and KV benchmarks have been determined for these three configurations as a function of the number of cores actually used by the benchmarking processes. This number ranges from 2 to 40 for the ATLAS Tier-3 bare metal and for the [NEMO]{}VM, for which HT is enabled, and from 2 to 20 for the [NEMO]{}bare metal, for which HT is not implemented. The benchmarks have been run 20 times for each core multiplicity value, and the means and standard deviations of the corresponding distributions have been extracted.\
![Total score as a function of the core multiplicity for the HEP-SPEC06 (top) and KV (bottom) benchmarks for the ATLAS Tier-3 bare metal (blue open circles), the [NEMO]{}VMs (red full circles) and the [NEMO]{}bare metal (black open squares). The data points represent the average values of the benchmarks for each core multiplicity, and the vertical bars show the associated standard deviations.[]{data-label="bmk-total"}](benchmark-hepspec06-total.pdf "fig:"){width="44.00000%"} ![Total score as a function of the core multiplicity for the HEP-SPEC06 (top) and KV (bottom) benchmarks for the ATLAS Tier-3 bare metal (blue open circles), the [NEMO]{}VMs (red full circles) and the [NEMO]{}bare metal (black open squares). The data points represent the average values of the benchmarks for each core multiplicity, and the vertical bars show the associated standard deviations.[]{data-label="bmk-total"}](kv-BFG-BM-NEMO-VM-NEMO-BM-total.pdf "fig:"){width="44.00000%"}
The HEP-SPEC06 and KV results are presented in Figure \[bmk-total\] for the three configurations considered. The total scores of the two benchmarks are increasing until the maximum number of physical cores has been reached, and are characterized by a flattening increase afterwards. The scores of the virtual machines running on the [NEMO]{}cluster are only slightly lower than those obtained for the [NEMO]{}bare metal, and the loss of performance due to the virtualization does not exceed 10$\%$. For the VMs running on the [NEMO]{}cluster and the ATLAS Tier-3 bare metal, the interplay between the virtualization and the different operating systems leads to very similar scores for the two configurations, particularly for the KV benchmark, and the loss of performance is smaller than 10$\%$ as well.
Usage statistics
----------------
Fig. \[fig-frplots\] shows the utilization of virtual machines which were orchestrated by [[ROCED]{}]{}depending on the resource demands of the users of the KIT group. At peak times, up to 9000 virtual cores were filled with user jobs, consuming more than a half of the initial 16000 [NEMO]{}cores.
![Utilization of the shared HPC system by booted virtual machines. Up to 9000 virtual cores were in use at peak times. The fluctuations in the utilization reflects the patterns of the submission of jobs by the CMS users at the physics institute in Karlsruhe. The number of draining slots displays the amount of job slots still processing jobs while the rest of the node’s slot are already empty.[]{data-label="fig-frplots"}](NEMO_KIT_utiliztion.pdf){width="1.0\linewidth"}
The usage of the hybrid cluster model is presented in Fig. \[fig-nodeusage\]. The diagram shows the shared usage of [NEMO]{}’s cluster nodes running either bare-metal or virtualized jobs. The part of the cluster which runs virtualized jobs changes dynamically from job to job, since the VREs are started by a standard bare-metal job.
At the beginning the cluster was only containing the operating system and some basic development tools. Scientific software was added after the cluster was already in production mode. Since the VRE for the CMS project was already available when the [NEMO]{}cluster started, it could already use the whole cluster while other groups still had to migrate from other ressources. This explains the high usage by VREs in the first months of operation. With more and more software being available for bare-metal usage the fraction of VRE jobs decreased.
![Estimated usage of the [NEMO]{}cluster in the time from September 2016 to September 2018. The orange bars indicate the usage by jobs running directly in the hosts’ operating system, while the blue bars are jobs running in virtual machines. The decrease of VRE jobs is partially explained by an increasing number of bare metal jobs submitted.[]{data-label="fig-nodeusage"}](NodeUsage_2016-09_2018-09.pdf){width="\linewidth"}
Conclusions and Outlook
=======================
A novel system for the dynamic, on-demand provisioning of virtual machines to run jobs in a high energy physics context on an external, not dedicated resource as realized at the HPC cluster [NEMO]{}at the University of Freiburg has been implemented. An interface between the schedulers of the host system and the external systems from which requests are sent is needed to monitor and steer jobs in a scalable way. For this workflow the cloud meta-scheduler [[ROCED]{}]{}has been implemented and deployed for the described use-cases. The approach can be adapted to work with other platforms and could be extended to container technologies like Singularity [@VRE2017].
The CPU performance and usage of the setup have been analyzed for the job execution environment. The expected performance loss due to the virtualization has been found to be sufficiently small to be compensated by the added flexibility and other benefits of this setup.
A possible extension of such a virtualized setup is the provisioning of functionalities for snapshots and migration of jobs. This would facilitate the efficient integration of long-running monolithic jobs into HPC clusters.
The provided solution extends the available compute ressources for HEP calculations and could be one possibilitly to cope with new data from the upcoming High-Luminosity upgrade of the LHC. Since HEP VREs are perfect for backfilling this could be used on various cluster ressources.
This research is supported by the Ministry of Science, Research and the Arts Baden-Württemberg through the bwHPC grant and by the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG for the bwForCluster NEMO. The work of F.B. and T.H. was supported by the Virtual Open Science Collaboration Environment (ViCE) project MWK 34-7547.221 funded by the Ministry of Science, Research and the Arts Baden-Württemberg. The work of U.S. was supported by the Federal Ministry of Education and Research within the project 05H15VFCA1 “Higgs-Physik mit dem und Grid-Computing für das ATLAS-Experiment am LHC”. The work of T.H. was supported by the Federal Ministry of Education and Research within the project 05H15VKCCA “Physik bei höchsten Energien mit dem CMS-Experiment am LHC”.
ATLAS Collaboration (2018) Estimation of ATLAS computing needs vs predictable increase on a flat budget. ATLAS Public results <https://twiki.cern.ch/twiki/pub/AtlasPublic/ComputingandSoftwarePublicResults/cpuHLLHC.pdf>, accessed 2018-09-19
Eck C, Knobloch J, Robertson L, Bird I, Bos K, Brook N, Düllmann D, Fisk I, Foster D, Gibbard B, Grandi C, Grey F, Harvey J, Heiss A, Hemmer F, Jarp S, Jones R, Kelsey D, Lamanna M, Marten H, Mato-Vila P, Ould-Saada F, Panzer-Steindel B, Perini L, Schutz Y, Schwickerath U, Shiers J, Wenaus T (2005) LHC Computing Grid: Technical Design Report, CERN-LHCC-2005-024
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von Suchodoletz D, Wiebelt B, Meier K, Janczyk M (2016) Flexible HPC: bwForCluster [NEMO]{}, Proceedings of the 3rd bwHPC-Symposium Janczyk M, Wiebelt B, von Suchodoletz D (2017) Virtualized Research Environments on the bwForCluster [NEMO]{}, Proceedings of the 4th bwHPC Symposium
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|
---
abstract: 'We present a multi-wavelength study of the energetic interaction between the central active galactic nucleus (AGN), the intra-cluster medium, and the optical emission line nebula in the galaxy cluster Sérsic 159-03. We use X-ray data from , high resolution X-ray spectra and UV images from , H$\alpha$ images from the [*SOAR*]{} telescope, optical imaging, and and radio data. The cluster center displays signs of powerful AGN feedback, which has cleared the central regions ($r<7.5$ kpc) of dense, X-ray emitting ICM. X-ray spectral maps reveal a high pressure ring surrounding the central AGN at a radius of $r\sim15$ kpc, indicating an AGN driven weak shock. The cluster harbors a bright, 44 kpc long H$\alpha$+\[[N]{}\] filament extending from the centre of the cD galaxy to the north. Along the filament, we see low entropy, high metallicity, cooling X-ray gas. The gas in the filament has most likely been uplifted by ‘radio mode’ AGN activity and subsequently stripped from the galaxy due to its relative southward motion. Because this X-ray gas has been removed from the direct influence of the AGN jets, part of it cools and forms stars as indicated by the observed dust lanes, molecular and ionized emission line nebulae, and the excess UV emission.'
author:
- |
N. Werner$^1$[^1], M. Sun$^2$, J. Bagchi$^3$, S. W. Allen$^1$, G. B. Taylor$^4$[^2], S. K. Sirothia$^5$, A. Simionescu$^1$[^3], E. T. Million$^6$, J. Jacob$^7$, M. Donahue$^8$\
$^1$Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305-4085, USA\
and SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA\
$^2$Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA\
$^3$Inter University Center for Astronomy and Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune 411 007, India\
$^{4}$Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA\
$^5$National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Post Bag 3, Ganeshkhind, Pune 411007, India\
$^6$Department of Physics and Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL 35487, USA\
$^7$Department of Physics, Newman College, Thodupuzha 685 585, India\
$^8$Physics&Astronomy Department, Michigan State University, East Lansing, MI 48824-2320, USA
bibliography:
- 'clusters.bib'
title: 'Violent interaction between the AGN and the hot gas in the core of the galaxy cluster Sérsic 159-03'
---
X-rays: galaxies: clusters – galaxies: individual: Sérsic 159-03 – galaxies: intergalactic medium – cooling flows
Introduction
============
Supermassive black holes (SMBH) play a crucial role in galaxy formation. The correlation between the mass of the central SMBH and the bulge mass of their host galaxies suggests that their evolution is tightly coupled [@magorrian1998]. Outbursts of accreting SMBH (referred to as active galactic nuclei or AGN) disturb and heat the surrounding gas, lowering the accretion rate and, in some cases, driving so much gas out of the galaxy that the star formation is drastically reduced.
The most dramatic portrayal of the interaction between AGN and their surroundings can be seen in the distribution of X-ray emitting gas in nearby giant ellipticals and clusters of galaxies. X-ray images have revealed giant cavities and shocks in the hot gas produced by repeated outbursts of the central AGN [e.g. @fabian2003; @nulsen2005; @forman2005]. These AGN outbursts in principle provide enough power to offset radiative losses, suppress cooling, and prevent further star formation in the dense cores of clusters of galaxies [see e.g. @mcnamara2007]. Detailed X-ray imaging and 2D spectral mapping of cooling cores with cavities, X-ray bright filaments, and shock fronts, provides the most reliable means of measuring the energy injected into the hot intra-cluster medium (ICM) by AGN, and is a powerful tool to study the physics of the AGN feedback in general [see e. g. @randall2010; @million2010; @werner2010; @simionescu2009b].
Here we present a detailed multi-wavelength study of AGN feedback processes in the core of the nearby [z=0.0564; @maia1987], relatively poor, low mass cluster of galaxies Sérsic 159-03 (A S1101). The thermal properties and chemical enrichment of the hot ICM in this cooling core cluster have been studied in detail using [@kaastra2001; @kaastra2004; @deplaa2006]. Compared to other cooling core clusters, where the central temperature drops to $\sim$1/3 of the peak ambient value, Sérsic 159-03 has a relatively modest (factor of 1.5) temperature drop in the core [@sun2009]. The radio properties of this cluster have been reported by @birzan2008, in the study of a sample of 24 cooling cores. The bright optical emission line nebulae in the core of the cluster have been studied in detail by @crawford1992 and more recently by @oonk2010 using integral field spectroscopy.
Throughout the paper we adopt a flat $\Lambda_{\mathrm{CDM}}$ cosmology with $H_{0}=70$ km$\, $s$^{-1}\, $Mpc$^{-1}$ and $\Omega_{M}=0.3$, which implies a linear scale of 1.2 kpc arcsec$^{-1}$ at the cluster redshift of $z=0.0564$ [@maia1987]. Throughout the paper, abundances are given with respect to the solar values by @grevesse1998. All errors are quoted at the 68 per cent confidence level.
Frequency (MHz) Array Resolution (arcsec) P.A. (deg) rms Noise ($\mu$Jy beam$^{-1}$) Obs. date
----------------- ------- --------------------- ------------ --------------------------------- --------------------
244 GMRT $20.04\times8.92$ 11.0 1420 2007 November 2, 3
325 GMRT $17.67\times9.08$ -179.4 352 2009 May 16,17
617 GMRT $8.29\times3.13$ 9.7 250 2009 May 09,11
1400 VLA-B $10\times4$ 0.0 64 2006 August 8
8400 VLA-B $3.1\times0.72$ 8.3 21 2006 August 8
\[obs\]
Data reduction and analysis
===========================
Optical, H$\alpha$, and UV data
-------------------------------
To study the optical properties of the central cluster galaxy, we analyzed a 600 s observation available in the [*HST*]{} archive (proposal ID: 8719). The image was taken with WFPC2 using the F555W filter and placing the galaxy onto the WF3 chip.
Narrow-band optical imaging was performed with the 4.1 m [*SOAR*]{} telescope on September 13, 2009 (UT) using the [*SOAR*]{} Optical Imager (SOI). The night was photometric with a seeing of around 1. We used the 6916/78 CTIO narrow-band filter for the H$\alpha$+\[[N]{}\] lines, and the 6738/50 filter for the continuum. We took four 1200 s exposures with the 6916/78 filter and four 750 s exposures with the 6738/50 filter. We reduced each image using the standard procedures in the [IRAF MSCRED]{} package using EG 274 as a spectroscopic standard. The pixels were binned 2$\times2$, for a scale of 0.154 per pixel. More details on the SOI data reduction can be found in @sun2007.
We also analyzed near Ultra Violet data obtained by the Optical Monitor on 2002 November 20–21. Exposures were taken with the UVW1 (2400–3600 Å) and UVW2 (1800–2400 Å) filters.
Radio data
----------
The summary of radio observations with [^4] and , including the observation date, obtained beam, and rms noise, is shown in Table \[obs\]. The strong, stable calibrator 3C48 was used for absolute flux density and bandpass calibration at all frequencies. We tied the final flux density scale to the standard ‘Baars-scale’ [@baars1977]. The VLA data were obtained in the B configuration. Their calibration was performed using [AIPS]{} [@greisen2003] in the standard fashion, while imaging and self-calibration were performed using [Difmap]{} [@shepherd1995].
The data were reduced using [AIPS++]{} (version: 1.9). After applying bandpass corrections to the phase calibrator, gain and phase variations were estimated, and flux density, bandpass, gain and phase calibration were applied. The data for antennas with high errors in antenna-based solutions were examined and flagged over certain time ranges. Some baselines were also flagged based on closure errors on the bandpass calibrator. Channel and time-based flagging of data points corrupted by radio frequency interference (RFI) was applied using a median filter with a $6\sigma$ threshold. Residual errors above $5\sigma$ were also flagged after a few rounds of imaging and self calibration. The system temperature ($T_{\mathrm{sys}}$) was found to vary with antenna, the ambient temperature and elevation [@sirothia2009]. In the absence of regular $T_{\mathrm{sys}}$ measurements for antennas, this correction was estimated from the residuals of corrected data with respect to the model. The corrections were then applied to the data. The final images were made after several rounds of phase self calibration, and one round of amplitude self calibration, where the data were normalized by the median gain for all the data.
X-ray data
-----------
The observations of Sérsic 159-03 were taken in August 2009 using the Advanced CCD Imaging Spectrometer (ACIS). The total net exposure time after cleaning is 89.6 ks. We follow the data reduction procedure described in @million2009 and @million2010. Background spectra were extracted from the appropriate recent blank-sky fields available from the Chandra X-ray Center. These were normalized by the ratio of the observed and blank-sky count rates in the 9.5–12 keV band.
Background subtracted images were created in 13 narrow energy bands, spanning 0.5–7.5 keV. These were flat fielded with respect to the median energy for each image. The blank sky background fields were processed in an identical way to the source observation and reprojected to the same coordinate system. The background subtracted and exposure corrected images were co-added to create the broad band images.
For the spectral analysis, point sources were excluded. The individual regions for the 2D spectral mapping were determined using the Contour Binning algorithm [@sanders2006b], which groups neighboring pixels of similar surface brightness until a desired signal-to-noise threshold is met. In order to have small enough regions to resolve substructure and still have enough counts to achieve better than 7 per cent accuracy in the temperature determination, we adopted a signal-to-noise ratio of 35 ($\sim$1230 counts per region). Separate photon-weighted response matrices and effective area files were constructed for each region analyzed.
Spectral modeling has been performed with the [SPEX]{} package [[SPEX]{} uses an updated version of the MEKAL plasma model with respect to [XSPEC]{}, @kaastra1996] in the 0.6–7.0 keV band. To each bin we fitted a model consisting of absorbed [Galactic $N_{\mathrm{H}}=1.14\times10^{20}$ cm$^{-2}$, @kalberla2005] collisionally ionized equilibrium plasmas with temperature, spectral normalization (emission measure), and metallicity as free parameters.
RGS data
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![H$\alpha$+\[[N]{}\] image obtained with the 4.1 m SOAR telescope. The image shows a large 44 kpc long north-south filament, a smaller western filament, and a separate nebula at the northeast. The northern end of the large filament separates into two parallel structures. The contours of the 8.4 GHz radio emission are over-plotted in red. []{data-label="Halpha"}](HalphaN.pdf){width="0.85\columnwidth"}
The hot plasma in the core of Sérsic 159-03 exhibits a complex multi-temperature structure [@deplaa2006]. In order to determine the amount of low-temperature cooling X-ray gas in the core and obtain upper limits on the mass deposition rate, we analyze high spectral resolution Reflection Grating Spectrometer (RGS) X-ray data.
The RGS data were obtained on November 20–21 2002, with a net exposure time of 86.3 ks. The data were processed as described in @deplaa2006. Spectra were extracted from a 4 wide extraction region, centred on the optical centre of the galaxy. Because the RGS operates without a slit, it collects all photons from within the 4$\times\sim$12 field of view. Line photons originating at angle $\Delta\theta$ (in arcminutes) along the dispersion direction are shifted in wavelength by $\Delta\lambda=0.138\Delta\theta$ Å. Therefore, every line is broadened by the spatial extent of the source. To account for this spatial broadening in our spectral model, we produce a predicted line spread function (LSF) by convolving the RGS response with the surface brightness profile of the cluster derived from the EPIC/MOS1 image along the dispersion direction. Because the radial profile of a particular spectral line can be different from the overall radial surface brightness profile, the line profile is multiplied by a scale factor $s$, which is the ratio of the observed LSF to the expected LSF. This scale factor is a free parameter in the spectral fit. We fit the spectra in the 8–28 Å band.
Results
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Optical and UV properties
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The cooling core of Sérsic 159-03 harbors a remarkable H$\alpha$+\[[N]{}\] emission line filament system shown in Fig. \[Halpha\]. This system of emission line nebulae consists of a large 44 kpc long north-south filament extending to the radius of 35 kpc in the north and a smaller western structure extending out to 16 kpc. The total H$\alpha$+\[[N]{}\] flux is $5.9\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$. Assuming \[[N]{}\]$_{6583\AA}$/H$\alpha$ = 1 and \[[N]{}\]$_{6548\AA}$/\[[N]{}\]$_{6583\AA}$ = 0.35, the total H$\alpha$ flux is $\sim$$2.5\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$. The total luminosity of the H$\alpha$ line emission (with the \[[N]{}\] flux excluded) is $L_{\mathrm{H\alpha}}=2\times10^{41}$ ergs s$^{-1}$. Our measured H$\alpha$ flux is higher than the flux $f_{\mathrm{H}\alpha}=1.79\pm0.15\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$ measured by @mcdonald2010, who used a filter with a 10 times narrower bandpass targeting only the H$\alpha$ emission lines. A separate nebula can be seen 25 kpc to the northeast of the core.
Assuming a volume filling fraction of unity, a cylindrical geometry in the H$\alpha$ bright regions, optically thin ionized gas, isotropic radiation, and case B recombination at 10$^4$ K [@bhm1999], the total mass of the H$\alpha$ emitting gas is about $2.7\times10^9$ $M_\odot$. However, these optical filaments most likely consist of many thin threads with very small volume filling fractions [e.g. @fabian2008]. The inferred mass is therefore likely to be significantly overestimated.
The giant elliptical galaxy in the core of Sérsic 159-03 has its major axis aligned in the northeast-southwest direction (left panel of Fig. \[HST\]). The HST data reveal a 4.5 kpc long dust lane extending from the centre to the north, coincident with the bright H$\alpha$ emitting gas. It is displaced to the east from the core of the galaxy by about 1.2 ($\sim$1.4 kpc). The dust lane is clearly visible in the middle panel of Fig. \[HST\], where the HST image is divided by the best fit 2D elliptical de Vaucouleurs model. This ratio image also uncovers a fainter dust lane to the south of the core. The over-subtraction of the emission from center of the galaxy by the model shows that the surface brightness of the stellar core is flatter than the de Vaucouleurs profile. We do not detect optical emission from the AGN.
Near Ultra Violet images (NUV) from the Optical Monitor show diffuse emission which is co-spatial with the dust lanes and the emission line nebulae, providing a strong indication for current star-formation (right panel of Fig. \[HST\]). After correcting for coincidence, dead-time loss, and time sensitivity degradation, the total UVW1 luminosity within a radius of 7 arcsec is 3.7$\times10^{42}$ ergs s$^{-1}$. Based on @cardelli1989, the Galactic extinction in the UVW1 band was assumed to be 0.07 mag and the unknown internal extinction was set to zero. We measured the 2MASS J-band luminosity within the same aperture and subtracted the contribution from the passive stellar population from the empirical $L_{\mathrm{UVW1}} - L_{\mathrm{J}}$ relation derived by @hicks2005. We find a net NUV luminosity excess of $\sim$2.3$\times10^{42}$ ergs s$^{-1}$. Without correcting for internal extinction, we obtain a UVW2-UVW1 color $\sim$0.4 mag.
Radio morphology
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Frequency Array Flux density
(MHz) (Jy)
244 GMRT $3.360\pm0.20$
325 GMRT $2.024\pm0.20$
408 MOST $1.390\pm0.10$
617 GMRT $0.786\pm0.05$
843 MOST $0.472\pm0.014$
1425 VLA-B $0.235\pm0.020$
4752 ATCA $0.0469\pm0.007$
8400 VLA-B $ 0.0313\pm0.005$
\[radiofluxes\]
----------------- ------- -------------------
: Radio flux densities. The flux densities obtained by GMRT and VLA are from this work, the values obtained at 408 MHz and 843 MHz by MOST are from the MRC [@large1981] and SUMSS surveys [@bock1999; @mauch2003]. The flux density at 4.75 GHz obtained by [*ATCA*]{} is from Hogan et al. in preparation.
The contours of the 8.4 GHz radio emission in Fig. \[Halpha\] and \[radiospec\] resolve the source into an S-shape, oriented primarily north-south, with a compact central feature of flux density 6.3 mJy/beam, which is approximately the fifth of the total radio flux at this frequency. The jets seem to start out in the northeast-southwest direction, before bending clockwise. At 1.4 GHz the total flux density is $\sim$230 mJy and shows a core with an extension $\sim$12 arcseconds to the east (see the white contours in Fig. \[images\]). The centers of the radio and optical emission are well aligned. At the 617 MHz radio map, the morphology of the radio emission is nearly identical to 1.4 GHz. The extension to the east is robustly detected at 617 MHz and seems to be divided into two filaments. The lower frequency GMRT radio maps do not significantly resolve the radio source.
In the left panel of Fig. \[radiospec\] we show a red+green color image, produced from the radio data at 617 MHz (red) and 1.4 GHz (green) with the contours of the 8.4 GHz radio emission overplotted. This combined radio image clearly shows the extension to the east. The central panel of Fig. \[radiospec\] shows the spectral index map of the core of Sérsic 159-03 produced by combining the 617 MHz and the 1.4 GHz radio data. Only data above 2 mJy/beam at 617 MHz and 0.2 mJy/beam at 1.4 GHz were included. To the east, north-east and north-west of the core the radio emission has a very steep spectrum described by a power-law $S_{\nu} \propto \nu^{-\alpha}$ with index $\alpha>2.2$. The spectrum of the central region is flatter with index $\alpha\sim1$.
In Table \[radiofluxes\] we show the integrated radio flux densities at five different frequencies obtained with and . We also list the published flux densities obtained by the MRC survey at 408 MHz [@large1981] and by the SUMSS survey at 843 MHz [@bock1999; @mauch2003] performed with the Molonglo Observatory Synthesis Telescope (MOST). The integrated flux density at 4.75 GHz obtained by the Australia Telescope Compact Array (ATCA) is from Hogan et al. in preparation. The broad band radio spectrum is shown in the right panel of Fig. \[radiospec\]. Between 244 MHz and 1.4 GHz the integrated spectrum can be described by a power-law $S_{\nu} \propto \nu^{-\alpha}$ with $\alpha=1.49$. At high frequencies, between 4.8 GHz and 8.4 GHz, however, the best fit power-law is significantly flatter with index $\alpha=0.71$.
X-ray imaging: a disturbed morphology
-------------------------------------
\
\
The X-ray data show no evidence for a point source associated with the central radio source. Assuming a power-law like spectrum with a photon index $\Gamma=2.0$, a conservative upper limit on the X-ray flux in the 2–10 keV band is $f_{\mathrm{X}}=2.3\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$. The large scale X-ray emission from the cluster is elongated in the same northeast-southwest direction as the major axis of the cD galaxy. The X-ray emission does not show obvious sharp surface brightness discontinuities, indicative of cold fronts due to sloshing gas.
While the large scale X-ray morphology of Sérsic 159-03 is relatively relaxed, its core is strongly disturbed. The brightest, densest X-ray emitting gas is displaced northward from the centre of the cD galaxy to a radius of $r\sim8$ kpc. In the top left panel of Fig. \[images\], we show the smoothed image of the cluster core in the 0.5–7.5 keV band. The black cross indicates the position of the central AGN. In the top right panel, we show the same image, with the contours of the H$\alpha$+\[[N]{}\] emission over-plotted in black, and the contours of the 8.4 GHz and 1.4 GHz radio emission over-plotted in blue and white, respectively. The lower panels of Fig. \[images\] show the same images divided by their best fit 2D elliptical double beta model. The images shows a bright ridge of dense thermal X-ray gas displaced by about 8 kpc to the north of the AGN and a clumpy X-ray filament extending along the H$\alpha$ filament to 31 kpc beyond this ridge. Another X-ray filament extends to the west and coincides with the western optical emission line nebula.
The ridge of dense, thermal X-ray emitting gas to the north of the AGN seems to be interacting with, and confining, the 8.4 GHz radio emitting plasma (blue contours in Fig. \[images\]). The jets appear distorted by the interaction with the dense cooling gas. The 1.4 GHz radio plasma also appears deflected by the ridge of dense gas to the east, where it fills a gap - an elongated cavity - in the X-ray surface brightness distribution. This elongated cavity is possibly composed of two cavities. The X-ray surface brightness drops sharply to the southeast of the AGN, forming an apparent cavity with a radius of $\sim$11 kpc, filled by 1.4 GHz radio emission. This drop in surface brightness is spatially coincident with the sharp southeastern edge of the bright emission line nebulae. A possible ghost cavity with a radius of $\sim$10 kpc, with no associated radio emission, can be seen about 30 kpc to the east of the nucleus.
Thermodynamic properties of the core {#sect:thermo}
------------------------------------
The disturbed morphology of the cluster core is also reflected in the 2D maps of density, temperature, entropy, and pressure shown in Fig. \[thermo\]. The ICM spatially associated with the large north-south H$\alpha$ filament has a relatively low projected temperature (top right panel of Fig. \[thermo\]). The high density (top left panel of Fig. \[thermo\]) and low temperature of this feature translate into low entropy (lower left panel of Fig. \[thermo\]). The western filament, on the other hand, has a significantly higher temperature and entropy.
The projected thermal pressure peaks in an approximate ring surrounding the core (lower right panel of Fig. \[thermo\]). The pressure in this ring is approximately 20 per cent higher than in the center. The maps reveal that the relatively over-pressured gas to the southeast of the core has a higher temperature than the surrounding gas, which strongly indicates that it has been compressed and heated by an AGN driven shock.
The metallicity of the ICM along the H$\alpha$+\[[N]{}\] filaments is higher than that of the surrounding plasma (see Fig. \[metallicity\]). The apparent metallicities of the bright northern ridge of cooling plasma and of the core are, however, low. It has been shown that the metallicity is sensitive to the modeling of the underlying temperature structure and, if multi-temperature plasma is modeled with a single-temperature spectral model, the metallicity of gas can be significantly underestimated [e.g. @buote2000b]. The low metallicities of these features thus indicate the presence of multi-phase gas.
![Metallicity map of the core of the cluster with the contours of the H$\alpha$+\[[N]{}\] optical line emission over-plotted. The metallicity of the ICM along the filament is higher than that of the surrounding medium. The apparent metallicity of the bright ‘northern ridge’ and of the core is low, indicating the presence of multi-phase gas. []{data-label="metallicity"}](FeSN35.pdf){width="0.95\columnwidth"}
High resolution X-ray spectra
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Parameter 4T-model CIE+c.f.+c.f. model
---------------------------- --------------- --------------------- -- -- -- --
$Y_{\mathrm{0.25keV}}$ $0.05\pm0.04$ –
$Y_{\mathrm{0.75keV}}$ $0.19\pm0.03$ –
$Y_{\mathrm{1.5keV}}$ $0.7\pm0.3$ –
$Y_{\mathrm{3.0keV}}$ $12.1\pm0.3$ –
$Y_{CIE}$ – $12.3\pm0.2$
$\dot{M}_{\rm1.9-0.5keV}$ – $82\pm11$
$\dot{M}_{\rm 0.5-0.1keV}$ – $<25$
kT (keV) – $3.05\pm0.16$
s $1.12\pm0.13$ $1.13\pm0.11$
O $0.39\pm0.04$ $0.43\pm0.04$
Ne $0.43\pm0.12$ $0.44\pm0.11$
Fe $0.78\pm0.04$ $0.82\pm0.06$
\[RGStable\]
: Best fit parameters for a four-temperature fit and a CIE+two-cooling-flow model fit to the high-resolution RGS spectra extracted from a 4 wide region centred on the core of Sérsic 159-03. Emission measures, $Y=\int n_{\mathrm{H}}n_{\mathrm{e}}\mathrm{d}V$, are given in 10$^{66}$ cm$^{-3}$. Radiative cooling rates are given in $M_{\odot}$ yr$^{-1}$. The scale factor $s$ is the ratio of the observed LSF to the expected LSF based on the overall radial surface brightness profile. The upper limits are quoted at their 95 per cent confidence level. Abundances are quoted with respect to the values of @grevesse1998.
![The first order RGS spectrum extracted from a 4 wide region centred on the core of Sérsic 159-03. The continuous line represents the best fit model to the spectrum. [Fe]{} lines emitted by plasma with $kT<0.9$ keV are clearly visible in the spectrum. []{data-label="fig:rgs"}](RGS.pdf){width="1.1\columnwidth"}
In order to determine whether cooling X-ray gas with $kT<1$ keV is present in the core of the cluster, we examine the RGS spectra (see Fig. \[fig:rgs\]) to search for low temperature line emission (primarily [Fe]{}) tracing rapidly cooling gas. We fit the RGS data with a model consisting of collisionally ionized equilibrium plasmas at four fixed temperatures (0.25 keV, 0.75 keV, 1.5 keV, and 3 keV) with variable normalizations and common metal abundances. The abundances of O, Ne, Mg, and Fe are free parameters in the fit. Our fit confirms the presence of relatively cool, $kT<1$ keV plasma. Based on the lack of [O]{} line emission however, which is emitted at $kT<0.4$ keV, we place a strong upper limit on the amount of gas cooling radiatively to very low temperatures. Our 90% confidence upper limit on the emission measure of gas with $kT=0.25$ keV is $Y=\int n_{\mathrm{H}}n_{\mathrm{e}}dV=9\times10^{64}$ cm$^{-3}$, which is about 50% of the best fit emission measure for the 0.75 keV gas.
In order to place constraints on the amount of cooling in the cluster core we fit the RGS spectra with a separate model consisting of thermal plasma in collisional ionization equilibrium and two isobaric cooling flow models. The first cooling flow model is used to account for gas with temperatures between $kT_\mathrm{upper}=1.9$ keV and $kT_\mathrm{Lower}=0.5$ keV, which appears to be the ‘temperature floor’ in several cooling core clusters [@sanders2009; @sanders2009b; @werner2010]. The second cooling flow model is used to place an upper limit on the radiative cooling rate from 0.5 keV down to cold gas. While in the temperature range of 1.9–0.5 keV the data are formally consistent with a radiative cooling rate of $\dot{M}=82\pm11~M_{\odot}$ yr$^{-1}$, the 95% confidence upper limit for radiative cooling from 0.5 keV to lower temperatures is only 25 $M_{\odot}$ yr$^{-1}$.
Discussion
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Displacement of gas from the cD galaxy
--------------------------------------
The core of Sérsic 159-03 has a remarkably complex and rich morphology at all wavelengths. It displays signs of a powerful AGN feedback. The central regions of the galaxy ($r<7.5$ kpc) are cleared of the densest, X-ray emitting ICM and the cluster core displays a massive, bright H$\alpha$ filament extending northward from the centre of the cD galaxy to a radius of 35 kpc. This long filament is reminiscent of that in Abell 1795 [@fabian2001; @crawford2005; @mcdonald2009].
While the densest cooling X-ray plasma has been pushed away and uplifted from the cooling core by the radio jets to a radius of at least $r\sim8$ kpc, the brightest core of the H$\alpha$ emission is not displaced from the galactic nucleus. The most likely reason is that the H$\alpha$ emitting gas is only a thin layer on an underlying large reservoir of dense atomic and molecular gas at the base of the galaxy potential, which is difficult to uplift entirely. Observations of K-band emission lines of molecular and ionized hydrogen with the SINFONI integral field spectrograph reveal extended filaments in the core of Sérsic 159-03, tracing closely each other and the H$\alpha$ emission [@oonk2010]. The line-of-sight velocities, as measured by @oonk2010, show a striking east/west dichotomy, suggesting that part of the velocity vector of the jets, which push and uplift the gas, is oriented along our line of sight. The gas to the east of the stellar core has a radial velocity of $\sim$200 km s$^{-1}$ with respect to the cD galaxy, which seems to be decreasing with the distance from the nucleus.
The radial velocity distribution of the gas along the western filament goes smoothly from about -200 km s$^{-1}$ to 50 km s$^{-1}$ with increasing distance from the core. These velocity gradients indicate that the gas has been pushed out by the jets of the AGN.
The morphology of the cooling filamentary gas indicates relative gas motions with the ambient ICM moving toward the northwest, dragging and bending the H$\alpha$ filament. The gas uplifted by the jet in the southwestern direction has been dragged to the west and the gas uplifted towards northeast has been displaced in the north-northwestern direction. These relative gas motions are most likely the result of the south-southeastward peculiar motion of the cD galaxy.
Relatively cool X-ray gas is present along the whole northern filament of Sérsic 159-03, but the correlation between the H$\alpha$ and soft X-ray emission is not perfect. We see clumps of dense, cooling X-ray emitting plasma at $\sim$8 kpc and at $\sim$17 kpc, but no obvious H$\alpha$ peaks at the same locations. The ‘northern ridge’ at $r\sim8$ kpc has the coldest projected temperature and lowest entropy. The X-ray filament extends to the north by about 3 kpc further than the observed H$\alpha$ filament. The western H$\alpha$+\[[N]{}\] filament is also surrounded by bright, dense X-ray emitting gas, but its projected temperature is $\sim$0.8 keV higher. The relatively high metallicity of the filament compared to the ambient plasma (see Fig. \[metallicity\]) indicates that the filamentary system has been uplifted and stripped from the cD galaxy.
Gas uplift by buoyant radio-emitting plasma, supplied by the jets of the AGN, has also been observed in other cooling core clusters. Detailed spectroscopic mapping of the cooling core of the Virgo Cluster, centered on M87, shows clearly that radio mode AGN feedback is highly efficient in stripping the core of the galaxy of its lowest entropy gas [@werner2010]. The total mass of uplifted gas in M87 is 6–9$\times10^8$ $M_{\odot}$, which is similar to the current gas mass within its innermost $r\sim3.8$ kpc region. The disruption of the core of Sérsic 159-03 is considerably larger than that of M87, but by no means extreme. The AGN feedback in the Hydra A cluster is responsible for the uplift of a few times $10^9$ $M_{\odot}$ of low entropy plasma [@simionescu2009a]. Recently, @ehlert2010 presented a multi-wavelength study of the cluster MACS J1931.8-2634 where extreme AGN feedback with a jet power of $P_{\mathrm{jet}}\sim10^{46}$ erg s$^{-1}$ and the sloshing gas disrupted the core, separating it into two X-ray bright ridges, which are currently at a distance of $\sim$25 kpc from the core.
Cooling of the displaced gas and star-formation
-----------------------------------------------
To the north of the AGN, we see cooling X-ray plasma displaced by about 8 kpc from the nucleus, producing a prominent bright ridge which confines the high frequency radio plasma. The northern X-ray filament extends 31 kpc beyond this ridge. The relatively dense uplifted X-ray emitting gas, which is removed from the direct influence of the AGN jets, will cool in the absence of heating and eventually form stars. Multiphase cooling X-ray gas, displaced from the center of the cD galaxy and spatially coincident with H$\alpha$ emission, has also been seen in the Ophiuchus Cluster, in Abell 2052, and in MACS J1931.8-2634 [@million2010; @edwards2009; @deplaa2010; @ehlert2010].
Assuming that the filament has been uplifted at half of the sound speed (at 400 km s$^{-1}$), the age of the filament will be $\sim10^8$ yr. That is approximately equal to the cooling time of 1 keV plasma in pressure equilibrium with the ambient ICM at the observed location. The RGS spectra clearly reveal [Fe]{} line emission associated with gas cooling to $kT<1$ keV. The 95 per cent confidence upper limit on radiative cooling below 0.5 keV is 25 $M_{\odot}$ yr$^{-1}$.
The densest and coolest X-ray emitting clumps, in particular the bright multiphase ‘northern ridge’, will cool and form narrow H$\alpha$ emitting filaments [e.g. see simulations by @sharma2010]. It is possible that a significant fraction of the H$\alpha$ emitting gas in the northern filament at large radii is due to the cooling of the uplifted X-ray emitting plasma. The cold gas is likely to go on to eventually form stars. The OM UVW2 and images [@mcdonald2010] indicate that the UV emission is extended along the brightest regions of the northern H$\alpha$ filament, indicating ongoing star formation.
The HST images show that the brightest regions of the filament are dusty. An infrared survey with the [*Spitzer Space Telescope*]{}, however, did not detect the system at 70$\mu$m [@quillen2008] indicating that the filaments do not contain an exceptionally large amount of warm dust. The excess UV emission from the cD galaxy corresponds to a star-formation rate of $\sim$2.3 $M_\odot$ yr$^{-1}$ for the assumed Salpeter IMF. Assuming the @kennicutt1998 relation, SFR $(M_{\odot}$ yr$^{-1})=7.9\times10^{-42}$ ($L_{\mathrm{H}\alpha}$/erg s$^{-1}$), and neglecting intrinsic extinction due to dust, the measured H$\alpha$ luminosity corresponds to a star-formation rate of 1.5 $M_{\odot}$ yr$^{-1}$. Accounting for internal extinction would further increase the intrinsic H$\alpha$ and NUV fluxes. Star-formation, however, is most likely not the only heating and ionizing source in the cluster center.
The large turbulent velocities and the elevated \[[N]{}\]/H$\alpha$ ratio of $\sim$1.5 in the nucleus [@crawford1992] indicates that the interaction with the radio jets contributes strongly to the heating of the gas in the center. The \[[N]{}\]/H$\alpha$ line ratios of 0.6 in the western filament are relatively high as well [@crawford1992], indicating the presence of an additional non-ionizing source of energy at larger radii [see e.g. @ferland2009].
Based on the data for M87, @werner2010 proposed that the H$\alpha$ filaments in its core may be powered by shock induced mixing of cold gas with the surrounding ICM [see @begelman1990]. By bringing the hot thermal particles into contact with the cool gas, mixing can supply the power and ionizing particles to explain the observed spectra. Hot ICM electrons that penetrate into the cold gas excite the molecular hydrogen and deposit heat. This scenario has been explored theoretically by @ferland2008 [@ferland2009], who studied heating by cosmic-rays, which affect the cooler ionized and neutral components in a similar way to hot ICM electrons. The fact that the soft X-ray emission traces the H$\alpha$ emitting gas suggests that this process could be responsible for part of the H$\alpha$+\[[N]{}\] line emission, part of the UV emission, and for the non-radiative cooling of the coldest X-ray gas in Sérsic 159-03.
The most similar known X-ray/H$\alpha$ filament system is observed in Abell 1795. In that cluster the filament extends for 50 kpc. The cD galaxy at the head of this filament is moving with respect to the ICM and the emission line nebula may originate from a runaway cooling of the hot X-ray emitting gas in the wake of that motion [@fabian2001; @markevitch2001; @mcdonald2009]. The filamentary system in Sérsic 159-03, however, is most likely the result of AGN driven uplift and ram pressure stripping from the cD galaxy. The filament in Abell 1795 is composed of a pair of thin $w<1$ kpc intertwined H$\alpha$+\[[N]{}\] filaments [@mcdonald2009], spatially coincident with relatively cool X-ray emitting gas [@fabian2001], and with chains of FUV-bright young star clusters condensing from the cooling gas in the filament [@crawford2005; @mcdonald2009]. The northern end of the large H$\alpha$ filament in Sérsic 159-03 separates into two narrow, parallel structures. Given a better spatial resolution, we would most likely resolve the filament into more thin threads. As discussed by @fabian2008 for the emission line nebulae in the core of the Perseus Cluster, the thin thread-like filamentary structures are most likely stabilized by magnetic fields. These magnetic fields might be possible to detect using Faraday Rotation against the polarized jet emission [as has been done for the filaments in the Centaurus Cluster by @taylor2007].
The powerful radio mode AGN
---------------------------
The core of Sérsic 159-03 harbors a powerful radio mode AGN. Inflating the southern cavity required a $4pV$ work of about $7\times10^{58}$ ergs, indicating powerful jets. However, no optical or X-ray point source is presently seen at the position of the radio bright AGN. Using the ‘Black hole fundamental plane’ relation of @merloni2003, for a 10 mJy core flux at 4.75 GHz and for a black hole mass of $6\times10^8$ $M_{\odot}$ [determined from the K-band bulge luminosity by @rafferty2006], the expected X-ray core flux is $2.4\times10^{-13}$ erg s$^{-1}$ cm$^{-2}$. This value is two orders of magnitude higher than our conservative upper limit of $2.3\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$. The observed scatter around the ‘Black hole fundamental plane’ is, however, large. The radio luminosities of some other well known brightest cluster galaxies (e.g. NGC 1275) also show similar offsets with respect to the expected relation.
The radio morphology is similar to 4C26.42 in Abell 1795 [@liuzzo2009] and it is also reminiscent of PKS 1246-410 in the Centaurus Cluster [@taylor2007]. The distortion of the jets, and the change in the axis from N-S to E-W, is most likely due to the strong interactions with the dense gas. The velocity dispersion of the NIR emission lines, which sharply increases in the nuclear region [@oonk2010] shows that the interaction of the jets with the cold gas is the strongest within the innermost 2 kpc region. The presence of cold, high density material in the cluster core is likely to decelerate the jets, which might entrain thermal gas, slow down to subsonic velocities, and continue to rise buoyantly. Such deceleration due to strong interaction with dense gas on small scales has been seen using VLBA observations in Hydra A [@taylor1996] and Abell 1795 [@liuzzo2009]. While the 1.4 GHz and 617 MHz radio plasmas appear to be deflected by the dense ‘northern ridge of cooling plasma’ to the east, forming the ‘eastern elongated X-ray dark cavity’, the younger radio jet seen at 8.4 GHz is being deflected to the northwest where it is likely to continue to buoyantly rise along the short axis of the cluster. The 8.4 GHz southern jet is being deflected by the H$\alpha$ emitting gas to the southeast where it is partly filling the ‘southern X-ray cavity’.
Between 244 MHz and 1.4 GHz the integrated radio emission in the cluster core has a remarkably steep power-law spectrum $S_{\nu} \propto \nu^{-\alpha}$ with index $\alpha=1.49$. At higher frequencies, however, the radio spectrum flattens to index $\alpha=0.71$. At the 4.75 GHz [*ATCA*]{} radio map (Hogan et al. in prep.), the radio morphology is very similar to that at 8.4 GHz, indicating that the flatter part of the integrated radio emission originates in the $\sim$10 kpc scale inner lobes (see the contours in the left panel of Fig. \[radiospec\]) where active jets are injecting relativistic particles. At the lower frequencies the radio emission is more extended, indicating an older population of electrons. The spectral index map produced from the 617 MHz and 1.4 GHz radio data (central panel of Fig. \[radiospec\]) shows that while in the central regions $\alpha\sim1$, to the east, where the radio plasma seems to be filling the elongated cavity, the spectral index steepens to $\alpha>2$. The spectral index also steepens in the northwest, where the plasma fills a region with a relative deficit in X-ray surface brightness. This radio plasma is most likely buoyant and exerting work on the surrounding medium.
The steep integrated spectrum of the radio source is similar to radio mini-halos found within the cooling radius of some clusters [e.g @ferrari2008]. The morphology of the extended radio emission, however, indicates that it is due to the plasma from the radio jets which have been decelerated, deflected and confined by the interaction with the dense gas. A steep radio spectral index has been found in other cooling core clusters [e.g., PKS 0745-191, A2029, A4059, A2597 @taylor1994; @pollack2005] and taken as an indicator of confinement.
Our thermodynamic maps show that the projected thermal pressure peaks in an approximate ring surrounding the AGN. The pressure in the ring is approximately 20 per cent higher than in the center. The high pressure region to the south of the AGN is hotter than the plasma both outside and inside this feature, strongly suggesting that it has been shock heated. The observed projected temperature jump of $\Delta T\sim1.2$ suggests a shock with a Mach number of $M\sim1.5$. The complex X-ray morphology with a cavity inside the shock front unfortunately prevents a more precise modeling of the shock. The relatively modest, factor of 1.5, temperature drop in the core of this cluster [@sun2009] also points towards a relatively strong AGN feedback activity. AGN induced shocks, which may be the most significant channel for heating of the ICM near to the AGN, have also been observed around M87 in the center of the Virgo Cluster and in Hydra A, [@forman2005; @million2010; @nulsen2005; @simionescu2009b].
Conclusions
===========
We performed a multi-wavelength study of the energetic interaction between the central active galactic nucleus (AGN), the intra-cluster medium, and the optical emission line nebula in the galaxy cluster Sérsic 159-03. We conclude that:
- Powerful ‘radio mode’ AGN feedback and possible ram pressure cleared the central region of the cD galaxy ($r<7.5$ kpc) of the cooling low entropy X-ray gas.
- This low entropy, high metallicity, relatively cool X-ray gas lies along the bright, 44 kpc long H$\alpha$+\[[N]{}\] filament extending from the centre of the cD galaxy to the north.
- As indicated by the observed dust lanes, molecular and ionized emission line nebulae, and the excess UV emission, part of this displaced gas, which is removed from the direct influence of the AGN, cools and forms stars.
- The pressure map shows evidence for an AGN induced weak shock at a radius of $r=15$ kpc and the X-ray images reveal cavities indicating past powerful AGN activity with an energy of $\sim7\times10^{58}$ ergs. At low frequencies, the radio source has an unusually steep spectrum with $\alpha=1.5$ indicating aging and confinement by the ambient gas.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Alastair Edge and Michael Hogan for providing the reduced 4.75 GHz [*ATCA*]{} radio data and for helpful discussions. We thank Seth Bruch and Emily Wang for their help with the SOAR observations. Support for this work was provided by the National Aeronautics and Space Administration through Chandra/Einstein Postdoctoral Fellowship Award Number PF8-90056 and PF9-00070 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics and Space Administration under contract NAS8-03060, and by the grants GO0-11019X and GO0-11139X. This work was supported in part by the U.S. Department of Energy under contract number DE-AC02-76SF00515. The Very Large Array is part of the National Radio Astronomy Observatory. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under a cooperative agreement by Associated Universities, Inc. We thank the staff of the that made these observations possible. The SOAR Telescope is a joint project of Conselho Nacional des Pesquisas Cientficas e Tecnologicas CNPq-Brazil, The University of North Carolina Chapel Hill, Michigan State University, and the National Optical Astronomy Observatory.
[^1]: Chandra/Einstein fellow, E-mail: [email protected]
[^2]: Adjunct Astronomer at the National Radio Astronomy Observatory
[^3]: Einstein fellow
[^4]: The is a national facility operated by the National Centre for Radio Astrophysics of the TIFR, India.
|
---
author:
- 'Robert [Kamiński]{}'
title: '${\mib{\pi-\pi}}$ interaction amplitudes with chiral constraints'
---
Model
=====
The [$\pi\pi$ ]{}interactions are one of the main sources of information on low energy physics. Their knowledge in scalar-isoscalar channel allows us to investigate scalar meson spectroscopy and to test the predictions of many theoretical models.
Amplitudes of the [$\pi\pi$ ]{}interactions should fulfill some basic conditions like unitarity and crossing symmetry. Described in [@kll2] three-coupled channel amplitudes for [$\pi\pi$ ]{}, [$K\overline{K}$ ]{}and [$\sigma\sigma$ ]{}interactions were constructed using a $S$-matrix unitary model without any constraints on model parameters. Several soutions were obtained by fits to experimental data in effective two pion mass region $m_{\pi\pi}$ from the [$\pi\pi$ ]{}threshold to 1600 MeV.
Problem of the crossing symmetry for the [$\pi\pi$ ]{}scattering has been studied in the past using dispersion techniques [@roy; @basdevant] which led to the construction of Roy’s equations. Using them with parameters as presented in [@basdevant] we calculated new real parts (treated hereafter as output) for the [$\pi\pi$ ]{}amplitudes obtained in [@kll2]. The aim of our study was comparison of the output with real parts of given amplitudes (treated hereafter as input). Roy’s equations explicitly depend both on the energy behaviour of amplitudes and on theresold parameters $a_{IJ}$ (for $I=0$ and 2)so such comparison was done for various [$\pi\pi$ ]{}threshold parameters in isoscalar and isotensor waves.
Results
=======
In Fig. \[fig:0\] one can see comparison of the output calculated using Roy’s equations with input for solution A of [@kll2].
=12 cm =16 cm
=12 cm =16 cm
=12 cm =16 cm
=12 cm =16 cm
The scattering lengths $a_{IJ}$ and the so called slope parameters $b_{IJ}$ have been calculated using the threshold expansion: $$\begin{aligned}
Re f_{IJ}(s) = (\frac{s-4}{4})^J \left[a_{IJ} + b_{IJ}(\frac{s-4}{4}) + ...\right],
\label{expansion}\end{aligned}$$ where $I$ and $J$ denote isospin and total spin of the [$\pi\pi$ ]{}system, $s=(m_{\pi\pi}/m_{\pi})^2$. The curves in Fig. \[fig:0\] correspond to the case when both $a_{IJ}$ and $b_{IJ}$ were treated as free parameters (not imposed by hand). Significant differences between input and output can be seen for both isoscalar and isotensor wave.
First correction was the change of the scattering length in the isotensor waves. Its value $a_{02} = -0.045$ follows roughly the results of two loop calculations of ChPT [@bijnens]. As can be seen in Fig. \[fig:1\] such a constraint allows significant decrease of the differences between input and output in comparison with those seen in Fig. \[fig:0\]. The big differences that still exist near the [$\pi\pi$ ]{}threshold in isotensor wave indicated that the slope parameter $b_{20}$ had to be changed. Value $b_{20}=-0.07$ was chosen according to two loop predictions of ChPT (see [@bijnens]). As can be seen in Fig. \[fig:2\] introduction of such a constraint for $b_{20}$ allows to obtain much better agreement between input and output near the [$\pi\pi$ ]{} threshold in the isotensor wave. The last calculations were done for amplitudes with additional constraint for isoscalar scattering wave $a_{00} = 0.21$. As can be seen in Fig. \[fig:3\] inputs and outputs well agree in wide energy range up to about 1000 MeV for both isoscalar and isotensor waves.
Conclusions
===========
Additional constraints for isotensor and isoscalar scattering lengths and slope parameter for the isotensor wave are necessary to obtain reasonably good agreement of the [$\pi\pi$ ]{}scattering amplitudes with those calculated using Roy’s equations. Numerical values of those parameters predicted by ChPT may be used as constraints.
[99]{}
R. Kamiński, L. Leśniak and B. Loiseau, Europ. Phys. J. [**C 9**]{}, 141 (1999)
S. M. Roy, Phys. Lett. [**B36**]{}, 353 (1971)
J.-L. Basdevant, C. D. Froggatt and J.L Petersen, Nucl. Phys. [**B72**]{}, 413 (1974)
J. Bijnens [*et al.,*]{} Nucl. Phys. [**B508**]{}, 263 (1997)
|
---
abstract: 'We review the status and future of direct searches for light dark matter. We start by answering the question: ‘Whatever happened to the light dark matter anomalies?’ i.e. the fate of the potential dark matter signals observed by the CoGeNT, CRESST-II, CDMS-Si and DAMA/LIBRA experiments. We discuss how the excess events in the first two of these experiments have been explained by previously underestimated backgrounds. For DAMA we summarise the progress and future of mundane explanations for the annual modulation reported in its event rate. Concerning the future of direct detection we focus on the irreducible background from solar neutrinos. We explain broadly how it will affect future searches and summarise efforts to mitigate its effects.'
address: |
Institut d’Astrophysique de Paris\
98 bis boulevard Arago\
75014 Paris\
France\
[email protected]
author:
- 'JONATHAN H. DAVIS'
title: The Past and Future of Light Dark Matter Direct Detection
---
Introduction
============
In this review we summarise the progress which has been made by direct detection experiments in the past few years towards probing the interactions between light dark matter (DM) and nucleons, and the challenges faced by these experiments in the future. By ‘light’ dark matter we refer to particles with masses $m$ in the approximate range $3\,\mathrm{GeV} \lesssim m \lesssim 20\,\mathrm{GeV}$. Direct detection experiments aim to observe evidence of dark matter particles from the halo of our galaxy interacting with nuclei on Earth. Signals of such scattering events correspond to $\sim$ keV-energy recoils of nuclei in these detectors. However there are many other potential ‘background’ sources of nuclear recoils which could mimic a dark matter signal, and so these experiments seek to minimise these backgrounds as much as possible. For example the rates of cosmogenic events such as atmospheric muons are minimised by placing direct detection experiments deep underground and within layers of shielding. Radioactive decay of the materials surrounding the detector results in significant backgrounds, both from beta and gamma radiation inducing electronic recoils, and neutrons inducing nuclear recoils. As direct detection experiments get larger they will also have to contend with an irreducible background from neutrinos, particularly those from the Sun whose flux is large. The observation of extra events in addition to those from expected background sources would in principle constitute a discovery of dark matter. Hence the better these backgrounds are controlled and understood, the more sensitive the experiment will be to dark matter recoils. However conversely if backgrounds are underestimated this can lead to spurious claims of dark matter detection.
![Comparison of $90\%$ confidence upper limits on the DM-nucleon cross section set by the LUX [@Akerib:2013tjd] and SuperCDMS [@Agnese:2014aze] experiments with preferred regions (at various levels of statistical significance) claimed by the CoGeNT [@Aalseth:2012if], CRESST-II [@Angloher:2011uu], DAMA/LIBRA [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa] and CDMS-Si [@Agnese:2013rvf] experiments for positive signals of dark matter recoils. Hashed regions for the CoGeNT and CRESST-II experiments indicate that the excess events in these detectors, previously taken as tentative evidence for dark matter, have now been explained entirely by underestimated backgrounds (indicated in parentheses) [@Davis:2014bla; @Angloher:2014myn; @Kuzniak:2012zm]. The filled regions represent anomalies which may have a dark matter or mundane explanation.[]{data-label="fig:dm_exclusion"}](dd_review_plot.pdf){width="95.00000%"}
The experimental status of light dark matter direct detection in past few years has been dominated by competition between experiments claiming tentative signals of dark matter discovery, and those setting exclusion limits due to null results. However more recently it has become increasingly clear that the null scenario is the more likely, the reasons for which we will elucidate in this review. These potential signals are referred to generically as ‘anomalies’, in the form of excess events seen by the CoGeNT [@Aalseth:2012if], CRESST-II [@Angloher:2011uu; @Angloher:2014myn] and CDMS-Si [@Agnese:2013rvf] experiments above their expected backgrounds. Additionally there is the DAMA/LIBRA collaboration [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa], who have been claiming discovery of dark matter for over a decade due to an annual variation seen in their observed events.
The interest surrounding these anomalies stems from the fact that they could all be interpreted in terms of dark matter interacting with nuclei, leading to preferred regions such as shown in figure \[fig:dm\_exclusion\]. However null results from other experiments, particularly XENON100 [@Aprile:2012nq] (but also e.g. XENON10 [@Angle:2011th], EDELWEISS [@Armengaud:2012pfa], ZEPLIN-III [@Akimov:2011tj] and CDMS-II [@Ahmed:2009zw]), placed strong constraints on dark matter interpretations of these anomalies. Negative results from the LUX [@Akerib:2013tjd] and SuperCDMS [@Agnese:2014aze] experiments followed (and also e.g. CDEX [@Yue:2014qdu] and CDMSlite [@Agnese:2013jaa]), which made it impossible to explain these anomalies with dark matter elastically scattering with nuclei while evading upper bounds from all null searches, as can be seen in figure \[fig:dm\_exclusion\]. Hence there has been tension between two sets of experiments: those claiming discovery signals and those setting upper limits. Solutions to this tension fall into two main categories:
1. The first is to change how dark matter particles interact with nuclei. Indeed their interactions with nuclei are unknown and may be more complicated than simple elastic scattering, which can weaken the upper limits set by null results with respect to the preferred regions from positive searches [@Frandsen:2013cna; @Chang:2008gd; @Chang:2010yk; @TuckerSmith:2001hy]. Such explanations appeared initially promising, however due to the strength of bounds from LUX, SuperCDMS and other experiments most of these models are now still in some tension [@Chen:2014tka].
2. The second is that some or all of the positive results are not due to dark matter particles, but to underestimated backgrounds. This is particularly relevant for light dark matter particles as they should result in nuclear recoils with energies just above the typical threshold of a direct detection experiments, where the backgrounds are less well understood. Indeed much of the tension has now been resolved due to precisely this reason, as we discuss in section \[sec:past\] with respect to the CRESST-II [@Angloher:2011uu; @Angloher:2014myn] and CoGeNT [@Aalseth:2012if] experiments.
Much of this review focuses on the fate of the ‘anomalies’ observed by the CoGeNT, CRESST-II and DAMA/LIBRA experiments in the context of this second option. In section \[sec:past\] we discuss the explanations of the first two anomalies in terms of underestimated backgrounds, while section \[sec:dama\] deals with progress and challenges in the explanation of the DAMA/LIBRA data both with dark matter and mundane sources. We also discuss in section \[sec:nu\_bg\] the future challenges which all direct detection experiments will face in separating any potential light dark matter signals from the potentially large solar neutrino background. We present a short summary of the theory behind dark matter direct detection in \[sec:app1\], however the interested reader should consult other reviews such as refs. for more information.
Light dark matter anomalies in CoGeNT and CRESST-II explained with backgrounds \[sec:past\]
===========================================================================================
In this section we review the fate of the anomalous events observed by the CoGeNT [@Aalseth:2012if] and CRESST-II [@Angloher:2011uu] experiments. In both cases these anomalies took the form of a large number of additional low-energy recoils, which could not (at the time) be accounted for by the known backgrounds from e.g. natural and cosmogenic radioactivity in materials surrounding the detector. These were initially taken as evidence for dark matter interactions primarily due to their spectra, however they have both now been explained by underestimated backgrounds.
The CoGeNT excess from surface events
-------------------------------------
Here we summarise the analysis performed by the CoGeNT collaboration which led to an erroneous preference for dark matter recoils in their data, represented by the ‘region of interest’ in figure \[fig:dm\_exclusion\]. The CoGeNT experiment[@Aalseth:2012if; @Aalseth:2014eft] works using a p-type point-contact germanium detector. An event in the detector constitutes a change in the measured voltage over a time of a few microseconds. For each event the magnitude of this change is proportional to the recoil energy, while the duration is measured as the rise-time. The majority of the active volume of the CoGeNT detector is a p-type semiconductor with a charge collection efficiency $\epsilon$ of unity, referred to as the bulk of the detector. Towards the outer edge of the detector modules is the millimetre-thick transition layer where $0 < \epsilon < 1$. Events occurring here are denoted as surface events[@Aalseth:2012if; @Aalseth:2014eft].
Backgrounds (e.g. from radioactivity) induce events preferentially towards the outside of the detector volume. However dark matter particles are weakly-interacting and so make no distinction between the surface and bulk of the detector. As such they are considerably more likely to scatter in the bulk than the surface, whose volume is much smaller. Hence the surface population is dominated by background events and as such needs to be removed or accounted for before an analysis of the CoGeNT data-set for dark matter recoils. Additionally the partial charge collection means these surface events will be measured with typically lower energies, resulting in a spectrum with a low-energy rise which can mimic the recoil spectrum of light dark matter. This makes surface events particularly dangerous as a background.
![The effect on a search for dark matter particles with a mass of 8 GeV due to uncertainties in the relative sizes of the bulk and surface event populations in the CoGeNT data-set [@Aprile:2012nq], where the latter population is dominated by background events. Different parameterisations for the bulk fraction $\mathcal{R}$ (right panel) lead to different spectra for the bulk events (central panel) and therefore different levels of preference for a dark matter signal (right panel). The CoGeNT collaboration used only the function labelled as ‘CoGeNT exponential’ leading to a strong preference for dark matter recoils, however when marginalising over the full uncertainty in the choice of function for $\mathcal{R}$ the preference is less than $1 \sigma$ [@Davis:2014bla]. This figure has been reproduced under a Creative Commons Attribution 3.0 License from ref. .[]{data-label="fig:cogent"}](likes_807.pdf){width="99.00000%"}
In order to separate the bulk and surface event populations the CoGeNT collaboration employed the following method in ref. . Full details of the analysis can be found in refs. . The spectrum of the full CoGeNT data-set was converted to that of pure bulk events (which may contain a dark matter signal or only background events) by multiplication by an energy-dependent function called the bulk fraction $\mathcal{R}$. For an energy $E$ this gives the fraction of the total number of events with energies between $E - \Delta E / 2$ and $E + \Delta E / 2$ which occur within the bulk of the detector, where $E$ is the central energy value and $\Delta E$ is the bin size. The bulk and surface events are separated using their rise-times, with the surface events being slower on average, leading to longer rise-times. However on an event-by-event basis one does not know whether an energy-deposit occurred in the bulk or on the surface, and so $\mathcal{R}$ was determined by the CoGeNT collaboration for different energy bins by statistical methods. Specifically the distributions of bulk and surface events in rise-time were fit with two separate log-normal distributions (other distributions have also been considered [@Davis:2014bla]). This six-parameter fit leads to significant uncertainties on the form of $\mathcal{R}$.
Values of the bulk fraction $\mathcal{R}$ as determined by the CoGeNT collaboration for their $807$ live days data-set are shown as data-points on the left panel of figure \[fig:cogent\]. The large error bars arise from the uncertainties in the log-normal fits to the rise-times and are particularly large at low energies where the bulk and surface populations are more difficult to separate. The spectral shape which the data-points follow is partially due to energy-dependent cuts on the rise-time performed by the CoGeNT collaboration. The CoGeNT collaboration fit this data with a one-parameter exponential function of form $\mathcal{R}(E)_{\mathrm{CoGeNT}} = 1 - \mathrm{exp}(- \alpha E)$, with the constant $\alpha$ determined from this fit to be $\alpha = 1.21 \pm 0.11$. This is labelled as ‘CoGeNT exponential’ in the left-panel of figure \[fig:cogent\] for $\alpha = 1.21$.
By multiplying the pure CoGeNT spectrum with $\mathcal{R}(E)_{\mathrm{CoGeNT}}$ (and applying corrections for the detector efficiency and known excitation peaks etc.) the collaboration obtained the spectrum of events labelled in the central panel of figure \[fig:cogent\] as ‘CoGeNT exponential’. This was obtained using $\alpha = 1.21$, though either $\alpha = 1.10$ or $\alpha = 1.32$ give very similar spectra. If $\mathcal{R}(E)_{\mathrm{CoGeNT}}$ represents the correct bulk fraction then this spectrum should be that of purely bulk events. As shown in the central panel of figure \[fig:cogent\] when using this function for the bulk fraction there is a significant excess of low-energy events above the expected background in the bulk which fits well to the expected spectrum from light dark matter recoils. Indeed the likelihood function shown in the right panel of figure \[fig:cogent\] shows a clear preference for cross sections just above $10^{-41}$ cm$^2$ for dark matter with a mass of 8 GeV i.e. the centre of the region of interest corresponding to the CoGeNT anomaly.
However this is not the only possible choice for $\mathcal{R}(E)$. As shown in the left panel of figure \[fig:cogent\] there are a wide-range of functions which fit well to the data for the bulk fraction within the large error bars. Since there is no theoretical motivation for choosing $\mathcal{R}(E)_{\mathrm{CoGeNT}} = 1 - \mathrm{exp}(- \alpha E)$ any function which fits the data for $\mathcal{R}$ is equally acceptable *a priori*. However the CoGeNT collaboration did not publish results using any other function for $\mathcal{R}(E)$. Indeed if we instead use cubic splines for the form of $\mathcal{R}(E)$ as labelled in the left panel of figure \[fig:cogent\] then it is clear from the central panel that the spectrum of bulk events looks radically different, compared to the version derived by the CoGeNT collaboration. Indeed for the solid blue spline the data are consistent with the expected background.
Hence the uncertainties in the amount of contamination from surface events are much larger than assumed by the CoGeNT collaboration through their assumption that the bulk fraction took the form of $\mathcal{R}(E)_{\mathrm{CoGeNT}} = 1 - \mathrm{exp}(- \alpha E)$. When the full uncertainty in the functional form of $\mathcal{R}(E)$ was taken account of in ref. , through marginalising over all functions, it was found that there was less than one-sigma evidence for dark matter in the CoGeNT data-set. Indeed any preference for dark matter was a result of a bias in the analysis from an underestimation of the uncertainties in the ratio of surface to bulk events. Said differently: the low-energy rise typical of surface events mimicked a low-mass dark matter signal, as seen in the central panel of figure \[fig:cogent\] for the ‘CoGeNT exponential’.
This result has been confirmed with the more recent 1129 live days data-set from CoGeNT [@Aalseth:2014eft] in analyses by refs. and and also by similar experiments such as CDEX [@Yue:2014qdu]. It is clear that the CoGeNT data is now fully consistent with known backgrounds from the bulk and surface of the detector, and so there is no preference for a dark matter recoil signal. The issue of the CoGeNT excess highlights the dangers of underestimated backgrounds for direct detection experiments, especially those which mimic the low-energy rise expected from light dark matter.
Surface roughness and CRESST-II
-------------------------------
In 2011 a group within the CRESST-II (Cryogenic Rare Event Search with Superconducting Thermometers) collaboration claimed evidence for a population of low-energy nuclear recoils in their detector which could not be explained by any known background source [@Angloher:2011uu]. This low-energy rise was similar to that observed by the CoGeNT experiment and so could be interpreted broadly with the same dark matter mass and cross section (although the agreement was not perfect, as shown in figure \[fig:dm\_exclusion\]). Additionally this result was potentially more credible since the CRESST-II experiment has the ability to distinguish nuclear and electronic recoils. Indeed the CRESST-II experiment is constructed from scintillating CaWO$_4$ crystals, for which recoil events generate signals in both phonons and scintillation light. The ratio of these two signals determines which type of recoil the event originates from [@Angloher:2011uu; @Angloher:2014myn].
However the authors of ref. showed that the study which lead to this claim of a tentative dark matter signal [@Angloher:2011uu] underestimated a particular background originating from the interactions of $^{206}$Pb nuclei (which themselves originate from the decay of $^{210}$Po) on the surface of the clamps holding the detector in place. The interactions of these $^{206}$Pb nuclei near the surface of the clams generated cascades of secondary recoils in the CRESST-II detector. These were modelled in ref. , however the authors assumed that the surface of the clamps was perfectly smooth. When the more realistic scenario of a rough surface for the clamps was taken into account this background was shifted partially to lower energies, as some of the energy from these cascades was absorbed by the rough surface instead of reaching the detector [@Kuzniak:2012zm]. This lead to a low-energy rise in the spectrum of nuclear recoil events from this background [@Kuzniak:2012zm], giving a good fit to the ‘excess’ of events in the study of ref. which was originally taken as tentative evidence for dark matter recoils.
Indeed a more recent study by the CRESST-II collaboration [@Angloher:2014myn] using a new set-up without these troublesome clamps found no evidence for dark matter recoils. Hence the collaboration set an upper limit on the DM-nucleon cross section which excluded the preferred region found in ref. . As with the case of CoGeNT this demonstrates the extreme difficulty in modelling backgrounds for direct searches, and the limitations of simulations in correctly capturing the physics of the real system.
Towards an understanding the origin or the DAMA annual modulation \[sec:dama\]
==============================================================================
The DAMA/LIBRA and the former DAMA/NaI experiment (referred to hereafter together as DAMA) have accumulated over ten years worth of data in the search for dark matter recoils [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa]. The DAMA apparatus is composed of approximately 250 kg of NaI target surrounded by layers of shielding and located underground at the Gran Sasso lab [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa]. The DAMA collaboration currently claim to have observed an annual modulation in their data at a level of around $2\%$ (the modulation fraction) of the total event rate and with a statistical significance of $9.2\sigma$. This modulation is approximately sinusoidal and peaks in late May. Additionally the collaboration claim to observe a modulation only at low energies, below approximately 6 keV$_{\mathrm{ee}}$, and only in the single-hit event population (i.e. events which occur alone within a small time-interval, in contrast to multiple-hit events).
Due to the Earth’s orbit around the Sun the relative velocity between dark matter particles in the galactic halo and terrestrial direct detection experiments varies over the year, leading to an approximately sinusoidal modulation for the recoil rate of DM particles at keV-energies which peaks in early June (see \[sec:app1\] and also e.g. refs. ). Hence due to its similarity with the annual modulation expected from a dark matter recoil signal the DAMA collaboration have claimed that their data is evidence for dark matter particles scattering in their detector. However the most basic scenario of dark matter scattering elastically (and coherently) with nuclei requires a cross section which is excluded by other direct detection experiments such as LUX [@Akerib:2013tjd] and SuperCDMS [@Agnese:2014aze]. This is because the rate of modulated events observed by DAMA is approximately 100 counts per day, and so a large interaction cross section is needed to provide this many events (there are also issues with the unmodulated event rate, which we discuss later). Hence in this section we start by discussing the challenges of explaining the DAMA signal with dark matter, before following with a discussion of models for the DAMA modulation which employ only Standard Model particles.
Challenges for dark matter
--------------------------
A popular method of alleviating the tension between a dark matter explanation for the DAMA signal and the exclusion limits from null experiments is to alter how the dark matter interacts with nuclei. The standard assumption is that dark matter particles scatter elastically and coherently with nuclei. ‘Elastic’ implies that the dark matter particle does not change state upon scattering, however it could instead transition to either a heavier state (inelastic) or a lighter state (exothermic) [@Frandsen:2013cna; @Chang:2008gd; @Chang:2010yk; @TuckerSmith:2001hy]. In this case the recoil spectrum is altered in a way which depends upon the mass of the target nucleus, allowing the rate at DAMA to be enhanced relative to the xenon-based experiments (e.g. LUX and XENON100), which set the strongest exclusion limits.
Coherent scattering means that the dark matter scatters identically with protons and neutrons, resulting in the scattering rate scaling with the square of the atomic number (see \[sec:app1\]). If this assumption is relaxed then the proton and neutron terms can be made to interfere destructively for e.g. xenon-based targets only, thereby weakening the limits on the DAMA region. Alternatively the scattering rate can also be altered such that it depends on the velocity of the dark matter, due to the dark matter interacting with nucleons via a pseudo-vector coupling [@Dienes:2013xya]. Since the dark matter velocity is non-relativistic this coupling is therefore heavily suppressed and again depends on the mass of the target nucleus.
However most of these models still find it difficult to evade all of the constraints from e.g. LUX [@Akerib:2013tjd] and SuperCDMS [@Agnese:2014aze], especially since the latter uses a germanium target which is not as heavy as xenon, and also those from e.g. flavour experiments [@Dolan:2014ska]. For the case of spin-dependent scattering (see \[sec:app1\]) constraints from the PICO [@Amole:2015lsj] and KIMS [@Lee.:2007qn] experiments are also difficult to evade [@DelNobile:2015lxa]. Hence models which explain DAMA while evading all other constraints face the problem of being strongly fine-tuned or else of becoming increasingly complex e.g. composite models of dark matter [@Wallemacq:2014sta] or mirror matter [@Foot:2014xwa].
A dark matter explanation (and indeed any model invoking a new population of events) faces an additional obstacle: the unmodulated spectrum of events in DAMA is well-fit by known radioactive backgrounds [@Pradler:2012qt; @Pradler:2012bf]. This means that even at low-energy (where the annual modulation is observed) there is little room for extra events in the data above background and so any new population will need a large modulation fraction to explain the DAMA signal i.e. if the DAMA data shows an $\sim 2\%$ modulation and the dark matter forms only $10\%$ or less of the unmodulated signal then the dark matter itself needs a $\sim 20\%$ modulation fraction or larger. So even if a dark matter signal could provide the required number of modulated events, if its modulation fraction is too small then it will over-shoot the unmodulated spectrum.
The modulation fraction from dark matter depends to some extent on the distribution of dark matter velocities in the galactic halo $f(v)$ (see \[sec:app1\]), which is *a priori* unknown. Generally this distribution is assumed to take the form of a Maxwell-Boltzmann distribution with a velocity dispersion of $\sigma_v = \sqrt{3/2} v_0$ where $v_0 = 220$kms$^{-1}$, cut off at the escape velocity. This is called the Standard Halo Model (SHM). In this case the modulation fraction is expected to be of a few percent in size. However there are many other forms which this distribution can take, for example those motivated by numerical simulations such as in refs. or with streams of dark matter [@Savage:2006qr].
It has been suggested that using one of these alternative forms for $f(v)$ could result in a different (and potentially larger) modulation fraction [@Freese:2012xd], thereby making a dark matter scenario more favourable for the DAMA data. However many authors have developed techniques to analyse the DAMA data and how it compares to null results from other experiments without having to make assumptions about the astrophysical velocity distribution [@Fox:2010bz; @McCabe:2011sr; @Anderson:2015xaa; @Feldstein:2014ufa]. Such velocity-independent methods have shown that it is very difficult to explain DAMA with dark matter for any choice of halo function, however arbitrary. Indeed the DAMA modulation is even inconsistent with its own unmodulated event rate for any velocity distribution [@HerreroGarcia:2012fu]. Hence although the possibility to explain the DAMA data with dark matter may remain, such models are no longer the most simple option.
Progress for non-DM models of the DAMA events
---------------------------------------------
Dark matter recoils are not the only method of explaining the DAMA signal. Models which explain the DAMA modulation without dark matter fall into two main categories:
1. A source of modulated events on top of the known un-modulated backgrounds. In this respect these models are similar to those which invoke dark matter, however they generally rely on cosmogenic particles such as muons or neutrinos to provide the modulation instead [@Davis:2014cja; @Blum:2011jf; @Ralston:2010bd].
2. The modulation is explained as an artefact of the detector itself or an error introduced when the raw data is processed. In this case there is no modulated population of events, but instead the un-modulated event rate appears to modulate artificially.
For option (1) it is possible to get the right phase of the modulation and make strong predictions for other experiments, however such models face the same difficulties regarding the unmodulated spectrum of events as dark matter. The mechanism for generating these modulated events also needs to be unique to DAMA, in order to get past the null results from other modulation searches [@2012arXiv1203.1309C]. Models which fall into this category are rather few compared to the number of dark matter explanations for the DAMA signal. They generally have the events seen by DAMA generated in some way by atmospheric muons alone [@Blum:2011jf; @Ralston:2010bd] or in combination with solar neutrinos [@Davis:2014cja], since the rates of these events are known to modulate with an annual period. Indeed the production of atmospheric muons by decaying cosmic ray particles in the stratosphere is correlated with temperature, with a maximum for the annual mode approximately 30 days later than the observed DAMA modulation [@Ambrosio:1997tc; @FernandezMartinez:2012wd; @D'Angelo:2011fs]. The muon signal also possesses significant power at periods longer than one year, including an approximately $1\%$ modulation at a period of 11 years from solar activity [@Ambrosio:1997tc; @FernandezMartinez:2012wd; @D'Angelo:2011fs]. Solar neutrinos also modulate with a period of one year due to the changing distance between the Earth and Sun, and so their flux is largest around January 4th [@Bellini:2013lnn; @PhysRevD.78.032002]. Alone the atmospheric muon and solar neutrino modulations have the wrong phase to fit the DAMA signal, however in combination they can partially interfere to give the required phase, period and modulation fraction [@Davis:2014cja]. Hence based purely on timing information the fit of such cosmogenic models to the DAMA data is as good as from dark matter, and so the closeness of the phase of the DAMA data to that expected from the dark matter rate is not a ‘smoking-gun’ for the latter.
These cosmogenic sources may not directly be responsible for the DAMA modulated events, but likely through a secondary such as neutrons, which are liberated from material surrounding the detector by the muons and neutrinos [@Blum:2011jf; @Ralston:2010bd; @Davis:2014cja]. The spectrum and rate of the produced neutrons can depend strongly on the detector environment, and so could be larger at DAMA than other experiments such as LUX. However the exact mechanism to generate the DAMA events from cosmogenic sources is unknown, and current simulations show that it may be difficult to produce enough neutrons in this way [@Klinger:2015vga; @Jeon:2015sya], at least if the neutrons are scattering elastically in the detector. Hence for now the actual rate of events which could be generated by these cosmogenic sources is unknown. Indeed as for dark matter recoils any new source must also fit well to the unmodulated DAMA spectrum, which likely means that it must have a large modulation fraction or that some of the backgrounds at low-energy experience a modulation themselves.
An important point for such models is that if the rate at which neutrons, or any alternative secondary, are generated from muons or neutrinos is non-linear then their modulation fraction could be much larger than for the individual cosmogenic sources themselves (which is around a few percent). For example if the secondary rate $R_2$ (e.g. neutrons) depends on the primary rate $R_1$ (e.g. muons or neutrinos) as $R_2 \sim R_1^{\alpha}$ then the modulation residual $\Delta R_2 / R_2 \approx \alpha \Delta R_1 / R_1$. Hence until we know all of the methods by which neutrons (or other secondaries which could lead to the DAMA events) are generated from cosmogenic sources we will not know their modulation fraction precisely, and so it is difficult to rule them out based on the modulation fraction of the primary source. Alternatively the mechanism for generating the DAMA events may not involve nuclear recoils. Since the DAMA detector can not distinguish whether an event is due to a nuclear recoil or some other source (e.g. a photon) it is possible even that photon emission induced by electron capture, responsible for a large population of events in the DAMA detector at low-energy [@Pradler:2012qt; @Pradler:2012bf], could be induced to modulate by cosmogenic sources and so explain the DAMA events. However this is purely speculative at this stage and it is clear that there is much scope for an improved understanding of annually modulated sources for the DAMA events.
Regarding option (2) this is the most attractive explanation from the perspective of the spectrum of the unmodulated events seen by DAMA, since it does not require any additional population of events. However it is not clear why the phase of any artificial effect would be close to that seen in the DAMA modulation, and why this would not also affect the multiple-hit event population (for example). Additionally it is harder to use such a model to make predictions for other direct detection experiments looking to replicated the DAMA signal, though to some extent any null result lends credence to this claim [@2012arXiv1203.1309C].
In all of the above cases: either dark matter, a new mundane event source or a detector effect, new results from other DAMA-like experiments will be highly beneficial. Examples of such experiments are DM-Ice [@Cherwinka:2014xta], ANAIS [@Amare:2013lca], KIMS [@kims_pres] and SABRE [@sabre_pres]. We summarise the different potential results at a second DAMA-like experiment in a different lab and its implications for DAMA below:
- *A second experiment observes a modulation with the same phase and modulation amplitude as DAMA.* This would constitute strong evidence for a dark matter interpretation of the DAMA signal, especially if the second experiment is in the southern hemisphere where effects correlated with the seasons (e.g. the modulation of atmospheric muons) have their phases shifted by half a year.
- *A second experiment observes a modulation with a different phase from DAMA.* Such a result would imply that the DAMA modulation is due to an additional source of mundane events i.e. not dark matter. In this case the phase of this second signal combined with that from DAMA could be used to pin down the exact model, especially if it is due to an interference effect between two sources [@Davis:2014cja].
- *A second experiment observes no modulation in its event rate.* In this case the DAMA modulation is likely due to a detector effect (i.e. option (2) above) rather than an extra source of events. There is already a null result from a modulation search by CDMS [@2012arXiv1203.1309C], however a definitive result can only really come from an experiment constructed from NaI (like DAMA) for which the interactions between dark matter and nuclei in the detector should be identical.
Solar neutrino backgrounds for light dark matter searches \[sec:nu\_bg\]
========================================================================
![Values of the $90\%$ upper limit for a ten tonne low-threshold xenon experiment as a function of running time, and for a dark matter mass of 6 GeV. Initially the limiting cross section scales as the inverse square root of running time, as expected from Poisson statistics. Using only information on the spectrum of events the limiting cross section reaches a saturation value where it remains mostly constant with increasing running time. However with the different temporal variation of the solar neutrino and dark matter signals used as an additional discrimination parameter the upper limit returns to the Poisson-dominated case, once enough statistics have been accumulated. This figure has been reproduced under a Creative Commons Attribution 3.0 License from ref. .[]{data-label="fig:lims_nu"}](lims_6.pdf){width="90.00000%"}
We have seen that a full understanding of backgrounds is crucial for a direct search for light dark matter, especially if these backgrounds have a similar spectrum to that expected from dark matter recoils. For the current generation of experiments all such backgrounds are in principle ‘reducible’ i.e. they can be brought to a nominal level using shielding, for example. One such example is natural radioactivity of the material surrounding the detector. This can be reduced through techniques such as volume fiducialisation for xenon experiments [@Akerib:2013tjd; @Aprile:2011dd], which employ the self-shielding properties of xenon by using only the innermost volume of the detector for a dark matter search. If the number of background events remains constant for increasingly large detectors then their limiting sensitivity scales linearly with exposure (which is running time multiplied by fiducial volume) i.e. $\sigma_{\mathrm{lim}} \sim (\mathrm{exposure})^{-1}$. Indeed most projections for future experiments assume ‘zero-background’ which is in principle attainable provided all of the background events arise from reducible sources.
However future multi-tonne experiments will be large enough such that solar neutrinos will lead to non-negligible numbers of nuclear-recoil events in these detectors, with recoil energies of a few keV. Since neutrinos are weakly interacting they can not be shielded against and so constitute an ‘irreducible’ background. Additionally their spectrum is almost identical to that expected from light dark matter [@Billard:2013qya; @Davis:2014ama], which makes such a background particularly troublesome. The DM-nucleon cross sections at which this neutrino background is of similar size to (or larger than) the dark matter recoil signal is referred to as the ‘neutrino floor’. Despite its name the neutrino floor does not represent an absolute limit on the sensitivity of direct detection experiments. Instead it represents broadly two effects: The *first and most fundamental* is that the sensitivity of future experiments is limited fundamentally by Poisson uncertainties on the size of the neutrino background i.e. if a DM signal is smaller than these uncertainties then it is difficult to either discover or exclude with confidence. This results from the fact that DM and neutrino events can only be distinguished on a statistical level, not event-by-event, and so even if this discrimination of the two populations is clear the limit will scale at best as $\sigma_{\mathrm{lim}} \sim (\textrm{exposure})^{-1/2}$.
However this separation is not always clear, leading to the second effect of the neutrino floor. This arises for DM masses where the recoil spectrum is indistinguishable from that expected for solar neutrinos. In this case the limiting cross section $\sigma_{\mathrm{lim}}$ improves even slower than $\sim (\textrm{exposure})^{-1/2}$, due to the systematic uncertainties on the size of the solar neutrino flux [@Billard:2013qya; @Davis:2014ama]. Indeed $\sigma_{\mathrm{lim}}$ reaches a saturation value set by the size of these uncertainties, at which it stays effectively constant even with increasing exposure. Assuming that the DM velocity distribution $f(v)$ is a Maxwell-Boltzmann distribution then the strongest effect is for DM masses around 6 GeV, however this extends to heavier masses when the uncertainties in $f(v)$ are taken into account [@Davis:2014ama]. Fortunately this second effect of the neutrino floor exists only when using a single experiment with only information on the spectrum used to separate DM and neutrino recoils. Indeed it is strongly reduced if either a second experiment is used in combination with the first [@Ruppin:2014bra], or if the different annual modulation [@Davis:2014ama] or directional dependence [@Grothaus:2014hja; @O'Hare:2015mda] of the solar neutrinos and DM is employed as an additional discrimination parameter[^1].
Examples of experiments for which this neutrino background will be important are the new SuperCDMS at SNOlab [@cdms_sno_pres], particularly due to its low threshold, and multi-tonne xenon experiments such as LZ [@2011arXiv1110.0103M], XENONnT [@2012arXiv1206.6288A] and DARWIN [@2012JPhCS.375a2028B]. There is additionally another background from atmospheric and diffuse supernovae neutrinos which will be important for the sensitivity of these experiments to heavier DM, with masses around 15 GeV and above. However since the flux of such backgrounds is much smaller than that from solar neutrinos they will be far less problematic.
Conclusion
==========
We have reviewed the recent progress which has been made by direct detection experiments in searching for light dark matter, with mass $m$ in the range $3\,\mathrm{GeV} \lesssim m \lesssim 20\,\mathrm{GeV}$, interacting with nucleons. Much of the recent history of this field has been concerned with tension between null results from experiments such as LUX [@Akerib:2013tjd], SuperCDMS [@Agnese:2014aze] and XENON100 [@Aprile:2012nq] and potential signals of dark matter in the CoGeNT [@Aalseth:2012if], CDMS-Si [@Agnese:2013rvf], CRESST-II [@Angloher:2011uu] and DAMA/LIBRA [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa] experiments (see figure \[fig:dm\_exclusion\]). The first three of these ‘signal’ experiments observe additional events (‘excesses’) above their background predictions, which could be interpreted as nuclear recoils induced by dark matter from the halo of our galaxy. The DAMA experiment instead claims to observe an annual variation in their data consistent with the expectation from dark matter [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa].
We discussed in section \[sec:past\] recent re-analyses of data from CoGeNT and CRESST-II which showed that these excess events are entirely attributable to underestimated backgrounds, and hence there is in fact no significant preference for dark matter recoils in these data [@Davis:2014bla; @Angloher:2014myn]. For CoGeNT this took the form of an underestimated background from events on the surface of the detector, whose spectrum mimics the low-energy rise expected from light dark matter [@Davis:2014bla]. While for CRESST-II the background originated from secondary products of $^{206}$Pb nuclei scattering near the surface of the clamps holding the detector [@Kuzniak:2012zm]. The spectrum of this background was softer than initially predicted due to the rough surface of the clamps which the original analysis [@Angloher:2011uu] had assumed this to be perfectly smooth [@Kuzniak:2012zm]. New results from a run without these clamps confirm this null result [@Angloher:2014myn].
In section \[sec:dama\] we discussed the interpretations of the data from the DAMA/LIBRA experiment [@Bernabei:2013xsa; @Bernabei:2008yi; @Bernabei:2013cfa] either in terms of dark matter or a more mundane source. The DAMA experiment observes an annually varying rate of events whose phase is close to that expected from dark matter. However in order to produce such a modulation the dark matter needs to scatter off nuclei with a large cross section. We discussed difficulties in creating a model for dark matter which can provide such a cross section while evading the upper limits from null searches. We also focused on explanations for this modulation which do not invoke dark matter. In this context we discussed models which can give the same modulation as a dark matter signal by combining signals from atmospheric muons and solar neutrinos, whose rates also vary throughout the year with different phases [@Davis:2014cja]. Finally we summarised the three different results which a second DAMA-like experiment in a different location could observe, and what they would imply for the DAMA signal. A second result with the same phase as DAMA would favour dark matter, a result with a different phase favours a mundane explanation such as muons or neutrinos while no modulation at all favours an explanation in terms of a detector effect in DAMA.
The recent progress in the field of light dark matter direct detection can be summarised as such (see also the review of ref. ):
- The CoGeNT and CRESST-II ‘excesses’ have been fully explained as due to underestimated backgrounds, and not dark matter recoils [@Davis:2014bla; @Angloher:2014myn; @Kuzniak:2012zm].
- The CDMS-Si excess events are still unexplained (at least publicly). However since the SuperCDMS experiment [@Agnese:2014aze] is more sensitive and finds no evidence for excess events above background then the former is unlikely to be due to dark matter.
- The DAMA annual modulation remains unexplained, however it is unlikely to be due to dark matter considering the strong constraints from null searches. Much progress has been made recently on explaining the data from DAMA without dark matter, and experiments aimed at replicating this experiment such as DM-Ice will hopefully bring this issue to a close soon [@Davis:2014cja; @Cline:2015yza].
In section \[sec:nu\_bg\] we discussed the future of light dark matter direct detection in terms of the background from solar neutrinos. Future multi-tonne direct detection experiments face the prospect of distinguishing a potential dark matter signal from the large and irreducible background due to solar neutrinos. Such a background will fundamentally limit the sensitivity of these experiments to signals larger than the Poisson uncertainties on the neutrino background. Sensitivity for light dark matter can be optimised by combining results from multiple experiments [@Ruppin:2014bra], or by using the different annual modulation [@Davis:2014ama] or directional dependence [@Grothaus:2014hja; @O'Hare:2015mda] of the dark matter and neutrino signals.
The direct detection of light dark matter faces an exciting future with larger and more sensitive experiments currently under construction or in development [@Cushman:2013zza]. This is especially true for light dark matter due to the development of experiments with low energy thresholds such as SuperCDMS [@Agnese:2014aze] and improvements in the understanding of how xenon-based experiments (e.g. XENON100 [@Aprile:2012nq] and LUX [@Akerib:2013tjd]) respond to recoils with energies below 3 keV [@lux_leff_pres]. The presence of a background from solar neutrinos will reduce the rate at which future sensitivity improves and more work needs to be done to understand its full effects, however it should never realistically present an absolute limit on this sensitivity.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research of the author is supported at IAP by ERC project 267117 (DARK) hosted by Université Pierre et Marie Curie - Paris 6.
Dark matter scattering in direct detection experiments \[sec:app1\]
===================================================================
The spectrum of dark matter recoils in a given detector (in units of counts per day per kg per keV) takes the form of [@Cerdeno:2010jj], $$\frac{\mathrm{d}R}{\mathrm{d}E} = \frac{\rho_{\chi}}{m_N m} \int_{v_{\mathrm{min}}}^{\infty} v f(v + u_e) \frac{\mathrm{d}\sigma}{\mathrm{d}E} \mathrm{d}^3 v ,
\label{eqn:rate1}$$ where $m_N$ is the mass of the nucleus in the detector, $m$ is the DM particle mass, $\rho_{\chi}$ is the local DM density, generally taken to be around 0.3 GeVcm$^{-3}$ [@Widrow:2008yg], $v_{\mathrm{min}} = \sqrt{E m_N / 2 \mu_N^2}$, $\mu_N$ is the DM-nucleus reduced mass, and $ \frac{\mathrm{d}\sigma}{\mathrm{d}E}$ is the differential interaction cross section. The velocity integral accounts for the fact that a DM particle does not have to deposit all of its energy in the detector upon collision, and so any particle with a velocity greater than $v_{\mathrm{min}}$ can impart a kinetic energy of $E$ to the nucleus. All velocities in equation \[eqn:rate1\] are in the Earth’s rest frame, hence we use $u_e$ to boost the distribution of galactic DM velocities $f(v)$ into the correct frame. Since the relative direction of the Earth’s velocity with respect to the DM wind varies over the year the rate ${\mathrm{d}R} / {\mathrm{d}E}$ exhibits an annual modulation, which is what the DAMA/LIBRA collaboration claim to observe in their data.
Since the dark matter in the galactic halo moves at non-relativistic velocities this formula is simplified by expanding the differential cross section in terms of recoil velocity $v$ and taking only the lowest-order term. This leads to the expression for the spin-independent scattering cross section, $$\frac{\mathrm{d}\sigma}{\mathrm{d}E} = \frac{\sigma m_N F(E)}{2 \mu_N^2 v^2},
\label{eqn:rate2}$$ where $\sigma$ is the ‘zero-momentum’ DM-nucleus cross section. The function $F(E)$ is the nuclear form-factor, which for spin-independent interactions is essentially a Fourier transform of the nucleus [@Lewin199687]. Hence the expression for the DM-nucleus recoil rate simplifies to $$\frac{\mathrm{d}R}{\mathrm{d}E} = \frac{\sigma \rho_{\chi} F(E)}{2 \mu_N^2 m} \int_{v_{\mathrm{min}}}^{\infty} \frac{f(v + u_e)}{v} \mathrm{d}^3 v .
\label{eqn:rate3}$$
The final step is to express this in terms of the scattering cross section between dark matter and *nucleons*, which is the cross section all direct detection experiments set upper limits on (or preferred regions for positive results). If we assume that the DM couples equally to protons and neutrons we obtain for the cross section, $$\sigma(E) = \sigma_0 \left( \frac{\mu_N}{\mu_p} \right)^2 A^2 ,
\label{eqn:rate4}$$ where $\sigma_0$ is the zero-momentum DM-nucleon cross section, $\mu_p$ is the DM-proton reduced mass and $A$ is the atomic number of the nucleus with which the DM interacts. Assuming equal couplings to protons and neutrons is not necessary, and relaxing this assumption may reduce or enhance the rate depending on the particular nuclear target as discussed in section \[sec:dama\].
If instead the dark matter couples to nuclei via their spin then the scattering is said to be ‘spin-dependent’. In this case instead of equation (\[eqn:rate2\]) we have for the differential cross section the expression, $$\frac{\mathrm{d} \sigma}{\mathrm{d} E} = \frac{16 m_N}{\pi v^2} \Lambda^2 G_F^2 J(J+1) \frac{S(E)}{S(0)},$$ where $G_F$ is the Fermi constant, $J$ is the total spin of the nucleus, $S(E)$ is the spin form factor and $\Lambda = \frac{1}{J} [a_p \langle S_p \rangle + a_n \langle S_n \rangle]$, with $\langle S_p \rangle$ and $\langle S_n \rangle$ being the expectation values for the spin of the proton and neutron respectively, and $a_p$ and $a_n$ are coupling constants for the proton and neutron.
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[^1]: An interesting point is that when these additional discrimination parameters take effect $\sigma_{\mathrm{lim}}$ can improve faster than $\sim (\textrm{exposure})^{-1/2}$. However this is only when the limiting cross section is far from the fundamental Poisson limit, which can not be surpassed. Upon approaching this limit the Poisson scaling of $\sigma_{\mathrm{lim}} \sim (\textrm{exposure})^{-1/2}$ is regained.
|
---
author:
- 'Lev S. Bishop'
- 'J. M. Chow'
- Jens Koch
- 'A. A. Houck'
- 'M. H. Devoret'
- 'E. Thuneberg'
- 'S. M. Girvin'
- 'R. J. Schoelkopf'
title: Nonlinear response of the vacuum Rabi resonance
---
Circuit QED realizes the coupling between a superconducting qubit and the microwave field inside an on-chip transmission line resonator[@blais_cavity_2004; @wallra_strong_2004]. The drastic reduction in mode volume for such a quasi-1d system [@devoret_circuit-qed:_2007; @schoelkopf_wiring_2008], the recent improvement in coherence times[@schreier_suppressing_2008], and the absence of atomic motion render circuit QED an ideal system for studying the strong-coupling limit. In the customary linear-response measurement of the transmitted microwave radiation, the vacuum Rabi splitting manifests itself as an avoided crossing between qubit and resonator.
It may be argued that the observation of the splitting is not yet a clear sign for quantum behaviour [@zhu_vacuum_1990; @tian_quantum_1992], since avoided crossings are also ubiquitous in classical physics. However, quantum mechanics gives rise to a distinct nonlinearity of these splittings when initializing the resonator in a higher photon Fock state: when the resonator mode is occupied by $n$ photons, the splitting is enhanced by a factor $\sqrt{n+1}$ as compared to the vacuum Rabi situation. This nonlinearity has been observed with single atoms in both microwave[@brune_quantum_1996] and optical cavities[@birnbaum_photon_2005; @schuster_nonlinear_2008]. In circuit QED, this characteristic trait has been observed very recently both spectroscopically[@wallraff], as well as in time-domain measurements[@simmonds; @martinis].
Here, we present a theoretical analysis and experimental investigation of the vacuum Rabi resonance in a circuit QED system, where we study the $\sqrt{n}$ nonlinearity up to $n=5$ by exploring the power dependence of the heterodyne transmission (see Methods). This type of detection is particularly simple, since it is a continuous measurement involving only a single driving frequency. We find that, surprisingly, each vacuum Rabi peak develops into a new doublet (supersplitting), which can be simply explained by the saturation of a two-level system comprising both photon and qubit degrees of freedom. For even stronger driving, a series of additional peaks due to multiphoton transitions[@solano] up the Jaynes–Cummings ladder appear, but the complete shape of the spectrum closely matches the predictions from a master equation treatment including only relaxation, dephasing, and the known multilevel spectrum of the superconducting qubit. The quantitative agreement confirms the validity of the extended Jaynes–Cummings Hamiltonian for our system, rendering circuit QED an excellent tool for probing the fundamental interaction of matter and light.
![[**Extended Jaynes–Cummings level diagram of the resonator–transmon system**]{}. [**a**]{}, Bare levels in the absence of coupling. The states are denoted ${\left\lvert n,j \right\rangle}$ for photon number $n$ and occupation of transmon level $j$. [**b**]{}, Spectrum of the system including the effects of transmon–resonator coupling. The effective two-level system relevant to describing the lower vacuum Rabi peak comprises the ground state and the antisymmetric combination of qubit and photon excitations. \[fig:fig1\]](sfigure1.eps){width="1.0\columnwidth"}
We consider a system consisting of a superconducting transmon qubit[@koch_charge-insensitive_2007; @schreier_suppressing_2008] whose multiple quantum levels are coupled to a single mode of a transmission-line resonator. Our arguments, however, will be general and should apply to any other realization of cavity QED with sufficiently large coupling and atom coherence times. Taking into account the relatively small anharmonicity of the transmon qubit, the system is modelled in terms of an extended Jaynes–Cummings hamiltonian including a coherent drive at frequency $\omega_\text{d}/2\pi$, $$\begin{aligned}
H&=\hbar\omega_\text{r} a^\dag a + \sum_j \hbar\omega_j {\left\lvert j \right\rangle}{\left\langle j \right\rvert}
\\\nonumber
&+\sum_{j}\hbar g_j\Bigl (a {\left\lvert j+1 \right\rangle}{\left\langle j \right\rvert}+\text{h.c.} \Bigr)+\hbar\xi (a^\dag e^{-i\omega_\text{d}t}+\text{h.c}).\end{aligned}$$ Here, the operator $a$ ($a^\dag$) annihilates (creates) a photon in the resonator mode with frequency $\omega_\text{r}/2\pi$. The energies corresponding to the transmon eigenstates are $\hbar \omega_j$ ($j=0,1,\ldots$), and they are determined by the Cooper pair box hamiltonian[@bouchiat_quantum_1998] for given Josephson energy $E_J$ and charging energy $E_C$. In the transmon regime considered here, $E_J/E_C\gg1$, the levels form a weakly anharmonic ladder with anharmonicity $\alpha=\omega_{12}-\omega_{01}\sim -E_C$ ($\omega_{ij}=\omega_j-\omega_i$ is the transition frequency between levels $i$ and $j$). The transmon–resonator coupling strengths are $g_j\sim g_0\sqrt{j+1}$, where $g_0/2\pi$ is the vacuum Rabi frequency. The resulting energy level diagram for vanishing drive, $\xi=0$, is schematically shown in Fig. \[fig:fig1\].
The interaction of the cavity–transmon system with its environment enables control and measurement by means of microwave drive and detection[@blais_cavity_2004; @wallra_strong_2004], but also leads to decoherence. The appropriate framework to capture all these effects is a description in terms of the reduced density matrix $\rho$ for the cavity and transmon. The Markovian master equation which governs the evolution of $\rho$ is given by $$\begin{aligned}
\label{master2}
\dot\rho=&-\frac{i}{\hbar}{\left[H,\rho\right]}+\kappa{\mathcal{D}}[a]\rho
+\gamma_{1}{\mathcal{D}}\Bigl[\sum_{j}\alpha_{j}{\left\lvert j \right\rangle}{\left\langle j+1 \right\rvert}\Bigr]\rho\nonumber\\
&+\frac{\gamma_\varphi}{2} {\mathcal{D}}\Bigl[\sum_j \beta_j{\left\lvert j \right\rangle}{\left\langle j \right\rvert}\Bigr]\rho ,\end{aligned}$$ where ${\mathcal{D}}[A]$ is the usual Lindblad damping superoperator defined by ${\mathcal{D}}[A]\rho=\left({\left[A\rho,A^\dag\right]}+{\left[A,\rho
A^\dag\right]}\right)/2$. The three damping terms on the right-hand side model the loss of cavity photons at rate $\kappa$, the intrinsic relaxation of transmon excitations at rate $\gamma_{1}$, and the pure dephasing of transmon state superpositions at rate $\gamma_\varphi$ and relative strengths $\alpha_j$, $\beta_j$ (see Methods).
![[**Supersplitting of the vacuum Rabi resonance when probing heterodyne transmission beyond linear response.**]{} The experimental data are obtained with a circuit QED system in the strong-coupling regime, where the vacuum Rabi splitting is observed to exceed 260 linewidths, see [**a**]{}. All plots show the squared heterodyne amplitude $A^2$ in arbitrary units. [**b**]{}, Measured transmission (colour scale) for the left vacuum Rabi peak, as a function of drive frequency and power. The plot reveals the supersplitting of a single Lorentzian into a doublet of peaks. [**c–f**]{}, Cuts for constant power at the values indicated in [**b**]{}. In linear response, [**f**]{}, the vacuum Rabi peak is Lorentzian; as the power increases a central dip develops, [**e**]{}, leading to supersplitting of the peak, [**d**]{}, and eventually becoming asymmetric at the largest powers, [**c**]{}. The experimental data (red line) is in excellent agreement with theory (black line). The results from the 2-level approximation are shown for comparison (green dashed line). \[fig:exp\]](sfigure2.eps){width="1.0\columnwidth"}
The quantity of interest for detection of the vacuum Rabi splitting is the transmission amplitude $A(\omega_\text{d})$ of a microwave signal at frequency $\omega_\text{d}/2\pi$. Here, we employ a heterodyne measurement of the field quadratures $I=\langle b_\text{out}+b_\text{out}^\dagger\rangle$ and $Q=\langle ib_\text{out}^\dagger-ib_\text{out}\rangle$, where $b_\text{out}$ refers to the output field at the drive frequency. Input-output theory[@walls_quantum_1995] enables us to express the steady-state transmission amplitude as $$\label{ampl}
A=\sqrt{I^2+Q^2}=2\sqrt{\kappa}\lvert\langle a\rangle\rvert = 2\sqrt{\kappa}\lvert \operatorname{tr}(a\,\rho_\text{s}) \lvert.$$ Our calculations thus only require the steady-state solution $\rho_\text{s}$ of the master equation . We explore the transmission intensity $A^2$ as a function of the drive strength $\xi$, and compare our results to experimental data obtained for a transmon qubit in resonance with a $\omega_\text{r}/2\pi=6.92\,\text{GHz}$ coplanar waveguide (CPW) resonator with coupling strength $g_0/\pi=347\,\text{MHz}$ (see Methods for experimental details). In the linear-response regime, the vacuum Rabi peaks have the characteristic Lorentzian line shape. Their separation and width are given by $2g_0$ and $(\gamma_1+2\gamma_\varphi+\kappa)/2$, respectively. For our specific sample, the splitting is observed to exceed 260 linewidths, shown in Fig. \[fig:exp\]a. Strikingly, when increasing the drive power beyond the linear-response regime, the shape of the transmission curve changes drastically as shown in Figs. \[fig:exp\]b–f. Each vacuum Rabi peak develops a central dip and eventually supersplits into a doublet of peaks. We find excellent agreement between our prediction based on Eqs. and and the measured transmission.
The physics of this supersplitting can be elucidated within a reduced two-level model. This model takes into account only the Jaynes–Cummings ground state and either the symmetric or antisymmetric superposition of transmon and photon excitation. These two states, ${\left\lvert \downarrow \right\rangle}={\left\lvert 0,0 \right\rangle}$ and ${\left\lvert \uparrow \right\rangle}=({\left\lvert 1,0 \right\rangle}\pm{\left\lvert 0,1 \right\rangle})/\sqrt{2}$, form an effective two-level system composed of both transmon and cavity degrees of freedom. Within the effective two-level subspace, the photon operators are mapped[@carmichael_statistical_2008] to Pauli operators $a\to\Sigma_-/\sqrt{2}$, $a^\dag\to\Sigma_+/\sqrt{2}$, so that the microwave tone acts as a drive on the effective two-level system, i.e.$$\label{Htilde}
H_\text{eff} = \frac{\hbar\Delta}{2}\Sigma_z+\frac{\hbar\Omega}{2}\Sigma_x,$$ a scenario that Carmichael et al.[@tian_quantum_1992; @carmichael_statistical_2008] have referred to as “dressing of dressed states." The hamiltonian $H_\text{eff}$ refers to the frame rotating at the drive frequency, $\Delta=\omega_{01}\pm g_0-\omega_\text{d}$ is the detuning between drive and one vacuum Rabi peak, and $\Omega=\sqrt{2}\xi$ is the effective drive strength. With the notable exception of the recent work by I. Schuster et al.[@schuster_nonlinear_2008], previous investigations were primarily concerned with effects on photon correlations and fluorescence, as observed in photon-counting measurements [@tian_quantum_1992; @birnbaum_photon_2005]. According to the operator mapping, photon counting can be related to the measurement of $\langle \Sigma_z \rangle$, whereas detection of the heterodyne transmission amplitude $A$ corresponds to $|\langle \Sigma_-\rangle |$. As a result, heterodyne detection fundamentally differs from photon counting and the vacuum Rabi supersplitting is a characteristic of heterodyne detection only.
After restricting the master equation to the two-level subspace, the system evolution can be expressed in terms of simple Bloch equations for the three components of the reduced density matrix $\rho=(\openone+x\Sigma_x+y\Sigma_y+z\Sigma_z)/2$, $$\begin{aligned}
\dot{x}&=-x/T'_2-\Delta y ,\qquad
\dot{y}=\Delta x-y/T'_2-\Omega z ,\nonumber\\
\dot{z}&=\Omega y -(z+1)/T'_1 .\end{aligned}$$ (An intuitive approach avoiding the Bloch equations is discussed in the Supplementary Information, Discussion.) Here, $T'_1$ and $T'_2$ are the effective relaxation and dephasing times, which are related to $\gamma_1$, $\gamma_\varphi$, and $\kappa$ via ${T'_1}^{-1}=(\gamma_1+\kappa)/2$, and ${T'_2}^{-1}=(\gamma_1+2\gamma_\varphi+\kappa)/4$. From the steady-state solution of the Bloch equations we find the transmission amplitude $$\label{blochsol} A = \frac{T'_2\Omega\sqrt{\kappa(\Delta^2{T'_2}^2+1)/2}}
{\Delta^2{T'_2}^2 + T'_1 T'_2 \Omega^2+1} .$$ This expression correctly describes the crossover from linear response at small driving strength, $ \Omega\ll
(T'_1 T'_2)^{-1/2}$, producing a Lorentzian of width $2 {T'_2}^{-1}$, to the doublet structure observed for strong driving. Specifically, as the drive power is increased, the response saturates and the peak broadens, until at $\Omega=(T'_1 T'_2)^{-1/2}$ the peak undergoes supersplitting with peak–peak separation $2{T'_2}^{-1}\sqrt{T'_1 T'_2 \Omega^2-1}$. The fact that we use heterodyne detection is indeed crucial for the supersplitting. It is easy to verify within the two-level approximation that photon-counting always results in a Lorentzian. For photon counting, probing beyond the linear-response regime merely results in additional power-broadening; specifically, the width of the Lorentzian is given by $2{T'_2}^{-1}\sqrt{T'_1T'_2\Omega^2+1}$.
In Figs. \[fig:exp\]c–e, the analytical expression is plotted for comparison with the full numerical results and the experimental data. We find good agreement for low to moderate drive power, confirming that the supersplitting can be attributed to driving the vacuum Rabi transition into saturation while measuring the transmission with the heterodyne technique. For higher drive power a left-right asymmetry appears in the true transmission spectrum, which is not reproduced by Eq. , and which is partly due to the influence of levels beyond the two-level approximation.
![[**Emergence of $\sqrt{n}$ peaks under strong driving of the vacuum Rabi transition.**]{} [**a**]{}, The extended Jaynes–Cummings energy spectrum. All levels are shown to scale in the left part of the diagram: black lines represent levels ${\left\lvert n,\pm \right\rangle}\simeq({\left\lvert n+1,0 \right\rangle}\pm{\left\lvert n,1 \right\rangle})/\sqrt{2}$ with only small contributions from higher ($j>1$) transmon states; grey lines represent levels with large contributions from higher transmon states. In the right part of the diagram, the $\sqrt{n}$ scaling of the splitting between the ${\left\lvert n,\pm \right\rangle}$ states is exaggerated for clarity, and the transitions observed in plots [**b–e**]{} are indicated at the $x$-coordinate $E_{n\pm}/n\hbar$ of their $n$-photon transition frequency from the ground state. [**b**]{}, Measured transmission ($A^2$, heterodyne amplitude squared) in colour scale as a function of drive frequency and power. The multiphoton transitions shown in [**a**]{} are observed at their calculated positions. [**c–e**]{}, Examples of cuts for constant power, at the values indicated in [**b**]{} (results from the master equation in black; experimental results in red), demonstrating excellent agreement between theory and experiment, which is reinforced in the enlarged insets. Good agreement is found over the full range in drive power from $-45\,\text{dB}$ to $+3\,\text{dB}$, for a single set of parameters (also see Supplementary Movie 1). \[fig:powsweep\]](sfigure3.eps){width="1.0\columnwidth"}
Higher levels of the extended Jaynes–Cummings hamiltonian become increasingly important when the drive power is raised further. Figure \[fig:powsweep\] shows the emergence of additional peaks in the transmission spectrum. Each of the peaks can be uniquely identified with a multiphoton transition from the ground state to an excited Jaynes–Cummings state. For simplicity, we consider the situation where the anharmonicity $\alpha$ and the coupling strength $g_0$ are sufficiently different that mixing between higher transmon levels and the regular Jaynes–Cummings states ${\left\lvert n,\pm \right\rangle}=({\left\lvert n+1,0 \right\rangle}\pm{\left\lvert n,1 \right\rangle})/\sqrt{2}$ is minimal for the low-excitation subspaces. Accordingly, the experiments are carried out using a different transmon qubit in the same sample, with a smaller coupling of $g_0/\pi=94.4\,\text{MHz}$ (see Methods). In this case, the $n$-photon transitions to the $n$-excitation subspace occur at frequencies $E_{n\pm}/nh=(\omega_\text{r}\pm n^{-1/2}g_0)/2\pi$, and thus reveal the nonlinearity of the Jaynes–Cummings ladder. The detailed comparison between experimental data and numerical simulation in Fig. \[fig:powsweep\]c–e shows superb agreement down to the narrowest features observed.
The possibility of multiphoton transitions at sufficiently large drive powers also affects the shape of the vacuum Rabi splitting when tuning the qubit frequency $\omega_{01}$ through resonance with the cavity, shown in Fig. \[fig:wolverine\]. Instead of the simple avoided crossing commonly observed at low drive powers[@wallra_strong_2004], the presence of multiphoton transitions leads to a fan-like structure where individual branches can again be identified one-to-one with the possible transitions in the Jaynes–Cummings ladder. In the experimental data of Fig. \[fig:wolverine\]a, processes up to the 5-photon transition are clearly visible. Detailed agreement with the theory verifies that the more general situation of non-zero detuning between qubit and resonator is correctly described by our model.
![[**Qubit–cavity avoided crossing at high drive power.**]{} Transmission measurement when tuning the qubit frequency through resonance for a drive power of $+1\,\text{dB}$. [**a**]{}, Measured transmission as a function of drive frequency and magnetic field. As the field is increased, the qubit frequency is tuned through resonance with the cavity, and anticrossing behaviour is observed. The multiphoton transitions shown in Fig. \[fig:powsweep\]a are visible. The anomaly at $B\simeq
15.59$ is most likely due to the crossing of a higher level of the second qubit present in the same cavity. [**b–c**]{}, Example cuts at constant magnetic field, at the values indicated in [**a**]{} (master equation results, calculated using the same parameters as for Fig. \[fig:powsweep\], are shown in black; measured results in red). (Also see Supplementary Movie 2). \[fig:wolverine\]](sfigure4.eps){width="1.0\columnwidth"}
In summary, we have shown the supersplitting and the emergence of multiphoton transitions in the nonlinear response of the vacuum Rabi resonance, presenting both theoretical predictions and their experimental verification. This enables the observation of the characteristic $\sqrt{n}$ nonlinearities in the Jaynes–Cummings ladder in a heterodyne transmission measurement, and is direct evidence for the strong coupling between the superconducting qubit and the quantized microwave field. The high precision of agreement between predictions and experiment demonstrates the robustness of the Jaynes–Cummings physics in a circuit QED system. This opens up pathways for further investigations in quantum optics and quantum information, such as generating number-squeezed states or employing multi-level quantum logic.
METHODS {#methods .unnumbered}
-------
[THEORY]{}\
The modelling of the transmon follows ref. . While some expressions given in the main text are asymptotic results valid for $E_J/E_C\gg1$, our calculations are based on a full diagonalization of the Cooper pair box Hamiltonian.
The description of the transmon–cavity system in terms of a master equation requires a model for the relaxation and dephasing of higher transmon levels. As detailed studies of the microscopic origin of the dominant relaxation and dephasing channels are still outstanding, we have chosen plausible superoperators for our master equation . Assuming that relaxation of higher transmon levels may arise due to a coupling of external degrees of freedom to the charge on the superconducting island, we take the relative strengths of relaxation to be related to the coupling parameters as $\alpha_j=g_j/g_0$. Dephasing of higher levels is likely to be due to charge noise. Denoting the charge dispersion[@koch_charge-insensitive_2007] of level $j$ by $\epsilon_j=\omega_j(n_g=0)-\omega_j(n_g=1/2)$, we therefore take the relative dephasing strengths to be $\beta_j=2\epsilon_j/(\epsilon_1-\epsilon_0$). (The normalisation of $\alpha_j$ and $\beta_j$ is chosen to allow the usual interpretation of $\gamma_1$ and $\gamma_\varphi$ as relaxation and dephasing rates for the first two levels of the transmon spectrum.) In fact, the pure dephasing rate is sufficiently small for our qubits that we set $\gamma_\varphi=0$. A comparison of additional simulations with auxiliary experimental results at increased temperatures allows us to place an approximate upper bound of $0.003$ on the number of thermal photons in the cavity.
For the steady-state solution of Eq. , the Hilbert space is truncated to a subspace with maximum number of excitations $N$, using the projector $P_N=\sum_{0\le n+j\le N}{\left\lvert n,j \right\rangle}{\left\langle n,j \right\rvert}$. In our simulations, we keep up to $N=7$ excitations. To reach agreement with the experimentally measured signal for the strongest drive powers, it is necessary to account for a small amount ($\sim-55\,\text{dB}$) of leakage of the drive past the cavity. In addition, there is a small bias introduced by measuring the transmission as the square of the $I$ and $Q$ quadratures, each of which is subject to noise. Accordingly, the quantity that corresponds to the experimental signal is $A^2=\lvert 2 \sqrt{\kappa}\operatorname{tr}(\rho_s a) + b \xi \rvert^2 + 2 \sigma_n^2$, where $b$ describes the leakage of the drive bypassing the cavity, and $\sigma_n$ is the measurement noise in each of the $I$ and $Q$ channels.
Fits are obtained by minimizing the mean squared deviation between experiment and calculation over the full power range, with fit parameters being $b$ and the two scaling factors describing the signal attenuation and amplification for input and output signals. To obtain optimal agreement, we also make small adjustments to the system parameters $\gamma_1$, $\kappa$, $g_0$, $\omega_\text{r}$, $\omega_{01}$, and $E_C$. These parameters are confined by separate experiments to narrow ranges, and all values used in fits are consistent within the experimental uncertainties. Once obtained, the same set of parameters was used in generating Figs. \[fig:powsweep\], \[fig:wolverine\], and the Supplementary Movies.\
[EXPERIMENT]{}\
Measurements are performed in a dilution refrigerator at $15\,\text{mK}$. The sample consists of two superconducting transmon qubits[@koch_charge-insensitive_2007; @schreier_suppressing_2008], coupled to an on-chip coplanar waveguide (CPW) cavity. Fabrication of the sample followed the description given in ref.. The CPW resonator has a half-wavelength resonant frequency of $\omega_\text{r}/2\pi = 6.92\,\text{GHz}$ and a photon decay rate of $\kappa/2\pi = 300\,\text{kHz}$. Transmission measurements are performed using a heterodyne detection scheme. The transmitted RF voltage signal through the cavity is mixed down to a 1MHz carrier signal, and then digitally mixed down to dc to obtain the transmitted voltage amplitude as a function of frequency. The vacuum Rabi coupling strengths for the two qubits are obtained as $g_0/\pi =347\,\text{MHz}$ (qubit 1) and $g_0/\pi = 94.4\,\text{MHz}$ (qubit 2). Time domain measurements of the qubits show that they are Purcell-limited and completely homogenously broadened at their flux sweet spots[@houck_controlling_2008]. The coherence times are $T_1 = 1.7\,\mu\text{s}$ and $T_2 =
0.7\,\mu\text{s}$ (qubit 1, away from the flux sweet spot) and $T_1 = 1.4\,\mu$s and $T_2 = 2.8\,\mu$s (qubit 2, at flux sweet spot). The charging energies of the two qubits are measured to be $E_C/h = 400\,\text{MHz}$ and $E_C/h=340\,\text{MHz}$. (See Supplementary Information, Methods, for further details.)
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Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M., and Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits:An architecture for quantum computation. , 062320 (2004).
Devoret, M., Girvin, S., and Schoelkopf, R. J. Circuit-QED: How strong can the coupling between a Josephson junction atom and a transmission line resonator be? , 767 (2007).
Schoelkopf, R. J. and Girvin, S. M. Wiring up quantum systems. , 664 (2008).
Schreier, J. A., Houck, A. A., Koch, J., Schuster, D. I., Johnson, B. R., Chow, J. M., Gambetta, J. M., Majer, J., Frunzio, L., Devoret, M. H., Girvin, S. M., and Schoelkopf, R. J. Suppressing charge noise decoherence in superconducting charge qubits. , 180502 (2008).
Zhu, Y., Gauthier, D. J., Morin, S. E., Wu, Q., Carmichael, H. J., and Mossberg, T. W. Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations. , 2499 (1990).
Tian, L. and Carmichael, H. J. Quantum trajectory simulations of the two-state behavior of an optical cavity containing one atom. , R6801 (1992).
Brune, M., Schmidt-Kaler, F., Maali, A., Dreyer, J., Hagley, E., Raimond, J. M., and Haroche, S. Quantum Rabi Oscillation: A Direct Test of Field Quantization in a Cavity. , 1800 (1996).
Schuster, I., Kubanek, A., Fuhrmanek, A., Puppe, T., Pinkse, P. W. H., Murr, K., and Rempe, G. Nonlinear spectroscopy of photons bound to one atom. , 382 (2008).
Birnbaum, K. M., Boca, A., Miller, R., Boozer, A. D., Northup, T. E., and Kimble, H. J. Photon blockade in an optical cavity with one trapped atom. , 87 (2005).
Fink, J. M., Göppl, M., Baur, M., Bianchetti, R., Leek, P. J., Blais, A., and Wallraff, A. Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system. , 315 (2008).
Hofheinz, M., Weig, E. M., Ansmann, M., Bialczak, R. C., Lucero, E., Neeley, M., O’Connell, A. D., Wang, H., Martinis, J. M., and Cleland, A. N. Generation of Fock states in a superconducting quantum circuit. , 310 (2008).
Simmonds, R. et al. (unpublished)
Deppe, F., Mariantoni, M., Menzel, E., Marx, A., Saito, S., Kakuyanagi, K., Tanaka, H., Meno, T., Semba, K., Takayanagi, H., Solano, E., and Gross, R. Two–photon Probe of the Jaynes-Cummings Model and Symmetry Breaking in Circuit QED. (2008).
Koch, J., Yu, T. M., Gambetta, J., Houck, A. A., Schuster, D. I., Majer, J., Blais, A., Devoret, M. H., Girvin, S. M., and Schoelkopf, R. J. Charge-insensitive qubit design derived from the Cooper pair box. , 042319 (2007).
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Carmichael, H. J. . Springer (2008).
Houck, A. A., Schreier, J. A., Johnson, B. R., Chow, J. M., Koch, J., Gambetta, J. M., Schuster, D. I., Frunzio, L., Devoret, M. H., Girvin, S. M., and Schoelkopf, R. J. Controlling the spontaneous emission of a superconducting transmon qubit. (2008).
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work has been supported by Yale University via a Quantum Information and Mesoscopic Physics Fellowship (AAH, JK), the LPS/NSA-ARO grant W911NF-05-1-0365, NSF grants DMR-0653377, DMR-0603369, and PHY-0653073, and by Academy of Finland. We thank J. Gambetta, A. Blais, and A. Wallraff for helpful discussions, and L. Frunzio and B. Johnson for fabrication of the sample.
Author contributions {#author-contributions .unnumbered}
--------------------
JMC led the experimental effort. LSB and JK performed the calculations and did most of the writing. AAH gave technical support and conceptual advice. ET contributed to the early theory. MHD, SMG, and RJS provided support and supervised the project.
Author information {#author-information .unnumbered}
------------------
Correspondence and requests for materials should be addressed to RJS.
|
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abstract: 'I present an approach for modeling areal spatial covariance by considering the stationary distribution of a spatio-temporal Markov random walk. This stationary distribution corresponds to an intrinsic simultaneous autoregressive (SAR) model for spatial correlation, and provides a principled approach to specifying areal spatial models when a spatio-temporal generating process can be assumed. I apply the approach to a study of spatial genetic variation of trout in a stream network in Connecticut, USA, and a study of crime rates in neighborhoods of Columbus, OH, USA.'
author:
- |
Ephraim M. Hanks[^1]\
Department of Statistics, The Pennsylvania State University\
bibliography:
- '/home/ephraim/Dropbox/Research/library.bib'
title: '**A Constructive Spatio-Temporal Approach to Modeling Spatial Covariance**'
---
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[**A Constructive Spatio-Temporal Approach to Modeling Spatial Covariance**]{}
[*Keywords:*]{} SAR models, Diffusion, Autoregressive models.
Introduction {#sec:intro}
============
Almost all spatial data can be viewed as arising from a spatio-temporal generating process. For example, a spatial survey of infectious disease prevalence is a snapshot of a dynamic epidemic process occuring in space and time. Similarly, spatial genetic data are the result of spatio-temporal dispersal, mating, and survival processes at the population level. When these spatial processes are observed at multiple successive time points, the known science behind the spatio-temporal process is often used to motivate a spatio-temporal statistical model [e.g., @Wikle2010; @CressieWikleBook].
In contrast, consider the case of “spatial” data, where only one temporal realization of the spatio-temporal process is observed. In this case, spatial autocorrelation is often modeled by including a spatial random effect [e.g., @Diggle2007] in the fitted statistical model. The prior distribution for this spatial random effect is almost always modeled semiparametrically using a Gaussian process model with covariance function chosen based on the support of the data, irrespective of the spatio-temporal generating process. For example, when the spatial data are point-referenced, the Matern class of covariance functions [e.g., @Cressie1993] are often used, while if the spatial data have areal or lattice support, then either conditional autoregressive [CAR; e.g., @Besag1974; @Besag1995; @RueText] or simultaneous autoregressive [SAR: e.g., @Wall2004; @CressieWikleBook] models are common. In either case, the choice of prior distribution for the spatial random effect is almost always made based solely on the support of the data, without consideration of an underlying generating process.
Spatial data are poor in information relative to spatio-temporal data; however, we are increasingly able to collect large amounts of spatial data. The increased information present in large spatially-correlated data provides an opportunity for more realistic modeling of spatial covariance than has been possible in the past. Additionally, recent recognition of the potential for spatial confounding [@HodgesReich2010; @Paciorek2010; @Hughes2013; @HanksRSR] highlights the need to choose a spatial model with care, as the structure of a spatially-correlated random effect can influence inference on fixed effects.
I propose a general constructive approach to modeling spatial correlation based on considering the stationary distribution of a spatio-temporal generating process. This spatio-temporal generating process can either be specified based on scientific knowledge, or can be thought of simply as a device to construct a spatial correlation with desired properties, such as anisotropy and nonstationarity. In Section 2, I describe the proposed general approach, and link it to current spatial models for continuous (geostatistical) random fields. In Section 3, I focus on areal spatial models and show that the stationary distribution for a spatio-temporal random walk model results in a spatial SAR model, which provides a principled approach for choosing areal neighbors and SAR weights when spatial data can be seen as arising from a spatio-temporal random walk. In Section 4, I use this development to model spatial genetic data based on a spatio-temporal random walk generating process. I apply this model to genetic data collected from trout in the Jefferson-Hill Spruce Brook in Connecticut, USA. In Section 5 I present a second example by modeling crime rates for areal neighborhoods in Columbus, Ohio, USA. This second example illustrates how a spatio-temporal generating process can be used to jointly model fixed and random spatial effects. In Section 6 I close with discussion of the proposed approach.
A Constructive Spatio-Temporal Approach to Modeling Spatial Covariance
======================================================================
The proposed approach is as follows.
1. Define a deterministic spatio-temporal generating model for the spatio-temporal process $\mathbf{y}(s,t)$, where $s$ indexes space and $t$ indexes time $$\frac{\partial}{\partial t} \mathbf{y}(s,t)=\mathcal{F}\left(
\mathbf{y}(s,t)\right).$$ For example, $\mathcal{F}$ could be a differential operator (e.g., $\frac{\partial^2}{\partial s^2}$) in which case (1) is a partial differential equation (PDE).
2. Drive the spatio-temporal process defined by (1) with time-homogeneous spatial noise $\mathbf{W}(s)$ $$\frac{\partial}{\partial t} \mathbf{y}(s,t)=\mathcal{F}\left(
\mathbf{y}(s,t)\right)+\mathbf{W}(s)\quad,\quad \mathbf{W}(s) \sim
N(\cdot,\cdot).$$ The process (2) is now a random (stochastic) process in contrast to (1), which is deterministic.
3. The stationary distribution $\boldsymbol\pi(s)=\lim_{t\rightarrow \infty} \mathbf{y}(s,t)$ of (2) provides a spatial model capturing the dynamics of the spatio-temporal process. $$\frac{\partial}{\partial t} \mathbf{y}(s,t)=0 \quad \Rightarrow\quad
\mathcal{F}\left(
\boldsymbol\pi(s)\right) = -\mathbf{W}(s)$$
Solving (3) for the stationary distribution $\boldsymbol\pi(s)$ can be done analytically in some cases, but in many others a numerical approximation will be required.
Spatio-Temporal Generating Models for Continuous-Space Spatial Models
---------------------------------------------------------------------
I first consider this approach in the context of continuous-space processes, and restrict attention to spatial processes in $\mathcal{R}^2$, with the two dimensions $\mathbf{s}=(x_1,x_2)$. The generalization to higher (or lower) spatial dimensions is straightforward [@Lindgren2011]. The most common spatial covariance function used in continuous space is the Matern class, with covariance function given by $$\text{cov}(\mathbf{s}_i,\mathbf{s}_j)=\sigma^2\frac{1}{\Gamma(\nu)2^{\nu-1}}
\left(\sqrt{2\nu}\frac{d_{ij}}{\phi}\right)^\nu
K_\nu\left(\sqrt{2\nu}\frac{d_{ij}}{\phi}\right)$$ where $d_{ij}=\sqrt{(x_{i1}-x_{j1})^2+(x_{i2}-x_{j2})^2}$ is the Euclidean distance between the spatial locations of the $i$-th and $j$-th observations, $\sigma^2$ is the partial sill parameter, $\nu$ is the Matern smoothness parameter, $\phi$ is a range parameter, and $K_\nu(\cdot)$ is the modified Bessel function of the second kind [e.g., @Cressie1993].
As a special case of the constructive spatio-temporal approach proposed in the previous section, consider the random partial differential equation $$\frac{\partial}{\partial t}
\mathbf{y}(x_1,x_2,t)=(\Delta-\kappa^2)^{\alpha/2}\mathbf{y}(x_1,x_2,t)
+\mathbf{W}(x_1,x_2),$$ where $\Delta=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}$ is the Laplacian and $\mathbf{W}(x_1,x_2)$ is time-homogeneous spatial Gaussian white noise. Note that while equation (4) has been termed a stochastic partial differential equation [SPDE; @Lindgren2011], I follow [@KloedenPlaten] and reserve SPDE to refer to a differential equation model driven by noise which varies over time (e.g., $\mathbf{W}(x_1,x_2,t)$ could be a spatial Wiener process), while a random partial differential equation (RPDE) is a differential equation driven by time-homogeneous noise, as in (2) and (4).
The stationary distribution of (4) satisfies the RPDE $$(\kappa^2-\Delta)^{\alpha/2}\boldsymbol\pi(x_1,x_2)=\mathbf{W}(x_1,x_2),$$ whose solution is a random field of the Matern class [@Whittle1954; @Lindgren2011]. As a concrete example, consider (4) when $\kappa^2=0$ and $\alpha=2$ $$\frac{\partial}{\partial t}
\mathbf{y}(x_1,x_2,t)=\left(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right)\mathbf{y}(x_1,x_2,t)+\mathbf{W}(x_1,x_2).$$ The spatio-temporal generating process (5) is a two-dimensional diffusion with time-homogeneous spatial sources and sinks defined by $\mathbf{W}(s)$. The corresponding stationary spatial distribution is an intrinsic Matern random field with smoothness parameter $\nu=2$ [See @Lindgren2011 for details].
The novelty introduced in this section is the temporal aspect of the RPDE approach, which is not necessary to use spatial models motivated by RPDEs. However, the interpretation of spatial models as stationary distributions of spatio-temporal RPDEs opens up the possibility of scientific modeling of spatial random effects when a spatio-temporal generating process, such as diffusion (5), can be assumed. Having shown how the Matern class of spatial models can be seen as the stationary distributions of the continuous space spatio-temporal RPDE (4), I now consider spatio-temporal models with discrete (areal) spatial support.
Discrete Space Random Walk Models for Spatial Covariance {#sec:disc}
========================================================
[@Lindgren2011] consider discrete (areal) spatial models in the context of numerically approximating the solution to the RPDE (4) using a finite element basis set. Approximating a continuous spatial random field with a finite element approximation leads to increases in computational efficiency, as the finite element basis solution results in a Gaussian Markov random field (GMRF) with sparse precision matrix.
Instead of continuous spatial effects, consider modeling areal spatial processes directly. That is, consider modeling a spatial random effect $\mathbf{y}=[y_1,y_2,\ldots,y_n]'$ on $n$ spatial locations which constitute the full spatial support of the random effect. While both CAR and SAR models have been used extensively to model spatial autocorrelation in areal spatial models, there is little in the way of guidance on when to use one or the other, and little guidance on how to define the neighborhood structure that defines the CAR and SAR models [e.g., @Wall2004; @Assuncao2009].
By considering a Markov random walk on a discrete spatial support, I will first derive a population-level diffusion RPDE based on a large-population approximation to the spatial movements of many individuals. I will then show that the stationary distribution to this RPDE is a SAR model. This result will then be used in Section 5 and Section 6 to propose spatial covariance models based on population-level spatial random walk or diffusion processes.
Population-level Markov random walks
------------------------------------
Let $\mathbf{G}=(\mathbf{V},\mathbf{E})$ be a graph with vertices $\mathbf{V}=\{V_1,V_2,\ldots,V_M\}$ and directed edges $\mathbf{E}=\{\alpha_{ij},i=1,2,\ldots,M;j=1,2,\ldots,M\}$. In particular, consider the case where $\alpha_{ij}$ is the exponential rate at which a random walker in node $i$ transitions to node $j$. As in a standard continuous-time Markov chain (CTMC) model for a random walk, the time $T_i$ spent by a random walker in node $i$ before transitioning to any other node is exponentially-distributed with rate $\alpha_i=\sum_{k=1}^n \alpha_{ik}$.
Consider population-level processes on the graph $\mathbf{G}$ in which there are $N$ members of the population, all behaving as a random walk. If there are $n_i(t)$ individuals at node $i$ and time $t$, then the rate at which individuals move from node $i$ to node $j$ is given by $n_i\alpha_{ij}$. Following [@Kurtz1978] and [@Baxendale2011], the normalized population process $\mathbf{z}(t)=[z_1(t) \ z_2(t)\ \ldots z_M]$ can be defined as $z_i(t)=n_i(t)/N$.
{width="5.1in"}
In an open population model, individuals may enter (birth) or leave (death) the system continuously in time at any node (Figure 1). It is common to model the birth and death rates at node $i$ as being density dependent, with birth rate of $n_i b$ and death rate of $n_i d$, for constant rates $b$ and $d$ shared across space. Instead, I will allow the birth and death rates to vary spatially, as this will provide a convenient mechanism for accounting for unmodeled spatial variation. To this end, consider birth and death rates that scale with the total population size ($N$). Let $N
b_i$ be the rate at which individuals are introduced into node $i$ and let $N d_i$ be the rate at which individuals in node $i$ are removed from the system.
To write a spatio-temporal model for the normalized population process $\mathbf{z}(t)$, it will be helpful to write each of the potential jumps (movement between nodes, births, and deaths) possible in this discrete system. If an individual is introduced at node $i$, then the population at $i$ increases by 1. Notationally, represent this transition in the population process $\mathbf{n}$ as $\mathbf{n} \rightarrow
\mathbf{n}+\mathbf{e}_i$, where $\mathbf{e}_i$ is the canonical vector with $M$ componants, all of which are zero except for the $i$-th element, which is equal to 1. The jump in this birth transition is given by $\mathbf{e}_i$. Similarly, a death (removal) at node $i$ decreases the population at node $i$ by 1 and is given by the jump $-\mathbf{e}_i$. Spatial movement (transitions) from node $i$ to node $j$, which occur with rate $n_i\alpha_{ij}$, have jumps given by $\mathbf{e}_i-\mathbf{e}_j$. The possible transitions with their rates are given in Table 1.
-------------------------------------------------------------------------------------------------------------------------
Description Transition Jump Rate
-------------------------------- --------------------------------------- ----------------------------- ------------------
Birth at node $i$ $\mathbf{n} \rightarrow $\mathbf{e}_i$ $Nb_i$
\mathbf{n}+\mathbf{e}_i$
Death at node $i$ $\mathbf{n} \rightarrow $-\mathbf{e}_i$ $Nd_i$
\mathbf{n}-\mathbf{e}_i$
Move from node $i$ to node $j$ $\mathbf{n} \rightarrow $\mathbf{e}_j-\mathbf{e}_i$ $n_i\alpha_{ij}$
\mathbf{n}+\mathbf{e}_j-\mathbf{e}_i$
-------------------------------------------------------------------------------------------------------------------------
: Transitions and Poisson rates in the continuous-time Markov population process. \[tab:tabone\]
Given an initial unnormalized population state $\mathbf{n}(0)$ at time zero, the transient distribution $\mathbf{n}(t)$ is given by [e.g., @Baxendale2011] $$\mathbf{n}(t) = \mathbf{n}(0)+ \sum_{ij \neq 0} (\mathbf{e}_{j}-\mathbf{e}_i)
P_{ij}\left[ \int_0^t n_i(s)\alpha_{ij}\text{d}s\right]
+ \sum_i \left( \mathbf{e}_{i} P_{0i}\left[ \int_0^t Nb_i\text{d}s\right] -\mathbf{e}_i
P_{i0}\left[ \int_0^t Nd_i\text{d}s\right]\right)$$ where $$P_{ij}(a) \sim Pois(a),\quad i=0,1,\ldots,M;\
j=0,1,\ldots,M; \ i\neq j.$$ The transient distribution for the normalized density $\mathbf{z}=\mathbf{n}/N$ is given by $$\mathbf{z}(t) = \mathbf{z}(0)+ \sum_{ij \neq 0}
(\mathbf{e}_{j}-\mathbf{e}_i) \frac{1}{N}
P_{ij}\left[ \int_0^t n_i(s)\alpha_{ij}\text{d}s\right]
+ \sum_i \mathbf{e}_i \left(\frac{1}{N}P_{0i}\left[ Nb_i t\right] - \frac{1}{N}P_{i0}\left[ Nd_i t\right]\right).$$ Taking the large population limit as $N\rightarrow\infty$ [@Kurtz1978; @Baxendale2011] gives the integral equation for the normalized density $$\mathbf{z}(t) = \mathbf{z}(0)+ \sum_{i \neq j } (\mathbf{e}_{j}-\mathbf{e}_i)
\int_0^t z_i(s)\alpha_{ij}\text{d}s + \sum_i
\mathbf{e}_{i} (b_i-d_i) t.$$ Details of this calculation are given in Appendix A.
The differential equation associated with (7) is $$\frac{\partial \mathbf{z}(t)}{\partial t} = \sum_{i \neq j } \alpha_{ij}(\mathbf{e}_{j}-\mathbf{e}_i)
z_i(t) + \sum_i \mathbf{e}_{i} (b_i-d_i)$$ which has vectorized form $$\frac{\partial \mathbf{z}(t)}{\partial t} = -\mathbf{Q}'\mathbf{z}(t) +(\mathbf{b} -\mathbf{d})$$ where $\mathbf{b}=[b_1\ b_2\ \ldots \ b_M]'$, $\mathbf{d}=[d_1\ d_2\
\ldots \ d_M]'$, and $\mathbf{Q}$ is the infinitessimal generator of the CTMC or the Laplacian matrix of the graph $$\mathbf{Q}=\left(\begin{array}{ccccc}
\sum_k \alpha_{1k} & -\alpha_{12} & -\alpha_{13} & \cdots &
-\alpha_{1m} \\
-\alpha_{21} & \sum_k \alpha_{2k} & -\alpha_{23} & \cdots &
-\alpha_{2m} \\
-\alpha_{31} & -\alpha_{32} &\sum_k \alpha_{3k} & \cdots &
-\alpha_{3m} \\
\vdots & & &\ddots & \vdots \\
-\alpha_{m1} & -\alpha_{m2} & -\alpha_{m3} & \cdots &
\sum_k \alpha_{mk}
\end{array}\right).$$
Equation (8) specifies a graph diffusion process where $\mathbf{b}-\mathbf{d}$ is a vector of net inputs and outputs to the system and $-\mathbf{Q}'$ is a matrix describing proportional rates of transfer between spatial locations.
Spatial Models From Random Walks
--------------------------------
To specify a spatial model motivated by the differential equation (8), consider modeling the spatial birth and death rates as spatial white noise $$\boldsymbol\gamma=\mathbf{b}-\mathbf{d} \sim
N(\mathbf{0},\sigma^2\mathbf{I})$$ subject to the constraint that $\mathbf{1}'\boldsymbol\gamma=0$. This sum-to-zero constraint on $\boldsymbol\gamma$ is necessary to ensure the existence of a stationary distribution $\boldsymbol\pi$ for (8). The spatio-temporal differential equation (8) can then be written as the RPDE $$\frac{\partial}{\partial t} \mathbf{z}(t) =
-\mathbf{Q}'\mathbf{z}(t)+\boldsymbol\gamma,\quad
\boldsymbol\gamma\sim N(\mathbf{0},\sigma^2\mathbf{I}).$$ The stationary distribution $\boldsymbol\pi$ for the normalized population process $\mathbf{z}$ satisfies the balance equation that $\frac{\partial}{\partial t} \mathbf{z}(t) =\mathbf{0}$, which implies that $$\mathbf{Q'}\boldsymbol\pi = \boldsymbol\gamma,\quad
\boldsymbol\gamma\sim N(\mathbf{0},\sigma^2\mathbf{I})$$ and thus the stationary distribution for (10) is given by $$\boldsymbol\pi \sim N(\mathbf{0}, (\mathbf{QQ}')^-), \text{ with
}\mathbf{1}'\boldsymbol\pi=0.$$
This stationary distribution is a random field on the discrete spatial support of the population process $\mathbf{z}(t)$ with spatial covariance defined by the spatio-temporal CTMC random walk with infinitessimal generator $\mathbf{Q}$ (9).
### Links to Intrinsic Simultaneous Autoregressive Random Fields
The random field in (11) corresponds to an intrinsic simulataneous autoregressive (SAR) model for spatial correlation. This correspondence provides an intuitive approach for specifying the SAR neighborhood structure in situations where some information is known about the spatio-temporal dynamics of the system being modeled.
The standard SAR model can be written [see e.g., Section 4.2.7 of @CressieWikleBook] as $$\mathbf{y}\sim
N(\mathbf{0},(\mathbf{I}-\mathbf{B})^{-1}\boldsymbol\Lambda(\mathbf{I}-\mathbf{B}')^{-1}
)$$ where $\mathbf{B}$ has zeroes on the diagonal and $\boldsymbol\Lambda$ is a diagonal matrix with $i$-th diagonal $\Lambda_{ii}$. Then setting $$B_{ij}=\frac{\alpha_{ji}}{\sum_k \alpha_{ik}} \text{ and }
\Lambda_{ii}=\frac{1}{\left(\sum_k \alpha_{ik}\right)^2}$$ expresses (6) as an intrinsic SAR model. As in standard SAR models, the matrix $\mathbf{Q}$ from (4) does not have to be symmetric, but rather can incorporate models for asymmetric random walks. Additionally, if $\mathbf{Q}$ is sparse (many of the $\{\alpha_{ij}\}$ are zero), then sparse matrix methods [e.g., @RueText] can be employed to sample from and evaluate the density in (6). The SAR models (and related CAR models) have been viewed as unintuitive [@Wall2004]. The spatio-temporal random walk motivation for the spatial model in (11) provides a principled framework for incorporating knowledge about the spatio-temporal spread of a system into a model for spatial autocorrelation.
The random field $\boldsymbol\pi$ in (11) is an intrinsic random field, in that only linear combinations are proper [@Besag1995]. An alternative formulation is that the density for $\boldsymbol\pi$ is proper under the constraint that $\boldsymbol\pi$ sums to zero over the spatial domain. Intrinsic random fields are often used as prior distributions, where the posterior distribution is proper. For example, consider modeling a Gaussian response $\mathbf{y}$ as $$\mathbf{y} = \mu\mathbf{1} + \boldsymbol\pi +
\boldsymbol\epsilon,\quad \boldsymbol\epsilon \sim N(\mathbf{0},\tau^2\mathbf{I})$$ where $\boldsymbol\pi \sim N(\mathbf{0}, (\mathbf{QQ}')^-), \text{ with
}\mathbf{1}'\boldsymbol\pi=0.$ Under this formulation, $\boldsymbol\pi$ is constrained to sum to zero, but $\mu\mathbf{1}+\boldsymbol\pi$ is not. This formulation can be seen as a form of restricted spatial regression [@Hughes2013; @HanksRSR] where the spatial random effect $\boldsymbol\pi$ is constrained to be orthogonal to the intercept $\mu\mathbf{1}$.
### Identifiability
The likelihood of (11) $$f(\boldsymbol\pi|\mathbf{Q}) \propto |\mathbf{QQ}'|^{-1/2}\exp\left\{-\frac{1}{2}\boldsymbol\pi'\mathbf{QQ}'\boldsymbol\pi\right\}$$ is a function of $\mathbf{QQ}'$, rather than purely a function of the infinitessimal generator $\mathbf{Q}$. Thus, if there are two generator matrices $\mathbf{Q}$ and $\mathbf{W}$ such that $\mathbf{QQ}'=\mathbf{WW}'$, then $\mathbf{Q}$ is not identifiable. However, the special structure required for a generator matrix of a CTMC allows us to prove that $\mathbf{Q}$ is identifiable in all but pathological situations.
If $\mathbf{Q}$ and $\mathbf{W}$ are both generator matrices (9) for irreducible $M$-state CTMCs, and at least one row of $\mathbf{Q}$ has more than one nonzero off-diagonal entry, then $\mathbf{QQ}'$=$\mathbf{WW}'$ if and only if $\mathbf{Q}=\mathbf{W}$.
The proof is given in Appendix B. The significance of this result is that the only forms for $\mathbf{Q}$ that are unidentifiable come when the embedded chain of the irreducible CTMC governed by $\mathbf{Q}$ is deterministic and topologically the graph given by $\mathbf{Q}$ is a loop, with flow only possible in one direction (either clockwise or counter-clockwise). In all other graph topologies, identifiability is guaranteed.
Example 1: Random walk models for spatial genetic variation on stream networks
==============================================================================
I now present two examples of spatial analyses using the assumption of a spatio-temporal random walk generating process leading to a spatial random effect. The first example comes from landscape ecology, where a common goal is to understand how the landscape influences spatial connectivity or correlation. Random walk models are among the most common models for gene flow, both in theory and in practice. [@McRae2006] showed that under a random walk model for migration, a common formulation of genetic dissimilarity (the linearized fixation index) was proportional to the circuit resistance distance [@Klein1993] between the nodes in question. Under the formulation of [@McRae2006], the spatial domain is envisioned as a graph of spatial nodes with symmetric edge weights $\alpha_{ij}$ proportional to the (symmetric) rate of random walkers between nodes. The resistance distance is the effective resistance in an electric circuit where the nodes are connected by resistors with resistance $1/\alpha_{ij}$ equal to the inverse of the migration rate. This approach to studying gene flow is known as the isolation by resistance approach, and is often used to explore the relationship between landscape features and gene flow.
While most studies addressing isolation by resistance choose between a finite number of pre-specified edge weights (resistances), [@HanksHootenJASA] modeled the observed pairwise genetic distance matrix using the generalized Wishart distribution of [@McCullagh2009] with symmetric precision matrix $\mathbf{Q}$ (9) and made inference on the edge weights $\alpha_{ij}$ as a function of landscape covariates. Instead of using the RPDE stationary distribution approach that gives rise to (11), [@HanksHootenJASA] considered a variogram argument, as follows. Using links between symmetric random walks and electric circuits [@Doyle1984], [@McRae2006] showed that under a random walk model for migration, a common formulation of genetic dissimilarity (the linearized fixation index) was proportional to the resistance distance [@Klein1993]. [@HanksHootenJASA] showed that the resistance distance was exactly the variogram (expected squared difference) of an intrinsic Gaussian spatial random field with precision matrix $\mathbf{Q}$. While this provides an interesting link between random walks and variograms, our goal in this analysis is to directly motivate a spatial model by the stationary distribution of a spatio-temporal model, something not explicitly considered by [@HanksHootenJASA].
The isolation by resistance approach assumes symmetric edge weights (and thus symmetric migration rates), though often it would be more realistic to assume asymmetric migration rates reflecting source and sink dynamics. As an example, consider the system studied by [@Kanno2011], consisting of trout in the Jefferson-Hill Spruce Brook in Connecticut, USA. 470 trout were captured at 173 spatial locations along the brook (Figure 2) and genotyped, with microsatellite allele data obtained at 15 loci. An isolation by resistance approach would require symmetric migration rates between upstream and downstream locations, but a more realistic model (which I will propose) would consider asymmetric migration rates reflecting the potentially increased difficulty in moving upstream relative to moving downstream.
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Additionally, [@Kanno2011] examine the effect of two seasonal blockages of the brook - two locations where the brook dries up and is seasonally impassible to the trout. The hypothesized drivers of gene flow and genetic connectivity among the trout population on Jefferson Hill Spruce Brook are both directional (differential rates of movement upstream and downstream) and non-directional (decreased connectivity between stream locations on opposite sides of the seasonal blockages). If spatio-temporal trout movement data were available, modeling these directional and non-directional responses to covariates would be straightforward [@Hooten2010b; @HanksCTDS]. For example, movement could be envisioned as occuring on a graph with a node at each spatial location where trout were sampled, and edge weights equal to random walk transition rates between nodes could be modeled as $$\alpha_{ij}=\begin{cases}
\frac{1}{d_{ij}}\text{exp} \left\{\beta_0+\beta_1 u_{ij}+ \beta_2 v_{ij}\right\} & \text{if nodes }i \text{ and }j \text{ are neighbors}\\
0 & \text{otherwise}\end{cases}$$ where $\{u_{ij}\}$ and $\{v_{ij}\}$ are indicator variables with $u_{ij}=1$ if node $j$ is downstream from node $i$ and $v_{ij}=1$ if a seasonal blockage is located between nodes $i$ and $j$. In this formulation, each node on a branch of the stream network has two neighbors, one upstream and one downstream, and edge weights $\alpha_{ij}$ are zero for all other non-neighboring nodes. Each node at a confluence of two stream branches will have three neighbors, one downstream and two upstream. The rate at which a random walker at a node $i$ on a branch of the stream network transitions to the nearest upstream node $j$ is $\alpha_{ij}=1/d_{ij}\text{exp}\{\beta_0\}$ if there is not a seasonal blockage between nodes $i$ and $j$. Similarly, the rate at which the random walker transitions from $i$ to the nearest downstream node $k$ is $\alpha_{ik}=1/d_{ik}\text{exp}\{\beta_0+\beta_1\}$. The parameter $\beta_2$ models the additive effect that a seasonal blockage has on log-transition rates. Together, this simple random walk model allows for transition rates that vary with direction and location based on known spatial stream characteristics.
A spatial model for the observed microsatellite allele data could then be specified with a latent spatial autocorrelation modeled using the stationary distribution (11) of the random walk model (12) when driven by time-homogeneous white Gaussian noise, as described in Section 3.2.
Microsatellite allele data were observed at $L=15$ distinct loci for each spatially referenced trout captured. At the $\ell^{\text{th}}$ locus, $\ell=1,2,\ldots,L$, denote the list of all distinct observed alleles from all individuals in the study as $\{a_{\ell 1},a_{\ell 2},\ldots,a_{\ell K_\ell}\}$. Following [@Guillot2005] and others, I model the two observed alleles for each (diploid) individual as arising from a multinomial distribution with spatially varying allele probabilities $\mathbf{p}_{s\ell}=(p_{s\ell 1}\ p_{s\ell 2}\ \ldots \ p_{s\ell
K_\ell})'$, where $s \in \{1,2,\ldots,S\}$ indexes the spatial location.
Let $y_{sip\ell k}=1$ if the $p^{\text{th}}$ (indexing ploidy) observed allele at the $\ell^{\text{th}}$ locus is $a_{\ell k}$ for the $i^{\text{th}}$ individual at the $s^{\text{th}}$ spatial location, and $y_{sip\ell k}=0$ otherwise. Then the multinomial probit model [e.g., @Albert1993] for categorical data is often specified in terms of latent variables, $\mathbf{z}$, as follows. Let
$$y_{sip\ell k}=\begin{cases}
1 & \ ,\ z_{sip\ell k}=\max\{z_{sip\ell a},\ a=1,\ldots,K_\ell\}\\
0 & \ ,\ \text{otherwise}
\end{cases}$$
where
$$z_{sip\ell k} \sim N(\mu_{\ell k}+\eta_{s\ell k},1).$$
Then the allele $a_{\ell k}$ makes up a fraction $p_{s\ell
k}$ of the genetic makeup of the subpopulation at location $s$, where
$$p_{s\ell k}=\text{Prob}\left(z_{sip\ell k}=\max\{z_{sip\ell a},\
a=1,\ldots,K_\ell\}\right)$$
{width="7in"}
The mean of the latent variable $z_{sip\ell k}$ in (14) consists of the sum of two effects. The first is $\mu_{\ell k}$, an allele specific intercept which determines the relative frequency of the $k^{\text{th}}$ allele at the $\ell^{\text{th}}$ locus across the entire population being studied. Large values of $\mu_{\ell k}$, relative to $\mu_{\ell k'}$ make it more likely that $z_{sip\ell k}$ will be larger than $z_{sip\ell k'}$, and so the $k^{\text{th}}$ allele will be more prevalent than the $(k')^{\text{th}}$ allele. Note that the model (13)-(14) is invariant to a shift in all $\mu_{\ell k}$, as the likelihood is a function of the contrasts $z_{sip\ell k}-z_{sip\ell k'}$, and not the actual values of $z_{sip\ell k}$. Thus, if $\mu_{\ell
k}$ were replaced by $\mu_{\ell k}+c$ for $k=1,2,\ldots,K_\ell$ and some constant $c$, the likelihood of the observed allele data would remain unchanged. To maintain model identifiability, fix $\mu_{\ell 1}=0$ for $\ell=1,2,\ldots,L$, as only the relative differences (contrasts) in $\mu_{\ell k}$ are identifiable.
The second term in the mean of (14) is $\eta_{s \ell k}$, which is a spatially varying random effect that allows the allele frequencies $\mathbf{p}_{s\ell}$ to vary over the stream network. Following the reasoning in Section 3.2, the spatial random effects are modeled as the stationary distribution of a random walk process driven by time-homogeneous noise. Let
$$\boldsymbol\eta_{\ell k}=\left[\begin{array}{c}
\eta_{1\ell k}\\
\eta_{2\ell k}\\
\vdots\\
\eta_{n\ell k}
\end{array}\right]
\sim
N(\mathbf{0},(\mathbf{QQ}')^{-1}),\quad\mathbf{1}'\boldsymbol\eta_{\ell k}=0$$
where $\mathbf{Q}$ is the infinitessimal generator (9) of the random walk with transition rates (12).
The model is completed by specifying diffuse Gaussian priors for the random walk parameters $\beta_0,\ \beta_1,\ \beta_2$ and the allele specific intercepts $$\beta_j \sim N(0,10^2),\quad j=0,1,2$$ $$\mu_{\ell k} \sim N(0,10^2),\quad \ell=1,2,\ldots,L;\quad k=2,3,\ldots,K_\ell.$$
{width="6in"}
A Markov chain Monte-Carlo (MCMC) algorithm was constructed to sample from the posterior distribution of model parameters, given the observed microsatellite allele data. Fifteen chains were run with different starting values. Each chain was run for $10^6$ iterations, with the first $10^5$ samples discarded as burn in. Convergence was assessed by comparing posterior histograms obtained from only the first half of each chain with posterior histograms obtained from only the second half of each chain. Histograms of the marginal posterior distributions of the random walk parameters are given in Figure 3. The posterior distribution for $\beta_1$ is greater than zero, indicating that the data support the anisotropic hypothesis that gene flow is more rapid downstream than upstream. The posterior distribution for $\beta_2$, which captures the effect of the seasonal blockages, overlaps zero (Figure 3(c)), with the $95\%$ equal-tailed credible interval being bounded by $(-2.4,2.2)$. This indicates only weak support (if any) for the hypothesis that the seasonal blockages affect gene flow.
Posterior distributions for the allele specific intercepts are not shown, and posterior mean values for the intercepts ranged from $-2.2$ to $0.7$. To qualitatively illustrate the genetic correlation structure implied by the estimated parameters, four realizations of random fields on the stream network were simulated using the posterior mean parameter values. These random fields are shown in Figure 4. The constructive spatio-temporal approach proposed here provides a valid autoregressive spatial model for data collected on a stream network. In contrast, [@VerHoef2010] present a moving average (convolution) approach to modeling spatial autocorrelation on stream networks.
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{width="2.61in"} {width="2.61in"}
Example 2: Crime rates in Columbus, OH.
=======================================
A second example illustrates how considering a spatio-temporal generating process can provide insights into modeling the interplay between mean and covariance structure in spatial models. As mentioned previously, recent recognition of the potential for spatial confounding [e.g., @Hughes2013; @HanksRSR] suggests that correctly modeling the relationship between the fixed and random effects in a model is important, even if we only desire to interpret the relationship between fixed effects and the response.
Consider the case of 1980 crime rates in 49 neighborhoods in Columbus, Ohio, USA [@Anselin1988]. Figure 5(a) shows the number of residential burglaries and vehicle thefts per thousand housholds in each of the 49 neighborhoods. Figure 5(b) shows the average value of homes in each neighborhood, in thousands of dollars. These data are freely available in the ‘spdep’ package [@Bivand2015] of the R statistical computing environment [@R].
A preliminary linear regression with crime rates as response and average home values as predictor variable indicates a negative correlation between average home values and crime rates. However, the residuals from this simple linear regression are shown in Figure 5(c) and show clear spatial autocorrelation. A standard spatial analysis might consider the following spatial linear model $$\mathbf{c}=\mu\mathbf{1}+\beta\mathbf{h}+\sigma\boldsymbol\eta+\boldsymbol\epsilon,\quad
\boldsymbol\epsilon\sim N(\mathbf{0},\tau^2\mathbf{I})$$ $$\boldsymbol\eta\sim N(\mathbf{0},(\mathbf{QQ}')^{-1})
\text{ with } \mathbf{1}\boldsymbol\eta=0$$ where $\mathbf{c}$ is a vector of the 1980 crime rates, $\mathbf{h}$ is a vector of average home values, $\boldsymbol\eta$ is a spatial random effect with SAR structure defined by $\mathbf{Q}$, and $\boldsymbol\epsilon$ is nonspatial error. A symmetric neighborhood graph was defined with edges between all polygons that share a polygon edge, as shown in Figure 5(d). If neighborhoods $i$ and $j$ are neighbors, say that $i\sim j$ or, equivalently in this symmetric relationship, $j\sim
i$. The matrix $\mathbf{Q}$ in (19) then has elements $$Q_{ij}=\begin{cases}
-1&\ , \text{ if }i\neq j,\ i\sim j\\
0&\ , \text{ if }i\neq j,\ i\nsim j\\
\sum_{j\sim i}1&\ , \text{ if }i=j\end{cases}.$$ Thus $\boldsymbol\eta$ is an intrinsic spatial random effect with precision matrix $\mathbf{Q}^2$. Heuristically, $\boldsymbol\eta$ is a missing covariate that is spatially smooth on the support of the 49 neighborhoods in Columbus.
Now contrast this purely spatial approach with an approach based on considering a spatio-temporal graph diffusion generating process. As noted in Section 3.1, the differential equation (8) resulting from the large $N$ limit of the population-level random walk process is a diffusion process defined by a vector of inputs to the system and a matrix $-\mathbf{Q}'$ encoding rates of transfer between spatial nodes in the graph. In this spirit, consider a process where the inputs (sources and sinks) are random variables with mean defined by the predictor variable (average home value) and spatial diffusion rates defined by the spatial neighborhood graph. Note that while this is not a science-based mechanistic model for crime in Columbus, it does provide two competing models for how crime rates are related to average home values. In the standard spatial model, the spatial random effect $\boldsymbol\eta$ is a missing covariate unrelated to average home values $\mathbf{h}$. In the graph diffusion based model presented below, a diffusion process spatially smooths the effect of $\mathbf{h}$, similar to a moving average (or convolution-based) spatial model [e.g., @Lee2005].
As an alternative to the standard spatial mixed effect model in (18)-(19), consider modeling crime rates ($\mathbf{c}$) as $$\mathbf{c}=\mu\mathbf{1}+\boldsymbol\pi+\boldsymbol\epsilon$$ where $\boldsymbol\pi$ is the stationary distribution of the spatio-temporal graph diffusion process $\mathbf{z}(t)$ defined elementwise as $$\frac{\partial z_i(t)}{\partial t} = -\kappa n_i z_i(t)
+\sum_{j\sim i} \kappa z_j(t) +
\beta\cdot h_i+\delta_i.$$ The first term on the right hand side of (21) defines the flow out of node $i$ to the $n_i=\sum_{j\sim i} 1$ neighboring nodes. The second term defines the flow into node $i$ from other nodes. The net input/output from “births” and “deaths” into node $i$ is $\beta
h_i+\delta_i$. The intuition here is that the spatial source of crime in Columbus neighborhoods is correlated with home values, and that crime spreads spatially out from neighborhoods with high crime rates to neighboring regions, with a constant diffusion rate of $\kappa$ between all neighboring nodes.
If $\delta_i$ are modeled as independent zero mean Gaussian random variables, the RPDE can be written in vector form as $$\frac{\partial \mathbf{z}(t)}{\partial t} =
-\kappa\mathbf{Q}'\mathbf{z}(t)+ \beta\mathbf{h}+ \boldsymbol\delta,\quad
\boldsymbol\delta\sim N(\mathbf{0},\sigma^2\mathbf{I})$$ and the stationary distribution $\boldsymbol\pi$ satisfies $$\kappa\mathbf{Q}'\boldsymbol\pi=\beta\mathbf{h}+\boldsymbol\delta,$$ or, equivalently $$\boldsymbol\pi \sim
N\left(\frac{\beta}{\kappa}(\mathbf{Q}')^{-1}\mathbf{h},\frac{\sigma^2}{\kappa}(\mathbf{QQ}')^{-1}\right),\quad
\mathbf{1}'\boldsymbol\pi=0$$ where $(\mathbf{Q}')^{-1}$ is the Bott-Duffin constrained generalized inverse [@Bott1953] of $\mathbf{Q}'$. The data model (20) for the graph diffusion spatial model can then be written as $$\mathbf{c}=\mu\mathbf{1}+\tilde{\beta}(\mathbf{Q}')^{-1}\mathbf{h}+\tilde{\sigma}\boldsymbol\eta+\boldsymbol\epsilon$$ with $\tilde{\beta}=\beta/\kappa$, $\tilde{\sigma}=\sigma/\kappa$, and $\boldsymbol\eta$ a random effect defined as in (19). Without strong prior information, $\kappa$ will be unidentifiable. Instead, consider inference on $\tilde{\beta}=\beta/\kappa$ and $\tilde{\sigma}=\sigma/\kappa$, which are identifiable. In this formulation, the only difference between the standard spatial model in (18) and the graph diffusion based spatial model in (23) is that the fixed effect $\mathbf{h}$ in (18) is smoothed by $(\mathbf{Q}')^{-1}$ in (23).
Within a Bayesian framework for inference, I assigned flat Gaussian priors to the regression parameters $\mu$, $\beta$, and $\tilde{\beta}$. Flat half-normal priors were chosen for the spatial random effect variance parameters $\sigma$ and $\tilde{\sigma}$, and an inverse-gamma prior was chosen for the non-spatial error variance $\tau^2$. Inference on the parameters in (19) and (23) was obtained by a Markov chain Monte Carlo sampler. In each case, the MCMC sampler was run for $10^5$ iterations. Convergence was assessed by comparing histograms of samples from the first half of the Markov chain with histograms of samples from the second half of the Markov chain.
[rrrr]{} Parameter & Post. Mean & Post. 0.025 Quantile & Post. 0.975 Quantile\
\
$\mu$ & 35.12 & 32.09 & 38.12\
$\beta$ & -9.28 & -12.48 & -6.16\
$\sigma$ & 1.81 & 0.31 & 3.50\
$\tau$ & 10.75 & 8.86 & 13.04\
\
$\mu$ & 35.13 & 31.89 & 38.33\
$\tilde{\beta}$ & -9.38 & -12.89 & -5.92\
$\tilde{\sigma}$ & 0.94 & 0.03 & 2.67\
$\tau$ & 11.51 & 9.68 & 13.75\
Posterior means and 95$\%$ credible interval bounds are shown in Table 2. To compare models, I computed the Deviance information criterion [DIC, @Spiegelhalter2002]. Posterior distributions for $\mu$ and $\beta$ from the spatial model (18) are similar to those of $\mu$ and $\tilde{\beta}$ from the graph diffusion model (23); however, the standard deviation $\sigma$ of the spatial random effect $\boldsymbol\eta$ in the spatial model (18) is larger than the corresponding standard deviation $\tilde{\sigma}$ in the graph diffusion model (23). This indicates that the need for the spatial random effect is greater in the spatial model than in the graph diffusion model where the home value covariate was smoothed by $(\textbf{Q}')^{-1}$. The DIC of the graph diffusion model (DIC=411) was lower than that of the standard spatial model (DIC=442), indicating that in this case, considering a spatio-temporal generating process resulted in a better model fit than would be obtained by the inclusion of a standard spatial random effect.
Discussion
==========
While we have focused on discrete space models, this general approach has potential for application in continuous space as well. Spatial deformation approaches to nonstationary covariance [e.g., @Schmidt2003; @Lindgren2011] can be viewed as stationary distributions of diffusion processes with spatially heterogeneous diffusion rates. Reaction-diffusion models are common in ecology and other fields [e.g., @Keeling2004; @Hu2013] and would provide a natural spatio-temporal generating process basis for spatial random effect models in a wide variety of systems. Finite element basis and grid-based approaches to approximating continuous spatial fields have a long history in spatio-temporal [e.g., @Wikle2010] and spatial [e.g., @Lindgren2011] analysis, and could be used to approximate the stationary distribution of a continuous (infinite-dimensional) spatio-temporal generating process with a finite number of basis functions.
Current standard approaches to modeling spatial correlation focus on nonparametric random effect models. This work proposes a parametric constructive approach to modeling spatial random effects based on an assumed spatio-temporal generating process. The two examples give some indication of how this approach may be used. In the first example, existing scientific knowledge about the system (gene flow on a stream network) was used to specify a spatio-temporal generating model (a population-level random walk), and the stationary distribution of this spatio-temporal process defined the distribution of the spatial random effect used to model genetic correlation. In the second example, a descriptive approach was taken to compare multiple models for spatial variation. In particular, for the Columbus crime data, the graph diffusion model provided a better model fit than was obtained using a standard spatial random effect model. Modeling spatial random effects nonparametrically is the current standard practice; however, there are benefits to parametric modeling of spatial random effects when the existing science can suggest a spatio-temporal generating mechanism.
Appendix A: Large population limits of population processes {#appendix-a-large-population-limits-of-population-processes .unnumbered}
===========================================================
The interested reader is referred to [@Kurtz1981] for a full treatment of stochastic population processes. This derivation follows the spirit of [@Kurtz1981] and [@Baxendale2011], but with the novelty of birth and death rates that are not density dependent.
Following from (6) in Section 3.1, the transient distribution for the normalized density $\mathbf{z}=\mathbf{n}/N$ is given by $$\mathbf{z}(t) = \mathbf{z}(0)+ \sum_{ij \neq 0}
(\mathbf{e}_{j}-\mathbf{e}_i) \frac{1}{N}
P_{ij}\left[ \int_0^t n_i(s)\alpha_{ij}\text{d}s\right]
+ \sum_i \mathbf{e}_i \left(\frac{1}{N}P_{0i}\left[ Nb_i t\right] - \frac{1}{N}P_{i0}\left[ Nd_i t\right]\right)$$ where $$P_{ij}(a) \sim Pois(a),\quad i=0,1,\ldots,M;\
j=0,1,\ldots,M; \ i\neq j.$$
Note that $$\begin{aligned}
P_{ij}(a) &= a +(P_{ij}(a)-a)\\
&=a+W_{ij}(a)\ ,\quad \quad W_{ij}(a)\sim(0,a)\end{aligned}$$ where each $W_{ij}$ has mean zero on constant variance. Applying this to the transient distribution gives $$\begin{aligned}
\mathbf{z}(t) &= \mathbf{z}(0)+ \sum_{ij \neq 0}
(\mathbf{e}_{j}-\mathbf{e}_i) \frac{1}{N}
\left[ \int_0^t n_i(s)\alpha_{ij}\text{d}s\right]
+ \sum_i \mathbf{e}_i \left(b_i t-d_i t\right) \\
&+\frac{1}{N}\left(\sum_{i \neq j} (\mathbf{e}_{j}-\mathbf{e}_i)
W_{ij}\left[ \int_0^t n_i(s)\alpha_{ij}\text{d}s\right] + \sum_i
\mathbf{e}_{i} \left(W_{0i}\left[ Nb_i t\right] - W_{i0}\left[ Nd_i t\right]\right)\right).\end{aligned}$$
Consider a fixed $t>0$ and note that $N \geq n_i(s)$ for all $s\in
(0,t)$. This gives the result that $$\int_0^t n_i(s)\alpha_{ij}\text{d}s \leq N\alpha_{ij}t.$$ Then to show that all terms above including random variables $W_{ij}$ disappear in the limit as $N\rightarrow \infty$, it is enough to consider the behavior of $$\frac{1}{N}W(Na),\quad W(a)\sim (0,a)$$ for a constant $a>0$. It is trivial to note that $$E\left[\frac{1}{N}W(Na)\right]=0$$ and that $$Var\left[\frac{1}{N}W(Na)\right]=\frac{1}{N^2}Na$$ which vanishes in the limit as $N\rightarrow \infty$.
Then, in the large population limit, the transient distribution of the normalized population $\mathbf{z}(t)$ will be given by $$\mathbf{z}(t) = \mathbf{z}(0)+ \sum_{i \neq j}
(\mathbf{e}_{j}-\mathbf{e}_i) \frac{1}{N}
\left[ \int_0^t n_i(s)\alpha_{ij}\text{d}s\right]
+ \sum_i \mathbf{e}_i \left(b_i t-d_i t\right).$$
Appendix B: Proof of Theorem 3.1 {#appendix-b-proof-of-theorem-3.1 .unnumbered}
================================
In this appendix, we prove Theorem 3.1. The proof follows from the fact that $\mathbf{QQ}'$ is a Gramian matrix [e.g., @GentleText] and thus $\mathbf{QQ}'=\mathbf{WW}'$ if and only if $\mathbf{W}=\mathbf{QU}'$ for a real unitary matrix $\mathbf{U}'$. As $\mathbf{W}$ and $\mathbf{Q}$ are both generators for CTMC random walks, their rows sum to zero ($\mathbf{Q1}=\mathbf{W1}=\mathbf{0}$), with negative diagonal entries ($q_{ii}<0$, $w_{ii}<0$) and non-negative off-diagonal entries ($q_{ij}\geq0$, $w_{ij}\geq 0$ for $i\neq j$). If $\mathbf{Q}$ and $\mathbf{W}$ are both generators for irredicible CTMCs, then both matrices have rank $n-1$ and their null spaces are both spanned by the $\mathbf{1}$ vector. As $\mathbf{W1}=\mathbf{0}$, it follows that $\mathbf{QU}'\mathbf{1}=\mathbf{0}$ and thus $\mathbf{U}'\mathbf{1}=\lambda\mathbf{1}$ for some $\lambda$. The eigenvalues of any unitary matrix $\mathbf{U}'$ have absolute value equal to 1, so $\lambda$ either equals $1$ or $-1$. If $\mathbf{u}'_i$ is the $i$-th row of $\mathbf{U}'$, then $\mathbf{u}'_i\mathbf{1}$ equals either 1 or $-1$, but since $\mathbf{U}$ is unitary, $\mathbf{u}'_i\mathbf{u}_i=1$. These requirements both hold if and only if $\mathbf{u}_i=\lambda \mathbf{e}_k$, where $\mathbf{e}_k$ is the canonical vector with $k$-th element equal to 1 and all other elements equal to zero. As $\mathbf{U}$ is of full rank, the rows of $\mathbf{U}'$ must contain a full set of canonical vectors spanning $\mathcal{R}^n$.
First consider the case where $\lambda=1$. Then $\mathbf{U}'$ is a permutation matrix, with the columns of $\mathbf{W}$ being permuted columns of $\mathbf{Q}$. However, as $\mathbf{W}$ and $\mathbf{Q}$ are generator matrices, each diagonal entry of $\mathbf{W}$ and $\mathbf{Q}$ must be negative, while all off-diagonal entries are non-negative. This can only hold for $\mathbf{W}$ if the permutation matrix $\mathbf{U}'$ is the identity matrix, and thus $\mathbf{W}=\mathbf{Q}$.
Now consider the case where $\lambda=-1$. Again $\mathbf{U}'$ permutes the columns of $\mathbf{Q}$, but now the sign of all entries is changed through multiplication by $\lambda=-1$. So $w_{ii}=-q_{ik}$ and $w_{ik}=-q_{ii}$ for some $k$. As $\mathbf{W}$ is a generator matrix, $w_{ii}=-\sum_{j\neq i} w_{ij}$, which is only possible if $q_{ik}$ is the only non-zero off-diagonal entry in the $i$-th row of $\mathbf{Q}$. This completes the proof.
[^1]: The author gratefully acknowledges conversations with John Fricks, whose suggestions were instrumental in the analysis of Section 3.1
|
---
abstract: 'Reducible representations of CAR and CCR are applied to second quantization of Dirac and Maxwell fields. The resulting field operators are indeed operators and not operator-valued distributions. Examples show that the formalism may lead to a finite quantum field theory.'
---
\[firstpage\]
Reducible representation of CAR
===============================
The main objective of the paper is to discuss a reducible representation of canonical anticommutation relations (CAR) which generalizes to Dirac electrons the construction previously employed in [@I; @II; @III] to electromagnetic fields. The case of canonical commutation relations (CCR) is briefly discussed in the last section.
Beginning with the CAR operators $b_\pm$, $d_\pm$ of an [*irreducible*]{} representation we introduce the following four operators $$b(\vec p,\pm)
=
|\vec p\rangle\langle \vec p|\otimes b_\pm=c_1(\vec p,\pm),\quad
d(\vec p,\pm)
=
|\vec p\rangle\langle \vec p|\otimes d_\pm=c_2(\vec p,\pm)$$ satisfying a reducible representation of CAR. The momentum eigenvectors are normalized by $\langle \vec p|\vec p\,'\rangle
=
\delta_{\Gamma_m}(\vec p,\vec p\,')
=
(2\pi)^3 2\sqrt{\vec p^2+m^2}\delta^{(3)}(\vec p-\vec p\,')
$.The reducible representation of CAR can be written in a compact form as $$\big\{c_j(\vec p,s),c_{j'}(\vec p\,',s')^{\dag}\big\}
=
\delta_{jj'}\delta_{ss'}\delta_{\Gamma_m}(\vec p,\vec p\,')
|\vec p\rangle\langle \vec p|\otimes 1
=
\delta_{jj'}\delta_{ss'}\delta_{\Gamma_m}(\vec p,\vec p\,')
I_{\vec p},\label{nCAR}$$ the remaining anti-commutators vanishing. The identity 1 in (\[nCAR\]) is the one occuring in the CAR relations $\{b_\pm,b_{\pm}^{\dag}\}=\{d_\pm,d_{\pm}^{\dag}\}=1$ and the RHS of (\[nCAR\]) is in the center of the CAR algebra. Similarly to [@I; @II; @III] we have introduced the operator $
I_{\vec p}=|\vec p\rangle\langle \vec p|\otimes 1
$ satisfying the resolution of unity $
\int d \Gamma_m(\vec p)
I_{\vec p}=
\int d \Gamma_m(\vec p)
|\vec p\rangle\langle \vec p|\otimes 1
=I.
$ We define the single-oscillator Dirac field operator by $$\begin{gathered}
\Psi(x)=\sum_s\int d \Gamma_m(\vec p)\Big(
u(\vec p,s) b(\vec p,s)e^{-ip\cdot x}
+
v(\vec p,s) d(\vec p,s)^{\dag}e^{ip\cdot x}
\Big).\end{gathered}$$ In order to perform the second step of quantization we introduce $
I_0=\int d \Gamma_m(\vec p)|\vec p\rangle\langle \vec p|\otimes 1_0
$ where $1_0$ is a Hermitian operator satisfying $1_0^2=1$, $\{b_\pm,1_0\}=\{d_\pm,1_0\}=0$. A fermionic $N$-oscillator Jordan-Wigner-type extension is defined by $$\begin{gathered}
\underline{b}(\vec p,s)
=
\frac{1}{\sqrt{N}}
\sum_{n=1}^{N}
\overbrace{
\underbrace{I_0\otimes\dots\otimes I_0}
_{n-1} \otimes\, b(\vec p,s)\otimes I \otimes \dots
\otimes I}^{N}
=
\underline{ c}_1(\vec p,s),\\
\underline{ d}(\vec p,s)
=
\frac{1}{\sqrt{N}}
\sum_{n=1}^{N}
\overbrace{
\underbrace{I_0\otimes\dots\otimes I_0}
_{n-1} \otimes\, d(\vec p,s)\otimes I \otimes \dots
\otimes I}^{N}
=
\underline{ c}_2(\vec p,s),\\
\underline{\Psi}(x)
=
\sum_s\int d \Gamma_m(\vec p)\Big(
u(\vec p,s) \underline{ b}(\vec p,s)e^{-ip\cdot x}
+
v(\vec p,s) \underline{ d}(\vec p,s)^{\dag}e^{ip\cdot x}
\Big).\end{gathered}$$ The reducible representation of CAR reads $$\begin{gathered}
\big\{\underline{ c}_j(\vec p,s),\underline{ c}_{j'}(\vec p\,',s')^{\dag}\big\}
=
\delta_{jj'}\delta_{ss'}\delta_{\Gamma_m}(\vec p,\vec p\,')
\underline{ I}_{\vec p},\\
\underline{ I}_{\vec p}
=\frac{1}{N}
\sum_{n=1}^{N}
\overbrace{
\underbrace{I\otimes\dots\otimes I}
_{n-1} \otimes I_{\vec p}\otimes I \otimes \dots
\otimes I}^{N},\quad
\int d \Gamma_m(\vec p)
\underline{I}_{\vec p}=\underline{I}.
\label{uu nCAR}\end{gathered}$$ In order to verify that $\underline{\Psi}(x)$ is an operator it is sufficient to check this property for $\Psi(x)$. The choice of the representation implies that $$\begin{gathered}
\Psi(x)
=
\sum_s\Big(
u(\hat{\vec p},s) e^{-i\hat p\cdot x}\otimes b_s
+
v(\hat{\vec p},s) e^{i\hat p\cdot x} \otimes d_{s}^{\dag}
\Big)\nonumber\end{gathered}$$ where $
\hat{p}_a=\sum_s\int d \Gamma_m(\vec p)
p_a|\vec p\rangle\langle \vec p|
$ is the spectral representation of the unbounded operator, and $u(\hat{\vec p},s)$, $v(\hat{\vec p},s)$ are functions of the operator $\hat{\vec p}$ in the sense of spectral theory. All these objects are well defined and there is no problem with products of fields taken at the same point $x$ of the configuration space. The difference between fields taken in our reducible representation and those arising from the standard irreducible one is analogous to this between the unitary operator $e^{-i\hat p\cdot x}=
\int d \Gamma_m(\vec p)e^{-ip\cdot x}|\vec p\rangle\langle \vec p|$ and the distribution $\int d \Gamma_m(\vec p)e^{-ip\cdot x}$. Unitary representation of the Poincaré group $$U_{\Lambda,y}^{-1}\underline{\Psi}_\alpha(x)U_{\Lambda,y}=
S_{\alpha}{^\beta} \underline{\Psi}_\beta\big(\Lambda^{-1}(x-y)\big)$$ is explicitly constructed in [@IV]. Our representation of CAR written in terms of Dirac fields reads (the superscripts $(+)$ or $(-)$ denote the parts of field operators containing only creation or annihilation operators, respectively) $$\begin{gathered}
\underline{ S}^{(\pm)}_{\alpha\beta}(x-y)
\stackrel{\rm def}{=}
i\{\underline{\Psi}^{(\mp)}_\alpha(x),\underline{\bar\Psi}^{(\pm)}_\beta(y)\}
=
\frac{1}{N}
\sum_{n=1}^{N}
\overbrace{
\underbrace{I\otimes\dots\otimes I}
_{n-1} \otimes
S^{(\pm)}_{\alpha\beta}(x-y)
\otimes I \otimes \dots
\otimes I}^{N}
\nonumber\\
S^{(\pm)}_{\alpha\beta}(x-y)
\stackrel{\rm def}{=}
i\{\Psi^{(\mp)}_\alpha(x),\bar\Psi^{(\pm)}_\beta(y)\}
=
i(\gamma \cdot \hat p\pm m)_{\alpha\beta}e^{\mp i\hat p\cdot(x-y)}\otimes 1.\end{gathered}$$ Neither anti-commutators nor products of fields evaluated at $x=y$ lead to any difficulty. The essence of the new representation is in the replacement of ordinary integrals over momenta by spectral integrals (e.g. regularization of zero-point energy in a finite volume would take the form $\sum n \to \sum n|n\rangle\langle n|$). The limit $N\to\infty$ provides a correspondence principle analogous to $\hbar\to 0$ or $c\to\infty$. For physical aspects of the modification cf. [@I; @II].
Vacuum and multi-electron states
================================
The vacuum consists of a Hilbert space of all the states which are annihilated by all annihilation operators. We begin with a “single-oscillator vacuum" $$\begin{gathered}
|O\rangle
=
\int d \Gamma_m(\vec p)O(\vec p)|\vec p\rangle\otimes|0\rangle,\end{gathered}$$ where $b_s|0\rangle=d_s|0\rangle=0$, i.e. $|0\rangle$ is the vacuum of the irreducible representation of CAR. The multi-oscillator vacuum is defined at the $N$-oscillator level as the tensor product of one-oscillator vacua $
|\underline{O}\rangle
=
|O\rangle\otimes\dots\otimes|O\rangle.
$ As expected $
\underline{b}(\vec p,s)|\underline{O}\rangle=
\underline{d}(\vec p,s)|\underline{O}\rangle=0.
$ Let us stress that the vacuum [*space*]{} as a whole is Poincaré invariant, whereas a concrete vector $|\underline{O}\rangle$ is only covariant: The wave function transforms as a mass-$m$ scalar field which carries the unitary representation $O(\vec p)\mapsto e^{-2i p\cdot y}O(\overrightarrow {\Lambda^{-1}p})$ [@II; @IV]; the exponent comes from the vacuum part of 4-momentum and can be removed by a unitary transformation. The vacuum space plays a role of a base space in a bundle whose fibers are Fock spaces generated in the usual way in terms of the representation of CAR. The structure of the fibers is characterized by a theorem on the thermodynamic limit $N\to\infty$ (the proof is exactly analogous to the bosonic case explained in detail in [@II; @III]). Let $
\underline{c}_j(f)
=
\sum_{s}\int d \Gamma_m(\vec p) \overline{f(\vec p,s)}
\underline{c}_j(\vec p,s).
$ The scalar product of two unnormalized one-electron states is $$\begin{gathered}
\langle \underline{O}|\underline{c}_j(f) \underline{c}_{j'}(g)^{\dag}
|\underline{O}\rangle
=
\delta_{jj'}\sum_{s}\int d \Gamma_m(\vec p)Z(\vec p)
\overline{f(\vec p,s)}g(\vec p,s)
\stackrel{\rm def}{=}
\delta_{jj'}\langle f|g\rangle_Z\label{<of|og>}\end{gathered}$$ where $Z(\vec p)=|O(\vec p)|^2$. The scalar product $\langle f|g\rangle_Z$ reappears in the thermodynamic limit $N\to\infty$ for arbitrary multi-electron states.
Let $|\underline{O}\rangle
=
\underbrace{|O\rangle\otimes\dots\otimes|O\rangle}_N$. Then $$\begin{gathered}
\lim_{N\to\infty}
\langle \underline{O}|\underline{c}_j(f_1)\dots \underline{c}_j(f_M)
\underline{c}_j(g_1)^{\dag}\dots
\underline{c}_j(g_M)^{\dag}|\underline{O}\rangle
=
\sum_{\sigma}\delta_\sigma
\langle f_1|g_{\sigma(1)}\rangle_Z
\dots
\langle f_M|g_{\sigma(M)}\rangle_Z\nonumber\\
=
\sum_{\sigma}\delta_\sigma\sum_{s_1\dots s_M}
\int d\Gamma_m(\vec p_1)\dots
\int d\Gamma_m(\vec p_M)
\nonumber\\
\phantom{=}
\times
Z(\vec p_1)\dots Z(\vec p_M)\overline{f_1(\vec p_1,s_1)}\dots
\overline{f_m(\vec p_M,s_M)}
g_{\sigma(1)}(\vec p_1,s_1)
\dots
g_{\sigma(m)}(\vec p_M,s_M)\nonumber\end{gathered}$$ where $\delta_\sigma$ is the sign of the permutation $\sigma$.
Let us note that the weights $Z(\vec p)=\langle \underline{O}|\underline{I}_{\vec p}|\underline{O}\rangle$ occur here automatically and regularize ultraviolet divergences since $\int d \Gamma_m(\vec p)Z(\vec p)=
\langle \underline{O}|\int d \Gamma_m(\vec p)
\underline{I}_{\vec p}|\underline{O}\rangle=
\langle \underline{O}|\underline{I}|\underline{O}\rangle=1$ implies that $Z(\vec p)$ decays at infinity. Furthermore, the Poincaré transformation $Z(\vec p)\mapsto Z(\overrightarrow {\Lambda^{-1}p})$ implies that $Z=\max_{\vec p}\{Z(\vec p)\}$ is a nonvanishing invariant. This allows us to introduce the cut-off functions $\chi(\vec p)=Z(\vec p)/Z$, $0\leq \chi(\vec p)\leq 1$. Quantum electrodynamic regime is defined as a condition on supports of wave packets: $f(\vec p,s)\chi(\vec p)=f(\vec p,s)$. For any two such wave packets one finds $\langle f|g\rangle_Z=Z\langle f|g\rangle$ and the thermodynamic limit simply redefines a bare charge by a multiple of the invariant $Z$, as we shall see explicitly in the next section.
Interaction Hamiltonian
=======================
For Dirac electrons interacting with classical electromagnetic fields the departure point is the interaction-picture Hamiltonian $$H(x_0)
=
e_0 \int d^3x
\underline{\bar\Psi}(x)\gamma_a\underline{\Psi}(x)
A^a(x)$$ Let us note that $H(x_0)
=
-i(2\pi)^3e_0 \underline{S}^{(-)}_{\alpha\beta}(0)
\gamma_a^{\alpha\beta}\tilde A^a(x_0,0)
+
:H(x_0):\,
$where $\tilde A^a(x_0,\vec k)$ is the 3-dimensional Fourier transform. The difference between $:H(x_0):$ and $H(x_0)$ is a well behaved element of the center of CAR if the Fourier transform is not singular at $\vec k=0$. This shows that the Coulomb field leads to an infrared vacuum divergence. However, if one restricts $A_a(x)$ to solutions of free Maxwell equations then $\tilde A^a(x_0,0)=0$ since the origin of the light cone is excluded. We shall return to this question in the next section.
As a first purely fermionic modification consider the vacuum average $$\langle\underline{O}|\underline{S}^{(\pm)}(x)|\underline{O}\rangle
=
\int d\Gamma_m(\vec p)
\langle \underline{O}|\underline{I}_{\vec p}|\underline{O}\rangle
(\gamma \cdot p\pm m)e^{\mp ip\cdot x}
=
Z
\int d\Gamma_m(\vec p)\chi(\vec p)
(\gamma \cdot p\pm m)e^{\mp ip\cdot x}.\nonumber$$ This is the usual-looking expression but with the cut-off function $\chi(\vec p)$ and the “renormalization constant" $Z$ automatically built-in. The regularization by $\chi(\vec p)$ is a straightforward consequence of reducibility: For irreducible representations one finds $\langle \underline{O}|\underline{I}_{\vec p}|\underline{O}\rangle=1$.
A first-order correction in a $S$-matrix element evaluated between two single-electron wave packets reads $$\begin{gathered}
\langle\phi|S^{(1)}|\psi\rangle
=
-ie_0
\sum_{s,s'}
\int d\Gamma_m(\vec p)\int d\Gamma_m(\vec p\,')
\\
\phantom{\langle{\rm out}|S^{(1)}|{\rm in}\rangle
=}\times
\underbrace{
\langle \underline{O}|\underline{I}_{p\,'}
\underline{I}_p|\underline{O}\rangle}_{\rm modification}
\overline{\phi(\vec p\,',s')}
\psi(\vec p,s)\bar u(\vec p\,',s')\gamma_a u(\vec p,s)
\int d^4x
e^{i(p\,'-p)\cdot x} A^a(x).\nonumber\end{gathered}$$ The underbraced expression, equal to 1 in the standard irreducible representation, is the only modification we encounter. Explicitly $$\langle \underline{O}|\underline{I}_{p\,'}
\underline{I}_p|\underline{O}\rangle
=
Z\frac{1}{N}\delta_{\Gamma_m}(\vec p,\vec p\,')
\chi(\vec p)
+
Z^2\Big(1-\frac{1}{N}\Big)
\chi(\vec p)\chi(\vec p\,').\label{1/N'}$$ Keeping in mind that one power of $Z$ gets absorbed into normalization of single-electron states, assuming $|\int d^4x A^a(x)|<\infty$ we find, in the thermodynamic limit $N\to\infty$ and for wave packets satisfying the QED regime condition, that $\langle\phi|S^{(1)}|\psi\rangle=Z\langle\phi|S^{(1)}|\psi\rangle_{\rm standard}$. For finite $N$ there are additional corrections arising from the term proportional to $1/N$.
As a next exercise consider the vacuum polarization tensor which appears in second-order calculation of vacuum polarization. The standard expression is here replaced by $$\begin{gathered}
{\rm Tr\,}
\Big(
\langle\underline{O}|\underline{S}^{(-)}(x_2-x_1)
\gamma^a
\underline{S}^{(+)}(x_1-x_2)\gamma^b|\underline{O}\rangle
\Big)\nonumber\\
\phantom{{\rm Tr\,}}
=
-
\frac{Z}{N}
\int d\Gamma_m(\vec p)
\chi(\vec p)e^{2i{p}\cdot (x_2-x_1)}
{\rm Tr\,}
\Big((\gamma\cdot p-m)\gamma^a (\gamma\cdot p+m)\gamma^b\Big)
\nonumber\\
\phantom{{\rm Tr\,}=}
-Z^2
\Big(1-\frac{1}{N}\Big)
\int d\Gamma_m(\vec p) \int d\Gamma_m(\vec p\,')\nonumber\\
\phantom{{\rm Tr\,}==}\times
\chi(\vec p)\chi(\vec p\,')e^{i(p+p')\cdot (x_2-x_1)}
{\rm Tr\,}
\Big((\gamma\cdot p-m)\gamma^a (\gamma\cdot p'+m)\gamma^b\Big).
\nonumber\end{gathered}$$ The expression which survives the thermodynamic limit is a regularized version of the standard formula, multiplied by $Z^2$.
Reducible representation of CCR
===============================
The automatic appearance of the cut-off functions $\chi(\vec p)=|O(\vec p)|^2/Z$ was a consequence of replacing at the RHS of CAR the identity operator by $\underline{I}_{\vec p}$. The claim is that an analogous effect occurs for reducibly quantized electromagnetic fields [@I; @II; @III].
The strategy is similar to the CAR case. One starts with $$a(\vec k,\pm)
=
|\vec k\rangle\langle \vec k|\otimes a_\pm$$ where $[a_s,a^{\dag}_{s'}]=\delta_{ss'}1$ is an irreducible representation of CCR and then performs a bosonic $\tilde N$-particle extension $a(\vec k,\pm)\mapsto
\underline{a}(\vec k,\pm)$. The parameters $N$ and $\tilde N$ representing the numbers of fermionic and bosonic oscillators are in principle unrelated. A theorem analogous to Theorem 1 holds for the $\tilde N\to\infty$ limit. Coherent states of light are defined in the standard way in terms of a displacement operator $$\begin{gathered}
\underline{\cal D}(f) = e^{\underline{a}(f)^{\dag}
-\underline{a}(f)},\quad
\underline{a}(f)=
\sum_{s}\int d \Gamma_0(\vec k) \overline{f(\vec k,s)}
\underline{a}(\vec k,s),\\
\underline{\cal D}(f)^{\dag}
\underline{a}(\vec k,\pm)
\underline{\cal D}(f)
=
\underline{a}(\vec k,\pm)+f(\vec k,\pm)\underline{I}_{\vec k}.\label{D}\end{gathered}$$ Let us note the important difference with respect to the irreducible representation: The shift in (\[D\]) is proportional to $\underline{I}_{\vec k}$ and not to the identity operator. For $\tilde N\to\infty$ the statistics of excitations of a coherent state is Poissonian, as it should be for physical reasons. Now assume we have a classical transverse current $J_a(x)$ and the interaction-picture Hamiltonian $$H(x_0)
=
\int d^3x
J_a(x)
\underline{A}^a(x)\label{HJ}$$ where $\underline{A}^a(x)$ is a reducibly quantized vector potential in a Lorenz gauge. A simple calculation shows that the $S$ matrix, up to a unitary operator which belongs to the center of CCR, is given by the displacement operator and $$\begin{gathered}
\underline{a}(\vec k,\pm)_{\rm out}
=
S^{\dag}
\underline{a}(\vec k,\pm)_{\rm in}
S
=
\underline{a}(\vec k,\pm)_{\rm in}+j(\vec k,\pm)\underline{I}_{\vec k}
=
\underline{\cal D}(j)^{\dag}
\underline{a}(\vec k,\pm)_{\rm in}
\underline{\cal D}(j)\end{gathered}$$ where $j(\vec k,\pm)$ are amplitudes of the two transverse components of the 4-dimensional Fourier transform of $J_a(x)$, restricted to the light cone, i.e. the same functions one finds in the standard formalism. But there is also a difference: The irreducible formalism would have produced $j(\vec k,\pm)\underline{I}$ and not $j(\vec k,\pm)\underline{I}_{\vec k}$.
The vacuum space of the representation is constructed in analogy to the CAR case. One begins with $
|\tilde O\rangle
=
\int d \Gamma_0(\vec k)\tilde O(\vec k)|\vec k\rangle\otimes|0\rangle,
$ where $a_s|0\rangle=0$, and the $\tilde N$-oscillator extension is $
|\underline{\tilde O}\rangle
=
|\tilde O\rangle\otimes\dots\otimes|\tilde O\rangle.
$ The “renormalization constant" $\tilde Z=\max_{\vec k}\{|\tilde O(\vec k)|^2\}$ is a nonvanishing Poincaré invariant and thus the cut-off function $\tilde \chi(\vec p)=|\tilde O(\vec p)|^2/\tilde Z$ is well defined. What is important, the field $\tilde O(\vec k)$ is now massless and therefore vanishes not only at infinity but also at $\vec k=0$.
Having the “out" fields we can compute their (“in"-)vacuum averages. The result is analogous to the standard one, but now the Fourier transform of the field involves the amplitude $f(\vec k,\pm)
=
\langle \underline{\tilde O}|\underline{I}_{\vec k}|\underline{\tilde O}\rangle
j(\vec k,\pm)
=|\tilde O(\vec k)|^2j(\vec k,\pm)
=\tilde Z\tilde \chi(\vec k)j(\vec k,\pm)
$ and not $f(\vec k,\pm)=j(\vec k,\pm)$ which would have occured in the irreducible formalism. The difference is subtle but of crucial importance since the property $\tilde\chi(0)=0$ can regularize the infrared divergence. In the irreducible case the radiation field produced by an accelerated pointlike charge involves the amplitude $j(\vec k,\pm)$ which instead of vanishing blows up at $\vec k=0$. In the reducible case we obtain a regularization which eliminates the infrared divergence if the vacuum $\tilde O(\vec k)$ is correctly chosen. The same concerns the average number of photons of the radiated field [@II; @III]. Let us mention that the operators $\underline{I}_{\vec k}$ appear also in reducibly quantized solutions of Maxwell equations with classical currents and thus regularize classical divergences.
To conclude, the reducible representations seem to produce the cut-off functions in exactly those places one expects them to occur. This is a consequence of the RHS of CAR and CCR where instead of identities one finds $\underline{I}_{\vec p}$ and $\underline{I}_{\vec k}$. At the level of amplitudes and in thermodynamic limits one finds the effective rule $\underline{I}_{\vec p}\to Z \chi(\vec p)$, $\underline{I}_{\vec k}\to \tilde Z \tilde \chi(\vec k)$. The analogy to renormalized fields[^1] whose CAR and CCR relations involve at RHS the renormalization constants $Z_2$ and $Z_3$ supplemented by cut-offs is probably not accidental. An argument in favor of our approach is that the RHS of CAR and CCR are Poincaré covariant [@II; @IV] since $U_{\Lambda,y}^{-1}\underline{I}_{\vec p}U_{\Lambda,y}=
\underline{I}_{\overrightarrow{\Lambda^{-1} p}}$, $U_{\Lambda,y}^{-1}\underline{I}_{\vec k}U_{\Lambda,y}=
\underline{I}_{\overrightarrow{\Lambda^{-1} k}}$, which would not be possible if $\underline{I}_{\vec p}$ and $\underline{I}_{\vec k}$ were simply replaced in CAR and CCR by the functions $Z \chi(\vec p)$ and $\tilde Z \tilde \chi(\vec k)$. Our theory is nonlocal [@Efimov] but in an unusual sense.
The above representations have a status of toy models. The main message we have tried to convey is that reducibility of some type may be crucial for a consistent QFT, a viewpoint advocated also by Dirac in his last paper [@Dirac]. Generalizations to more abstract formalisms may be essential for the issue of gauge invariance, which is beyond our reach at the moment. In this context let us note the recent paper [@Jan] where a similar representation of CCR is discussed at the level of correlation-function approach to photons.
My understanding of the problem was influenced by numerous discussions with Jan Naudts. I am grateful to UIA, Antwerp, for a financial support of this work.
[99]{}
Czachor M, Non-canonical quantum optics, [*J. Phys.*]{} [**A33**]{} (2000), 8081.
Czachor M and Syty M, Non-canonical quantum optics (II): Poincaré covariant formalism and thermodynamic limit, quant-ph/0205011.
Czachor M, States of light via reducible quantization, [*Phys. Lett.*]{} [**A313**]{} (2003), 380.
Czachor M, Reducible field quantization (II): Electrons, quant-ph/0212061.
Efimov G V, Nonlocal interactions of quantized fields, Nauka, Moscow, 1977 (in Russian).
Dirac P A M, The future of atomic physics, Int. J. Theor. Phys. [**23**]{}, 677 (1984).
Naudts J, Kuna M and De Roeck W, Photon fields in a fluctuating spacetime, hep-th/0210188.
\[lastpage\]
[^1]: I am indebted to prof. H. Grosse for drawing my attention to this point during our discussion in Bia[ł]{}owieża
|
Ł ł
=-15mm=-3mm amssym.def
**A Note on Normal Forms of Quantum States and Separability**
Ming Li$^{1}$, Shao-Ming Fei$^{1,2}$ and Zhi-Xi Wang$^{1}$
$~^{1}$ [Department of Mathematics, Capital Normal University, Beijing 100037, China]{}
[$~^{2}$ Institut für Angewandte Mathematik, Universität Bonn, D-53115, Germany]{}
Abstract
We study the normal form of multipartite density matrices. It is shown that the correlation matrix (CM) separability criterion can be improved from the normal form we obtained under filtering transformations. Based on CM criterion the entanglement witness is further constructed in terms of local orthogonal observables for both bipartite and multipartite systems.
PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w
One of the important problems in the theory of quantum entanglement is the separability: to decide whether or not a given quantum state is entangled. A multipartite state $\rho_{AB\cdots C}$ is called separable if it can be written as $\rho_{AB\cdots C}=\sum
\limits_{i}p_{i}\rho_{i}^{A}\otimes\rho_{i}^{B}\otimes\cdots\otimes\rho_{i}^{C},
$ where $\rho_{i}^{A}$, $\rho_{i}^{B},\cdots,\rho_{i}^{C}$ are density matrices on subsystems $A$, $B,\cdots, C$, and $p_{i}\geq
0,\sum\limits_{i}p_{i}=1$. There have been many separability criteria such as Bell inequalities [@bell], PPT (positive partial transposition) [@peres], entanglement witnesses [@witness; @zhang], realignment [@chen], local uncertainty relations [@guhne1; @hofmann] etc. In [[@julio]]{} by using the Bloch representation of density matrices the author has presented a separability criterion, which is further generalized to multipartite case [@hassan]. In [@leinaas] the normal form of a bipartite state has been obtained. We indicate that this normal form can be used to improve the separability criteria from Bloch representation and local uncertainty relations. In this note we study the normal form of multipartite density matrices. We show that the correlation matrix (CM) criterion can be improved from the normal form we obtained under filtering transformations. Based on CM criterion we further construct the entanglement witness in terms of local orthogonal observables (LOOs) [@sixia] for both bipartite and multipartite systems.
For bipartite case, $\rho\in{\mathcal {H}}={\mathcal {H}}_{A}\otimes {\mathcal
{H}}_{B}$ with $dim\,{\mathcal {H}}_{A}=M$, $dim\,{\mathcal {H}}_{B}=N$, $M\leq N$, is mapped to the following form under local filtering transformations [@verstraete]: $$\begin{aligned}
\label{FT}\rho\rightarrow \widetilde{\rho}=\frac{(F_{A}\otimes
F_{B})\rho(F_{A}\otimes F_{B})^{\dag}}{{\rm Tr}[(F_{A}\otimes
F_{B})\rho(F_{A}\otimes F_{B})^{\dag}]},\end{aligned}$$ where $F_{A/B}\in GL(M/N, \Cb)$ are arbitrary invertible matrices. This transformation is also known as stochastic local operations assisted by classical communication (SLOCC). By the definition it is obvious that filtering transformation will preserve the separability of a quantum state.
It has been shown that under local filtering operations one can transform a strictly positive $\rho$ into a normal form [@leinaas], $$\begin{aligned}
\label{NF2} \widetilde{\rho}=\frac{(F_{A}\otimes
F_{B})\rho(F_{A}\otimes F_{B})^{\dag}}{{\rm Tr}[(F_{A}\otimes
F_{B})\rho(F_{A}\otimes
F_{B})^{\dag}]}=\frac{1}{MN}(I+\sum\limits_{i=1}^{M^{2}-1}{\xi}_{i}
G_{i}^{A}\otimes G_{i}^{B}),\end{aligned}$$ where ${\xi}_{i}\geq 0$, $G_{i}^{A}$ and $G_{i}^{B}$ are some traceless orthogonal observables. The matrices $F_{A}$ and $F_{B}$ can be obtained by minimizing the function $$\begin{aligned}
f(A,B)=\frac{{\rm Tr} [\rho(A\otimes B)]}{(\det A)^{1/M}(\det
B)^{1/N}},\end{aligned}$$ where $A=F_{A}^{\dag}F_{A}$ and $B=F_{B}^{\dag}F_{B}$. In fact, one can choose $F_{A}^{0}\equiv |\det
(\rho_{A})|^{1/2M}(\sqrt{\rho_{A}})^{-1}$, and $F_{B}^{0}\equiv
|\det (\rho_{B}^{'})|^{1/2N}(\sqrt{\rho_{B}^{'}})^{-1}$, where $\rho_{B}^{'}={\rm Tr} _{A}(I\otimes (\sqrt{\rho_{A}})^{-1}\rho
I\otimes (\sqrt{\rho_{A}})^{-1})$. Then by the iteration one can get the optimal A and B. In particular, there is a matlab code available in [@verstraete2]. The normal form of a product state (if exists) must be proportional to identity.
For bipartite separable states $\rho$, the CM separability criterion [@julio] says that $$\begin{aligned}
\label{CM2}||T||_{KF}\leq \sqrt{MN(M-1)(N-1)},\end{aligned}$$ where $T$ is an $(M^{2}-1)\times (N^{2}-1)$ matrix with $T_{ij}=MN\cdot {\rm Tr} (\rho \lambda_{i}^{A}\otimes
\lambda_{j}^{B})$, $||T||_{KF}$ stands for the trace norm of $T$, $\lambda_{k}^{A/B}$s are the generators of $SU(M/N)$ and have been chosen to be normalized, ${\rm Tr}
\lambda_{k}^{(A/B)}\lambda_{l}^{(A/B)}=\delta_{kl}$.
As the filtering transformation does not change the separability of a state, one can study the separability of $\tilde{\rho}$ instead of $\rho$. Under the normal form (\[NF2\]) the criterion (\[CM2\]) becomes $$\begin{aligned}
\label{g1}
\sum\limits_{i}\xi_{i}\leq\sqrt{MN(M-1)(N-1)}.\end{aligned}$$
In [@guhne1] a separability criterion based on local uncertainty relation (LUR) has been obtained. It says that for any separable state $\rho$, \[hh\] 1-\_[k]{}G\_[k]{}\^[A]{}G\_[k]{}\^[B]{}-G\_[k]{}\^[A]{}I - IG\_[k]{}\^[B]{}\^[2]{}0, where $G_{k}^{A/B}$s are LOOs such as the normalized generators of $SU(M/N)$ and $G_{k}^{A}=0$ for $k=M^{2}+1, \cdots, N^{2}$. The criterion is shown to be strictly stronger than the realignment criterion [@chen]. Under the normal form ($\ref{NF2}$) criterion (\[hh\]) becomes $$\begin{aligned}
1&-&\sum\limits_{k}\la G_{k}^{A}\otimes G_{k}^{B}\ra-\frac{1}{2}\la
G_{k}^{A}\otimes
I - I\otimes G_{k}^{B}\ra^{2} \\
&=&1-\frac{1}{\sqrt{MN}}-\frac{1}{MN}\sum\limits_{k}\xi_{k}-\frac{1}{2}(\sum\limits_{k}\la
G_{k}^{A}\ra^{2} +\sum\limits_{k}\la
G_{k}^{B}\ra^{2}-2\sum\limits_{k}\la G_{k}^{A}\ra\la
G_{k}^{B}\ra)\\
&=&1-\frac{1}{MN}\sum\limits_{k}\xi_{k}-\frac{1}{2}(\frac{1}{M}+\frac{1}{N})\geq 0,\end{aligned}$$ i.e. $$\begin{aligned}
\label{g2}
\sum\limits_{k}\xi_{k}\leq MN - \frac{M+N}{2}.\end{aligned}$$ As $\sqrt{MN(M-1)(N-1)}\leq MN - \frac{M+N}{2}$ holds for any $M$ and $N$, from (\[g1\]) and (\[g2\]) it is obvious that the CM criterion recognizes entanglement better when the normal form is taken into account.
We now consider multipartite systems. Let $\rho$ be a strictly positive density matrix in ${\mathcal {H}}={\mathcal {H}}_{1}
\otimes {\mathcal {H}}_{2} \otimes \cdots \otimes {\mathcal
{H}}_{N}$, $dim\,{\mathcal {H}}_{i}=d_i$. $\rho$ can be generally expressed in terms of the $SU(n)$ generators $\lambda_{\alpha_{k}}$ [@hassan], \[OS\] &=&(\_[j]{}\^[N]{}I\_[d\_[j]{}]{} +\_[{\_[1]{}}]{}\_[\_[1]{}]{} \_[\_[1]{}]{}\^[{\_[1]{}}]{}\_[\_[1]{}]{}\^[{\_[1]{}}]{} +\_[{\_[1]{}\_[2]{}}]{}\_[\_[1]{}\_[2]{}]{} \_[\_[1]{}\_[2]{}]{}\^[{\_[1]{}\_[2]{}}]{}\_[\_[1]{}]{} \^[{\_[1]{}}]{}\_[\_[2]{}]{}\^[{\_[2]{}}]{}.\
&&+\_[{\_[1]{}\_[2]{}\_[3]{}}]{}\_[\_[1]{}\_[2]{}\_[3]{}]{} \_[\_[1]{}\_[2]{}\_[3]{}]{}\^[{\_[1]{}\_[2]{}\_[3]{}}]{}\_[\_[1]{}]{} \^[{\_[1]{}}]{}\_[\_[2]{}]{}\^[{\_[2]{}}]{}\_[\_[3]{}]{}\^[{\_[3]{}}]{}\
&&++\_[{\_[1]{}\_[2]{}\_[M]{}}]{}\_[\_[1]{}\_[2]{}\_[M]{}]{} \_[\_[1]{}\_[2]{}\_[M]{}]{}\^[{\_[1]{}\_[2]{}\_[M]{}}]{}\_[\_[1]{}]{} \^[{\_[1]{}}]{}\_[\_[2]{}]{}\^[{\_[2]{}}]{}\_[\_[M]{}]{}\^[{\_[M]{}}]{}\
&&.++\_[\_[1]{}\_[2]{}\_[N]{}]{} \_[\_[1]{}\_[2]{}\_[M]{}]{}\^[{1,2,,N}]{}\_[\_[1]{}]{} \^[{1}]{}\_[\_[2]{}]{}\^[{2}]{}\_[\_[N]{}]{}\^[{N}]{}), where $\lambda_{\alpha_{k}}^{\{\mu_{k}\}}=I_{d_{1}}\otimes
I_{d_{2}}\otimes\cdots\otimes \lambda_{\alpha_{k}}\otimes
I_{d_{\mu_{k}+1}}\otimes\cdots\otimes I_{d_{N}}$ with $\lambda_{\alpha_{k}}$ appears at the $\mu_k$th position and $$\begin{aligned}
{\mathcal{T}}_{\alpha_{1}\alpha_{2}\cdots\alpha_{M}}
^{\{\mu_{1}\mu_{2}\cdots\mu_{M}\}}=\frac{\prod_{i=1}^{M}
d_{\mu_{i}}}{2^{M}}{\rm Tr}[\rho\lambda_{\alpha_{1}}
^{\{\mu_{1}\}}\lambda_{\alpha_{2}}^{\{\mu_{2}\}}\cdots\lambda_{\alpha_{M}}^{\{\mu_{M}\}}].\end{aligned}$$
The generalized CM criterion says that: if $\rho$ in (\[OS\]) is fully separable, then $$\begin{aligned}
\label{hhh}
||{\mathcal{T}}^{\{\mu_{1},\mu_{2}, \cdots, \mu_{M}\}}||_{KF}\leq
\sqrt{\frac{1}{2^{M}}\prod_{k=1}^{M}d_{\mu_{k}}(d_{\mu_{k}}-1)},\end{aligned}$$ for $2\leq M \leq N, \{\mu_{1},\mu_{2}, \cdots, \mu_{M}\} \subset
\{1, 2, \cdots, N\}$. The KF norm is defined by $$\begin{aligned}
||{\mathcal{T}}^{\{\mu_{1},\mu_{2}, \cdots,
\mu_{M}\}}||_{KF}=max_{m=1, 2, \cdots,
M}||{\mathcal{T}}_{(m)}||_{KF},\end{aligned}$$ where ${\mathcal{T}}_{(m)}$ is a kind of matrix unfolding of ${\mathcal{T}}^{\{\mu_{1},\mu_{2}, \cdots, \mu_{M}\}}$.
The criterion (\[hhh\]) can be improved by investigating the normal form of (\[OS\]).
By filtering transformations of the form $$\begin{aligned}
\label{LFM}
\widetilde{\rho}=F_{1}\otimes F_{2}\otimes\cdots\otimes
F_{N}\rho F_{1}^{\dag}\otimes F_{2}^{\dag}\otimes F_{N}^{\dag},\end{aligned}$$ where $F_{i}\in GL(d_{i},\Bbb{C}), i=1, 2, \cdots N$, followed by normalization, any strictly positive state $\rho$ can be transformed into a normal form \[NF\] &=&(\_[j]{}\^[N]{}I\_[d\_[j]{}]{} +\_[{\_[1]{}\_[2]{}}]{}\_[\_[1]{}\_[2]{}]{} \_[\_[1]{}\_[2]{}]{}\^[{\_[1]{}\_[2]{}}]{}\_[\_[1]{}]{} \^[{\_[1]{}}]{}\_[\_[2]{}]{}\^[{\_[2]{}}]{} +\_[{\_[1]{}\_[2]{}\_[3]{}}]{}\_[\_[1]{}\_[2]{}\_[3]{}]{} \_[\_[1]{}\_[2]{}\_[3]{}]{}\^[{\_[1]{}\_[2]{}\_[3]{}}]{}\_[\_[1]{}]{} \^[{\_[1]{}}]{}\_[\_[2]{}]{}\^[{\_[2]{}}]{}\_[\_[3]{}]{}\^[{\_[3]{}}]{}.\
&&++\_[{\_[1]{}\_[2]{}\_[M]{}}]{}\_[\_[1]{}\_[2]{}\_[M]{}]{} \_[\_[1]{}\_[2]{}\_[M]{}]{}\^[{\_[1]{}\_[2]{}\_[M]{}}]{}\_[\_[1]{}]{} \^[{\_[1]{}}]{}\_[\_[2]{}]{}\^[{\_[2]{}}]{}\_[\_[M]{}]{}\^[{\_[M]{}}]{}\
&&.++\_[\_[1]{}\_[2]{}\_[N]{}]{} \_[\_[1]{}\_[2]{}\_[M]{}]{}\^[{1,2,,N}]{}\_[\_[1]{}]{} \^[{1}]{}\_[\_[2]{}]{}\^[{2}]{}\_[\_[N]{}]{}\^[{N}]{}).
Let $D_{1}, D_{2}, \cdots, D_{N}$ be the sets of density matrices of the $N$ subsystems. The cartesian product $D_{1}
\times D_{2} \times \cdots \times D_{N}$ consisting of all product density matrices $\rho_{1} \otimes \rho_{2} \otimes \cdots \otimes
\rho_{N}$ with normalization ${\rm Tr} \rho_{i}=1$, $i=1, 2, \cdots,
N$, is a compact set of matrices on the full Hilbert space $\mathcal
{H}$. For the given density matrix $\rho$ we define the following function of $\rho_{i}$ $$\begin{aligned}
f(\rho_{1}, \rho_{2}, \cdots, \rho_{N})=\frac{{\rm Tr}
[\rho(\rho_{1}\otimes \rho_{2} \otimes\cdots\otimes
\rho_{N})]}{\prod_{i=1}^{N}\det (\rho_{i})^{1/d_{i}}}.\end{aligned}$$ The function is well-defined on the interior of $D_{1} \times D_{2}
\times \cdots \times D_{N}$ where $\det \rho_{i}>0$. As $\rho$ is assumed to be strictly positive, we have ${\rm Tr}
[\rho(\rho_{1}\otimes \rho_{2} \otimes\cdots\otimes \rho_{N})]>0$. Since $D_{1} \times D_{2} \times \cdots \times D_{N}$ is compact, we have ${\rm Tr} [\rho(\rho_{1}\otimes \rho_{2} \otimes\cdots\otimes
\rho_{N})]\geq C>0$ with a lower bound C depending on $\rho$.
It follows that $f\rightarrow \infty$ on the boundary of $D_{1}
\times D_{2} \times \cdots \times D_{N}$ where at least one of the $\rho_{i}$s satisfies that $\det \rho_{i}=0$. It follows further that $f$ has a positive minimum on the interior of $D_{1} \times D_{2}
\times \cdots \times D_{N}$ with the minimum value attained for at least one product density matrix $\tau_{1} \otimes \tau_{2} \otimes
\cdots \otimes \tau_{N}$ with $\det \tau_{i}>0$, $i=1, 2, \cdots, N$. Any positive density matrix $\tau_{i}$ with $\det \tau_{i}>0$ can be factorized in terms of Hermitian matrices $F_{i}$ as $$\begin{aligned}
\label{tau}\tau_{i}=F_{i}^{\dag}F_{i}\end{aligned}$$ where $F_{i}\in GL(d_{i}, \Bbb{C})$. Denote $F=F_{1}
\otimes F_{2} \otimes \cdots \otimes F_{N}$, so that $\tau_{1}
\otimes \tau_{2} \otimes \cdots \otimes \tau_{N} =F^{\dag}F$. Set $\widetilde{\rho}=F\rho F^{\dag}$ and define $$\begin{aligned}
\widetilde{f}(\rho_{1}, \rho_{2}, \cdots \rho_{N}) &=&\frac{{\rm Tr}
[\widetilde{\rho}(\rho_{1}\otimes \rho_{2} \otimes\cdots\otimes
\rho_{N})]}{\prod_{i=1}^{N}\det (\rho_{i})^{1/d_{i}}}\\
&=&\prod_{i=1}^{N}\det (\tau_{i})^{1/d_{i}}\cdot\frac{{\rm Tr}
[\rho(F_{1}^{\dag}\rho_{1}F_{1}\otimes F_{2}^{\dag}\rho_{2}F_{2}
\otimes\cdots\otimes
F_{N}^{\dag}\rho_{N}F_{N})]}{\prod_{i=1}^{N}\det (\tau_{i})^{1/d_{i}}\det (\rho_{i})^{1/d_{i}}}\\
&=&\prod_{i=1}^{N}\det (\tau_{i})^{1/d_{i}}\cdot
f(F_{1}^{\dag}\rho_{1}F_{1}, F_{2}^{\dag}\rho_{2}F_{2}, \cdots,
F_{N}^{\dag}\rho_{N}F_{N}).\end{aligned}$$
We see that when $F^{\dag}_{i}\rho_{i}F_{i}=\tau_{i}$, $\widetilde{f}$ has a minimum and $$\begin{aligned}
\rho_{1} \otimes \rho_{2} \otimes \cdots \otimes
\rho_{N}=(F^{\dag})^{-1} \tau_{1} \otimes \tau_{2} \otimes \cdots
\otimes \tau_{N} F^{-1}=I.\end{aligned}$$
Since $\widetilde{f}$ is stationary under infinitesimal variations about the minimum it follows that $$\begin{aligned}
{\rm Tr}[\widetilde{\rho}\delta(\rho_{1} \otimes \rho_{2} \otimes
\cdots \otimes \rho_{N})]=0\end{aligned}$$ for all infinitesimal variations, $$\begin{aligned}
\delta(\rho_{1} \otimes \rho_{2} \otimes \cdots \otimes
\rho_{N})=\delta \rho_{1} \otimes I_{d_{2}} \otimes \cdots \otimes
I_{d_{N}}+ I_{d_{1}} \otimes \delta_{\rho_{2}} \otimes I_{d_{3}}
\otimes \cdots \otimes I_{d_{N}}\\
+ \cdots \cdots + I_{d_{1}} \otimes
I_{d_{2}} \otimes \cdots \otimes I_{d_{N-1}} \otimes \delta \rho_{N},\end{aligned}$$ subjected to the constraint $\det (I_{d_{i}}+\delta\rho_{i})=1$, which is equivalent to ${\rm Tr} (\delta\rho_{i})=0$, $i=1, 2, \cdots,
N$, using $\det (e^{A})=e^{{\rm Tr} A}$ for a given matrix $A$. Thus, $\delta\rho_{i}$ can be represented by the $SU$ generators, $\delta\rho_{i}=\sum\limits_{k}\delta
c_{k}^{i}\lambda_{k}^{i}$. It follows that ${\rm
Tr}(\widetilde{\rho}\lambda_{\alpha_{k}}^{\{\mu_{k}\}})=0$ for any $\alpha_{k}$ and $\mu_{k}$. Hence the terms proportional to $\lambda_{\alpha_{k}}^{\{\mu_{k}\}}$ in ($\ref{OS}$) disappear. $\hfill\Box$
The normal form of a product state in ${\mathcal {H}}$ must be proportional to the identity.
Let $\rho$ be such a state. From (\[NF\]), we get that $$\begin{aligned}
\label{RMONF}\widetilde{\rho}_{i}={\rm Tr}_{1, 2, \cdots, i-1, i+1,
\cdots, N}\rho=\frac{1}{d_{i}}I_{d_{i}}.\end{aligned}$$ Therefore for a product state $\rho$ we have $$\begin{aligned}
\rho=\rho_{1}\otimes\rho_{2}\otimes\cdots\otimes\rho_{N}=\frac{1}{\prod_{i=1}^{N}
d_{i}}\otimes_{i=1}^{N}I_{d_{i}}.\end{aligned}$$ $\hfill\Box$
To show the separability of multipartite states in terms of their normal forms ($\ref{NF}$) we consider the PPT entangled edge state [@acin] $$\begin{aligned}
\rho=\left(%
\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
0 & a & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & b & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & c & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & \frac{1}{c} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \frac{1}{b} & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{a} & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\
\end{array}%
\right)\end{aligned}$$ mixed with noises: $$\begin{aligned}
\rho_{p}=p\rho+\frac{(1-p)}{8}I_{8}.\end{aligned}$$ Select $a=2, b=3$, and $c=0.6$. Using the criterion in [@hassan] we get that $\rho_{p}$ is entangled for $0.92744< p \leq 1$. But after transforming $\rho_{p}$ to its normal form (\[NF\]), the criterion can detect entanglement for $0.90285< p \leq 1$.
Here we indicate that the filtering transformation does not change the PPT property. Let $\rho\in {\mathcal {H}}_{A}\otimes{\mathcal {H}}_{B}$ be PPT, i.e. $\rho^{T_{A}}\geq 0,$ and $ \rho^{T_{B}}\geq 0$. Let $\widetilde{\rho}$ be the normal form of $\rho$. From ($\ref{FT}$) we have $$\begin{aligned}
\widetilde{\rho}^{T_{A}}=\frac{(F_{A}^{*}\otimes F_{B}) \rho^{T_{A}}
(F_{A}^{T}\otimes F_{B}^{\dag})}{{\rm Tr}[(F_{A}\otimes
F_{B})\rho(F_{A}\otimes F_{B})^{\dag}]}.\end{aligned}$$ For any vector $|\psi \rangle$, we have $$\begin{aligned}
\langle \psi | \widetilde{\rho}^{T_{A}}| \psi \rangle
&=&\frac{\langle \psi |(F_{A}^{*}\otimes F_{B}) \rho^{T_{A}}
(F_{A}^{T}\otimes F_{B}^{\dag})| \psi \rangle}{{\rm
Tr}[(F_{A}\otimes
F_{B})\rho(F_{A}\otimes F_{B})^{\dag}]}
\equiv\langle \psi^{'} |\rho^{T_{A}} |\psi^{'} \rangle\geq 0,\end{aligned}$$ where $|\psi^{'} \rangle=\frac{(F_{A}^{T}\otimes F_{B}^{\dag})| \psi
\rangle}{\sqrt{{\rm Tr}[(F_{A}\otimes F_{B})\rho(F_{A}\otimes
F_{B})^{\dag}]}}.$ $\widetilde{{\rho}}^{T_{B}}\geq 0$ can be proved similarly. This property is also valid for multipartite case. Hence a bound entangled state will be bound entangled under filtering transformations.
For N-partite systems in ${\mathcal {H}}={\mathcal
{H}}_{1}\otimes {\mathcal {H}}_{2}\otimes\cdots\otimes {\mathcal
{H}}_{N}$ $(N\geq 2)$ with $dim\,{\mathcal {H}}_{i}=d_{i}$, $i=1, 2,
\cdots, N$, the local orthogonal observables (LOOs) can be given in the following way. Assume $d_n\equiv \max\{d_{i}, i=1, 2, \cdots,
N\}$. One can choose $d^{2}$ observables $G_{k}^{n}$ associated with the subsystem ${\mathcal {H}}_{n}$. For other subsystems with smaller dimensions, say ${\mathcal {H}}_{1}$, one can choose $d_{1}^{2}$ observables $G_{k}^{1}$, $k=1,2,\cdots, d_{1}^{2}$ and set $G_{k}^{1}=0$ for $k=d_{1}^{2}+1, \cdots, d^{2}$. Based on CM criterion we can further construct entanglement witness (EW) in terms of such LOOs. EW [@sixia] is an observable of the composite system that has (i) nonnegative expectation values in all separable states and (ii) at least one negative eigenvalue (or equivalently, can recognizes at least one entangled state).
We first consider bipartite systems in ${\mathcal {H}}_{A}^{M}\otimes
{\mathcal {H}}_{B}^{N}$ with $M\leq N$.
For any LOOs $G^{A}_{k}$ and $G^{B}_{k}$, $$W=I-\alpha\sum\limits^{N^{2}-1}_{k=0}G^{A}_{k}\otimes
G^{B}_{k}$$ is an EW, where $\alpha=\frac{\sqrt{MN}}{\sqrt{(M-1)(N-1)}+1}$ and $$\begin{aligned}
\label{th2c}
G^{A}_{0}=\frac{1}{\sqrt{M}}I_{M},~~~
G^{B}_{0}=\frac{1}{\sqrt{N}}I_{N}.\end{aligned}$$
Let $\rho=\sum\limits_{l,m=0}^{N^{2}-1}T_{lm}\lambda_{l}^{A}\otimes\lambda_{m}^{B}$ be a separable state, where $\lambda_{k}^{A/B}$ are normalized generators of $SU(M/N)$ with $\lambda^{A}_{0}=\frac{1}{\sqrt{M}}I_{M}$, $\lambda^{B}_{0}=\frac{1}{\sqrt{N}}I_{N}$. Any other LOOs $G^{A/B}_{k}$ fulfill ($\ref{th2c}$) can be obtained from these $\lambda$s through orthogonal transformations ${\mathcal{O}}^{A/B}$, $G^{A/B}_{k}=\sum\limits_{l=0}^{N^{2}-1}{\mathcal
{O}}^{A/B}_{kl}\lambda_{l}$, where ${\mathcal{O}}^{A/B}=\left(%
\begin{array}{cc}
1 & 0 \\
0 & {\mathcal {R}}^{A/B} \\
\end{array}%
\right)$, ${\mathcal {R}}^{A/B}$ are $(N^{2}-1)\times (N^{2}-1)$ orthogonal matrices. We have $$\begin{aligned}
{\rm Tr} \rho W&=&1-\alpha \frac{1}{\sqrt{MN}}- \alpha
\sum_{k=1}^{N^{2}-1}\sum_{l,m=1}^{N^{2}-1}{\mathcal {R}}_{kl}^{A}{\mathcal {R}}_{km}^{B}
{\rm Tr} \rho(\lambda^{A}_{l}\otimes \lambda^{B}_{m})\\
&=&\frac{\sqrt{(M-1)(N-1)}}{\sqrt{(M-1)(N-1)}+1}-\frac{1}{\sqrt{MN}(\sqrt{(M-1)(N-1)}+1)}
\sum_{k=1}^{N^{2}-1}\sum_{l,m=1}^{N^{2}-1}{\mathcal {R}}_{kl}^{A}T_{lm}{\mathcal {R}}_{km}^{B}\\
&=&\frac{\sqrt{(M-1)(N-1)}}{\sqrt{(M-1)(N-1)}+1}-\frac{1}{\sqrt{MN}(\sqrt{(M-1)(N-1)}+1)}
{\rm Tr} ({\mathcal {R}}^{A}T({\mathcal {R}}^{B})^{T})\\[2mm]
&\geq&\frac{\sqrt{MN(M-1)(N-1)}-||T||_{KF}}{\sqrt{MN}(\sqrt{(M-1)(N-1)}+1)}\geq
0,\end{aligned}$$ where we have used ${\rm Tr} ({\mathcal
{R}}T) \leq ||T||_{KF}$ for any unitary ${\mathcal {R}}$ in the first inequality and the CM criterion in the second inequality.
Now let $\rho=\frac{1}{MN}(I_{MN}+\sum\limits_{i=1}^{M^{2}-1}s_{i}\lambda_{i}^{A}\otimes
I_{N}+\sum\limits_{j=1}^{N^{2}-1}r_{j}I_{M}\otimes\lambda_{j}^{B}+\sum\limits_{i=1}^{M^{2}-1}
\sum\limits_{j=1}^{N^{2}-1}T_{ij}\lambda_{i}^{A}\otimes
\lambda_{j}^{B})$ be a state in ${\mathcal
{H}}_{A}^{M}\otimes{\mathcal {H}}_{B}^{N}$ which violates the CM criterion. Denote $\sigma_{k}(T)$ the singular values of $T$. By singular value decomposition, one has $T= U^{\dag}
\Lambda V^{*}$, where $\Lambda$ is a diagonal matrix with $\Lambda_{kk}=\sigma_{k}(T)$. Now choose LOOs to be $G_{k}^{A}=\sum_{l}U_{kl}\lambda^{A}_{l}$, $G_{k}^{B}=\sum_{m}V_{km}\lambda^{B}_{m}$ for $k=1,2,\cdots,N^{2}-1$ and $G_{0}^{A}=\frac{1}{M}I_{M}, G_{0}^{B}=\frac{1}{N}I_{N}$. We obtain $$\begin{aligned}
{\rm Tr} \rho W&=&1-\alpha \frac{1}{\sqrt{MN}}- \alpha
\sum_{k=1}^{N^{2}-1}\sum_{l,m=1}^{N^{2}-1}U_{kl}V_{km}
{\rm Tr} \rho(\lambda^{A}_{l}\otimes \lambda^{B}_{m})\\
&=&\frac{\sqrt{(M-1)(N-1)}}{\sqrt{(M-1)(N-1)}+1}-\frac{1}{\sqrt{MN}(\sqrt{(M-1)(N-1)}+1)}
{\rm Tr} (UTV^{T})\\
&=&\frac{\sqrt{MN(M-1)(N-1)}-||T||_{KF}}{\sqrt{MN}(\sqrt{(M-1)(N-1)}+1)}<
0\end{aligned}$$ where the CM criterion has been used in the last step. $\Box$
As the CM criterion can be generalized to multipartite form in [@hassan], we can also define entanglement witness for multipartite system in ${\mathcal {H}}_{1}^{d_{1}}\otimes {\mathcal
{H}}_{2}^{d_{2}}\otimes\cdots\otimes {\mathcal {H}}_{N}^{d_{N}}$. Set $d(M)=\max\{d_{\mu_{i}}, i=1, 2, \cdots, M\}$. Choose LOOs $G^{\{\mu_{i}\}}_{k}$ for $0\leq k\leq d(M)^{2}-1$ with $G^{\{\mu_{i}\}}_{0}=\frac{1}{d_{\mu_{i}}}I_{d_{\mu_{i}}}$ and define $$\begin{aligned}
\label{EWFM}
W^{(M)} = I-\beta^{(M)}
\sum_{k=0}^{d(M)^{2}-1}G_{k}^{\{\mu_{1}\}}\otimes
G_{k}^{\{\mu_{2}\}}\otimes \cdots \otimes G_{k}^{\{\mu_{M}\}},\end{aligned}$$ where $\beta^{(M)}=\frac{\sqrt{\prod_{i=1}^{M}d_{\mu_{i}}}}{1+\sqrt{\prod_{i=1}^{M}(d_{\mu_{i}}-1)}},
2 \leq M\leq N$. One can prove that ($\ref{EWFM}$) is an EW candidate for multipartite states. First we assume $||{\mathcal{T}}^{(M)}||_{KF}=||{\mathcal{T}}_{(m_{0})}||_{KF}$. Note that for any ${\mathcal{T}}_{(m_{0})}$, there must exist an elementary transformation $P$ such that $\sum\limits_{k=1}^{d(M)^{2}-1}{\mathcal{T}}_{kk\cdots
k}^{\{\mu_{1}\mu_{2}\cdots\mu_{M}\}} ={\rm Tr}
({\mathcal{T}}_{(m_{0})}P)$. Then for an N-partite separable state we have $$\begin{aligned}
{\rm Tr} \rho W^{(M)}&=&1-\beta^{(M)}
\frac{1}{\sqrt{\prod_{i=1}^{M}d_{\mu_{i}}}}-\beta^{(M)}
\frac{1}{\prod_{i=1}^{M}d_{\mu_{i}}}\sum\limits_{k=1}^{d(M)^{2}-1}{\mathcal{T}}_{kk\cdots
k}^{\{\mu_{1}\mu_{2}\cdots\mu_{M}\}}\\
&=&1-\beta^{(M)}
\frac{1}{\sqrt{\prod_{i=1}^{M}d_{\mu_{i}}}}-\beta^{(M)}
\frac{1}{\prod_{i=1}^{M}d_{\mu_{i}}}{\rm Tr} ({\mathcal{T}}_{(m_{0})}P)\\
&\geq&1-\beta^{(M)}
\frac{1}{\sqrt{\prod_{i=1}^{M}d_{\mu_{i}}}}-\beta^{(M)}
\frac{1}{\prod_{i=1}^{M}d_{\mu_{i}}}||{\mathcal{T}}_{(m_{0})}||_{KF}\\
&\geq&1-\beta^{(M)}
\frac{1}{\sqrt{\prod_{i=1}^{M}d_{\mu_{i}}}}-\beta^{(M)}
\frac{1}{\prod_{i=1}^{M}d_{\mu_{i}}}\sqrt{\prod_{k=1}^{M}d_{\mu_{k}}(d_{\mu_{k}}-1)}\\
&=&0\end{aligned}$$ for any $2\leq M\leq N$, where we have used that $P$ must be orthognal matrix and ${\rm Tr} (MU)\leq ||M||_{KF}$ for any unitary $U$ at the first inequality. The second inequality is due to the generalized CM criterion.
By choosing proper LOOs it is also easy to show that $W^{(M)}$ has negative eigenvalues. For example for three qubits case , taking the normalized pauli matrices as LOOs, one find a negative eigenvalue of $W^{(M)}$, $\frac{1-\sqrt{3}}{2}$.
We have studied the normal form of multipartite density matrices. It has been shown that separability criteria can be improved by transforming the states to their normal forms through filtering transformations. The entanglement witness has been constructed in terms of local orthogonal observables for both bipartite and multipartite systems. Here we considered only the strictly positive (full rank) states. Although full rank is a sufficient condition for the existence of the normal forms, in fact for many rank deficiency density matrices their normal forms can be also calculated.
[**Acknowledgments**]{} This work is supported by the NSFC 10675086, NSF of Beijing 1042004 and KM200510028022, NKBRPC(2004CB318000).
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---
abstract: 'Reconstructing complex networks from measurable data is a fundamental problem for understanding and controlling collective dynamics of complex networked systems. However, a significant challenge arises when we attempt to decode structural information hidden in limited amounts of data accompanied by noise and in the presence of inaccessible nodes. Here, we develop a general framework for robust reconstruction of complex networks from sparse and noisy data. Specifically, we decompose the task of reconstructing the whole network into recovering local structures centered at each node. Thus, the natural sparsity of complex networks ensures a conversion from the local structure reconstruction into a sparse signal reconstruction problem that can be addressed by using the lasso, a convex optimization method. We apply our method to evolutionary games, transportation and communication processes taking place in a variety of model and real complex networks, finding that universal high reconstruction accuracy can be achieved from sparse data in spite of noise in time series and missing data of partial nodes. Our approach opens new routes to the network reconstruction problem and has potential applications in a wide range of fields.'
author:
- Xiao Han
- Zhesi Shen
- 'Wen-Xu Wang'
- Zengru Di
title: Robust Reconstruction of Complex Networks from Sparse Data
---
Complex networked systems are common in many fields [@taming; @boccaletti; @newman]. The need to ascertain collective dynamics of such systems to control them is shared among different scientific communities [@hecker; @timmereview; @credit]. Much evidence has demonstrated that interaction patterns among dynamical elements captured by a complex network play deterministic roles in collective dynamics [@strogatz]. It is thus imperative to study a complex networked system as a whole rather than study each component separately to offer a comprehensive understanding of the whole system [@takeover]. However, we are often incapable of directly accessing network structures; instead, only limited observable data are available [@raredata], raising the need for network reconstruction approaches to uncovering network structures from data. Network reconstruction, the inverse problem, is challenging because structural information is hidden in measurable data in an unknown manner and the solution space of all possible structural configurations is of extremely high dimension. So far a number of approaches have been proposed to address the inverse problem [@hecker; @timmereview; @raredata; @connectivity; @topology; @catastrophe; @pg; @shen; @auroc; @indirect; @feizi]. However, accurate and robust reconstruction of large complex networks is still a challenging problem, especially given limited measurements disturbed by noise and unexpected factors.
In this letter, we develop a general framework to reconcile the contradiction between the robustness of reconstructing complex networks and limits on our ability to access sufficient amounts of data required by conventional approaches. The key lies in converting the network reconstruction problem into a sparse signal reconstruction problem that can be addressed by exploiting the lasso, a convex optimization algorithm [@hastie; @python]. In particular, reconstructing the whole network structure can be achieved by inferring local connections of each node individually via our framework. The natural sparsity of complex networks suggests that on average the number of real connections of a node is much less than the number of all possible connections, i.e., the size of a network. Thus, to identify direct neighbors of a node from the pool of all nodes in a network is analogous to the problem of sparse signal reconstruction. By using the lasso that incorporates both an error control term and an L1-norm, the neighbors of each node can be reliably identified from a small amount of data that can be much less than the size of a network. The L1-norm, according to the compressed sensing theory [@CS], ensures the sparse data requirement while, simultaneously, the error control term ensures the robustness of reconstruction against noise and missing nodes. The whole network can then be assembled by simply matching neighboring sets of all nodes. We will validate our reconstruction framework by considering three representative dynamics, including ultimatum games [@altruism], transportation [@transportation] and communications [@communication], taking place in both homogeneous and heterogeneous networks. Our approach opens new routes towards understanding and controlling complex networked systems and has implications for many social, technical and biological networks.
We articulate our reconstruction framework by taking ultimatum games as a representative example. We then apply the framework to the transportation of electrical current and communications via sending data packets.
In evolutionary ultimatum games (UG) on networks, each node is occupied by a player. In each round, player $i$ plays the UG twice with each of his/her neighbors, both as a proposer and a responder with strategy ($p_i$, $q_i$), where $p_i$ denotes the amount offered to the other player if $i$ proposes and $q_i$ denotes the minimum acceptance level if $i$ responds [@nowak; @szolnoki]. The profit of player $i$ obtained in the game with player $j$ is calculated as follows $$\begin{aligned}
U_{ij}=\left\{\begin{array}{cc}
p_j+1-p_i & p_i\geq q_j\ \hbox{and}\ p_j\geq q_i\\
1-p_i & p_i\geq q_j\ \hbox{and}\ p_j<q_i\\
p_j & p_i< q_j\ \hbox{and}\ p_j\geq q_i\\
0 & p_i< q_j\ \hbox{and}\ p_j< q_i\\
\end{array}\right.
\label{eq:UG}\end{aligned}$$ where $p_i,p_j\in[0,1]$. The payoff $g_i$ of $i$ at a round is the sum of all profits from playing UG with $i$’s neighbors, i.e., $g_i=\sum_{j\in\Gamma_i}U_{ij}$, where $\Gamma_i$ denotes the set of $i$’s neighbors. In each round, all participants play the UG with their direct neighbors simultaneously and gain payoffs. Players update their strategies ($p,q$) in each round by learning from one of their neighbors with the highest payoffs. To be concrete, player $i$ selects the neighbor with the maximum payoff $g_{\max}(t)$ and takes over the neighbor’s strategy with probability $W(i\leftarrow {\max})=g_{\max}(t)/[g_i(t)+\sum_{j\in\Gamma_i}g_j(t)]$ [@game:review]. To better mimic real situations, random mutation rates are included in each round: all players adjust their ($p,q$) according to $(p_i(t+1), q_i(t+1))=(p_i(t)+\delta, q_i(t)+\delta)$, where $\delta\in[-\varepsilon,\varepsilon]$ is a small random number [@kuperman]. Without loss of generality, we set $\varepsilon=0.05$ and $p,q\in[0,1]$. During the evolution of UG, we assume that only the time series of ($p_i(t),q_i(t)$) and $g_i(t)$ ($i=1,\cdots,N$) are measurable.
The network reconstruction can be initiated from the relationship between strategies $(p_i(t),q_i(t))$ and payoffs $g_i(t)$. Note that $g_i(t) = \sum_{j=1,j\neq i}^N a_{ij}U_{ij}$, where $a_{ij}=1$ if player $i$ and $j$ are connected and $a_{ij}=0$ otherwise. Moreover, $U_{ij}$ is exclusively determined by the strategies of $i$ and $j$. These imply that hidden interactions between $i$ and its neighbors can be extracted from the relationship between strategies and payoffs, enabling the inference of $i$’s links based solely on the strategies and payoffs. Necessary information for recovering $i$’s links can be acquired with respect to different time $t$. Specifically, for $M$ accessible time instances $t_1,\cdots , t_M$, we convert the reconstruction problem into the matrix form $\mathbf{Y}_i=\Phi_i\times \mathbf{X}_i$: $$\left[\begin{array}{c}
y_i(t_1) \\
y_i(t_2) \\
\vdots\\
y_i(t_M)
\end{array}\right]=
\left[\begin{array}{cccc}
\phi_{i1}(t_1) & \phi_{i2}(t_1) & ... & \phi_{iN}(t_1)\\
\phi_{i1}(t_2) & \phi_{i2}(t_2) & ... & \phi_{iN}(t_2)\\
\vdots & \vdots & \vdots & \vdots \\
\phi_{i1}(t_M) & \phi_{i2}(t_M) & ... & \phi_{iN}(t_M)\\
\end{array}\right]
\left[
\begin{array}{c}
x_{i1} \\
x_{i2} \\
\vdots \\
x_{iN} \\
\end{array}
\right],$$ where $\mathbf{Y}_i \in \mathbb{R}^{M\times 1}$ is the payoff vector of $i$ with $y_i(t_\mu)=g_i(t_\mu)$ $(\mu = 1,\cdots, M)$, $\mathbf{X}_i \in \mathbb{R}^{N\times 1}$ is the neighboring vector of $i$ with $x_{ij}=a_{ij}$ $(j=1,\cdots, N)$ and $\Phi_i \in \mathbb{R}^{M\times N}$ is the virtual-payoff matrix of $i$ with $\phi_{ij}(t_\mu)=U_{ij}(t_\mu)$.
Because $U_{ij}(t)$ is determined by $(p_i(t),q_i(t))$ and $(p_j(t),q_j(t))$ according to Eq. (\[eq:UG\]), $\mathbf{Y}_i$ and $\Phi_i$ can be collected or calculated directly from the time series of strategies and payoffs. Our goal is to reconstruct $\mathbf{X}_i$ from $\mathbf{Y}_i$ and $\Phi_i$. Note that the number of nonzero elements in $\mathbf{X}_i$, i.e., the number of the neighbors of $i$, is usually much less than length $N$ of $\mathbf{X}_i$. This indicates that $\mathbf{X}_i$ is sparse, which is ensured by the natural sparsity of complex networks. An intuitive illustration of the reconstruction method is shown in Fig. \[fig:illustration\]. Thus, the problem of identifying the neighborhood of $i$ is transformed into that of sparse signal reconstruction, which can be addressed by using the lasso.
The lasso is a convex optimization method for solving $$\min_{\mathbf{X}_i}\left \{\frac{1}{2M}\|\mathbf{Y}_{i}-\Phi_{i}\mathbf{X}_i\|_{2}^2+\lambda\|\mathbf{X}_{i}\|_1 \right\},$$ where $\lambda$ is a nonnegative regularization parameter [@hastie; @python]. The sparsity of the solution is ensured by $\|\mathbf{X}_{i}\|_1$ in the lasso according to the compressed sensing theory [@CS]. Meanwhile, the least square term $\|\mathbf{Y}_{i}-\Phi_{i}\mathbf{X}_i\|_{2}^2$ makes the solution more robust against noise in time series and missing data of partial nodes than would the $L_1$-norm-based optimization method.
The neighborhood of $i$ is given by the reconstructed vector $\mathbf{X}_i$, in which all nonzero elements correspond to direct neighbors of $i$. In a similar fashion, we construct the reconstruction equations of all nodes, yielding the neighboring sets of all nodes. The whole network can then be assembled by simply matching the neighborhoods of nodes. Due to the sparsity of $\mathbf{X}_i$, it can be reconstructed by using the lasso from a small amount of data that are much less than the length of $\mathbf{X}_i$, i.e., network size $N$. Although we infer the local structure of each node separately by constructing its own reconstruction equation, we only use one set of data sampling in time series. This enables a sparse data requirement for recovering the whole network.
We consider current transportation in a network consisting of resistors [@transportation]. The resistance of a resistor between node $i$ and $j$ is denoted by $r_{ij}$. If $i$ and $j$ are not directly connected by a resistor, $r_{ij}=\infty$. For arbitrary node $i$, according to Kirchhoff’s law, we have $$\sum_{j=1}^{N}\frac{a_{ij}}{r_{ij}}(V_i-V_j)=I_i,
\label{eq:Kirchhoff}$$ where $V_i$ and $V_j$ are the voltage at $i$ and $j$ and $I_i$ is the total electrical current at $i$. To better mimic real power networks, alternating current is considered. Specifically, at node $i$, $V_i=\bar{V}\sin[(\omega+\Delta\omega_i)t]$, where the constant $\bar{V}$ is the voltage peak, $\omega$ is frequency and $\Delta\omega_i$ is perturbation. Without loss of generality, we set $\bar{V}=1$, $\omega =10^3$ and the random number $\Delta\omega_i \in [0,20]$. Given voltages at nodes and resistances of links, currents at nodes can be calculated according to Kirchhoff’s laws at different time constants. We assume that only voltages and electrical currents at nodes are measurable and our purpose is to reconstruct the resistor network. In an analogy with networked ultimatum games, based on Eq. (\[eq:Kirchhoff\]), we can establish the reconstruction equation $\mathbf{Y}_i=\Phi_i\times \mathbf{X}_i$ with respect to time constants $t_1,\cdots,t_M$, where $y_i(t_\mu)=I_i(t_\mu)$, $x_{ij}=1/r_{ij}$ and $\phi_{ij}(t_\mu)=V_i(t_\mu)-V_j(t_\mu)$ with $\mu =1,\cdots,M$ and $j=1,\cdots,N$. Here, if $i$ and $j$ are connected by a resistor, $x_{ij}=1/r_{ij}$ is nonzero; otherwise, $x_{ij}=0$. Thus, the neighboring vector $\mathbf{X}_i$ is sparse and can be reconstructed by using the lasso from a small amount of data. Analogously, the whole network can be recovered by separately reconstructing the neighboring vectors of all nodes.
[p[1.3cm]{}<p[1.3cm]{}<p[1.3cm]{}<p[1.3cm]{}< p[3.8cm]{}<p[3.8cm]{}<p[3.8cm]{}<]{}
------------------------------------------------------------------------
& & & $n_m$ & UG & RN & CN\
& & & &\
------------------------------------------------------------------------
& 6 & 0 & 0 & 0.38 / 0.36 / 0.41 & 0.28 / 0.25 / 0.32 & 0.30 / 0.28 / 0.30\
& 6 & 0.05 & 0 & 0.44 / 0.43 / 0.47 & 0.29 / 0.26 / 0.37 & 0.34 / 0.31 / 0.34\
& 6 & 0.3 & 0 & 1.68 / 1.75 / 1.60 & 0.32 / 0.29 / 0.38 & 1.72 / 1.81 / 1.80\
& 6 & 0 & 0.05 & 0.61 / 0.55 / 0.64 & 1.61 / 1.65 / 1.60 & 1.33 / 1.19 / 1.32\
& 6 & 0 & 0.3 & 2.33 / 2.03 / 2.14 & 5.74 / 8.51 / 8.50 & 5.38 / 6.23 / 6.20\
& 12 & 0 & 0 & 0.46 / 0.47 / 0.52 & 0.37 / 0 / 35 / 0.42 & 0.42 / 0.40 / 0.42\
& 18 & 0 & 0 & 0.53 / 0.53 / 0.58 & 0.44 / 0.44 / 0.50 & 0.50 / 0.50 / 0.50\
500 & 6 & 0 & 0 & 0.120 / 0.116 / 0.132 & 0.094 / 0.080 / 0.120 & 0.094 / 0.088 / 0.100\
1000 & 6 & 0 & 0 & 0.071 / 0.068 / 0.078 & 0.058 / 0.049 / 0.079 & 0.055 / 0.050 / 0.055\
\[tab:simu\]
We propose a simple network model to capture communications in populations via phones, emails, etc. At each time, individual $i$ may contact one of his/her neighbors $j$ according to probability $w_{ij}$ by sending data packets. If $i$ and $j$ are not connected, $w_{ij}=0$. In a period, the total incoming flux $f_i$ of $i$ can be described as $$f_{i}= \sum_{j=1}^{N}w_{ij}\widetilde{f}_{j},
\label{eq:flux}$$ where $\widetilde{f}_{j}$ is the total outgoing flux from $j$ to its neighbors in the period and $\sum_{j=1}^{N}w_{ij}=1$. Equation (\[eq:flux\]) is valid because of the flux conservation in the network. In the real situation, $\widetilde{f}_{j}$ usually fluctuates with time, providing an independent relationship between incoming and outgoing fluxes for constructing the reconstruction equation $\mathbf{Y}_i=\Phi_i\times \mathbf{X}_i$. Here, $y_i(t_\mu)=f_{i}(t_\mu)$ is the total incoming flux of $i$ at time period $t_\mu$, $\phi_{ij}(t_\mu)=\widetilde{f}_{j}(t_\mu)$ is the total outgoing flux of $j$ at time period $t_\mu$, and $x_{ij}=w_{ij}$ captures connections between $i$ and its neighbors. Given the total incoming and outgoing fluxes of nodes that can be measured without the need of any network information and communication content, we can as well use the lasso to reconstruct the neighboring set of node $i$ and those of the other nodes, such that full reconstruction of the whole network is achieved from sparse data.
We simulate ultimatum games, electrical currents and communications on both homogeneous and heterogeneous networks, including random [@er], small-world [@ws] and scale-free [@ba] networks. For the three types of dynamical processes, we record strategies and payoffs of players, voltages and currents, and incoming and outgoing fluxes at nodes at different times, to apply our reconstruction method with respect to different amounts of Data (Data$\equiv M/N$, where $M$ is the number of accessible time instances in the time series). Figure \[fig:TprFpr\] shows the results of networked ultimatum games. For very small amounts of data, e.g., Data=0.1, links are difficult identify because of the mixture of reconstructed elements in $\mathbf{X}$, whereas for Data=0.4, there is a vast and clear gap between actual links and null connections, assuring perfect reconstruction (Fig. \[fig:TprFpr\](a)). Even with strong measurement noise, e.g., $\mathcal{N}(0,0.3^2)$, by increasing Data, full reconstruction can be still accomplished (Fig. \[fig:TprFpr\](b)). We use two standard indices, true positive rate (TPR) versus false positive rate (FPR), and Precision versus Recall to measure quantitatively reconstruction performance [@auroc] (see [@AUC_SM] for more details). We see that for Data=0.4, both the area under the receiver operating characteristic curve (AUROC) in TPR vs. FPR (Fig. \[fig:TprFpr\](c)) and the area under the precision-recall curve (AUPR) in Precision vs. Recall (Fig. \[fig:TprFpr\](d)) equal 1, indicating that links and null connections can be completely distinguished from each other with a certain threshold. Because high reconstruction accuracy can always be achieved, we explore the minimum data for assuring 0.95 AUROC and AUPR simultaneously for different types of dynamics and networks. As displayed in Table \[tab:simu\], with little measurement noise and a small fraction of inaccessible nodes, only a small amount of data are required, especially for large networks, e.g., $N=1000$. In the presence of strong noise and a large fraction of missing nodes, high accuracy can be still achieved from a relatively larger amount of data. We have also tested our method on several empirical networks (Table \[tab:real\]), finding that only sparse data are required for full reconstruction as well. These results demonstrate that our general approach offers robust reconstruction of complex networks from sparse data.
[p[1.3cm]{}<p[2.5cm]{}<p[1.3cm]{}<p[1.3cm]{}< p[1.3cm]{}<]{}
------------------------------------------------------------------------
& Networks & $N$ & $\langle k\rangle$ & Data\
------------------------------------------------------------------------
& Karate & 34 & 4.6 & 0.69\
& Dolphins & 62 & 5.1 & 0.50\
& Netscience & 1589 & 3.5 & 0.07\
------------------------------------------------------------------------
& IEEE39BUS & 39 & 2.4 & 0.33\
& IEEE118BUS & 118 & 3.0 & 0.23\
& IEEE300BUS & 300 & 2.7 & 0.10\
------------------------------------------------------------------------
& Football & 115 & 10.7 & 0.35\
& Jazz & 198 & 27.7 & 0.49\
& Email & 1133 & 9.6 & 0.10\
\[tab:real\]
In conclusion, we develop a general framework to reconstruct complex networks with great robustness from sparse data that in general can be much less than network sizes. The key to our method lies in decomposing the task of reconstructing the whole network into inferring local connections of nodes individually. Due to the natural sparsity of complex networks, recovering local structures from time series can be converted into a sparse signal reconstruction problem that can be resolved by using the lasso, in which both the error control term and the L1-norm jointly enable robust reconstruction from sparse data. Insofar as all local structures are ascertained, the whole network can be assembled by simply matching them. Our method has been validated by the combinations of three representative dynamical processes and a variety of model and real networks with noise and inaccessible nodes. High reconstruction accuracy can be achieved for all cases from relatively small amounts of data.
It is noteworthy that our reconstruction framework is quite flexible and not limited to the networked systems considered here. The crucial issue is to find a certain relationship between local structures and measurable data to construct the reconstruction form $\mathbf{Y}=\Phi \mathbf{X}$. Indeed, there is no general manner to establish the reconstruction form for different networked systems, implying that the application scope of our approach is yet not completely known. Nevertheless, our method could have broad applications in many fields due to its sparse data requirement and its advantages in robustness against noise and missing information. In addition, network reconstruction allows us to infer intrinsic nodal dynamics from time series by canceling the influence from neighbors [@SM], although this is beyond our current scope. Taken together, our approach offers deeper understanding of complex networked systems from observable data and has potential applications in predicting and controlling collective dynamics of complex systems, especially when we encounter explosive growth of data in the information era.
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---
abstract: 'We consider non-ergodic magnetic random Schr[ö]{}dinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of arguments from [@Klein-13], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [@BorisovTV-15]. This generalizes Klein’s result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian without magnetic field.'
author:
- Matthias Täufer
- Martin Tautenhahn
title: 'Wegner estimate and disorder dependence for alloy-type Hamiltonians with bounded magnetic potential'
---
Introduction
============
We study a class of magnetic, non-ergodic random Schrödinger operators on $L^2 ({\mathbb{R}}^d)$ of the type $$\label{eq:model}
H_\omega = H_0 + \lambda V_\omega ,$$ where $\lambda > 0$ is the disorder parameter, $H_0 = ( - {\mathrm{i}}\nabla + A_0)^2 + V_0$ with a bounded electric potential $V_0 \in L^\infty({\mathbb{R}}^d)$ and a bounded magnetic vector potential $A_0 \in L^\infty({\mathbb{R}}^d, {\mathbb{R}}^d)$ satisfying ${\operatorname{div}}(A_0)$ bounded, and where $$V_\omega(x)
=
\sum_{j \in {\mathbb{Z}}^d} \omega_j u_j(x - z_j ) .$$ The random variables $\omega_j$, $j \in {\mathbb{Z}}^d$, are independent and identically distributed with compactly supported, non-degenerate distribution, and the single-site potentials $( u_j )_{j \in {\mathbb{Z}}^d}$, are measurable and real-valued functions on ${\mathbb{R}}^d$ satisfying $$u_- \chi_{{B (\delta_-)}}
\leq
u_j
\leq
\chi_{\Lambda_{\delta_+}}$$ for some $u_- \in (0,1]$ and $\delta_\pm > 0$, where ${B (\delta_-)}$ and $\Lambda_{\delta_+}$ denote the $d$-dimensional ball of radius $\delta_-$ and the $d$-dimensional cube of side length $\delta_+$, centered at $0$. Furthermore, we assume that the centers $z_j$ of the single-site potentials are in a certain sense equidistributed in ${\mathbb{R}}^d$. Such operators are used to model quantum mechanical properties of disordered solids. While each configuration of the randomness corresponds to a particular realization of the solid, the law of the random variables models their distribution.
One distinctive feature of random operators is the phenomenon of localization, i.e. that parts of the spectrum consist only of pure point spectrum (spectral localization) or that the solutions of the Schrödinger equation stay almost surely trapped in a finite region of space for all time (dynamical localization). This is in contrast to periodic operators which exhibit only absolutely continuous spectrum. One method for proving localization is the so-called multiscale analysis introduced in [@FroehlichS-83; @FroehlichMSS-85] and further developed in [@DreifusK-89; @GerminetK-01; @GerminetK-03; @GerminetK-06]. The multiscale analysis is an induction argument. While the induction anchor is provided by the so-called initial-scale estimate, a so-called Wegner estimate is needed for the induction step. A Wegner estimate is an upper bound on the expected number of eigenvalues of a self-adjoint restriction $H_{\omega , L}$ of $H_\omega$ to a cube $\Lambda_L \subset {\mathbb{R}}^d$ of side length $L>0$ in an energy interval $[a,b] \subset {\mathbb{R}}$. More precisely, a Wegner estimate is an estimate of the form $$\label{eq:Wegner_introduction}
{\mathbb{E}}\bigl( \operatorname{Tr} \chi_{[a,b]} (H_{\omega , L}) \bigr)
\leq
C(\lambda) S (b-a) \lvert \Lambda_L \rvert^m ,$$ where $C(\lambda)$ is a constant depending on $\lambda$ and the various model parameters, $S (b-a)$ denotes the concentration function of the distribution of the random variables $\omega_j$, $j \in {\mathbb{Z}}^d$, and $m \geq 1$. A Wegner estimate is optimal if $m = 1$.
In this note, we prove optimal Wegner estimates for the family of operators defined in valid at all energies $[a,b] \subset {\mathbb{R}}$, see Theorems \[thm:Klein1\] and \[thm:Klein2\]. Furthermore, we study the dependence of $C(\lambda)$ on the disorder parameter $\lambda$. If $$b < E_0 (\infty) := \lim_{t \to \infty} \inf \sigma \biggl( H_0 + t \sum_{j \in {\mathbb{Z}}^d} u_j (\cdot - j) \biggr)
\in (0, \infty]$$ then the constant $C(\lambda)$ in the Wegner estimate will tend to zero if the disorder $\lambda$ tends to infinity, see Theorem \[thm:Klein3\]. In this case, the Wegner estimate can be used to obtain the initial length scale estimate at sufficiently large disorder, and localization follows via multiscale analysis, see [@DreifusK-89; @Kirsch-08]. Hence, it is natural to ask whether such a Wegner estimate (where $C(\lambda) \to 0$ if $\lambda \to \infty$) also holds above the threshold $E_0 (\infty)$. Until now there have been no results in this direction, cf. [@Klein-13]. We show in Theorem \[thm:counterexample\] that it is indeed impossible to obtain a Wegner estimate with $C(\lambda) \to 0$ if $\lambda \to \infty$ above $E_0 (\infty)$. This result is even new in the absence of a magnetic field. In the case where the family $H_\omega$ is ergodic, Theorem \[thm:counterexample\] has an interpretation in terms of the integrated density of states, see Theorem \[thm:conterexample\_ergodic\]. Theorems \[thm:counterexample\] and \[thm:conterexample\_ergodic\] show in particular that the spectral behaviour at large disorder changes drastically below and above the model-dependent constant $E_0(\infty) \in [0, \infty]$, see Remark \[rem:phase\_transition\].
Optimal and non-optimal Wegner estimates for ergodic and non-ergodic random operators with and without magnetic field have been studied by many authors before. Let us give a brief overview here. We refer to [@Klein-13] and [@Veselic-08] for further references. In [@CombesH-94], an optimal Wegner estimate at all energies is proved for the usual (ergodic) Anderson Hamiltonian under the additional assumption that a covering condition holds. In [@CombesHK-07], the authors remove the covering condition and consider operators $H_0$ with non-vanishing magnetic field. More precisely, they assume that the magnetic vector potential $A_0$ is either periodic (which allows only for a subset of the class of periodic magnetic fields) or $H_0$ is the Landau Hamiltonian in two dimensions, i.e. $A_0 = B/2 (-x_2, x_1)$ where $B$ is the magnetic field strength. The Landau Hamiltonian can be treated in spite of its unbounded vector potential $A_0$ since unique continuation properties for the Landau Hamiltonian are quite well understood, cf. [@CombesHKR-04]. However, this method strongly relies on the structure of the Landau Hamiltonian and breaks down for arbitrarily small perturbations of the Landau magnetic field. In the non-ergodic setting, optimal Wegner estimates have been proved, e.g., in [@RojasMolina-12] for the Landau Hamiltonian, and up to a logarithmic correction in [@RojasMolinaV-13] for non-magnetic Schrödinger operators. The logarithmic factor in the energy is removed in [@Klein-13].
The drawback of our results is that we assume boundedness of $A_0$ and ${\operatorname{div}}(A_0)$. This assumption stems from the quantitative unique continuation result from [@BorisovTV-15]. While the long-term goal is to treat also unbounded magnetic vector potentials $A_0$, we emphasize that already bounded $A_0$ include important physically relevant examples, cf. [@Krakovsky-96].
Our paper is organized as follows: In Section \[sect:model\_and\_results\] we introduce the notation and state our main results. In Section 3 we quote a quantitative unique continuation estimate for solutions of elliptic equations given in [@BorisovTV-15]. In Section \[sec:proofs\] we give the proofs of our main results, while a technical estimate is postponed to the appendix. The proof of the Wegner estimates from Theorems \[thm:Klein1\], \[thm:Klein2\] and \[thm:Klein3\] rely on the recent quantitative unique continuation result of [@BorisovTV-15]. By now there is a wealth of results pursuing the connection between unique continuation and spectral theory of random Schrödinger operators, see e.g. [@CombesHK-03; @CombesHKR-04; @BourgainK-05; @CombesHK-07; @Stollmann-10; @BoutetDeMonvelLS-11; @RojasMolina-12; @RojasMolinaV-13; @BourgainK-13; @Klein-13; @TaeuferV-15; @NakicTTV-15; @NakicTTV-16-arxiv; @TaeuferV-16]. While the proofs of Theorems \[thm:Klein1\], \[thm:Klein2\] and \[thm:Klein3\] use in particular the guiding thread of [@Klein-13], they show how this approach and the recent result of [@BorisovTV-15] complement each other in an efficient way. On the contrary, the proofs of Theorem \[thm:counterexample\] and \[thm:conterexample\_ergodic\] are original even for the usual (ergodic) Anderson Hamiltonian with vanishing magnetic field.
Model and results {#sect:model_and_results}
=================
For $x \in {\mathbb{R}}^d$ or ${\mathbb{C}}^d$ we denote by $\lvert x \rvert$ the Euclidean norm of $x$. For $L,r > 0$ we denote by $\Lambda_L = (- L/2, L/2)^d$ the open cube with side length $L$, and by ${B (r)} = \{y \in {\mathbb{R}}^d \colon \lvert y \rvert < r\}$ the open ball in ${\mathbb{R}}^d$ of radius $r$, centered at $0$. For $x \in {\mathbb{R}}^d$ we denote by $\Lambda_L (x) = \Lambda_L + x$ and ${B (r , x)} = {B (r)} + x$ its translates.
Let us define the class of random Schrödinger operators studied in this note. It is a generalization of the models studied in [@Klein-13]. The non-random part is given by the self-adjoint magnetic Schrödinger operator $$H_0 = ( - {\mathrm{i}}\nabla + A_0)^2 + V_0$$ on $L^2 ({\mathbb{R}}^d)$ with a bounded electric potential $V_0 \in L^\infty({\mathbb{R}}^d)$ and a bounded magnetic vector potential $A_0 \in L^\infty({\mathbb{R}}^d, {\mathbb{R}}^d)$ such that ${\operatorname{div}}(A_0)$ is bounded and $\inf \sigma(H_0) = 0$. Note that we can rewrite $H_0$ to $H_0 = - \Delta + b_0^T \nabla + c_0$ where $$\label{eq:definition_b_c}
b_0(x) = - 2 i A_0(x)
\quad
\text{and}
\quad
c_0(x) = V_0(x) + \lvert A_0(x) \rvert^2 - i {\operatorname{div}}(A_0)(x).$$ In order to introduce the random part, we define the probability space $(\Omega , \mathcal{A}, \allowbreak {\mathbb{P}})$ where $$\Omega = { \ensuremath\mathop{ \tikz[baseline,line width=0.8pt,line cap=round] \draw(0,-0.2em)--(0.9em,0.7em)(0.9em,-0.2em)--(0,0.7em); }}\limits_{k \in {\mathbb{Z}}^d} {\mathbb{R}}, \quad \mathcal{A} = \bigotimes_{k \in {\mathbb{Z}}^d} \mathcal{B} ({\mathbb{R}}), \quad \text{and} \quad {\mathbb{P}}= \bigotimes_{k \in {\mathbb{Z}}^d} \mu_k ,$$ where $\mu_k$, $k \in {\mathbb{Z}}^d$, are non-degenerate probability measures on ${\mathbb{R}}$ with $\operatorname{supp} \mu_k \allowbreak \subset [0,M]$ for some $M > 0$ and all $k \in {\mathbb{Z}}^d$. By non-degenerate we mean that for all $L > 0$ we have $S_L (t) \to 0$ as $t \to 0$, see below for the definition of $S_L$. As a consequence, the projections $\Omega \ni \omega \mapsto \omega_k$, $k \in {\mathbb{Z}}^d$ give rise to the independent sequence of random variables $(\omega_k)_{k \in {\mathbb{Z}}^d}$, each coordinate $\omega_k$ distributed according to the measure $\mu_k$. We write $S_\mu(t) := \sup_{a \in {\mathbb{R}}} \mu([a, a+t])$ for the concentration function of a probability measure $\mu$ and set for $t \geq 0$ $$S_L(t) := \sup_{j \in \Lambda_L \cap {\mathbb{Z}}^d} S_{\mu_j}(t) .$$ We use the symbol ${\mathbb{E}}$ for the expectation with respect to the probability measure ${\mathbb{P}}$.
Let now $\delta_- \in (0,1/2)$ and $Z = (z_j)_{j \in {\mathbb{Z}}^d} \subset {\mathbb{R}}^d$ such that $$\forall j \in {\mathbb{Z}}^d \colon \quad {B (\delta_- , z_j)} \subset \Lambda_1(j).$$ For each $\omega \in \Omega$, the *crooked alloy-type potential* $V_\omega : {\mathbb{R}}^d \to {\mathbb{R}}$ is defined by $$V_\omega(x)
=
\sum_{j \in {\mathbb{Z}}^d} \omega_j u_j(x - z_j ) ,$$ where the single-site potentials $( u_j )_{j \in {\mathbb{Z}}^d}$, are measurable and real-valued functions on ${\mathbb{R}}^d$ satisfying $$u_- \chi_{{B (\delta_-)}}
\leq
u_j
\leq
\chi_{\Lambda_{\delta_+}(0)}$$ for some $u_- \in (0,1]$ and $\delta_+ > 0$. For each $\omega \in \Omega$ and $\lambda > 0$ we define the self-adjoint operator $$H_\omega = H_0 + \lambda V_\omega$$ in $L^2 ({\mathbb{R}}^d)$, and call the family of operators $( H_\omega)_{\omega \in \Omega}$ the *magnetic crooked alloy-type Hamiltonian*. For $L > 0$ we denote by $H_{\omega,L}$ the restrictions of $H_\omega$ to $L^2 (\Lambda_L)$ subject to Dirichlet boundary conditions. Given a Borel set $B \subset {\mathbb{R}}$, we denote by $P_{\omega, L}(B) := \chi_B(H_{\omega, L})$ the spectral projection onto the set $B$ with respect to $H_{\omega,L}$.
\[thm:Klein1\] Let $E_0 > 0$ and set $$\label{eq:gamma_1}
\gamma_1^2
=
\frac{1}{2} \delta_-^{N_2 \left( 1 + \lvert E_0 \rvert^{2/3} + \lVert b_0 \rVert_\infty^2 + \lVert c_0 \rVert_\infty^{2/3} \right)}$$ where $M_d > 0$ is the constant from Theorem \[thm:qucp2\] and $b_0, c_0$ are as in Eq. . Then there is a constant $
C_1 = C_1 (d, \delta_\pm, u_-, \gamma_1, \lVert V_0 \rVert_\infty, E_0)
$ such that for any closed interval $I \subset (- \infty, E_0]$ with $\lvert I \rvert \leq 2 \gamma_1$, and $\lambda > 0$, and any $L \in {\mathbb{N}}_{\mathrm{odd}}$ with $L \geq 2 + \delta_+$, we have $${\mathbb{E}}\bigl( \operatorname{Tr} P_{\omega, L}(I) \bigr)
\leq
C_1 \left( 1 + (\lambda M)^{2^{2 + \frac{\log d}{\log 2}}} \right)
S_L( \lambda^{-1} \lvert I \rvert) \lvert \Lambda_L \rvert.$$
\[thm:Klein2\] Let $E_0 >0$ and set $$\label{eq:gamma_2}
\gamma_2
=
\frac{1}{2} \delta_-^{N_2 \left( 1 + \lvert E_0 \rvert^{2/3} + \lVert b_0 \rVert_\infty^2 + \left( \lVert c_0 \rVert_\infty + \lambda M (2 + \delta_+)^d \right)^{2/3} \right)}$$ where $N_2 > 0$ is the constant from Theorem \[thm:qucp2\] and $b_0, c_0$ are as in Eq. . Then there is a constant $
C_2 = C_2(d,\delta_+, \lVert V_0 \rVert_\infty)
$ such that for any closed interval $I \subset (- \infty, E_0]$ with $\lvert I \rvert \leq 2 \gamma_2$, any $\lambda > 0$, and any $L \in {\mathbb{N}}_{\mathrm{odd}}$ with $L \geq 2 + \delta_+$, we have $${\mathbb{E}}\bigl ( \operatorname{Tr} P_{\omega, L}(I) \bigr)
\leq
C_2 \left( u_-^{-2} \gamma_2^{-4} (1 + E_0) \right)^{2^{1 + \frac{\log d}{\log 2}}}
S_L(\lambda^{-1} \lvert I \rvert) \lvert \Lambda_L \rvert.$$
For $t \geq 0$ we define $$H_0(t)
:=
H_0
+
t \sum_{j \in {\mathbb{Z}}^d} u_j(\cdot - z_j),$$ and set $$E_0 (t) := \inf \sigma(H_0(t)) \quad \text{and} \quad
E_0(\infty)
:=
\lim_{t \to \infty} E_0(t)
=
\sup \left\{E_0(t) : t \geq 0 \right\} .$$
\[thm:Klein3\] We have $E_0(\infty) > 0$. Let $E_1 \in ( 0, E_0(\infty))$ and set $$\kappa_0
=
\sup_{s > 0 : E_0(s) \geq E_1}
\frac{E_0(s) - E_1}{s}
>
0 .$$ Then for any Borel set $B \subset (-\infty , E_1]$, any $\lambda > 0$, any $L \in {\mathbb{N}}_{\mathrm{odd}}$, and almost all $\omega \in \Omega$ we have $$\label{eq:Klein3_1}
P_{\omega, L} (B)
\Bigl(
\sum_{j \in \Lambda_L \cap {\mathbb{Z}}^d} u_j(\cdot - z_j)
\Bigr)
P_{\omega, L} (B)
\geq
\kappa_0
P_{\omega, L} (B) .$$ Moreover, for any closed interval $I \subset (- \infty, E_1]$, any $\lambda > 0$, and for any $L \in {\mathbb{N}}_{\mathrm{odd}}$ with $L \geq 2 + \delta_+$, we have $$\label{eq:Klein3_2}
{\mathbb{E}}\bigl( \operatorname{Tr} P_{\omega, L} (I) \bigr)
\leq
C_3
\left(
\kappa_0^{-2} ( 1 + E_1)
\right)^{2^{1 + \frac{\log d}{\log 2}}}
S_L(\lambda^{-1} \lvert I \rvert) \lvert \Lambda_L \rvert,$$ where $C_3 > 0$ is a constant depending on $d$, $\delta_+$, $\lVert V_0 \rVert_\infty$, $\lVert b_0 \rVert_\infty$, and $\lVert c_0 \rVert_\infty$.
The Wegner estimates from Theorems \[thm:Klein1\], \[thm:Klein2\] and \[thm:Klein3\] can be used as an ingredient for the multiscale analysis [@FroehlichS-83; @DreifusK-89; @GerminetK-01; @GerminetK-03; @GerminetK-06]. Let us emphasize that the multiscale analysis requires that the concentration functions $S_L$, $L \in {\mathbb{N}}$, are sufficiently regular, e.g. with a uniformly bounded density, or uniformly Hölder continuous, cf. the just mentioned references. The multiscale analysis is an induction argument to establish localization in its various manifestations (spectral, dynamical, etc.). While a Wegner estimate is required for the induction step, the so-called initial length scale estimate corresponds to the induction anchor. Hence, if the concentration functions $S_L$ are sufficiently regular, Theorems \[thm:Klein1\], \[thm:Klein2\] and \[thm:Klein3\] will imply localization at energies where an appropriate initial scale estimate is satisfied. If the upper bound in the Wegner estimate becomes small at large disorder, the initial scale estimate will follow from the Wegner estimate at sufficiently large disorder as observed in [@DreifusK-89], see also [@Kirsch-08].
The upper bounds in Theorems \[thm:Klein1\] and \[thm:Klein2\] grow as the disorder $\lambda$ increases. As discussed above, this is not sufficient to deduce an initial length-scale estimate and localization at large disorder. In contrast to that, the upper bound in Theorem \[thm:Klein3\] converges to $0$ as the disorder parameter $\lambda$ tends to $\infty$. Hence, an initial scale estimate and localization at large disorder follow, albeit only for energies below $E_0(\infty)$. Note that if a covering condition $$ \sum_{j \in {\mathbb{Z}}^d} u_j(\cdot - z_j)
\geq
{\varepsilon}> 0$$ is satisfied, then $E_0(\infty) = \infty$ and the statement of Theorem \[thm:Klein3\] holds at all energies, see [@CombesH-94] in the case of vanishing magnetic field. In contrast, $E (\infty)$ might be finite if we do not assume a covering condition. Since Wegner estimates with a disorder dependence as in Theorem \[thm:Klein3\] provide a relatively simple path to localization at large disorder, it is natural to ask if such a disorder dependence can be expected at all energies, even if no covering condition is assumed. However, so far one was not able to prove such a Wegner estimate for alloy-type models with and without magnetic field above the threshold $E (\infty)$, cf. [@Stollmann-10; @BoutetDeMonvelLS-11; @Klein-13].
Our next theorem shows that this is indeed not possible. A disorder dependence as in Theorem \[thm:Klein3\] holds if and only if we consider energy intervals below $E_0(\infty)$. In particular this shows that at high energies and at high disorder there is a fundamental difference between alloy-type models with and without a covering condition. This is a new result, even in the special case of vanishing magnetic potential ($A_0 = 0$) and ergodic potential ($V_0$ periodic, $z_j = j$, $u_j = u_0$, and $\mu_j = \mu_0$).
\[thm:counterexample\] Let $E_2 \in {\mathbb{R}}$. The following are equivalent:
(i) $E_2 \leq E_0(\infty)$.
(ii) For all sufficiently large $L>0$, and all closed intervals $I \subset (- \infty, E_2]$, we have $$\label{eq:equivalence}
{\mathbb{E}}\left( \operatorname{Tr} P_{\omega,L} (I) \right)
\to 0 \quad\text{as}\quad\lambda \to\infty .$$
For the rest of this section we assume that the family $H_\omega$, $\omega \in \Omega$, is ergodic, i.e. $V_0$ and $A_0$ are periodic, and for all $j \in {\mathbb{Z}}^d$ we have $\mu_j = \mu_0$, $u_j = u_0$ and $z_j = j$. In this situation, Theorem \[thm:counterexample\] has an interpretation in terms of the integrated density of states (IDS). Since the main argument in this case is rather instructive, we present it here: Let us denote by $N_\lambda : {\mathbb{R}}\to [0,\infty)$ the IDS of the family $H_\omega$, $\omega \in \Omega$. This is a distribution function satisfying $$N_\lambda(E)
=
\lim_{L \to \infty} \frac{\operatorname{Tr} P_{\omega, L}( (- \infty, E])}{\lvert \Lambda_L \rvert}.$$ at all continuity points of $N_\lambda$ and for almost all $\omega \in \Omega$, cf. [@Pastur-71; @Shubin-79; @KirschM-82c], see also [@Veselic-08] and the references therein. Note that the Wegner estimates from Theorems \[thm:Klein1\], \[thm:Klein2\] and \[thm:Klein3\] imply local Lipschitz continuity of $N_{\lambda}$ at all $E \in {\mathbb{R}}$ and for all $\lambda \in (0, \infty)$, if the measure $\mu_0$ is sufficiently regular. Moreover, if $\mathcal{U}^\infty := {\mathbb{R}}^d \backslash \operatorname{supp} \sum_{j \in {\mathbb{Z}}^d} u(\cdot - j)$ is non-empty we will denote by $H_0^\infty$ the corresponding Dirichlet operator on $\mathcal{U}^\infty $, i.e. the unique self-adjoint extension of the operator $(- i \nabla + A_0)^2 + V_0$ on $C_0^\infty(\mathcal{U}^\infty) \subset L^2(\mathcal{U}^\infty)$. For its IDS we use the notation $N_{0 , \infty}$.
By Floquet theory, cf. [@Zak-64; @Sjoestrand-91], we obtain $N_{0,\infty}(E) > 0$ for $E > E_0(\infty)$. Furthermore, $$ N_{0, \infty}(E)
\leq
N_\lambda(E)$$ for all $\lambda > 0$ and all $E \in {\mathbb{R}}$. This follows from $$\label{eq:inequality_eigenvalues_IDS}
\mu_k(H_{0,L}^\infty)
\geq
\mu_k(H_{\omega,L})$$ where $\mu_k$ denotes the $k$-th eigenvalue of the corresponding operator, ordered increasingly and counting multiplicities and $H_{0,L}^\infty$ is the Dirichlet restriction of $H_0^\infty$ to $L^2(\mathcal{U}^\infty \cap \Lambda_L)$. For the convenience of the reader, we give a proof of Ineq. in the appendix. Together with Theorem \[thm:Klein3\], we found the following theorem.
\[thm:conterexample\_ergodic\] Let $H_\omega$, $\omega \in \Omega$, be ergodic. Then $\lim_{\lambda \to \infty} N_\lambda(E) \geq N_{0, \infty}(E)$ for all $E \in {\mathbb{R}}$ and it holds that
(i) $\lim_{\lambda \to \infty} N_{\lambda}(E) = 0$ if $E < E_0(\infty)$,
(ii) $\lim_{\lambda \to \infty} N_\lambda(E) > 0$ if $E > E_0(\infty)$.
\[rem:phase\_transition\] Theorems \[thm:counterexample\] and \[thm:conterexample\_ergodic\] show that the difference between the alloy-type model with and without a covering condition is not merely a technical issue. Rather, we observe that the physical behaviour of the system at large disorder fundamentally differs between the phases $E < E_0 (\infty)$ and $E > E_0 (\infty)$.
Quantitative unique continuation
================================
In [@BorisovTV-15], the authors prove quantitative unique continuation principles for second order elliptic partial differential expressions with variable coefficients. Here, we formulate the special case where the leading term is the Laplacian. Let $${\mathcal{H}}u := - \Delta + b^{\mathrm{T}}\nabla u + c u$$ with $b \in L^\infty({\mathbb{R}}^d; {\mathbb{C}}^d)$ and $c \in L^\infty({\mathbb{R}}^d; {\mathbb{C}})$. For $L > 0$ we denote by ${\mathcal{D}}(\Delta_L)$ the domain of the Laplace operator in $L^2 (\Lambda_L)$ subject to Dirichlet boundary conditions. For $\Gamma \subset {\mathbb{R}}^d$ open and $\psi \in L^2 (\Gamma)$ we denote by $\lVert \psi \rVert = \lVert \psi \rVert_\Gamma$ the usual $L^2$-norm of $\psi$. If $\Gamma' \subset \Gamma$ we use the notation $\lVert \psi \rVert_{\Gamma'} = \lVert \chi_{\Gamma'} \psi \rVert_\Gamma$. The following theorem is a special case of Theorem 12 in [@BorisovTV-15].
\[thm:ucp\_eigenfunction\] For all $L \in {\mathbb{N}}_{\mathrm{odd}}$, all measurable and bounded $V : \Lambda_L \to {\mathbb{R}}$, all $\psi\in {\mathcal{D}}(\Delta_L)$ and $\zeta \in L^2 (\Lambda_L)$ satisfying $\lvert {\mathcal{H}}\psi \rvert \leq \lvert V\psi \rvert + \lvert \zeta \rvert$ almost everywhere on $\Lambda_L$, all $\delta \in (0,1/2)$ and all sequences $X = (x_j)_{j \in {\mathbb{Z}}^d}$ such that ${B (\delta , x_j)} \subset \Lambda_1(j)$ for all $j \in {\mathbb{Z}}^d$, we have $$\lVert \psi \rVert_{S_{\delta,X} (L)}^2 + \delta^2 G^2 \lVert \zeta \rVert_{\Lambda_L}^2
\geq C_{{\mathrm{sfuc}}} \lVert \psi \rVert_{\Lambda_L}^2 ,$$ where $$\label{eq:definition-Sdelta}
C_{{\mathrm{sfuc}}} = \delta^{N_1 \bigl( 1 + \lVert V \rVert_\infty^{2/3} + \lVert b \rVert_\infty^{2} + \lVert c \rVert_\infty^{2/3} \bigr)}
\quad\text{and}\quad
S_{\delta , X} (L) = \bigcup_{j \in {\mathbb{Z}}^d} {B (\delta , x_j)} \cap \Lambda_L .$$ Here $N_1 \geq 1$ is a constant depending only on the dimension.
For $L > 0$ we define the differential operator $H_L : {\mathcal{D}}(\Delta_L) \to L^2 (\Lambda_L)$ by $H_L \psi = {\mathcal{H}}\psi$. If $$\label{eq:sa}
b = \mathrm{i} \tilde b
\quad\text{and}\quad
c = \tilde c + \mathrm{i} {\operatorname{div}}\tilde b / 2$$ for some bounded $\tilde b , \tilde c \in L^\infty ({\mathbb{R}}^d)$, then $H_L$ is a self-adjoint operator in $L^2 (\Lambda_L)$. The following theorem is a special case of Theorems 13 and 14 in [@BorisovTV-15].
\[thm:qucp1\] Let be satisfied. Then for all $L \in {\mathbb{N}}_{\mathrm{odd}}$, all $E \in {\mathbb{R}}$, all $\delta \in (0,1/2)$, all sequences $X = (x_j)_{j \in {\mathbb{Z}}^d}$ such that ${B (\delta , x_j)} \subset \Lambda_1(j)$ for all $j \in {\mathbb{Z}}^d$, and all $\psi \in \operatorname{Ran} \chi_{[E-\gamma , E + \gamma]} (H_L)$ with $$\gamma^2 = \delta^{N_2\bigl( 1 + \lvert E \rvert^{2/3} + \lVert b \rVert_\infty^{2} + \lVert c \rVert_\infty^{2/3} \bigr)}$$ we have $$\lVert \psi \rVert_{S_{\delta,X} (L)}^2 \geq \gamma^2 \lVert \psi \rVert_{\Lambda_L}^2 .$$ Here $N_2\geq 1$ is a constant depending only on the dimension and $S_{\delta , X} (L)$ is as in Eq. .
As a corollary we obtain
\[thm:qucp2\] Let be satisfied, $E_0 \in {\mathbb{R}}$, $\delta \in (0,1/2)$, and $$\gamma^2 = \delta^{N_2\bigl( 1 + \lvert E_0 \rvert^{2/3} + \lVert b \rVert_\infty^{2} + \lVert c \rVert_\infty^{2/3} \bigr)} .$$ Then for all $I \subset (-\infty , E_0]$ with $\lvert I \rvert \leq 2\gamma$, and all sequences $X = (x_j)_{j \in {\mathbb{Z}}^d}$ such that ${B (\delta , x_j)} \subset \Lambda_1(j)$ for all $j \in {\mathbb{Z}}^d$, we have $$\chi_{I}(H_L) W_{\delta,X} (L) \chi_{I}(H_L)
\geq \gamma^2 \chi_{I}(H_L) .$$ Here, $W_{\delta,X} (L)$ denotes the operator of multiplication with the characteristic function of the set $S_{\delta , X} (L)$ defined in Eq. .
Proofs {#sec:proofs}
======
For $L \in {\mathbb{N}}_{\mathrm{odd}}$, we define $U_L : \Lambda_L \to {\mathbb{R}}$ and $W_L : \Lambda_L \to {\mathbb{R}}$ by $$U_L
:=
\sum_{j \in {\mathbb{Z}}^d \cap \Lambda_L} u_j(\cdot - z_j)
\quad\text{and}\quad
W_L
:=
\sum_{j \in {\mathbb{Z}}^d \cap \Lambda_L} \chi_{{B (\delta_- , z_j)}} .$$ Then $\lVert U_L \rVert_\infty \leq (2 + \delta_+)^d$ and $\lVert V_\omega \rVert_\infty \leq M \lVert U_L \rVert_\infty \leq M (2 + \delta_+)^d$ for almost all $\omega \in \Omega$. Note that $$0 \leq W_L \leq u_- U_L,
\quad
W_L^2 = W_L,
\quad
\text{and}
\quad
\lVert W_L \rVert_\infty = 1.$$
Proof of Theorem \[thm:Klein1\]
-------------------------------
We follow the proof of Theorem 1.4 in [@Klein-13] and assume $\lambda = 1$. The general case $\lambda > 0$ follows by scaling the random variables $\omega_k \mapsto \lambda \omega_k$ which leads to $M \mapsto \lambda M$ and $S_L(\lvert I \rvert) \mapsto S_L(\lambda^{-1} \lvert I \rvert)$. Rewriting $H_\omega = - \Delta + b_0^T \nabla + c_0 + V_\omega$, where $b_0, c_0$ are defined in Eq. and given $E_0 > 0$, we define $\gamma_1$ as in . Then, by Theorem \[thm:qucp2\], for all $L \in {\mathbb{N}}_{\mathrm{odd}}$ and all intervals $I \subset (- \infty, E_0]$ with $\lvert I \rvert \leq 2 \gamma_1$ we have $$\chi_I(H_{0,L})
\leq
\gamma_1^{-2} \chi_I(H_{0,L}) W_L \chi_I(H_{0,L})
\leq
u_-^{-1}
\gamma_1^{-2} \chi_I(H_{0,L}) U_L \chi_I(H_{0,L}).$$ Since $\sigma(H_0) \subset [0, \infty)$, we have $\sigma(H_{0,L}) \subset [0, \infty)$ and may assume $I \subset [0, E_0]$. Similarly to Theorem 1.4 in [@Klein-13], we can now carefully follow the proof in [@CombesHK-07], keeping in mind that the proof therein is formulated for magnetic Schrödinger operators $H_0 = (- i \nabla A_0)^2 + V_0^2$.
Note that the Combes-Thomas estimates (for magnetic Schrödinger operators) required in [@CombesHK-07] depend only on $d$, $\delta_+$ and on $\lVert V_0 \rVert_\infty$, but not on the magnetic potential $A_0$, see Theorem 4.6 of [@Shen-14]. Hence, the constant $C_1$ will only depend on $A_0$ via $\gamma_1$.
Proof of Theorem \[thm:Klein2\]
-------------------------------
We adapt the proof of [@Klein-13 Theorem 1.5] to the magnetic setting.
\[lem:Klein\_3.1\] Let $I \subset ( - \infty, E_0]$ be a closed interval and $L \in {\mathbb{N}}_{\mathrm{odd}}$, $L \geq 2 + \delta_+$. Suppose that there is $\kappa > 0$ such that $$P_{\omega,L}(I)
U_L
P_{\omega,L}(I)
\geq
\kappa
P_{\omega,L}(I)
\quad
\text{with probability one}.$$ Then there is a constant $$C_4 = C_4(d, \delta_+, \lVert V_0 \rVert_\infty)$$ such that $${\mathbb{E}}\bigl( \operatorname{Tr} P_{\omega,L}(L) \bigr)
\leq
C_4 \left( \kappa^{-2} ( 1 + E_0) \right)^{2^{1 + \frac{\log d}{\log 2}}} S_L(\lambda^{-1} \lvert I \rvert) \lvert \Lambda_L \rvert.$$
One can follow verbatim the proof of Lemma 3.1 in [@Klein-13] which partially relies on results from [@CombesHK-07]. The latter apply to the class of magnetic Schrödinger operators as considered in this note as well. The only issue to address is the dependence of $C_4$ on the various parameters. In [@Klein-13], Eqs. (3.8) and (3.17), constants from Combes-Thomas estimates (for non-magnetic Schrödinger operators) which depend only on $d$, $\delta_+$ and on $\lVert V_0 \rVert_\infty$ enter the final constant $C_4$. Combes-Thomas estimates for magnetic Schrödinger operators do not depend on the magnetic potential $A_0$, see Theorem 4.6 of [@Shen-14]. Therefore the constant $C_4$ will not depend on the magnetic potential $A_0$.
We follow the proof of Theorem 1.5 in [@Klein-13]. Given $E_0 > 0$, define $\gamma_2$ as in Eq. . Theorem \[thm:qucp2\] yields for all $\Lambda_L$ with $L \in {\mathbb{N}}_{\mathrm{odd}}$, all intervals $I \subset (- \infty, E_0]$ with $\lvert I \rvert \leq 2 \gamma_2$ and almost all $\omega \in \Omega$ the estimate $$\chi_I(H_{0,L})
\leq
\gamma_2^{-2} \chi_I(H_{0,L}) W_L \chi_I(H_{0,L})
\leq
u_-^{-1}
\gamma_2^{-2} \chi_I(H_{0,L}) U_L \chi_I(H_{0,L}).$$ The statement of the theorem now follows from Lemma \[lem:Klein\_3.1\].
Proof of Theorem \[thm:Klein3\]
-------------------------------
For the proof we shall need an abstract uncertainty relation for Schrödinger operators at the bottom of the spectrum which has been developed in [@BoutetDeMonvelLS-11]. The following lemma is a slight generalization thereof, see Lemma 4.1 of [@Klein-13].
\[lemma:abstract\_uncertainty\] Let $H$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, bounded from below, and let $Y \geq 0$ be a bounded operator on $\mathcal{H}$. Let $H (t) = H + tY$ for $t \geq 0$, and set $E (t) = \inf \sigma (H (t))$ and $E (\infty) = \lim_{t \to \infty} E (t) = \sup_{t \geq 0} E (t)$. Suppose that $E (\infty) > E (0)$. For $E_1 \in (E (0) , E (\infty))$ let $$\kappa = \kappa (H , Y , E_1) = \sup_{s > 0 \colon E (s) > E _1} \frac{E (s) - E_1}{s} > 0 .$$ Then for all bounded operators $V \geq 0$ on $\mathcal{H}$ and Borel sets $B \subset (-\infty , E_1]$ we have $$\chi_B (H + V) Y \chi_B (H + V) \geq \kappa \chi_B (H + V) .$$
We recall that $H_0 (t) = H_0 + t \sum_{j \in {\mathbb{Z}}^d} u_j(\cdot - z_j)$ for $t \geq 0$, $E_0 (t) = \inf \sigma (H_0 (t))$, and $E_0 (\infty) = \lim_{t \to \infty} E_0 (t) = \sup_{t \geq 0} E_0 (t)$.
\[lemma:positive\] For all $t \geq 0$ we have $$E_0 (t) \geq t u_- \delta_-^{N_1 \bigl( 1 + \bigl( \lVert V_0 \rVert_\infty + t u_- \bigr)^{2/3} + \lVert b_0 \rVert_\infty^{2} + \lVert c_0 \rVert_\infty^{2/3} \bigr)}.$$ Hence, $$E_0 (\infty) \geq \sup_{t \in [0,\infty)} t \delta_-^{N_1 \bigl( 1 + \bigl( \lVert V_0 \rVert_\infty + t \bigr)^{2/3} + \lVert b_0 \rVert_\infty^{2} + \lVert c_0 \rVert_\infty^{2/3} \bigr)} > 0 .$$
We define $\tilde H_0(t) := H_0 + t u_- W$ for $t \geq 0$, $\tilde E_0 (t) = \inf \sigma (\tilde H_0 (t))$, and $\tilde E_0 (\infty) = \lim_{t \to \infty} \tilde E_0 (t) = \sup_{t \geq 0} \tilde E_0 (t)$. By assumption we have $E_0 (0) = \tilde E_0(0) = 0$. Moreover, $E_0 (\infty)$ and $\tilde E_0(\infty)$ are both well defined in $[0,\infty]$ by monotonicity. Since $\tilde E_0(t) \leq E_0(t)$ for all $t \in [0, \infty]$, it suffices to show the statement of the lemma for $\tilde E_0(t)$. For $L \in {\mathbb{N}}_{\mathrm{odd}}$ we denote by $\tilde H_{0,L} (t)$ the restriction of $\tilde H_0 (t)$ to $L^2 (\Lambda_L)$ subject to Dirichlet boundary conditions with domain ${\mathcal{D}}(\Delta_L)$, and set $\tilde E_{0,L} (t) = \inf \sigma (\tilde H_{0,L} (t))$. Then $\tilde E_{0,L} (t) \geq \tilde E_0 (t) \geq 0$ for all $t \geq 0$. Since $\lVert W_L \rVert_\infty = 1$ we have $$\tilde E_{0,L} (t) \leq d \left( \frac{\pi}{L} \right)^2 + \lVert V_0 \rVert_\infty + t u_- .$$ Since $\tilde H_{0,L} (t)$ has purely discrete spectrum, there exists $\psi (t) \in {\mathcal{D}}(\Delta_L)$ with $\lVert \psi (t) \rVert = 1$ such that $\tilde H_{0,L} (t) \psi (t) = \tilde E_{0,L} (t) \psi (t)$. We apply Theorem \[thm:ucp\_eigenfunction\] and obtain for all $t \geq 0$ $$\begin{aligned}
\bigl\langle \psi (t) , W_L \psi (t) \bigr\rangle
&= \lVert \psi \rVert_{S_{\delta_- , Z} (L)}\\
&\geq \delta_-^{N_1 \bigl( 1 + \bigl( d \pi^2 / L^2 + \lVert V_0 \rVert_\infty + t u_- \bigr)^{2/3} + \lVert b_0 \rVert_\infty^{2} + \lVert c_0 \rVert_\infty^{2/3} \bigr)}
\lVert \psi (t) \rVert_{\Lambda_L}^2
\end{aligned}$$ where $$S_{\delta_- , Z} (L) = \bigcup_{j \in {\mathbb{Z}}^d} {B (\delta_- , z_j)} \cap \Lambda_L .$$ The first statement of the lemma follows since $\lim_{L \to \infty} \tilde E_{0,L} (t) = \tilde E_0 (t)$. The second statement follows immediately.
By Lemma \[lemma:positive\] we have $E_0 (\infty) > 0$ and hence $\kappa_0 > 0$. For $L \in {\mathbb{N}}_{\mathrm{odd}}$ we denote by $H_{0,L} (t)$ the restriction of $H_0 (t)$ to $L^2 (\Lambda_L)$ subject to Dirichlet boundary conditions with domain ${\mathcal{D}}(\Delta_L)$, and set $E_{0,L} (t) = \inf \sigma (H_{0,L} (t))$. Using $
0 \leq E_0 (t) \leq E_{0,L} (t)
$ we obtain $$\kappa_{0,L}
:=
\sup_{s > 0 : E_{0,L}(s) \geq E_1} \frac{E_{0,L}(s) - E_1}{s}
\geq \kappa_0
=
\sup_{s > 0 : E_0(s) \geq E_0(1)}
\frac{E_0(s) - E_1}{s} > 0 .$$ Hence, the assumptions of Lemma \[lemma:abstract\_uncertainty\] are satisfied with $H = H_{0,L}$ and $Y = \sum_{j \in {\mathbb{Z}}^d}u_j (\cdot - z_j)$, and we obtain Ineq. . Ineq. now follows from Ineq. and Lemma \[lem:Klein\_3.1\].
Proof of Theorem \[thm:counterexample\]
---------------------------------------
The implication (i) $\Rightarrow$ (ii) is the statement of Theorem \[thm:Klein3\]. In order to show the converse, we prove the contraposition: Let $E_2 > E_0(\infty)$, and $I = (- \infty, E_2]$. Note that for almost all $\omega \in \Omega$, we have $H_{\omega,L} \leq H_{\eta,L}$, where $\eta \in \Omega$, $\eta_k = M$ for all $k \in {\mathbb{Z}}^d$, hence $${\mathbb{E}}\left( \operatorname{Tr} P_{\omega,L} (I) \right)
\geq
\operatorname{Tr} P_{\eta,L} (I).$$ Since $$\lim_{t \to \infty} \lim_{L \to \infty} E_{0,L}(t) < E_2,$$ there are $L_0 > 0$ and $\lambda_0 > 0$ such that for all $\lambda > \lambda_0$ we have $$1
\leq
\operatorname{Tr} P_{\eta,L_0}(I)
\leq
{\mathbb{E}}\left( \operatorname{Tr} P_{\omega,L_0} (I) \right).$$ Hence, cannot hold.
Proof of Ineq.
===============
Recall that $H_{0,L}^\infty$ is the Dirichlet restriction of $H_{0,L}$ to $\mathcal{U}^\infty \cap \Lambda_L$, the complement of the support of the single-site potentials in $\Lambda_L$.
\[lem:forms\] Let $L > 0$, $\lambda > 0$, and $\mathcal{U}^\infty \cap \Lambda_L \neq \emptyset$. Then for all $\omega \in \Omega$ $$\mu_k(H_{0,L}^\infty)
\geq
\mu_k(H_{\omega, L}).$$
Let $H_{t, L}$ be the operator $H_{\omega, L}$ where all the random variables $\omega_j$ are set to $t$. Since $\mu_k(H_{t, L}) \geq \mu_k(H_{\omega, L})$ for all $t \geq \lambda M$, it suffices to show $\mu_k(H_{0,L}^\infty) \geq \mu_k(H_{t, L})$ for some $t \geq \lambda M$. We define a family of positive and closed quadratic forms $ \{ \mathcal{E}_t \}_{t > 0}$ on $W^{2,1}(\Lambda_L) \subset L^2(\Lambda_L)$ via $$\mathcal{E}_t(\phi)
=
\langle (- i \nabla - A_0) \phi, (- i \nabla - A_0) \phi \rangle
+
\langle \phi, ( V_0 + t \sum_j u_j(\cdot - j) )\phi \rangle.$$ The form $\mathcal{E}_t$ is the unique quadratic form associated with the self-adjoint operator $H_{t, L}$ and the family $\{ \mathcal{E}_t \}_{t > 0}$ is monotonously increasing in the sense of quadratic forms. By a version of Kato’s monotone convergence theorem for quadratic forms, see [@Simon-78 Theorem 4.1], there is a closed form $\mathcal{E}_\infty$ which is defined as $$\begin{aligned}
\mathcal{D} (\mathcal{E}_\infty) &= \left\{\phi \in \mathcal{D} (\mathcal{E}_1) \colon \sup_{t > 0} \mathcal{E}_t (\phi) < \infty \right\}, \\
\mathcal{E}_\infty (\phi) &= \lim_{t \to \infty} \mathcal{E}_t (\phi)
\end{aligned}$$ such that $\mathcal{E}_t \nearrow \mathcal{E}_\infty$ as $t \to \infty$ in strong resolvent sense.
There is a caveat here concerning the notion of convergence in strong resolvent sense, cf. [@Simon-78]: Since the form $\mathcal{E}_\infty$ is not densely defined on $L^2(\Lambda)$, one cannot define the corresponding resolvent in the usual manner. However, the form $\mathcal{E}_\infty$ can be restricted to the closed subspace $\overline{\mathcal{D}(\mathcal{E}_\infty)} \subset \Lambda_L$ on which it yields a densely defined, closed form $\mathcal{\tilde E}_\infty$. In fact, $\mathcal{\tilde E}_\infty$ is the unique form corresponding to $H_{0,L}^\infty$. Let now $T$ be the operator on $L^2(\Lambda_L)$ which is $(H_{0,L}^\infty + 1)^{-1}$ on $\overline{\mathcal{D}(\mathcal{E}_\infty)}$ and $0$ on its orthogonal complement. The nonzero eigenvalues of $T$ are the eigenvalues of $(H_{0,L}^\infty + 1)^{-1}$. Then the notion of strong resolvent convergence of $\mathcal{E}_t$ to $\mathcal{E}_\infty$ means in this situation that $(H_{t,L} + 1)^{-1} \phi \to T \phi$ for all $\phi \in L^2(\Lambda_L)$.
This implies that the $k$-th eigenvalue (counted from above) of $(H_{t,L} + 1)^{-1}$ converges from above to the $k$-th eigenvalue (counted from above) of $(H_{0,L}^\infty + 1)^{-1}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is our privilege and honor to thank our mentor Ivan Veselić for the idea to pursue this research direction. Furthermore, we gratefully acknowledge helpful discussions with Peter Stollmann.
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|
---
abstract: |
We discuss the hadroproduction of charmed mesons in the framework of the constituent cascade model taking into account the valence quark annihilation. It is shown that the small valence quark annihilation process dominates the leading particle production at large Feynman $x$ and explains the recent experimental data on the asymmetry between $D^0$ and $\bar{D}^0$ at 350 GeV/c.\
PACS\
13.75.Gx(Pion-baryon interactions)\
13.87.Fh(Fragmentation into hadrons)\
14.40.Lb(Charmed mesons)
---
=0.75mm =0.75mm
****
------------------------------------------------------------------------
Valence quark annihilation effect on charmed meson production in $\pi N$ collisions\
Tsutomu Tashiro$^1$, Hujio Noda$^2$, Kisei Kinoshita$^3$ and Shin-ichi Nakariki$^1$\
$^1$Department Simulation Physics, Okayama University of Science, Ridai-cho, Okayama 700-0005, Japan\
$^2$Department of Mathematical Science, Faculty of Science, Ibaraki University, Bunkyou, Mito 310-0056, Japan\
$^3$Physics Department, Faculty of Education, Kagoshima University, Korimoto, Kagoshima 890-0065, Japan\
------------------------------------------------------------------------
$^1$ email: [email protected], [email protected], Fax: +81 86 256 8006\
$^2$ email: [email protected], Fax: +81 29 228 8407\
$^3$ email: [email protected], Fax: +81 99 285 7735\
Introduction {#intro}
============
Recently experiments at CERN [@wa92] measured the neutral $D$ mesons in $\pi^-$ nucleus collisions and observed much smaller values of the leading/non-leading asymmetry than those of charged $D$ mesons i.e. less than 0.2 and even a negative value around $x=0.8$. The experimental data on the asymmetry of charged $D$ mesons increases from zero to nearly one with Feynman variable $x$ in the $\pi^-$ fragmentation region [@wa92; @wa82; @e769; @e791; @e769_2]. The leading particle contains the same type of quark as one of the valence quarks in the incident hadron, while the non-leading one does not contain the projectile valence quarks. For example, the asymmetry of $D^0(c\bar{u})/\bar{D}^0(u\bar{c})$ in $\pi^-(d\bar{u}) $ interaction with nucleon is defined as $$\begin{aligned}
A_{\pi^- N}(D^0,\bar{D}^0) = \frac{\sigma(D^0)-\sigma(\bar{D}^0)}{\sigma(D^0)+\sigma(\bar{D}^0)}.\end{aligned}$$ In the perturbative QCD at leading order, the factorization theorem predicts that $ c $ and $ \bar{c} $ quarks are produced with the same distributions and then fragment independently. In this case the asymmetry $A_{\pi^- N}(D^-,D^+)$ is equal to zero [@qs]. Even in the case of next to leading order, the predicted asymmetry is much smaller than the data [@nde]. The asymmetry $A_{\pi^- N}(D^-,D^+)$ has been investigated and explained by means of many approaches: string fragmentation [@PYTHIA], intrinsic charm contributions [@vb; @anzivino; @Armesto; @piskounova; @arakelyan_volko; @arakelyan], recombination process [@bednyakov; @a_h_magnin_s; @c_h_magnin], recombination using valon concept [@hwa; @das_hwa] and so on. We have proposed the constituent quark-diquark cascade model and explained the leading/non-leading asymmetry of charged $D$ mesons $A_{\pi^- N}(D^-,D^+)$ successfully [@tnnik]. The model, however, gives rather large values of $A_{\pi^- N}(D^0,\bar{D^0})$ at $ 0.5 \stackrel{<}{\sim} x $ as expected from the leading particle effect but deviating from the experimental data. Although several models have been applied more or less satisfactorily to $A_{\pi^- p}(D^0,\bar{D^0})$ at $ x \stackrel{<}{\sim} 0.6$ [@arakelyan; @c_h_magnin], the asymmetry problem about the charmed hadron productions is an open question.
In the present paper we investigate the leading/non-leading $D$-meson asymmetry in the framework of the constituent quark-diquark cascade model by taking into account the valence quark annihilation.
Model description
=================
We consider an inclusive reaction $ A+B \rightarrow C+X$ in the centre of mass system of $ A $ and $ B$. The light-like variables of $ A $ and $ B $ are defined as follows: $$\begin{aligned}
x_{0\pm}^A=\frac{E^A \pm p_{cm}}{\sqrt{s_0}},~~~x_{0\pm}^B=\frac{E^B \mp p_{cm}}{\sqrt{s_0}} ,\end{aligned}$$ where $ \sqrt{s_0} $ is the centre of mass energy of the incident hadrons $A$ and $B$. We briefly review our model and then introduce the valence quark annihilation mechanism into the model.
When the collision between $A$ and $B$ occurs, the incident hadrons break up into two constituents with a probability $ (1 - P_{gl})$; otherwise they emit wee gluons with $ P_{gl}$ followed by a quark-antiquark pair creation. We assume four interaction types: a) non-diffractive dissociation, b) and c) single-diffractive dissociations of $ B$ and $A$, and d) double-diffractive dissociation types as shown in Fig. \[fgr:intrctn type\]. The probabilities of these types to occur are $ (1 - P_{gl})^2 , P_{gl} (1 - P_{gl}), P_{gl} (1 - P_{gl})$ and $ P_{gl}^2$, respectively. Here we denote the quark-antiquark pair emitted from $ A $ ($ B $) via the wee gluons as $ M_A $ ( $ M_B $). The probabilities of $ M_A $ ( $ M_B $) to be $u\bar{u}, d\bar{d}, s\bar{s} $ and $ c\bar{c} $ are denoted as $ P_{u\bar{u}},~ P_{d\bar{d}},~ P_{s\bar{s}} $ and $ P_{c\bar{c}} $, respectively.
The momentum fraction of $M_A$ is fixed by the distribution function $$\begin{aligned}
H_{M_A/A}(z) = z^{\beta_{gl}-1}(1-z)^{\beta_{ld}-1}/B(\beta_{gl},\beta_{ld}) ,
\label{eqn:Hmaa}\end{aligned}$$ and the uniform distribution $R$ in the interval from zero to one as, $$\begin{aligned}
x_+^{M_A}=x_{0+}^Az,~~~x_-^{M_A}=x_{0-}^AR.
\label{eqn:Hmaa_2}\end{aligned}$$ Then the incident particles $A$ and $B$ have the following momentum fractions: $$\begin{aligned}
x_+^{A}=x_{0+}^A(1-z),~~x_-^{A}=m_A^2/(x_+^{A}~s_0),
\nonumber\end{aligned}$$ $$\begin{aligned}
x_-^B=x_{0-}^B-(x_{-}^A-x_{0-}^A(1-R)),~~x_+^B=x_{0+}^B,
\label{eqn:Hmaa_2_leading}\end{aligned}$$ where the mass shell condition is considered and transverse momenta are neglected. The momentum fraction of $M_B$ is treated similarly, exchanging the role of $A$ and $B$.\
In the centre of mass system of incidents $A(M_A) $ and $ B(M_B)$, we define the light-like fractions of these hadrons and fix the light-like fractions of the projectile constituents. The distribution functions of the constituents in the projectile $ A $ composed of $ a $ and $ a' $ are described as $$\begin{aligned}
H_{a/A}(z) = H_{a'/A}(1-z) = \frac{z^{\beta_a-1}(1-z)^{\beta_{a'}-1}}{B(\beta_a,\beta_{a'})}.
\label{eqn:Ha/A}\end{aligned}$$ Then the light-like fractions of $ a $ and $ a' $ are $ ~~x_+^a=x_{0+}^Az~,\\
x_-^a=x_{0-}^AR,\ x_+^{a'}=x_{0+}^A-x_+^a$ and $x_-^{a'}=x_{0-}^A-x_-^a $, respectively. The distribution functions of the constituents in $M_A, B$ and $M_B$ are similarly defined. In our model hadrons are produced on the chain between a valence quark(anti-quark) from $A(M_A)$ and the valence diquark(quark) from $B(M_B)$ via the cascade processes
$$\begin{aligned}
q & \rightarrow & M(q \bar{q}')+q', \nonumber \\
& & B(q[q'q''])+\overline{[q'q'']} , B(q\{q'q''\})+\overline{\{q'q''\}} , \nonumber \\
\overline{[q'q'']} & \rightarrow & \overline{B}(\bar{q}\overline{[q'q'']})+q, \nonumber \\
& & M(q\bar{q}')+\overline{[qq'']} , M(q\bar{q}')+\overline{\{qq''\}}, \nonumber \\
% & & M(q\bar{q}'')+\overline{[qq']} , M(q\bar{q}'')+\overline{\{qq'\}}, \nonumber \\
\overline{\{q'q''\}} & \rightarrow & \overline{B}(\bar{q}\overline{\{q'q''\}})+q, \nonumber \\
& & M(q\bar{q}')+\overline{[qq'']} ,M(q\bar{q}')+\overline{\{qq''\}},\nonumber\\
% & & M(q\bar{q}'')+\overline{[qq']} ,M(q\bar{q}'')+\overline{\{qq'\}}, \nonumber \\\end{aligned}$$ where $[\ ]$ and $\{\}$ denote the flavour antisymmetric and symmetric diquarks, respectively [@qdq]. Meson production probabilities from $q, \overline{[q'q'']} $ and $ \overline{\{q'q''\}}$ are $ 1-\epsilon, \eta_{[~]} $ and $ \eta_{\{\}} $, respectively.
We redefine the light-like fractions of the incident constituents in the rest frame of the cascade chain. The momentum sharing of the cascade process $ q + \bar{q}' \rightarrow M(q\bar{q}'') + q''+
\bar{q}' $ from a $q $ with $ x_\pm^q $ and $ \bar{q}' $ with $ x_\pm^{\bar{q}'} $ takes place as follows [@qdqA; @km]: First, using the emission function $$\begin{aligned}
F_{q''q}(z) = z^{\gamma\beta_q-1}(1-z)^{\beta_q+\beta_{q''}-1} /B(\gamma\beta_q,\beta_q+\beta_{q''}) ,
\label{eqn:Fqq} \end{aligned}$$ we fix the lightlike fractions of $ q'' $ and $ M $ as $ x^{q''}_{+} = x^q_{+} z $ and $ x^{M}_{+} = x^q_{+} - x^{q''}_{+} $, respectively and put $ x^{q''}_{-} = x^q_{-} $. Second, the transverse momentum of $ M $ is determined from the probability function $$\begin{aligned}
G(\mbox{\boldmath$p$}_{T}^2)=\frac{\sqrt{m}}{C}\exp(-\frac{C}{\sqrt{m}}\mbox{\boldmath$p$}_{T}^2)
\label{eqn:pT2} \end{aligned}$$ in $ p_{T}^2 $ space. Then, from the onshell condition, $ x^{M}_{-} $ is fixed as $ x^{M}_{-} = (m_M^2 + {{\mbox{\boldmath$p$}}_{T}}^2)/x^M_{+}s'$, where $ \sqrt{s'} $ is the subenergy of the cascade chain. The transverse momentum of $ q'' $ is $ {\mbox{\boldmath$p$}}^{q''}_T = {\mbox{\boldmath$p$}}^q_T - {\mbox{\boldmath$p$}}_{T} $. The lightlike fraction of $ \bar{q}' $ is decreased to $ \tilde{x}^{\bar{q}'}_{-} = x^{\bar{q}'}_{-} - x^{M}_{-}$. If the energy of $\bar{q}'$ is enough to create another hadron, the cascade such as $ q'' + \bar{q}' \rightarrow q'' + \bar{q}''' + M(q'''\bar{q}') $ takes place in the opposite side. Finally recombined hadrons are put on-shell by a two body decay process as explained in [@tnnik].
The dynamical parameters $ \beta$’s in (\[eqn:Ha/A\]) and (\[eqn:Fqq\]), which determine the momentum sharings of the constituents, are related to the intercepts of the Regge intercepts as $
\beta_u=\beta_d=1-\alpha_{\rho-\omega}(0),~~ \beta_s=1-\alpha_\phi(0),
\beta_c=1-\alpha_{J/\psi}(0)$ [@mnkt; @cthkp]. From previous analyses [@qdq; @qdqA], we determine the values for diquarks as $\beta_{[ij]}=\gamma_{[~]}(\beta_i+\beta_j),~~\beta_{\{ij\}}=\gamma_{\{\}}(\beta_i+\beta_j)$. We consider lower lying hadrons: pseudoscalar($PS$), vector ($V$), tensor($T$) mesons, octet($O$) and decuplet($D$) baryons composed of $ u,d, $ and $ s $ flavours and the correspondings with charm flavour. We assume the production probabilities for them to be $ P_{PS}, P_V, P_T $ ($=1-P_{PS}-P_V),
P_O $ and $ P_D$ ($=1-P_O$), respectively. Octet and decuplet baryons are described as $$\begin{aligned}
|8 > = \cos\theta|q[q'q'']> + \sin\theta|q\{q'q''\}>,
\label{eqn:octB} \end{aligned}$$ $$\begin{aligned}
|10 > = |q\{q'q''\}>.
\label{eqn:dcpB} \end{aligned}$$ Directly produced resonances decay into stable particles. Details of our model are explained in [@tnnik; @qdq; @qdqA].
We modify our model to include valence quark annihilation in the non-diffractive dissociation type interaction. Let us consider the case of $\pi^-p$ collision. We take account of the annihilation of the valence $\bar{u}$ from $\pi^-$ and valence $u$ from proton target in a slightly different way from the one pointed out in [@bednyakov; @c_h_magnin]. There are annihilation processes such as shown in Fig.2. The annihilation process in Fig.2a is considered soft process and its contribution is related to the magnitude of $\sigma_{inel}^{\pi^-}
- \sigma_{inel}^{\pi^+}$ by unitarity. This process, however, may be negligible for the charm-pair production. Here we only take into account the annihilation process in Fig.2b and assume that the process $$\begin{aligned}
\bar{u}u \rightarrow \bar{q}q
\label{eqn:anni} \end{aligned}$$ occurs with the probability $P_{anni}(1-P_{g\ell})^2$ and the non-diffractive type occurs with the probability $(1-P_{anni})(1-P_{g\ell})^2 $. Branching ratios of $\bar{u}u \rightarrow
\bar{u}u, \bar{d}d, \bar{s}s,$ and $\bar{c}c$ are chosen to be equal to each other for the channels allowed energetically.
This process is considered as a semi-hard scattering process and produced $q$ and $\bar{q}$ are supposed to be non-free. It seems natural to assume that $\bar{q}$ has tendency to be produced in the forward direction of the $\bar{u}$ in the centre of mass system of $\bar{u}$ and $u$, due to the confinement force between the valence $d$ quark and the produced $\bar{q}$ quark. Here we choose the distributions for $z=\cos\phi$ as $$\begin{aligned}
D(z)=\frac{3}{8}(1+z)^2
\label{eqn:annidrctn} \end{aligned}$$ in the region $-1 < z < 1$, where $\phi$ is the angle between the directions of $\bar{u}$ and $\bar{q}$ in the centre of mass system of $\bar{u}$ and $u$. After this annihilation mechanism, the non-diffractive type production of hadrons occurs as shown in Fig. 2b.
Comparison with the data {#sec:3}
========================
In this section, we give the results of our model for the inclusive hadron productions in $\pi^- N$ collisions. The value of the dynamical parameter $\beta_c$ is changed from the value $\beta_c=8.0$ used in [@tnnik] to $\beta_c=4.0$. This is in accord with the argument that the slopes of Regge trajectories of charmed mesons are smaller than those of light mesons as discussed in [@Burakovsky_g]. The parameter of $p_T^2$ distribution $ C=1.2$ and the probability $P_{c \bar{c}}=0.00016 ( P_{s\bar{s}}=0.09984)$ in [@tnnik] are changed into $C=1.0$ and $P_{c \bar{c}}=0.00025 ( P_{s\bar{s}}=0.09975)$. Furthermore the meson production probabilities are assumed as $P_{PS}=0.4, P_{V}=0.4 $ and $P_T=0.2$ for light mesons and $P_{PS}=1/9, P_V=1/3$ and $P_T=5/9$ for charmed mesons. We put the parameter $P_{anni}$ so as to reproduce the asymmetry $A_{\pi^-N}(D^0,\bar{D^0})$. We choose the value $P_{anni}=0.0005$. This value scarcely changes the features of the spectra of light hadrons with $u, d$ and $ s$. For other parameters, we use the same values used in the previous analysis [@tnnik].
In Fig.3 we show the results of $D^{*}$ productions and compare the two cases: case(1) $3(1+z)^2/8$ distribution for $\cos \alpha $ and case(2) without annihilation ($P_{anni}=0$:the same as in [@tnnik] except for the values of $\beta_c=4.0, C=1.0, P_{c\bar{c}} = 0.00025$, and $ P_{PS}=1/9, P_V=1/3$ and $ P_T=5/9$ for charmed meson productions). The annihilation effect of Fig.2b is seen in $\bar{D}^{*0}(u\bar{c})$ and $D^{*-}(d\bar{c})$. Our model gives a satisfactory description of $x$ dependence for production of $ D^{*+}$ or $D^{*-}$ in $\pi^-N$ collision. The result for production of $D^{*0}$ or $\bar{D}^{*0}$ at $x \approx 0.1$ is small as compared with experimental data [@aguilar].
The annihilation effect on meson productions in $\pi^- p$ is twice as much as in $\pi^- n$ collision. The annihilation mechanism have a considerable effect on the non-leading particle $\bar{D}^{*0}(u\bar{c})$. However, the shape of the leading particle $D^{*-}(d\bar{c})$ is affected little by the annihilation mechanism. Therefore our model gives larger differences between $A_{\pi^- p}(D^{*0},\bar{D}^{*0})$ and $A_{\pi^- n}$ ($D^{*0},\bar{D}^{*0})$ than those between $A_{\pi^- p}(D^{*-},D^{*+})$ and $A_{\pi^- n}(D^{*-},D^{*+})$. Fig.4a shows the results in case(1) for $A_{\pi^- p}(D^{*0},\bar{D}^{*0})$, $~A_{\pi^- n}(D^{*0},\bar{D}^{*0})$ and $A_{\pi^- N}(D^{*0},\bar{D}^{*0})$. Hereafter we show the average results of $\pi^-$ beam on proton and neutron targets. Fig.4b shows the results of $A_{\pi^- N}(D^{*0},\bar{D}^{*0})$ and $A_{\pi^- N}(D^{*-},
D^{*+})$ in case(1) and case(2).
Smaller values of $A_{\pi^- N}(D^{*-},D^{*+})$ as compared with $A_{\pi^- N}(D^{*0},\bar{D}^{*0})$ at $x\approx 0.5$ are due to the difference between the hadron productions on quark-diquark and antiquark-quark chains in non-diffractive dissociation type mechanism (Fig.1a) [@tnnik]. The leading particle $D^{*-}(d\bar{c})$ is produced on the chain between the valence $d $ quark in the beam and the valence diquark in the target. The proton target has a tendency to break into an energetic valence diquark and a wee valence quark. Then the $ D^{*-}$ meson produced in the first cascade step is energetic. However, in the case in which a baryon is produced from the valence diquark in the first cascade step, the momentum of the valence $d$ quark is decreased. Furthermore the total momentum is shifted to the diquark side. Then the leading spectrum of $D^{*-}$ on the quark-diquark chain tends to have a small momentum and the asymmetry is reduced in $0.3 \stackrel{<}{\sim} x \stackrel{<}{\sim} 0.7$.
On the other hand, the leading particle $D^{*0}(c\bar{u})$ is produced on the chain between the valence $\bar{u} $ quark in the $ \pi^- $ beam and the valence quark in the target. The momentum reduction of the $\bar{u}$ quark is small even in a case a meson is produced in the first cascade step from the valence quark in the target fragmentation. Therefore the asymmetry $ A_{\pi^- N}(D^{*0},\bar{D}^{*0}) $ is not so suppressed as compared with $ A_{\pi^- N}(D^{*-},D^{*+})$ in $ 0.3 \stackrel{<}{\sim} x \stackrel{<}{\sim} 0.7$. In case(1), the annihilation effect compensates the quark-diquark effect and the asymmetry $A_{\pi^- N}(D^{*0},\bar{D}^{*0})$ becomes negative at large $x$.
In Fig. 5, we show the result of $x$ dependence of $D$ mesons in $\pi^- N$ collisions for case(1) and case(2). The annihilation effect is also seen in $\bar{D}^0$ and a little in $D^-$ spectra as in $D^*$ spectra. In Fig. 6, we show the result of $p_T^2$ dependence of $D$ mesons in $\pi^- N$ collisions for case(1) and case(2). Our calculation shows smaller values of $ D^0$ than those of $\bar{D}^0$ at large $p_T^2$ deviating from the data. The newly introduced annihilation process explains the large $p_T^2$ charmed meson productions in part. Our model is in good agreement with the data except for $D^0,~\bar{D}^0$ at large $p_T^2$. In Fig. 7 the results of $x$ dependence of $A_{\pi^- N}(D^-,D^+)$ and $ A_{\pi^- N}(D^0,\bar{D}^0)$ are compared with the experimental data at 350 GeV/c [@wa92]. The agreement of $x$ dependence with the experimental data is satisfactory. The negative value of $ A_{\pi^- N}(D^0,\bar{D}^0)$ at large $x$ is well explained by introducing a small amount of the valence quark annihilation process to the model. Fig.8 presents the results of $p_T^2$ dependence of $A_{\pi^- N}(D^-,D^+)$ and $ A_{\pi^- N}
(D^0,$$\bar{D}^0)$. The $p_T^2$ dependence of $ A_{\pi^- N}(D^0,\bar{D}^0)$ disagrees with the data in sign as seen in Fig.6b. The discrepancy increases with the annihilation effect. The results of the $x$ and $p_T^2$ dependences of $A_{\pi^- N}(D^-,D^+)$ and $ A_{\pi^- N}(D^0,\bar{D}^0)$ at 500 GeV/c are compared with the experimental data [@e791] in Fig.9. The agreement with the data is fairly good except for the negative value of the data at $x \approx -0.15$. There is a valence $d$ quark in the incident nucleon and our model predicts a positive value of $A_{\pi^- N}(D^-,D^+)$ for $ x < 0$ as seen in Fig.9. However, in the case of proton target with $\cos \theta =1$ in (\[eqn:octB\]), there is no valence $d$ quark and the model gives a small value of $A_{\pi^- N}(D^-,D^+)$ in the target fragmentation region.
The results of the $x$ and $p_T^2$ dependence for production of $ D_S^-(s\bar{c})$ or $ D_S^{+}(c\bar{s}) $ are compared with the experimental data [@wa92] in Fig.10. The features of the spectra are in good agreement with the data except for the small discrepancy in normalization. Fig.11 shows the calculated results of asymmetries $A_{\pi^- N}(D_S^-,D_S^+)$ with respect to $x$ and $p_T^2$. The values are small because both $D_S^\pm$ mesons are non-leading particles. But in case(1) $A_{\pi^- N}(D_S^-,D_S^+)$ increases with $x$ at large $x$ due to the valence quark annihilation.
Discussions {#sec:4}
===========
The large leading/non-leading asymmetry $A_{\pi^- N}(D^-,D^+)$ is naturally explained by our constituent quark cascade model as in the case of light hadron productions. In [@tnnik], we noticed that the cascade chain properties are different between the antiquark-quark and the quark-diquark chains. The leading particle $D^-(d\bar{c})$ produced in the quark-diquark chain in $\pi^- p$ collision is less energetic when a baryon is produced in the target fragmentation in the first cascade process as compared with $D^+(c\bar{d})$ produced in the quark-antiquark chain in $\pi^+ p$ collision. We have different leading particle effects between $ \pi^+ $ and $ \pi^- $ beams i.e. $A_{\pi^+ N}(D^+,D^-) > A_{\pi^- N}(D^-,D^+)$ around $x \sim 0.6 $ [@tnnik]. Although the difference between the asymmetries $A_{\pi^- N}(D^0,\bar{D}^0)$ and $A_{\pi^- N}(D^-,D^+)$ disappears due to the decay effect and the valence quark annihilation effect,this quark-diquark chain effect is seen in the difference between the asymmetries $A_{\pi^-}(D^{*0},\bar{D}^{*0})$ and $A_{\pi^- N}(D^{*-},D^{*+})$\
around $ x \sim 0.5 $ as shown in Fig.4b.
The small valence quark annihilation process (\[eqn:anni\]) explains the negative value of $A_{\pi^- N}(D^0,\bar{D}^0)$ at large $x$ observed in the recent experiment [@wa92]. This process is considered as semi-hard interaction. All channels occur with equal probability and the produced constituents tend to have the forward direction of the incident valence constituents. This is a part of the hadron production process at large $p_T^2$. Our model can explain the spectra up to the region $ p_T^2 \stackrel{<}{\sim} 10$ (GeV/c)$^2$. As noticed in the previous section, the results of calculated behaviours of the $p_T^2$ dependence of $A_{\pi^- N}(D^0,\bar{D}^0)$ disagree with the experimental data. This implies that there may be considerable contribution from soft interactions which maintain the leading particle effect even at large $p_T^2$ region. The negative values of experimental data on $A_{\pi^- N}(D^0,\bar{D}^0)$ at $ p_T^2 \stackrel{<}{\sim} 3$ (GeV/c)$^2$ suggests that the $\bar{q}$ quark in the annihilation process (\[eqn:anni\]) is produced in the very small cone around the direction of incident $\bar{u}$ as compared with (\[eqn:annidrctn\]). It is interesting to investigate the leading/nonleading asymmetry in $K N$ collisions, since in $K^+ N$ collision, there is no annihilation process. These points will be discussed elsewhere.
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---
abstract: 'We present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier-Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.'
author:
- 'E.BRAND'
- 'J.F.GIBSON'
bibliography:
- 'doublylocal.bib'
title: 'A doubly-localized equilibrium solution of plane Couette flow'
---
intro numerics solutions conclusions
[**Acknowledgments.**]{} The authors thank Tobias Schneider, Greg Chini, and Bruno Eckhardt for helpful discussions, Hecke Schrobsdorff and Tobias Kreilos for their work on parallellizing [channelflow]{}, and the Max Planck Institute for Dynamics and Self-Organization for computer time.
|
---
abstract: 'An electric power distribution system is operated in several distinct radial topologies by opening and closing of system’s sectionalizing and tie switches. The estimation of the system’s current operational topology is a precursor to implementing any optimal control actions (during normal operation) or restorative actions (during outage condition). This paper presents a mathematical programming approach to estimate the operational topology of a three-phase unbalanced power distribution system under both outage and normal operating conditions. Specifically, a minimum weighted least absolute value estimator is proposed that uses the line flow measurements, historical/forecasted load data, and ping measurements and solves a mixed-integer linear program (MILP) to estimate the operational topology and any outaged sections simultaneously. We thoroughly validate the accuracy of the proposed approach using IEEE 123-bus and a 1069-bus three-phase multi-feeder test system with the help of numerous simulations. It is observed that the approach performs well even at high percentages of measurement errors.'
author:
- 'Anandini Gandluru, , Shiva Poudel, , and Anamika Dubey, [^1]'
bibliography:
- 'references.bib'
title: Joint Estimation of Operational Topology and Outages for Unbalanced Power Distribution Systems
---
Topology Estimation, Three-phase Power Distribution System, Mixed Integer Linear Programming.
Nomenclature {#nomenclature .unnumbered}
============
Set of buses in distribution system
Set of buses connected to capacitors
Set of distribution lines
Set of distribution lines with switches
Set of measurements
Binary variables representing status of the switch between buses $i$ and $j$
Per-phase real load demand at bus $j$
Per-phase reactive load demand at bus $j$
Vector of per-phase real and reactive load demands at bus $j$, $[p_{Lj}^\phi; q_{Lj}^\phi]$
Per-phase active power flow in line connecting buses $i$ and $j$
Per-phase reactive power flow in line connecting buses $i$ and $j$
Vector of per-phase active and reactive power flows in line connecting buses $i$ and $j$, $[P_{ij}^\phi; Q_{ij}^\phi]$
Load section connectivity variable for section $l$
Per-phase capacitor connectivity variable for capacitor bank connected at bus $j$.
Sum of error in ping measurements
Per-phase forecasted real load demand at bus $j$
Per-phase forecasted reactive demand at bus $j$
Per-phase rated reactive power for capacitor at $j$
Vector of per-phase forecasted values of real and reactive load demands at bus $j$; $[\hat{p}_{{Lj}}^\phi; \hat{q}_{{Lj}}^\phi]$
Vector of variances in per-phase forecasted real and reactive load demands at bus $j$
Per-phase measured value of real power flow in line connecting buses $i$ and $j$
Per-phase measured value of reactive power flow in line connecting buses $i$ and $j$
Vector of per-phase measured values of power flows in line connecting buses $i$ and $j$; $[\hat{P}_{ij}^\phi; \hat{Q}_{ij}^\phi]$
Vector of variances in per-phase real and reactive power flow measurements
Smart meter ping measurement of load at bus $j$
Introduction
============
typical distribution system includes tie-switches and sectionalizing switches and is operated in several distinct radial topologies. The problem of identifying the current operational topology from the given planned distribution system model using real-time measurements is termed as [*[topology estimation problem]{}*]{}. Since knowing the operational topology is the initial requirement for state-estimation (in conventional state estimation algorithms) and for implementing any optimal control action (during normal operation) or restorative actions (during outage condition), an incorrect estimate may lead to incorrect states and sub-optimal decisions. Note that a vast body of literature is available on the real-time topology estimation algorithm for transmission systems using techniques such as weighted least squares state estimation, generalized state estimation, and least median of squares estimation [@monticelli1993modeling; @da2016simultaneous; @caro2010breaker; @mili1996robust; @phaniraj1991least; @vempati2005topology; @kekatos2012joint]. However, unlike transmission systems, a typical distribution system is radially operated and incurs unbalanced and lossy system conditions. Therefore, existing topology estimation algorithms from transmission systems literature are not directly applicable to distribution systems. In this paper, we present a scalable optimization-based approach to simultaneously estimate the operational topology and any outaged sections for an unbalanced three-phase power distribution system.
Traditionally, a network topology processor (NTP) identifies the grid’s operational topology using the statuses of the switching devices by forming a linked list to capture the radial distribution structure [@korres2012state; @he2001efficient]. The inputs to traditional NTP are the bus-switch model and the user-defined, measured, scheduled or normal status of the switching devices. The bus-switch model consists of all the branches of the system that can be opened/closed depending upon the corresponding switches being opened/closed. The traditional NTP processes the bus/switch model with the measured switch statuses and results in a bus-branch model of the system consisting of all system buses and ‘closed’ branches of the network. Assuming full confidence in measured switch statuses, the traditional NTP simply forms a linked list that captures the radial distribution structure. It means that the traditional network topology processors (NTP) assume that the measured status of switches are accurate (i.e. they have no errors). Owing to frequent reconfigurations and manual changes by operators, the available switch status measurements may not reflect the actual switch statuses. Therefore, if the switch status measurements are incorrect, the radial topology identified by the “traditional network topology processor” will also be incorrect.
To include switch status errors, researchers have proposed generalized state estimation formulations for both transmission and distribution systems that simultaneously identify both continuous state variables and switch statuses [@monticelli1993modeling; @da2016simultaneous; @caro2010breaker; @mili1996robust; @alsac1998generalized; @phaniraj1991least; @vempati2005topology; @kekatos2012joint; @monticelli2000testing; @lourenco2004bayesian; @1489505; @singh2010recursive; @compareGSE; @korres2006substation; @korres2012state]. Unfortunately, due to nonlinear power flow models, the joint estimation of the system’s states and topology leads to a nonlinear problem that does not scale well for large-scale unbalanced power distribution systems. In fact none of the above-mentioned methods have been demonstrated for a large unbalanced three-phase distribution system model; the following feeder models were used in the above referenced papers: IEEE 14-bus transmission test case, IEEE-24 bus transmission test system, RTS-96 bus transmission test system, 26-bus U.K. Generic Distribution System (UKGDS) model, and 17-bus artificial distribution system models. Furthermore, while in the transmission system, simplified/approximate power flow models can be used to simplify the generalized state estimation problem, the unbalances and losses in distribution systems prohibit one from simplifying power flow model for accurate state estimation. An unbalanced distribution system requires three-phase nonlinear power flow formulation that further increases the computational complexity making the generalized state estimation algorithms even more difficult to scale compared to the transmission systems. Therefore, for distribution systems, there is the need for a separate topology estimation stage where an approximate/linearized distribution power flow model can be used to identify the real-time operational tree/radial topology. Afterward, any standard state-estimation algorithm (with a non-linear power flow model) can be applied on the known topology to estimate system states.
To avoid non-linearity in topology estimation problems, in [@sharon2012topology], authors establish a linear relation between real-time measurements and power injection statistics. Simulated real-time measurements for each topology are compared against the actual measurements to determine the switch statuses. Although the algorithm is linear in the number of topologies, evaluation of all possible operational topologies poses computational challenges. In [@sevlian2015distribution], a polynomial complexity approximate Maximum a Posteriori (MAP) detector is proposed to identify the operational topology using only flow measurements by exhaustively searching for the correct operational topology. The worst-case scenario requires evaluation of all possible spanning trees and hence poses computational challenges for large systems. In [@zamani2015topology], the distribution system topology is estimated using the residual of branch current distribution state estimation. The algorithm, however, also requires an exhaustive search of all possible topologies posing computational challenges for large systems.
Another body of literature employs a data-driven approach. In [@cavraro2018power], authors claim that a topology change leaves a signature in $\mu$PMU measurements and compare this signature with a library of signatures pre-determined for all possible topologies to detect distribution system topology. The algorithm assumes only one switch change at a time thus restricting the applicability for a real-world distribution system. Furthermore, for better performance on a larger system, the approach requires a large number of expensive $\mu$PMU measurement units. In related work, authors present data-driven methods to identify the distribution system connectivity model using several days of historical data [@liao2015distribution; @deka2018topology]. These methods identify a [*[static network connectivity model]{}*]{} using a large set of voltage measurements. Alternatively, in this paper, we address a different problem where our objective is to estimate the [*[time-varying]{}*]{} operational topology of the distribution system with a known planning/connectivity model using real-time measurements.
Unfortunately, the aforementioned literature on operational topology estimation assume that there are no outages and, therefore, cannot be extended to estimate the operational topology during outage conditions. For operational topology estimation during outages, our objective is to estimate both energized radial feeder and the outaged sections. The existing literature on outage detection assumes that the normal radial operational topology (without fault) is known [@jiang2016outage; @sevlian2018outage; @liao2016urban]. Starting with a radial topology, these methods identify outaged feeder sections and hence, are inapplicable when the current radial operational topology is not known.
The existing literature on distribution grid topology estimation poses scalability concerns for large-scale unbalanced distribution systems, and cannot identify operational topology during outages when the normal radial operational topology is unknown. This calls for a generalized model for operational topology estimation that simultaneously takes the outage and normal operating conditions into account. In this paper, we propose a mixed-integer linear program (MILP) that simultaneously identifies the current operational topology for the unbalanced distribution grid and any outaged sections using line flow measurements, historical/forecasted load data, and ping measurements from a limited set of smart meters. The proposed framework can be thought of as an “advanced network topology processor”. The major contributions of this paper are as follows:
1. [*Simultaneous Identification of Normal and Outaged Topology from Planning Model:*]{} The proposed MILP formulation is a generalized framework that estimates the operational topology during both normal operation and outage condition without enumeration. The method incorporates the possibility of switching of system’s legacy devices.
2. [*Incorporate Errors in both Continuous and Discrete Measurements:*]{} The proposed formulation models both continuous and discrete measurements into one unified formulation. The measurement errors and their distributions are appropriately modeled and the estimation accuracy is calculated for varying percentages of measurement errors.
3. [*Scalable MILP formulation for large-scale unbalanced distribution systems:*]{} A computationally-tractable MILP formulation is proposed to solve the topology estimation problem for large-scale unbalanced distribution systems. To demonstrate the scalability, the approach is validated using a 1069-bus three-phase test system where on an average the operational topology is obtained within 30 sec.
![A simple distribution grid. The radial operational topology is dictated by the statuses of sectionalizing switches and tie switches. The operational topology changes as a result of reconfiguration during normal operation (spanning trees) or due to fault isolation during outages (subtrees).[]{data-label="fig:1"}](images/fig_1-eps-converted-to.pdf){width="48.50000%"}
Measurement-based Topology Estimation
=====================================
The measurement-based topology estimation problem uses distribution system measurements to infer the operational topology of the distribution grid from the available planning model. First, we present a graph-theoretic framework to represent the three-phase power distribution system and associated topology estimation problem. Next, we detail the discrete and continuous variables to define the topology estimation problem and present the description of the measurement set.
Graphical Representation of the Distribution System
---------------------------------------------------
A distribution system is comprised of interconnected distribution feeders and can be operated in numerous radial operational topologies as shown in Fig. 1. Moreover, several normally open tie switches and normally closed sectionalizing switches are employed to facilitate the restoration efforts leading to numerous possible topologies during an outage. Let, the planning model be represented by graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ where $\mathcal{V}$ represents the set of buses and $\mathcal{E}$ is the set of edges representing all distribution lines and switches that can be open/closed. Let, the operational topology is represented by a tree $\mathcal{T} =(\mathcal{V_T},\mathcal{E_T}$), where, $\mathcal{E_T}\subset\mathcal{E}$ and $\mathcal{V_T}\subseteq\mathcal{V}$. Note that during normal operation, all buses are supplied and hence the operational topology, $\mathcal{T}$, is a spanning tree of the graph, $\mathcal{G}$. However, during an outage, the operational topology will be a subtree of the planned topology graph. This is because, in the event of a fault, multiple loads within a faulted load section will be isolated by the opening of respective switches to isolate the faulted load sections, resulting in a subtree.
Topology Estimation: Problem Description
----------------------------------------
Theoretically, the topology estimation problem involves a combinatorial problem of searching for a feasible tree, $\mathcal{T}$, within the graph, $\mathcal{G}$, that satisfies the measurements. There are both continuous and discrete measurements available including line flow measurements (active and reactive branch power flows), historical or forecasted load measurements also called pseudo-measurements, and switch status and smart meter ping measurements. Note that if the switch statuses are known and assumed to be correct, the topology estimation is a trivial problem. Therefore, the traditional network topology processor assumes that switch measurements are unreliable and are not included in the measurement set.
A combinatorial search problem is not tractable for a large-scale system with a large number of possible tree configurations, especially, when outages are taken into consideration. This is because the enumeration-based methods require processing not only all spanning trees but also all subtrees of the original graph to detect operational topology during outages. We formulate a computationally tractable optimization model to identify switch statuses by minimizing the weighted least absolute error in measured and predicted values of system variables subject to power flow constraints. Mathematical expressions for predicted line flows are obtained using a three-phase linearized power flow model coupled with binary switch variables. This results in an MILP model that estimates the switch statuses while incorporating the measurement errors.
System Variables
----------------
As stated in the problem description, switch statuses are the unknown variables of interest. To aid us in finding the switch statuses, we define load demand and power flow variables associated with each load and line in the distribution system. Also, to include possible outages, we define binary variables indicating outaged load sections.
### Switch Status Variable
A binary variable $\delta_{ij}=\{0,1\}$ is associated with each line section including switches, where $\delta_{ij}=1$ implies that switch connecting buses $i$ and $j$ is closed, while $\delta_{ij}=0$ implies that the corresponding switch is open. The switch status variables are used to formulate power flow equations and connectivity model for the distribution system.
### Load Section Connectivity Variable
A load section is comprised of a minimum set of loads that must be disconnected to isolate a fault. During an outage, one/more load sections are disconnected from the main feeder by automated operation of protection devices. All loads within a load section are disconnected during a fault within that section. We define a binary variable, $y_{l}$ corresponding to each load section, where, $y_{l} = 0$ implies load section $l$ is outaged and vice versa. The estimation of load section connectivity variables leads to the identification of both operational topology and outaged sections for the given distribution system planning model.
### Power Flow and Load Demand Variables
Per-phase active and reactive load demands at bus $j$ are represented by $p_{Lj}^\phi$ and $q_{Lj}^\phi$. Similarly, per-phase active and reactive power flows in line connecting buses $i$ and $j$, directed from $i$ to $j$, are represented by $P_{ij}^\phi$ and $Q_{ij}^\phi$, respectively.
### Capacitor Status Variables
Single-phase and three-phase capacitor banks are usually connected to the distribution system. Let, $y_{Cj}^\phi$ be the binary variable indicating status (ON/OFF) of the capacitor bank at bus $j$ in phase $\phi$, where, $y_{Cj}^\phi=1$ implies the capacitor bank is ON and vice-versa.
System Measurements
-------------------
The measurements required for the proposed formulation include: (1) forecasted active and reactive load demands of all loads in the system, (2) active and reactive power flow measurements at a few selected distribution lines, (at least one flow measurement from each fundamental cycle [@sevlian2015distribution]), and (3) smart meter ping measurements from at least one load in each load section. We adopt the conventional notation of real and pseudo-measurements [@schweppe1974static] to categorize the telemetered flow measurements and forecasted load measurements, respectively. Note that the forecasted load measurements can be replaced by corresponding AMI data, if available. Measurement devices and forecasting techniques have errors that cannot be avoided. Therefore, measurements are modeled as random variables. The following section details the error models for different measurement variables and corresponding distributions for the associated measurement random variables.
### Power Flow Measurements
Active and reactive power flow measurements for phase $\phi$ of the line between buses $i$ and $j$, directed from $i$ to $j$, are represented as random variables $\hat{P}_{ij}^\phi$ and $\hat{Q}_{ij}^\phi$, respectively. The corresponding errors in these measurements are modeled as Gaussian distribution with zero mean and prespecified variance, i.e. $e(\hat{P}_{ij}^\phi) \sim \mathcal{N}(0,\sigma^2_{{P}_{ij}^\phi})$ and $e(\hat{Q}_{ij}^\phi)=\mathcal{N}(0,\sigma^2_{{Q}_{ij}^\phi})$, respectively. Since these are telemetered measurements, they have low errors in the range of 1% to 5%.
The power flow measurements are related to the power flow variables. Notice that, $\hat{P}_{ij}^\phi$ = ${P}_{ij} + e(\hat{P}_{ij}^\phi)$ and $\hat{Q}_{ij}^\phi$ = ${Q}_{ij} + e(\hat{Q}_{ij}^\phi)$. Therefore, the random variables corresponding to flow measurements follow the following distributions: $\hat{P}_{ij}^\phi \sim \mathcal{N}(P_{ij}^\phi,\sigma^2_{{P}_{ij}^\phi})$ and $\hat{Q}_{ij}^\phi \sim \mathcal{N}(Q_{ij}^\phi,\sigma^2_{{Q}_{ij}^\phi})$.
### Forecasted (Pseudo) Load Demand Measurements
Active and reactive load demand measurements at bus $j$ in phase $\phi$ are represented by random variables $\hat{p}_{Lj}^\phi$ and $\hat{q}_{Lj}^\phi$ respectively. The corresponding errors in the load measurements are modeled as Gaussian random variables with zero mean and prespecified variance i.e. $e(\hat{p}_{Lj}^\phi) \sim \mathcal{N}(0,\sigma^2_{{p}_{Lj}^\phi})$ and $e(\hat{q}_{Lj}^\phi) \sim \mathcal{N}(0,\sigma^2_{{q}_{Lj}^\phi})$. As the load values are forecasted and used as pseudo measurements, they may incur high errors.
The load demand measurements are related to the load demand variables. Notice that, $\hat{p}_{Lj}^\phi$ = ${p}_{Lj} + e(\hat{p}_{Lj}^\phi)$ and $\hat{q}_{Lj}^\phi$ = ${q}_{Lj} + e(\hat{q}_{Lj}^\phi)$. Therefore, the random variables corresponding to load demand measurements follow the following distributions: $\hat{p}_{Lj}^\phi \sim \mathcal{N}(p_{Lj}^\phi,\sigma^2_{{p}_{Lj}^\phi})$ and $\hat{q}_{Lj}^\phi \sim \mathcal{N}(q_{Lj}^\phi,\sigma^2_{{q}_{Lj}^\phi})$.
### Smart Meter Ping Measurements
During an outage, section(s) of the distribution grid is(are) isolated and hence the operational topology will be a subtree of $\mathcal{G}$ instead of a spanning tree. Therefore, we require additional measurements carrying the information regarding outaged sections. Existing outage management systems (OMS) include smart meter ping measurements to detect the connectivity status of a load. A smart meter, when pinged, sends out its load consumption, if energized, and does not respond when it is disconnected due to an outage. We consider smart meter communication as an additional measurement for the topology estimation problem. The smart meter ping measurement is given by $\hat{y}_j$ where, if the smart meter connected at $j^{th}$ load bus responds to the ping request, $\hat{y}_j=1$, implying that the load is energized; otherwise, $\hat{y}_j=0$, implying load is disconnected.
We consider error in ping measurements, where, error is modeled using Bernoulli distribution i.e. $e(\hat{y}_j) \sim Bernoulli(q)$, where $q$ is the probability that the ping measurement is inaccurate, i.e. with $q$ probability, the difference between the ping measurement and the actual load section energization variable is 1 and with $(1-q)$ probability the difference is 0. This implies that $(\hat{y}_j - {y}_l) \sim Bernoulli(q)$, where load connected at bus $j$ is supplied by the load section $l \in L$. Let, a total of $l_{p}$ number of loads are pinged for each load section $l \in L$. Then total number of ping measurements are given by $n_{p} = \sum_{l = 1}^L l_{p}$.
Next, we identify the distribution of the sum of errors in smart meter ping measurements. Note, that the sum of $n$ identically distributed Bernoulli random variables with parameter $k$ is given by a binomial distribution with parameters $n$ and $k$ i.e. $B(n,k)$. Therefore, the following statement (1) is true for the sum of error in ping measurements, $S_{n_{p}}$. Also, the mean and variance of $S_{n_{p}}$ is $n_{p}q$ and $n_{p}q(1-q)$. $$S_{n_{p}} = \sum_{l = 1}^L \left(\sum_{j =1}^{l_{p}}|\hat{y}_j - {y}_l|\right) \sim B(n_{p},q)$$
Topological Observability
-------------------------
A brief discussion on conditions to ensure topological observability is presented here. Observability, as per control theory, is a measure of how well the internal states of a system can be inferred from the knowledge of the outputs. In similar terms, topological observability is defined as a measure of how well the system topology can be inferred (identified) from the knowledge of measured flows. The minimum number of measurements that ensure topological identifiability, during normal operation (without outages), with error-free measurements is stated in [@sevlian2015distribution]. The condition claims that the measurement set should be such that each possible spanning tree of the graph differs at least by one flow measurement to make the respective tree identifiable. This leads to the requirement of one flow measurement along each fundamental cycle [@sevlian2015distribution].
Following a similar line of thoughts, during an outage, the measurement set should be able to distinguish not only all spanning trees but also the sub-trees of the graph. Recall that the isolation of load sections results in different outaged topologies/sub-trees. Smart meter probing provides an obvious and economical approach to detect isolated load sections of the distribution system. To ensure each sub-tree is identifiable, at least one smart meter ping is required from each load section in addition to the flow measurements along the fundamental cycles. But due to the error in smart meter ping measurements and the associated cost of pinging a smart meter [[@tram2008technical]]{}, in this paper, we assume 10% smart meter ping measurements from each load section (that can be erroneous) and one flow measurement per fundamental cycle are available.
*Practical Considerations regarding Smart Meter Ping Measurements for Topology Estimation:* The capability to ping smart meters using the existing AMI has been used in the existing literature on outage management systems (OMS) for fault location and isolation functions; see references [@jamali2017fault; @majidi2014fault; @itron]. In fact, a modern OMS includes several AMI functions including outage notification (via smart meter last gasp measurements), outage confirmation (via smart meter pings), system restoration (AMI reports restoration), and restoration verification (by checking the status using smart meter ping). Therefore, under the current conditions, it is realistic to ping 10% smart meters for topology estimation.
- [*Asynchronous Measurements:*]{} One critical aspect when including smart meter pings in topology estimation problem is their time alignment with other measurements and pseudo-measurements. Note that asynchronous smart meter ping measurements may theoretically affect the accuracy of the proposed topology estimation algorithm. However, practically that is not the case. Since smart meter pings are operator initiated, operators have the autonomy to ping the meters at the convenient time steps to intentionally synchronize the smart meter pings with the other continuous measurements. Furthermore, the proposed approach performs well even with the errors in ping measurements; these errors could represent a few asynchronous smart meter pings.
- [*Bernoulli Distribution for Ping Measurements:*]{} A Bernoulli distribution is a discrete probability distribution of a random variable which takes the value $1$ with probability $q$ and the value $0$ with probability $(1-q)$. Basically, it models any single experiment with a yes-no question that results in the Boolean-valued outcome ($0/1$). When modeling error in ping measurement i.e., $e(\hat{y}_j) = |(\hat{y}_j - {y}_l)|$ as Bernoulli distribution, we are asking the question whether the measured and true status of the smart meters are the same. The outcome is Boolean-valued ($0/1$) that depends upon the error in measuring or reporting smart meter pings.
*Discussion on problems related to bad data:* As any LAV estimator, the proposed algorithm is susceptible to bad data errors. If enough redundant real-time measurements are available, related methods from transmission systems literature can be employed for bad data detection and elimination prior to implementing topology estimation algorithm [@singh2005topology]. However, recognizing that the distribution systems have relatively fewer measurements compared to the transmission networks, the topology estimation algorithm needs to be designed keeping in mind the scarcity of measurements in the distribution systems. If there are fewer measurements, a bad data detection and elimination problem is extremely difficult to solve. This is because eliminating even a single measurement (due to bad data error) can render the topology unobservable (see discussion regarding topological observability in Section II.E). Specialized algorithms are required for bad data detection and elimination in distribution systems with measurement scarcity. This, although an interesting direction for future research, is outside the scope of this work.
Problem formulation
===================
This section describes the problem formulation for distribution system topology estimation problem. First, we describe a maximum likelihood estimator (MLE) for topology estimation. Next, the MLE problem is reformulated into a computationally tractable optimization model.
Description of Likelihood Function
----------------------------------
Let, there be $|N|$ possible topologies in $\mathcal{G}$ and $\Delta_k$ be the set of all switch status variables corresponding to $k^{th}$ topology. Also let, the measurement set, $\mathcal{M}$ = $\{\mathcal{\hat{P}},\mathcal{\hat{Q}}, \hat{p},\hat{q},\hat{y}\}$, where, $\mathcal{\hat{P}}$ and $\mathcal{\hat{Q}}$, be the set of all active and reactive power flow measurements; $\hat{p}$ and $\hat{q}$, be the set of all active and reactive load demand measurements; and $\hat{y}$ be the set of all smart meter ping measurements. Then, the likelihood of observing a given topology, $\Delta_k$, based on the measurement set, $\mathcal{M}$, is defined as $\mathcal{L}(\Delta_k|\mathcal{M})$, for all $k\in\{1 ... |N|\}$. The aim is to estimate the most likely topology, $\Delta_k$, using the erroneous measurement set $\mathcal{M}$ by maximizing the likelihood function $\mathcal{L}(\Delta_k|\mathcal{M})$.
The optimization model requires an analytical expression for $\mathcal{L}(\Delta_k|\mathcal{M})$. It is difficult to analytically characterize the probabilistic model for categorical variables, i.e. $\Delta_k$. Note that each topology will induce a different set of system variables, $\mathcal{X}_{\Delta_k}$. Therefore, a new likelihood function, $\mathcal{L}(X_{\Delta_k}|\mathcal{M})$, can be defined that measures the likely system variables from the given measurement set and can be obtained using the error models for different measurement variables in $X_{\Delta_k}$. The problem objective is to obtain the most likely system variables by maximizing the expression for $\mathcal{L}(X_{\Delta_k}|\mathcal{M})$. The resulting topology induced by the most likely system’s variables is the most likely operational topology.
The variables set, $X_{\Delta_k}$, includes both continuous ($\mathcal{\hat{P}},\mathcal{\hat{Q}},\hat{p},\hat{q}$) and discrete variables ($\hat{y}$), thus making the joint characterization of the likelihood function difficult. One approach is to maximize the likelihood of both continuous and discrete variables is by defining a multi-objective optimization problem that maximizes a weighted sum of the respective likelihood functions. Unfortunately, tuning the weight parameters for the continuous and categorical variables is difficult to generalize. In the proposed approach, we formulate a single objective optimization problem where the estimation problem maximizes the likelihood of observing only continuous random variables. The probabilistic representation of discrete random variables is modeled in constraint formulation. The objective is to estimate most likely power flow and load demand variables while constraining the probability of observing discrete variables to lie within a pre-specified interval based on the known error distribution.
Objective Function Formulation
------------------------------
The MLE problem for Gaussian random variables is equivalent to the [*[method of least squares]{}*]{}. Therefore, the objective of maximizing the likelihood function for continuous Gaussian random variables leads to the problem of minimizing the squared errors between system measurements and respective variables weighted according to measurement variances as defined in (\[eq:nlnobj\]). The topology that has loads and flows closest to the measured values is the correct operational topology. $$\label{eq:nlnobj}
\small
\text{Min}. \displaystyle \sum_{\phi \in \{a,b,c\}}\Bigg({\displaystyle \sum_{j \in \mathcal{I}} \bigg(\frac{(\hat{s}_{Lj}^\phi-s_{Lj}^\phi)}{(\sigma_{s_{Lj}^\phi})}\bigg)^2+\displaystyle \sum_{ij \in \mathcal{B}} \bigg(\frac{(\hat{S}_{ij}^\phi-S_{ij}^\phi)}{(\sigma_{S_{ij}^\phi})}\bigg)^2\Bigg)}$$ where, $\mathcal{I}$ is the set of pseudo load measurements and $\mathcal{B}$ is the set of line flow measurements. Note that load and flow measurements are per-phase complex quantities; $\hat{s}_{Lj}^\phi$, $s_{Lj}^\phi$, $\hat{S}_{ij}^\phi$, $S_{ij}^\phi$ are vectors of corresponding active and reactive power components (see Nomenclature). Note that the least-squares problem leads to a nonlinear objective function. When coupled with binary variables (in constraints), this results in an MINLP problem that is difficult to scale. Here, we take a cue from Least Absolute Value state estimator that has been proved to be accurate for state estimation in past [@singh1994weighted]. Following this, we define a linearized objective function in (\[eq:obj\]) that minimizes the weighted absolute errors instead of squared errors. $$\label{eq:obj}
\small
\text{Min}. \displaystyle \sum_{\phi \in \{a,b,c\}}\Bigg({\displaystyle \sum_{j \in \mathcal{I}} \left|\frac{(\hat{s}_{Lj}^\phi-s_{Lj}^\phi)}{(\sigma_{s_{Lj}^\phi})}\right|+\displaystyle \sum_{ij \in \mathcal{B}} \left|\frac{(\hat{S}_{ij}^\phi-S_{ij}^\phi)}{(\sigma_{S_{ij}^\phi})}\right|\Bigg)}$$
The absolute value in the objective function is still nonlinear. However, this can be easily linearized by introducing new variables (vectors) $a$ and $b$ such that $\mid s_{Lj}^\phi-\hat{s}_{Lj}^\phi\mid=a(j)$ and $\mid S_{ij}^\phi-\hat{S}_{ij}^\phi)\mid=b(i)$ and by introducing the following additional inequality constraints in the problem formulation:
$$\begin{aligned}
\label{eq:abs}
s_{Lj}^\phi-\hat{s}_{Lj}^\phi\leq a(j), \hspace{0.3cm} &\text{and}& \hspace{0.3cm}
-(s_{Lj}^\phi-\hat{s}_{Lj}^\phi)\leq a(j)\\
S_{ij}^\phi-\hat{S}_{ij}^\phi \leq b(i), \hspace{0.3cm} &\text{and}& \hspace{0.3cm}
-(S_{ij}^\phi-\hat{S}_{ij}^\phi) \leq b(i)\end{aligned}$$
Constraint Formulation
----------------------
We identify three types of constraints: 1) power flow, 2) connectivity, and 3) error bounds on ping measurements.
### Power flow Constraints
A three-phase linearized AC power flow model for the unbalanced distribution system, proposed in [@low2014convex], is used. The linearized three-phase formulation is sufficiently accurate for topology estimation [@sevlian2015distribution]. Note that a linearized model ignores power losses. Therefore, the proposed approach can not distinguish between two topologies that only differ by losses in the flow measurements; however, this rarely happens for a practical distribution system [@sevlian2015distribution]. $$\begin{aligned}
v_j = v_i - (S_{ij}z_{ij}^H+z_{ij}S_{ij}^H) + z_{ij}l_{ij} z_{ij}^H \\
\text{diag}(S_{ij} - z_{ij}l_{ij}) = \sum_{k:j \rightarrow k}{\text{diag}(S_{jk})} + s_{L,j} \\
%[vi S_{ij} ; S_{ij}^H l_{ij}] = [V_i I_{ij}][V_i I_{ij}]^H
\left[
\begin{array}{cc}
v_i & S_{ij} \\
S_{ij}^H & l_{ij} \\
\end{array}
\right] = \left[
\begin{array}{c}
V_i\\
I_{ij}\\
\end{array}
\right]
\left[
\begin{array}{cc}
V_i\\
I_{ij}\\
\end{array}
\right]^H \\
\left[
\begin{array}{cc}
v_i & S_{ij} \\
S_{ij}^H & l_{ij} \\
\end{array}
\right] : - \text{Rank-1 PSD Matrix}\end{aligned}$$
First, a three-phase power flow formulation for a radial system based on branch flow relationship given in [@gan2014convex] is detailed. Let, there be directed graph $\mathcal{G} = (\mathcal{N}, \mathcal{E})$ where $\mathcal{N}$ denotes set of buses and $\mathcal{E}$ denotes set of lines. Each line connects ordered pair of buses $(i,j)$ between two adjacent nodes $i$ and $j$. Let, $\{a,b,c\}$ denotes the three phases of the system; $V_i := [V_i^{\phi}]_{\phi \in \{a,b,c\}}$ be the three phase voltage at node $i$; $I_{ij}^{\phi}$ be the $\phi$ phase current for line $(i,j)$ and define $I_{ij} := [I_{ij}^{\phi}]_{\phi \in \{a,b,c\}}$; and $z_{ij}$ be the phase impedance matrix. Then (6)-(8) represent nonlinear three-phase power flow equations.
Next, we describe the linear approximation for the three-phase branch flow equations obtained after approximating the nonlinear power flow model described in (6)-(9). The linear approximation assumes the branch power loss is relatively smaller as compared to the branch power flow [@gan2014convex]. Specifically, the impact of power loss on active and reactive power branch flow equations and on voltage drop equations is ignored. The linearized AC branch flow equations are shown in (10)-(11). Here, (10) corresponds to linearized active power flow, and (11) represents linearized reactive power flow equations. Note that we do not include voltage equations in topology estimation problem as they provide little inference in identifying the operational topology. This is because, in distribution systems, tight control of grid voltages results in approximately similar nodal voltage magnitudes for different feeder topologies [@sevlian2015distribution]. $$\label{eq:p1}
\small
\sum_{(i\rightarrow j) \in \mathpzc{E}} \textit{P}_{ij}^{\phi} = \ \textit{p}_{Lj}^{\phi} + \sum\limits_{\substack{(j\rightarrow c) \in \mathpzc{E}, i\neq c}}\textit{P}_{jc}^{\phi} \hspace{0.2 cm} \forall j\in \mathcal{V}$$ $$\label{eq:q1}
\small
\sum_{(i\rightarrow j) \in \mathpzc{E}} \textit{Q}_{ij}^{\phi} = \ \textit{q}_{Lj}^{\phi} + \sum\limits_{\substack{(j\rightarrow c) \in \mathpzc{E}, \\i\neq c}} \textit{Q}_{jc}^{\phi} \hspace{0.2 cm} \forall j\in \mathcal{V} \cap \mathcal{V}_c$$
The linearized power flow equations are only valid if the line and the corresponding loads are energized. To formulate topology-constrained power flow model, the branch flow equations are coupled with switch status, load section connectivity and capacitor connectivity variables. Note that $\delta_{ij}= 1$, if $(i\rightarrow j) \in \mathcal{E}$\\$\mathcal{E}_s$. Equations (\[eq:p\])-(\[eq:c\]) represent three-phase unbalanced linearized power flow equations coupled with switch status variable $\delta_{ij}$, load section connectivity variables $y_l$, where load $j$ belongs to load section $l \in L$ and capacitor connectivity variables $y_{Cj}^{\phi}$. Here (\[eq:p\])-(\[eq:c\]) define power flow constraints that must be satisfied for each energized line.
$$\label{eq:p}
\small
\sum_{(i\rightarrow j) \in \mathpzc{E}}\delta_{ij} \textit{P}_{ij}^{\phi} = y_l \ \textit{p}_{Lj}^{\phi} + \sum\limits_{\substack{(j\rightarrow c) \in \mathpzc{E}, i\neq c}} \delta_{jc} \textit{P}_{jc}^{\phi} \hspace{0.2 cm} \forall j\in \mathcal{V}$$
$$\label{eq:q}
\small
\sum_{(i\rightarrow j) \in \mathpzc{E}}\delta_{ij} \textit{Q}_{ij}^{\phi} = y_l \ \textit{q}_{Lj}^{\phi} + \sum\limits_{\substack{(j\rightarrow c) \in \mathpzc{E}, \\i\neq c}} \delta_{jc} \textit{Q}_{jc}^{\phi} \hspace{0.2 cm} \forall j\in \mathcal{V} \cap \mathcal{V}_c$$
For buses with capacitor banks, i.e. $j \in \mathcal{V}_c$, the reactive power flow equation need to be coupled with capacitor bank’s switching status as defined in (\[eq:c\]). $$\label{eq:c}
\small
\sum_{(i\rightarrow j) \in \mathpzc{E}}\delta_{ij} \textit{Q}_{ij}^{\phi} = y_l \ (\textit{q}_{Lj}^{\phi} - y_{Cj}^{\phi} \ \textit{q}_{Cj}^{\phi}) + \sum\limits_{\substack{(j\rightarrow c) \in \mathpzc{E},\\ i\neq c}} \delta_{jc} \textit{Q}_{jc}^{\phi} \hspace{0.0 cm} \forall j\in \mathcal{V}_c$$
Note that constraints (\[eq:p\])-(\[eq:c\]) involve product of variables and are linearized using big-M method.
### Connectivity Constraints
This section defines different connectivity constraints that must be satisfied to obtain a feasible radial operational topology.
- [*Cycle Constraints:*]{} The cycle constraints are to ensure that the grid operates in a radial topology with no cycles. Let, $\mathcal{C}$ be the set of all possible cycles in the grid. Then, (\[eq:cyc\]) is imposed on each cycle $k \in \mathcal{C}$ to ensure that all the loads in the network are supplied radially without forming loops. $$\label{eq:cyc}
\small
\sum_{ij\in \mathcal{C}(k)}\delta_{ij}\leq n_{sw}(k)-1; \hspace{0.5 cm} k=1....m$$ where, $n_{sw}(k)$ is the number of switches in cycle $k$.
- [*Load-switch Connectivity Constraints:*]{} These constraints ensure a feasible outage topology by establishing relationships between power flows in a load section and the statuses of all corresponding switches during outages.
- A load $s_{Lj}^\phi$ at bus $j$ in phase $\phi$ downstream from a single switch, $\delta_{ij}$, is connected to the power supply if and only if the corresponding switch is closed as described in (\[eq:bin\]). Note that for both single-phase and three-phase loads belonging to load section $l \in L$, we specify a single binary variable $y_l$ to represent the load connectivity status. $$\label{eq:bin}
\small
y_l=\delta_{ij}$$
- A load $s_{Lj}^\phi$ at bus $j$ in phase $\phi$ that can be supplied by a total of $n$ switches, $\{\delta_{ij}$....$\delta_{nj}\}$, due to open-loop distribution system configuration, is connected to the power supply if any one of the switches is closed. This constraint is defined using the set of equations in (\[eq:loop\]). $$\label{eq:loop}
\small
y_l \leq \sum_{i = 1}^n\delta_{ij} \hspace{0.2cm} \text{ and } \hspace{0.2cm} y_l \geq \delta_{ij} \hspace{0.2cm} \forall{i} \in \{1...n\}$$
- [*Ping Variable Constraints:*]{} If a smart meter responds to a ping request, the corresponding load cannot be in outage i.e., if $\hat{y}_j=1$, then $y_l=1$ where, load at bus $j$ is connected to load section $l\in L$ as defined in (\[eq:smart\]). $$\label{eq:smart}
\small
y_l \geq \hat{y}_j$$
- [*Switch-flow Constraints:*]{} If a switch $\delta_{ij}$ is open, then power flow, $\textit{S}_{ij}^\phi$, through the switch must be equal to zero. Otherwise the flow will be unconstrained as defined in (\[eq:swf\]). $$\label{eq:swf}
\small
-M\delta_{ij} \leq \textit{S}_{ij}^\phi \leq M\delta_{ij}; \hspace{0.5 cm} M >> 0$$ where, $M$ is a large positive number.
![Comparisons of CDF sum of Bernoulli variables and its Gaussian approximation for 2% and 5% error in ping measurements.[]{data-label="fig:2"}](images/fig_2-eps-converted-to.pdf){width="46.00000%"}
### Error Bounds in Smart Meter Ping Measurements
The objective function defined in (3) does not include the error in discrete smart meter ping measurements. As described before, unlike continuous variables, we model the errors in discrete measurements as constraints. The idea is to ensure that the estimated load section variables appropriately characterize the possible errors in smart meter pings.
As detailed in Section II.D, for a sufficiently large number of ping measurements, the sum of absolute errors in ping measurements, $ S_{n_{p}} $, can be modeled as Binomial distribution, see (1), with mean $ \mu_e = n_{p}q$ and variance $\sigma_e = \sqrt{n_{p}q(1-q)}$. Note that $S_{n_{p}}$ can be approximation as a Gaussian distribution even for modest sample size. We demonstrate this observation using Fig. \[fig:2\] where the PDF for $S_{n_{p}}$ for the multi-feeder test system having a total of $44$ load sections is obtained for 2% ($q = 0.02$) and 5% ($q = 0.05$) errors in ping measurements. The sample distribution is compared with a Gaussian distribution obtained using same mean and variance parameters ($\mu_e = n_{p}q$ and $\sigma_e = \sqrt{n_{p}q(1-q)}$).
Next, we derive constraints that ensure a 99.99% confidence in estimating $S_{n_{p}}$ for a given measurement set. For Gaussian random variables, a 99.99% confidence-level corresponds to 5$\sigma$ spread around the sample mean. Therefore, the bounds on $S_{n_p}$ are defined in (\[eq:ping\]) which imply that the sum of ping errors are bounded by $5\sigma$ spread of the associated distribution. $$\label{eq:ping}
\small
min(0,\mu_e-5\sigma_e) \leq S_{n_{p}} \leq \mu_e+5\sigma_e$$
The resulting optimization problem is an MILP with the objective function defined in (\[eq:obj\]) subject to (\[eq:abs\]), (5), (\[eq:p\])-(\[eq:ping\]).
Note that here we assume that a smart meter ping can respond inaccurately during both conditions when connected as well as when outaged. However, when smart meter is outaged it will not respond and $\hat{y}_j$ will be always 0, therefore, implying that ping errors have different distribution when $y_l$ is 0 or 1. We argue that a Bernoulli distribution is a valid assumption even with the case that outraged smart meter cannot respond erroneously for the following two reasons: 1) in a realistic outage scenario, there will be a smaller fraction of smart meters that will be disconnected and hence the distribution for $S_{n_{p}}$ will be dominated by the error in pings from connected smart meters; 2) even when actual sum of ping errors is lower as disconnected smart meters pings are never erroneous, the constraint formulation using $S_{n_{p}}$ for topology estimation given in (20) does not remove the correct topology from the search space of the proposed MILP algorithm. This can be observed by calculating the new constraint for the sum of error in ping measurement due to only those smart meters that are connected and comparing that to (20).
*Discussion on Accuracy of Secondary Feeder Models:* The proposed framework requires an accurate planning model including secondary feeder models particularly phase association for secondary transformer and customer loads for the underlying distribution system. For a distribution system, the primary feeder model (including connectivity and equipment models) are reasonably accurate. However, the secondary feeder models are known to be inaccurate. The secondary model inaccuracies include phase assignment errors for secondary loads, incorrect transformer phase connections, and incorrect triplex/secondary line codes, etc. Driven by the need to leverage secondary feeder data, lately, the secondary feeder model estimation problem has received significant attention. Several data-driven and model-based methods have been proposed to identify the accurate phase assignment for the customer loads, accurate transformer connections, and secondary line parameters [@arya2011phase; @short2012advanced; @8586483; @park2018exact]. Any of these methods can be used to estimate or improve secondary feeder models for the underlying distribution system. Afterward, the proposed approach can be used to model the topology estimation problem. We would like to emphasize that solving model estimation problem is outside of the scope of this work. Usually, model and parameter estimation algorithms are implemented at an earlier stage to obtain an accurate planning model for the underlying system. Topology estimation and state estimation problems are operational problems that are implemented on a known and reasonably accurate system model.
{width="45.00000%"}
\[fig:3\]
Results and Discussion
======================
topology estimation problem is formulated as an MILP which can be efficiently solved by using off-the-shelf solvers. We have used MATLAB R2018a to build the MILP model that is solved using inbuilt $intlinprog$ solver. The simulation is carried out on a PC with 3.4 GHz CPU and 16 GB RAM. In this study, modified IEEE 123-bus feeder [@ieee123] and multi-feeder test system consisting of four taxonomy feeder R3-12.47-2 [@multi_fed] are used to validate the proposed algorithm.
Metrics
-------
The accuracy of the proposed approach is thoroughly tested for different percentages of measurement errors during both normal and outage conditions. The following three metrics are used to quantify the accuracy of the proposed topology estimation algorithm.
- [[*Missed Detection Rate (MDR)*]{}: Out of $N$ possible topologies, if $N_{nc}$ is the number of topologies that are incorrectly estimated, then $\%MDR=\left(\frac{N_{nc}}{N}\times 100 \right)$. The incorrect estimation of a single switch status renders the given topology incorrectly identified.]{}
- [[*Mean Missed Switches (MMS)*]{}: MMS measures the performance of the algorithm in correctly estimating switch statuses. Out of $N$ possible topologies, if $S_i$ switches are identified incorrectly then $\%MMS=\left(\frac{S_i}{S\times N} \times 100 \right)$, where, $S$ is the total number of switches.]{}
- [[*Mean Missed Outages (MMO)*]{}: MMO measures the performance of the algorithm in correctly detecting all outaged sections. Out of $N$ possible topologies, if $L_i$ sections are incorrectly estimated then $\%MMO=\left(\frac{L_i}{L_o\times N} \times 100 \right)$, where, $L_o$ is the total number of outaged sections.]{}
Simulation Steps
----------------
Given a large number of possible operational topologies, numerous simulations are performed to realistically measure the performance of the algorithm. The simulation steps are detailed here. First, we randomly generate a radial topology for the given distribution test system that may or may not have outages. For the given topology, we run power flow using OpenDSS to obtain the measurement set. Next, we generate noisy measurements by adding Gaussian errors with zero mean and pre-specified variances to the flow and load measurements and Bernoulli errors to the smart meter ping measurements with a pre-specified error probability. For the simulation purpose, we study the performance of the proposed approach considering 1%, 10% & 20% error in load measurements and 0%, 2%, & 5% error in smart meter ping measurement for different cases. We solve the proposed MILP model to estimate the operational topology using the erroneous measurements. The performance metrics are calculated. The process is repeated several times until statistically significant results are obtained. On an average, it takes less than 30 sec to estimate the operational topology for the selected 1096-bus three-phase test system and less than 2 sec to estimate the operational topology for the selected IEEE 123-bus test case.
![Influence of R/X parameters in topology identification. Even for increased R/X ratio, the correct topology and set of incorrect topology are clearly distinguishable.[]{data-label="fig:loss"}](images/fig_4-eps-converted-to.pdf){width="50.00000%"}
Modified IEEE 123-bus Feeder
----------------------------
IEEE-123 bus system is three-phase unbalanced distribution system model supplying multiple single and three-phase loads. The planning model for the original IEEE 123-bus system is modified by adding five new sectionalizing switches to create several possible operating topologies. Among the switches, $s_5$ and $s_{10}$ are normally open and rest are normally closed (See Fig. \[fig:3\]). This system has 3 cycles and 20 possible normal radial operational topologies. To ensure topological observability, one power flow measurement unit is deployed in each fundamental cycle [@sevlian2015distribution]. Measurement locations in a cycle are randomly selected and shown in Fig. \[fig:3\].
{width="99.00000%"}
### Topology Estimation During Normal Operation
During the normal operation, all topologies are correctly detected for the pre-specified error probabilities, i.e., for load measurement errors of 1%, 10% and 20% and ping measurements errors of 2% and 5%. Note that the formulation discussed above approximates the power flow equations in the topology constrained power flow formulation. Therefore, we study the performance of the proposed approach for varying R/X ratio for distribution lines to evaluate the impact of power losses on the accuracy of the proposed topology estimation algorithm. To simulate these cases, we increase the R/X parameter for the distribution lines and observe whether the correct operational topology is identified or not. Also, we study how the change in R/X value affects the objective function value used in the topology estimation problem. Note that the correct topology should result in the minimum value for the defined objective function that measures the difference between the measured and calculated power flow and load demand variables. Also, note that on increasing the R/X ratio, the losses in the network increases as shown in Table \[table:loss\]; the percentage losses are calculated as the fraction of the total substation power demand.
The proposed topology estimation algorithm is tested for all simulated test cases with varying R/X ratio shown in Table I. It is observed that all topologies are still correctly identified on increasing the losses in the distribution system. We further elaborate on the observations using three randomly selected topologies. In this discussion, for a specific unknown topology, we calculate and plot the objective function value (i.e. minimization of errors, equation (\[eq:obj\])) for all possible topologies. Fig. \[fig:loss\] shows the objective function values all possible topologies for the three selected unknown topologies. Note that the correct topology leads to a minimum objective function value. The topology estimation is, therefore, able to identify the correct topology if the objective function value for the correct topology does not overlap with the objective function value for any of the incorrect topologies. As can be seen in Fig. 4, the objective function value for all incorrect topologies are much larger than for the correct topology. Therefore, we can identify the correct topology.
\[table:loss\]
----------------- ------ ---------------- ----------------- -------------- --------------
R/X ratio $\to$ Base 1.1$\times$R/X 1.15$\times$R/X 2$\times$R/X 3$\times$R/X
% Loss$\to$ 4.25 4.92 5.08 7.47 11.37
----------------- ------ ---------------- ----------------- -------------- --------------
: Percentage loss for different R/X ratio. The average R/X ratio for base case is 0.4343.
Next, to study the effect of increasing R/X ratio, we again plot the objective function values for all topologies for the three selected cases. The objective function values for different R/X ratios are shown as parallel plots in Fig. \[fig:loss\]. It is observed that even for increased R/X ratio, the objective function value for the correct topology is less than the values for incorrect topologies making it easily distinguishable from the rest of the topologies. Even if R/X ratio is increased by 200%, the topologies are still distinguishable as the objective function value for correct and incorrect topologies do not overlap. Although the objective function values for correct and incorrect topologies move close to each other on increasing the losses, there is still a big margin and losses do not make the correct topology indistinguishable from incorrect topologies.
\[table:out\]
--- ------- ------- ------- ------- ------- ------- ------- ------- -------
1% 10% 20% 1% 10% 20% 1% 10% 20%
0 0 0 0 0 0 0 - - -
2 0.047 0.143 0.238 0.004 0.013 0.022 0.015 0.015 0.015
5 0.095 0.190 0.333 0.008 0.017 0.03 0.015 0.015 0.015
--- ------- ------- ------- ------- ------- ------- ------- ------- -------
: %MDR, %MMS and %MMO for tested topologies with outages IEEE for 123-bus test system
### Topology Estimation During Outage Condition
For each topology, one fault is randomly simulated and the algorithm is repeatedly tested 10 times for each scenario. Table \[table:out\] shows the performance of the approach in the presence of errors in load and flow measurements. For 1% error in load measurements and no error in ping measurements, %MDR is 0 indicating that there are no misdetected topologies and all switches and outages are accurately identified. While for the worst case of 20% error in load measurements and 5% error in smart meter measurements, %MDR is 0.333 implying that out of 2100 tested scenarios, 7 topologies are misdetected. This amounts to 0.03% misdetection of switches (%MMS) and 0.015% of missed outages (%MMO).
One particular scenario for misdetection is detailed here. Let us consider the case where the operational topology (sub-tree) has switches $s_2$, $s_6$, and $s_{10}$ open with other switches closed. However, the optimization detected the correct operation topology having switch $s_5$ open instead of $s_2$. As observed, the algorithm detected $s_2$ as closed and supplying the load section downstream when it shouldn’t and meanwhile switch $s_5$ is detected as open while it’s actual status is closed and is supplying the section of load between $s_6$ and $s_5$. To further elaborate the reason for this misdetection, we observed the sectional loads that are supplied downstream from $s_2$ and between switches $s_5$ and $s_6$. It is observed that both sections are supplying for the approximately equal loads and therefore, the meter on line 13-18 (which supplies for both load sections) is unable to distinguish the correct section as it reads almost the same power flow for both topologies. Therefore, for the cases when flow meters cannot distinguish the load sections (due to similar load demands), the proposed approach detects an incorrect operational topology.
Multi-Feeder Test System
------------------------
The proposed topology estimation algorithm is also demonstrated using a three-phase unbalanced 1069-bus multi-feeder test system. Four taxonomy feeders R3-12.47-2 [@multi_fed] are replicated to obtain the four-feeder 1069-bus distribution system connected using seven normally open tie switches (See Fig. \[fig:4\]a). Detailed model of one of the four feeders is shown in Fig. \[fig:4\]b. Each feeder has 40 three-phase sectionalizing switches (normally-closed switches). Seven tie-switches and 40 sectionalizing switches lead to a large system with the number of operational topologies in the order of millions. Each feeder has three single-phase capacitors and one three-phase capacitor at locations shown in Fig. \[fig:4\]. The operational statuses of capacitor banks are unknown when solving the topology estimation problem. The measurement set is described next. Topological observability requires one flow measurement per fundamental simple cycle. A randomly selected measurement placement satisfying this criterion is generated as shown in Fig. \[fig:4\] and is used for testing. A total of 10% of smart meters are pinged from each load section. This measurement placement is tested with 1%, 10% and 20% errors in load measurements and for 0%, 2% and 5% errors in ping measurements.
{width="45.00000%"}
\[fig:5\]
### Topology Estimation During Normal Operation
In this section, the proposed algorithm is tested for its accuracy in detecting operational topology during normal operation (i.e., without any outages). For an exhaustive validation of the algorithm, a total of 3000 test scenarios are simulated by randomly sampling an operational topology as detailed in Section IV.A. To ensure that test set-up represents statistically significant test scenarios, a plot for the number of topologies tested vs. MDR for a scenario with worst-case error (20% error in load measurement and 5% error in ping measurements) is plotted (see Fig. \[fig:5\]). Note that MDR is almost constant for the number of tested scenarios greater than 2500 which justifies our choice of testing for 3000 scenarios.
\[table:2\]
--- ------ ------ -------- -------- ------ --------
%MDR %MMS %MDR %MMS %MDR %MMS
0 0 0 0 0 0.5 0.0182
2 0 0 0.0311 0.0011 1.50 0.0737
5 0 0 0.0667 0.0024 1.53 0.0752
--- ------ ------ -------- -------- ------ --------
: %MDR and %MMS for tested topologies without outages for 1096-bus test system
From results in Table \[table:2\], we see that as expected there is no missed detection when there is only 1% error in load measurements and no errors in smart meters. Note that the misdetection is still zero when the error in smart meters is increased to 10% while keeping the error in load measurement to 1%. The MDR increases on increasing the error in load measurements while keeping the errors in ping measurements as zero. For 10% error in load measurements, all the topologies are still distinguishable but when the errors increased to 20%, few topologies are incorrectly detected. On further investigation, it is observed that at most one switch pair is incorrectly identified for all topologies that are misdetected. Thus, MMS for all cases is significantly low as compared to MDR (see Table \[table:2\]).
Next, the error in ping measurements is also increased to 5% and 10%. It is observed from Table \[table:2\] that the %MDR is slightly increased on increasing the errors in ping measurement. The %MMS still remains small indicating that misdetection is largely due to the incorrect status of a small number of switch pairs. Based on small %MMS, we can conclude that the proposed algorithm is highly accurate in estimating the switch statuses even with high-levels of measurement errors.
### Topology Estimation During Outage Condition
We randomly select a possible normal topology and simulate any random number of faults between 1 to 3 at random locations; 3000 such random outaged topologies are simulated. For each outaged topology, the topology estimation algorithm is solved for different percentages of errors in load and smart meter measurements. The results are tabulated in Table \[table:3\].
As expected, for 1% error in load measurements and no error in smart meter measurements, all outaged topologies are detected. Unlike normal operation, the misdetection is no longer zero when the error in smart meters is increased while the error in load measurements is still at 1%. Next, the error in load measurements is increased while keeping smart meters error-free. For 10% error in load measurements, all the topologies are still distinguishable but when the error is increased to 20%, few topologies are incorrectly detected (see Table \[table:3\]).
\[table:3\]
--- ------ ------ ------ ------- ------ ------ ------- ------ ------
1% 10% 20% 1% 10% 20% 1% 10% 20%
0 0 0 1.03 0 0 0.04 0 0 0.07
2 0.53 2.80 7.10 0.019 0.11 0.35 0.041 0.32 0.73
5 0.91 4.35 7.23 0.041 0.17 0.37 0.067 0.40 0.78
--- ------ ------ ------ ------- ------ ------ ------- ------ ------
: %MDR, %MMS and %MMO for tested topologies with outages for 1096-bus test system
It can be observed from Table \[table:3\] that the %MDR during outages increases significantly with the increase in ping measurement errors. Note that MDR is an extremely conservative metric. For any topology, even if the single switch is incorrectly identified it is counted as a misdetected topology in MDR calculation. Therefore, we elaborate the strength of the algorithm in its capability to correctly estimate outaged sections and switch statuses using the other two metrics: MMS and MMO.
From Table \[table:3\], it can be noted that the %MMO is significantly low even for worst-case measurement errors in loads and pings implying that the algorithm can correctly estimate most of the outaged sections. Also, MMS is low indicating most topologies are misdetected by very few numbers of incorrect switch statuses. To further elaborate the result, we observe the number of switch pairs that are incorrectly estimated for each misdetected topology. For 5% error in smart meter measurement, out of the total misdetections, contribution factor ($C_f$) of misdetected switch pairs for different errors in load measurements is shown in Fig. \[fig:6\]. Note that $C_f$ is the fraction of misdetection contributed by different switch pairs to the total misdetection. It can be observed that most of the misdetections corresponded to only one pair of switches being incorrectly estimated. The proposed algorithm is, therefore, able to correctly estimate most of the switches and outaged sections even with high-levels of measurement errors.
![Contribution to %MDR from different number of switch pairs.[]{data-label="fig:6"}](images/fig_7-eps-converted-to.pdf){width="45.00000%"}
![Sectional load misdetection: 10% load measurement error.[]{data-label="fig:7"}](images/fig_8-eps-converted-to.pdf){width="45.00000%"}
We further explore the reason for misdetection during an outage by identifying the load sections that are most frequently misdetected. We observe that the errors in smart meter measurements make some of the healthy and outaged sections with similar load demand indistinguishable. This is because the flow measurement in the upstream line is nearly the same for an outage in either load. This observation is elaborated using the case with 10% error in load measurements and 5% error in ping measurement. A histogram for the sectional loads with actual and erroneous data is plotted in Fig. \[fig:7\]. The histogram represents the frequency of observing a load section of given kW demand. Next, we calculate the percentage of times each sectional load is misdetected (bar plot in Fig. \[fig:7\]). From Fig. \[fig:7\] it can be observed that the most frequently misdetected sections are indeed the ones that are most frequently observed; the bar plot overlaps with the spikes of a histogram for sectional loads. Note that there are a few sections that are never misdetected even though they are frequently observed. This is because they are never supplied by the same upstream flow meter.
Conclusion
==========
The existing literature on topology estimation cannot simultaneously estimate the distribution grid’s operation topology and outage sections in a computationally tractable manner. This paper presents a generalized algorithm to estimate the operational topology during both normal and outage conditions. The problem is formulated as an MILP to minimize the weighted error between the measurements and the system variables subject to topology constrained three-phase linearized power flow equations. The method relies on the pseudo load measurements, one power flow measurement on each cycle, and at least one smart meter measurement from each load section. The algorithm is thoroughly tested for a large-scale 1069-bus multi-feeder three-phase unbalanced test system for different percentages of measurement errors. For the test system, an operational topology is obtained on an average within 30 sec validating the applicability of the approach as a real-time topology processor. Furthermore, the results validate that the proposed approach is sufficiently accurate even with high percentages of measurement errors.
\[[{width="1in" height="1.25in"}]{}\] [**Anandini Gandluru**]{} (S’16) received the B.Tech. degree in Electrical Engineering from G. Narayanamma Institute of Technology and Science, Hyderabad, India in 2013, and the M.Tech. degree from the Department of Electrical Engineering, National Institute of Technology, Warangal, Telangana, India. She received the M.S. degree from the School of Electrical and Computer Science, Washington State University, Pullman, WA, USA. Her research interests include state estimation and distribution system topology estimation.
[**Shiva Poudel**]{} (S’15) received the B.E. degree from the Department of Electrical Engineering, Pulchowk Campus, Kathmandu, Nepal, in 2013, and the M.S. degree from the Electrical Engineering and Computer Science Department, South Dakota State University, Brookings, SD, USA, in 2016. He is now pursuing the Ph.D. degree in the School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA. In 2018 and 2019, he was a summer intern with Mitsubishi Electric Research Laboratories, Cambridge, MA, USA and Electric Power Research Institute, Palo Alto, CA, USA respectively. His current research interests include distribution system restoration, resilience assessment, and distributed algorithms.
[**Anamika Dubey**]{} (M’16) received the M.S.E and Ph.D. degrees in Electrical and Computer Engineering from the University of Texas at Austin in 2012 and 2015, respectively. Currently, she is an Assistant Professor in the School of Electrical Engineering and Computer Science at Washington State University, Pullman.
Her research focus is on the analysis, operation, and planning of the modern power distribution systems for enhanced service quality and grid resilience. At WSU, her lab focuses on developing new planning and operational tools for the current and future power distribution systems that help in effective integration of distributed energy resources and responsive loads.
[^1]: A. Gandluru, S. Poudel, and A. Dubey are with the School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA, 99164 e-mail: [email protected], [email protected], [email protected].
|
[**A. P. Isaev**]{}
.2cm
Bogoliubov Laboratory of Theoretical Physics, JINR\
141980 Dubna, Moscow Region, Russia
.4cm
0.2 cm
Center of Theoretical Physics[^1], Luminy\
13288 Marseille, France
.8cm
**Abstract**
Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For the Hecke and BMW algebras, the (anti)-symmetrizers have simple expressions in terms of the multiplicative shuffles. The (anti)-symmetrizers can be expressed in terms of the highest multiplicative 1-shuffles (for the Hecke and BMW algebras) and in terms of the highest additive 1-shuffles (for the Hecke algebras). The spectra and multiplicities of eigenvalues of the operators of the multiplication by the multiplicative and additive 1-shuffles are examined.
Braid shuffles
==============
In this section we collect some necessary information on shuffle elements in the braid group rings.
.2cm In the Artin presentation, the braid group $B_{M+1}$ is given by generators $\sigma_i$, $1\leq i\leq M$, and relations $$\begin{aligned}
\lb{ryb1}\sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{j}
&{\mathrm{if}}& |i-j|=1\ ,\\[.5em]
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}&{\mathrm{if}}&|i-j|>1\ .\label{braidg}\end{aligned}$$ The inductive limit $B_\infty =\displaystyle{\lim_{\longrightarrow}}\, B_M$ is defined by inclusions $B_M\rightarrow B_{M+1}$, $B_M\ni\sigma_i\mapsto \sigma_i\in B_{M+1}$, $i=1,\dots ,M-1$.
.2cm We denote $w^{\uparrow\ell}$, as in [@OP], the image of an element $w\in B_\infty$ under the endomorphism of $B_\infty$, sending $\sigma_i$ to $\sigma_{i+\ell}$, $i=1,2,\dots$ (we keep the same notation for the Hecke and BMW quotients of the braid group rings).
.2cm Braid shuffle elements $\Sha_{m,n}$ ($m,n\in {\mathbb Z}_{\geq 0}$) are analogues of the binomial coefficients. The shuffle elements belong to the group ring of $B_{m+n}$ (and thereby of $B_\infty$); they can be defined inductively by any of the recurrence relations (braid analogues of the Pascal rule) $$\begin{aligned}
\lb{shufrec}\Sha_{m,n}^{\phantom{\uparrow}}&=&\Sha_{m-1,n}^{\phantom{\uparrow}}+
\Sha_{m,n-1}^{\phantom{\uparrow}}\sigma_{m+n-1}\cdots\sigma_{n}\ ,\\[.5em]
\lb{shufrec2}\Sha_{m,n}^{\phantom{\uparrow}}&=&
\Sha_{m,n-1}^{\uparrow 1}+\Sha_{m-1,n}^{\uparrow 1}\sigma_{1}\cdots\sigma_{n}\ ,\end{aligned}$$ together with the boundary conditions $\Sha_{0,n}=1$ and $\Sha_{n,0}=1$ for any non-negative integer $n$.
.2cm Let $\Sigma_n$ be the lift [@Ma] of the symmetrizer $\sum_{g\in S_n}g$ from the symmetric group ring ${\mathbb Z}{\mathbb{S}}_n$ to ${\mathbb Z}B_n$. The element $\Sigma_n$ is the braid analogue of $n!$; it satisfies \_[m+n]{}\^=\_[n,m]{}\^ \_m\^\_n\^[m]{} .Using the automorphism $\mathfrak a$ and the anti-automorphism $\mathfrak b$, ${\mathfrak{b}}(xy)={\mathfrak{b}}(y){\mathfrak{b}}(x)$, of the braid group $B_{n+1}$, defined on the generators by a:\_i\_[n+1-i]{},b:\_i\_i ,and their composition, one obtains three more decompositions of $\Sigma$.
.2cm Higher shuffles (braid analogues of the trinomial [*etc.*]{} coefficients) appear in the further decompositions of the elements $\Sigma_n$, \_[m+n+k]{}\^={
[l]{}\_[n+k,m]{}\^ \_m\^\_[n+k]{}\^[m]{} =\_[n+k,m]{}\^\_[k,n]{}\^[m]{}\_m\^ \_n\^[m]{}\_[k]{}\^[m+n]{} ,\
\_[k,m+n]{}\^\_[m+n]{}\^\_[k]{}\^[m+n]{} =\_[k,m+n]{}\^\_[n,m]{}\^\_m\^ \_n\^[m]{}\_[k]{}\^[m+n]{} .
.Due to the existence of a (one-sided) order on the braid groups [@D], the braid group rings ${\mathbb Z}B_n$ have no zero divisors. Equating the two expressions for $\Sigma_{m+n+k}$ in (\[hsh\]) and simplifying, one finds \_[n+k,m]{}\^\_[k,n]{}\^[m]{}= \_[k,m+n]{}\^\_[n,m]{}\^ .A direct verification of (\[ctsh\]) is a good exercise. Any of the expressions in (\[ctsh\]) is the braid trinomial coefficient $\Sha_{k,n,m}$. The element $\Sigma_n$ is the shuffle $\Sha_{1,1,\dots ,1}$.
.2cm We shall later use the following identity \_[1,n-1]{}\^\_[1,n-2]{}\^…\_[1,n-k]{}\^= \_[k,n-k]{}\^\_[k]{}\^[n-k]{} ,which is verified by induction. For $k=1$ there is nothing to prove. The induction step uses (\[ctsh\]) and then (\[symsh\]):
[rcl]{} \_[k,n-k]{}\^\_[k]{}\^[n-k]{}\_[1,n-k-1]{}\^&=& \_[k,n-k]{}\^\_[1,n-k-1]{}\^\_[k]{}\^[n-k]{}\
&=&\_[k+1,n-k-1]{}\^\_[k,1]{}\^[n-k-1]{}\_[k]{}\^[n-k]{} =\_[k+1,n-k-1]{}\^\_[k+1]{}\^[n-k-1]{} .
.4cm Shuffle elements find numerous applications in the theories of free Lie algebras, polylogarithms and multiple zeta values, Hopf algebras, differential calculus on quantum groups, homology of quantum Lie algebras, braidings of tensor spaces [*etc.*]{} [@R], [@BB], [@N], [@Andr], [@Rosso2], [@Wor], [@IO], [@IOG], [@GO].
B-shuffle algebras
==================
In this section we recall the definition of the Nichols–Woronowicz algebras and construct, with the help of the baxterized elements, another family of graded associative algebras in the tensor spaces of local representations of the braid groups.
#### 1.
Let $V$ be a vector space over a field ${\mathfrak k}$. For an operator $X\in {\mathrm{End}}(V^{\otimes j})$ we denote by the same symbol the operator $X\otimes {\mathrm{Id}}^{\otimes l}\in {\mathrm{End}}(V^{\otimes (j+l)})$ for any $l\in {\mathbb Z}_{\geq 0}$; $X^{\uparrow l}$ denotes the operator ${\mathrm{Id}}^{\otimes l}\otimes X\in {\mathrm{End}}(V^{\otimes (j+l)})$.
.2cm Let $\{ {\cal{T}}_{m,n}\}_{_{m,n\in {\mathbb Z}_{\geq 0}}}$ be a collection of operators, ${\cal{T}}_{m,n}\in {\mathrm{End}}(V^{\otimes (m+n)})$, such that \_[n+k,m]{}\^ \_[k,n]{}\^[m]{}= \_[k,m+n]{}\^ \_[n,m]{}\^ m,n,k\_[0]{} .For tensors $u\in V^{\otimes m}$ and $v\in V^{\otimes n}$ let uv:=\_[n,m]{}(uv)V\^[(m+n)]{} .Due to (\[ctshr\]), the space $\bigoplus_j V^{\otimes j}$ with the composition law $\odot$ is an associative graded algebra. Assume, in addition, that \_[m,0]{}=\_[0,m]{}= m\_[0]{} (then $1\in {\mathfrak k}\equiv V^{\otimes 0}$ is the identity element of the algebra). By (\[ctshr\]) and (\[ico\]), the following collection $\{ {\cal{S}}_m\}_{_{m\in {\mathbb Z}_{\geq 0}}}$ of operators, ${\cal{S}}_m\in {\mathrm{End}}(V^{\otimes m})$, \_0=,\_1=,\_[m+n]{}\^=\_[n,m]{}\^ \_m\^ \_n\^[m]{} m,n\_[0]{} ,is well defined. The operation $\odot$ restricts on $\bigoplus_j {\mathrm{Im}}({\cal{S}}_j)$, the direct sum of images of the operators ${\cal{S}}_j$, making it an associative graded algebra as well.
.2cm Let $\hat{R}\in {\mathrm{End}}(V\otimes V)$ be a solution of the Yang-Baxter equation, that is, $\hat{R}\hat{R}^{\uparrow 1}\hat{R}=\hat{R}^{\uparrow 1}\hat{R}\hat{R}^{\uparrow 1}$. Denote by $\rho_{\hat{R}}$ the corresponding local representation of the braid groups $B_n$, $\rho_{\hat{R}}(\sigma_i):=\hat{R}^{\uparrow (i-1)}$. Then the collection ${\cal{T}}_{m,n}:=\rho_{\hat{R}}(\Sha_{m,n})$ obeys (\[ctshr\]) and (\[ico\]). The space $\bigoplus_j {\mathrm{Im}}\rho_{\hat{R}}(\Sigma_j)$ with the composition law $\odot$ is called Nichols–Woronowicz algebra.
#### 2.
The braid group rings admit a family of automorphisms $\sigma_i\mapsto t\,\sigma_i$, where $t\in {\mathfrak k}^*$ is an arbitrary parameter. The formal limits $\displaystyle{\lim_{t\rightarrow 0}}$ (the lowest power in $t$) and $\displaystyle{\lim_{t\rightarrow \infty}}$ (the highest power in $t$) of the elements $\Sigma_m$, $\Sha_{m,n}$ and the operation $\odot$ are well defined. For $t\rightarrow 0$ we obtain the usual tensor algebra, while for $t\rightarrow \infty$ the element $\Sigma_{n+1}$ becomes the lift of the longest element of the symmetric group ${\mathbb{S}}_{n+1}$ to $B_{n+1}$, |\_[n+1]{}=(\_1\_2…\_n)(\_1…\_[n-1]{}) …(\_1) .The shuffle elements, in the limit $\displaystyle{\lim_{t\rightarrow \infty}}$, become the elements $\bar{\Sha}_{m,n}$ which, in a representation in a vector space $V$, equip the tensor powers of $V$ with the standard braidings; the recurrency relations (\[shufrec\]) and (\[shufrec2\]) take the multiplicative form for $\bar{\Sha}_{m,n}$, |\_[m,n]{}=|\_[m,n-1]{}\_[m+n-1]{}…\_n , |\_[m,n]{}=|\_[m-1,n]{}\^[1]{}\_1…\_n .Explicitly, |\_[m,n]{}={
[l]{}(\_m\_[m+1]{}…\_[m+n-1]{}) (\_[m-1]{}\_[m]{}…\_[m+n-2]{})(\_1\_2…\_n ) ,\
(\_m\_[m-1]{}…\_1)(\_[m+1]{}\_m…\_2) (\_[m+n-1]{}\_[m+n-2]{}…\_n) .
.In addition to (\[muh1\]), the elements $\bar{\Sha}_{m,n}$ satisfy |\_[m,n]{}=\_m…\_1|\_[m,n-1]{}\^[1]{} , |\_[m,n]{}=\_m…\_[m+n-1]{}|\_[m-1,n]{} .
#### 3.
In this section we shall construct another collection ${\cal{T}}_{m,n}$ starting with the elements $\sigma_k(x,y)$, satisfying the Yang-Baxter equation with spectral parameters \_k(x\_[k+1]{},x\_[k+2]{})\_[k+1]{}(x\_k,x\_[k+2]{})\_k(x\_k,x\_[k+1]{}) =\_[k+1]{}(x\_k,x\_[k+1]{})\_k(x\_k, x\_[k+2]{})\_[k+1]{}(x\_[k+1]{},x\_[k+2]{})and the locality condition \_k(x\_[k]{},x\_[k+1]{})\_[l]{}(x\_l,x\_[l+1]{})= \_[l]{}(x\_l,x\_[l+1]{})\_k(x\_[k]{},x\_[k+1]{}) |k-l| > 1 .Here $x_k$ are variables (spectral parameters). Depending on the situation, the elements $\sigma_k(x,y)$ can live in certain quotients of the braid group rings or be realized as operators. We shall call $\sigma_k(x,y)$ baxterized elements (usually the term “baxterized” is applied when $\sigma(x,y)$ is a function of the solution $\sigma$ of the constant Yang-Baxter equation).
.2cm Let $\pi_k$ be the operator which permutes the variables $x_k$ and $x_{k+1}$, $$\pi_kf(\dots,x_k,x_{k+1},\dots)=f(\dots,x_{k+1},x_k,\dots)\pi_k\ .$$ Relations (\[sh1\]) and (\[sh2\]) acquire the braid form (\[ryb1\]) and (\[braidg\]) for the elements \_k:=\_k\_k(x\_k,x\_[k+1]{}) .The unitarity condition $\sigma_k(x_{k},x_{k+1})\sigma_k(x_{k+1},x_{k})=1$ (if imposed) for the baxterized elements takes the form $\underline{\sigma}_k^2 =1$ for the elements (\[sh3\]).
.2cm The operators $\pi_k$ obey the braid group relations; prepare the elements $\bar{\Sha}_{m,n}\{\pi\}$ and $\bar{\Sigma}_m\{\pi\}$ from $\pi$’s; the elements $\bar{\Sha}_{m,n}\{\underline{\sigma}\}$ and $\bar{\Sigma}_m\{\underline{\sigma}\}$ built from $\underline{\sigma}$’s can be written, after moving all $\pi$’s to the left, in the form |\_[m,n]{}{} = |\_[m,n]{}{}\_[m,n]{}(x\_1,…,x\_[m+n]{}),|\_m{} = |\_m{}\_m(x\_1,…,x\_m) ,where
[ccl]{}\_[m,n]{}(x\_1,…,x\_[m+n]{})&=& (\_m(x\_1,x\_[m+n]{})\_[m+1]{}(x\_2,x\_[m+n]{}) …\_[m+n-1]{}(x\_n,x\_[m+n]{}))\
&&(\_[m-1]{}(x\_1,x\_[m+n-1]{})\_[m]{}(x\_2,x\_[m+n-1]{}) …\_[m+n-2]{}(x\_n,x\_[m+n-1]{}))\
&…&(\_1(x\_1,x\_[n+1]{})\_2(x\_2,x\_[n+1]{}) …\_n(x\_n,x\_[n+1]{}))
and
[ccl]{}\_m(x\_1,…,x\_m)&=& (\_1(x\_[m-1]{},x\_m)\_2(x\_[m-2]{},x\_m)…\_[m-1]{}(x\_1,x\_m))\
&&(\_1(x\_[m-2]{},x\_[m-1]{})\_2(x\_[m-3]{},x\_[m-1]{}) …\_[m-2]{}(x\_1,x\_[m-1]{}))…(\_1(x\_1,x\_2)) .
The elements $\bar{\Sha}_{m,n}\{\pi\}$ and $\bar{\Sigma}_n\{\pi\}$ are invertible and obey the relations (\[symsh\]), (\[ctsh\]), (\[muh1\]) and (\[muh2\]); substituting (\[preti\]) into (\[symsh\]), (\[ctsh\]), (\[muh1\]) and (\[muh2\]), moving all $\pi$’s to the left and simplifying, we find relations for $\widetilde{\Sigma}$’s and $\widetilde{\Sha}$’s alone. The relations (\[muh1\]) and (\[muh2\]) take the form \_[m,n]{}(x\_1,…,x\_[m+n]{})={
[l]{} \_[m,n-1]{}(\_n) \_[m+n-1]{}(x\_n,x\_[m+n]{})\_[m+n-2]{}(x\_n,x\_[m+n-1]{})…\_n(x\_n,x\_[n+1]{}),\
\_[m-1,n]{}\^[1]{}(\_[n+1]{})\_1(x\_1,x\_[n+1]{}) \_2(x\_2,x\_[n+1]{})…\_n(x\_n,x\_[n+1]{}),\
\_m(x\_1,x\_[m+n]{})\_[m-1]{}(x\_1,x\_[m+n-1]{}) …\_1(x\_1,x\_[n+1]{})\_[m,n-1]{}\^[1]{}(\_1),\
\_m(x\_1,x\_[m+n]{})\_[m+1]{}(x\_2,x\_[m+n]{})…\_[m+n-1]{}(x\_n,x\_[m+n]{}) \_[m-1,n]{}(\_[m+n]{}}),
.where “$\hat{x}_j$” means that the argument $x_j$ is omitted. For a set $\overrightarrow{x}=\{ x_1,\dots,x_n\}$ of arguments, let $\overleftarrow{x}:=\{ x_n,\dots,x_1\}$ be the reversed set. The relation (\[symsh\]) becomes: \_[m+n]{}\^(,) =\_[n,m]{}\^(,) \_m\^() \_n\^[a]{}() ,where $\overrightarrow{x}=\{ x_1,\dots,x_m\}$ and $\overrightarrow{y}=\{ y_1,\dots,y_n\}$; the relation (\[ctsh\]) becomes \_[n+k,m]{}\^ (,,) \_[k,n]{}\^[m]{}(,)= \_[k,m+n]{}\^ (,,) \_[n,m]{}\^(,) ,where $\overrightarrow{x}=\{ x_1,\dots,x_m\}$, $\overrightarrow{y}=\{ y_1,\dots,y_n\}$ and $\overrightarrow{z}=\{ z_1,\dots,z_k\}$.
.2cm After the removal of all $\pi$’s, one can give values to the spectral variables. Each $\widetilde{\Sigma}_m$ can be evaluated on its own sequence $\vec{x}^{(m)}=(x_1^{(m)},\dots,x_m^{(m)})$. In the relation (\[wtsi\]), the beginning of the sequence for $\widetilde{\Sigma}_{m+n}$ becomes the beginning of the sequence for $\widetilde{\Sigma}_m$ while its end becomes the beginning of the sequence for $\widetilde{\Sigma}_n$. This is a strong restriction; if it is imposed on the sequences themselves, the general solution is that each $x_j^{(m)}$ is equal to one and the same number. However, assume that the baxterization is “trigonometric”, the baxterized elements depend on the ratio of the spectral parameters, $\sigma(x,y)=\sigma(x/y)$. The Yang-Baxter equation then reads \_n(x)\_[n-1]{}(xy)\_n(y)=\_[n-1]{}(y)\_n(xy)\_[n-1]{}(x) . Now $\widetilde{\Sigma}_m(\vec{x}^{(m)})=\widetilde{\Sigma}_m(\alpha\vec{x}^{(m)})$ for an arbitrary constant $\alpha\neq 0$ and the general solution of the restrictions imposed by (\[wtsi\]) for the projectivized sequences is $(x_1^{(m)},\dots,x_m^{(m)})=(1,s^{-1},s^{-2},\dots,s^{1-m})$, the geometric progression. Denote $\widetilde{\Sigma}_m(1,s^{-1},s^{-2},\dots,s^{1-m})$ by $^{^s}\!\Sigma_m$ and $\widetilde{\Sha}_{m,n}(s^{1-n},\dots,1,s^{1-m-n},\dots,s^{-n})$ by $^{^s}\!\Sha_{m,n}$. Explicitly: \^[\^s]{}\_m= (\_1(s)\_2(s\^2)…\_[m-1]{}(s\^[m-1]{})) (\_1(s)\_2(s\^2)…\_[m-2]{}(s\^[m-2]{}))…(\_1(s)) and \^[\^s]{}\_[m,n]{}=(\_m(s)…\_[m+n-1]{}(s\^n)) (\_[m-1]{}(s\^2)…\_[m+n-2]{}(s\^[n+1]{})) (\_1(s\^m)…\_n(s\^[m+n-1]{})) .
The elements $^{^s}\!\Sha_{m,n}$ obey the relation (\[ctsh\]). Therefore, in a local representation $\rho$, the collection ${\cal{T}}_{m,n}:=\rho_{\hat{R}}(^{^s}\!\Sha_{m,n})$ obeys (\[ctshr\]) and (\[ico\]) and defines a one parameter family of graded associative algebras on $\bigoplus_j V^{\otimes j}$ together with the subalgebras on $\bigoplus_j {\mathrm{Im}}({\cal{S}}_j)$ (now ${\cal{S}}_m=\rho_{\hat{R}}(^{^s}\!\Sigma_m)$), which we propose to call [*b-shuffle algebras*]{} (“b” from “baxterized”; maybe the term “buffle” would be an apt acronym).
.2cm It is known that the element $\bar{\Sigma}$ admits reduced expressions starting (or ending) with $\sigma_j$ for every $j=1,\dots,m-1$. In particular, $\bar{\Sigma}\{\underline{\sigma}\}$ can start (or end) with every $\underline{\sigma}_j$. It follows that $\widetilde{\Sigma}_m(x_1,\dots,x_m)$ can start (or end) with $\sigma_j(x_j,x_{j+1})$ for every $j$. We shall use this for the trigonometric $\sigma$’s: \^[\^s]{}\_m \_j(s)(…) (…)\_j(s) j=1,…,m-1 .
.2cm The baxterization is known for the Hecke and BMW quotients of the braid group rings; it is trigonometric. In the next section we discuss the baxterized collections for these quotients.
#### 4. Remarks.
[**(a)**]{} We suggest another natural source for collections ${\cal{T}}_{m,n}$ satisfying (\[ctshr\]) and (\[ico\]).
.2cm Let ${\cal{A}}$ be a Hopf algebra. Assume that ${\cal{A}}$ admits a twist ${\cal{F}}$, that is, an element ${\cal{F}}\in {\cal{A}}\otimes {\cal{A}}$ which satisfies ()()= \^[1]{}()() . Here $\Delta$ is the coproduct and $^\uparrow$ is the shift in the copies of ${\cal{A}}$ in ${\cal{A}}^{\otimes j}$. Define ${\cal{F}}_{m,0}:=1$, ${\cal{F}}_{0,m}:=1$, $m\in {\mathbb Z}_{\geq 0}$, and \_[m,n]{}:=\^[m-1]{}\^[n-1]{}(), m,n\_[1]{} .It is straightforward to verify that \_[k,m]{}\^\_[m+k,n]{}\^ =\_[m,n]{}\^[k]{}\_[k,m+n]{}\^(for $m=n=k=1$ this is (\[bai\]); by induction, ${\mathrm{Id}}^{i-1}\otimes\Delta
\otimes {\mathrm{Id}}^{m+n+k-i}$ increases $k$ by 1 for $1\leq i\leq k$, $m$ by 1 for $k<i\leq m+k$ and $n$ by 1 for $m+k<i\leq m+n+k$).
.2cm Therefore, given a representation $\rho$ of ${\cal{A}}$, the relations (\[ctshr\]) and (\[ico\]) hold for $${\cal{T}}_{m,n}:=\varpi\circ\rho^{\otimes (m+n)}({\cal{F}}_{n,m})\ ,$$ where $\varpi$ is any operation which reverses the order of terms in both sides of (\[utw\]) (it can be a transposition or, if ${\cal{F}}_{m,n}$ are invertible for all $m$ and $n$, an inversion).
.2cm It might be of interest to investigate this type of collections ${\cal{T}}_{m,n}$ for the twists [@IOt] corresponding to Belavin–Drinfeld triples.
#### (b)
Assume, in addition, that ${\cal{F}}$ satisfies ()()=\_[{1,3}]{}\_[{2,3}]{} , ()()=\_[{1,3}]{}\_[{1,2}]{} ,where ${\cal{F}}_{\{i,j\}}$ is the element ${\cal{F}}$ located in the copies number $i$ and $j$ in ${\cal{A}}\otimes {\cal{A}}\otimes\dots$ ; for example, for a quasi-triangular Hopf algebra, ${\cal{F}}$ can be the universal $R$-matrix. Then \_[m,n]{}=(\_[{1,m+n}]{}…\_[{m,m+n}]{} )(\_[{1,m+n-1}]{}…\_[{m,m+n-1}]{})…(\_[{1,m+1}]{}…\_[{m,m+1}]{})(in each bracket the first index increases from 1 to $m$, the second one is constant); this formula generalizes the formula $\Delta\otimes\Delta ({\cal{R}})={\cal{R}}_{\{1,4\}}
{\cal{R}}_{\{2,4\}}{\cal{R}}_{\{1,3\}}{\cal{R}}_{\{2,3\}}$ used in the theory of quasi-triangular Hopf algebras for establishing properties of the element giving the square of the antipode by conjugation, see, e.g., [@O], chapter 4. It follows from (\[fumn\]) that \_[m,n]{}=(\_[{1,m+n}]{} \_[{1,m+n-1}]{}… \_[{1,m+1}]{}) \_[m-1,n]{}\^[1]{} .Given a representation $\rho$ of ${\cal{A}}$ on a vector space $V$, let $P_i$ be the flip operator in the copies number $i$ and $i+1$ of the space $V$ in $V\otimes V\otimes\dots$ ; let $F:=\rho^{\otimes 2}({\cal{F}})$ and $\hat{F}:=P_1F$; for an operator $X\in {\mathrm{End}}(V\otimes V)$ denote by $X_{\{i,j\}}$ the operator $X$ acting in the copies number $i$ and $j$ of the space $V$ in $V\otimes V\otimes\dots$ and let $X_i:=X_{\{i,i+1\}}$. Then \^[(m+n)]{}(\_[m,n]{})= |\_[m,n]{}{ P}|\_[n,m]{}{} ,where $\bar{\Sha}_{m,n}\{ P\}$ are built from $P$’s and $\bar{\Sha}_{n,m}\{\hat{F}\}$ from $\hat{F}$’s. Indeed, by (\[inuf\]) and induction,
[l]{} \^[(m+n)]{}(\_[m,n]{})=(F\_[{1,m+n}]{}…F\_[{1,m+1}]{} )|\_[m-1,n]{}\^[1]{}{ P}|\_[n,m-1]{}\^[1]{}{}\
=(P\_[{1,m+n}]{}…P\_[{1,m+1}]{})(\_[m+n-1]{} \_[m+n-2]{}…\_[m+1]{}\_[{1,m+1}]{}) |\_[m-1,n]{}\^[1]{}{ P}|\_[n,m-1]{}\^[1]{}{} .
Use now (\_[m+n-1]{} \_[m+n-2]{}… \_[m+1]{}\_[{1,m+1}]{})|\_[m-1,n]{}\^[1]{}{ P}= |\_[m-1,n]{}\^[1]{}{ P}(\_n \_[n-1]{}… \_[1]{}) ,the first recursion relelation in (\[muh2\]) for $\bar{\Sha}_{n,m}\{\hat{F}\}$ and
[l]{} (P\_[{1,m+n}]{}…P\_[{1,m+1}]{}) |\_[m-1,n]{}\^[1]{}{ P} =|\_[m-1,n]{}\^[1]{}{ P} (P\_[{1,n+1}]{}P\_[{1,n}]{}…P\_[{1,2}]{})\
=|\_[m-1,n]{}\^[1]{}{ P}(P\_1P\_2…P\_n) =|\_[m,n]{}{ P}
(by the second recurrency relation in (\[muh1\]) for $\bar{\Sha}_{m,n}\{ P\}$) to finish the proof of (\[unf\]).
.2cm Thus the elements ${\cal{F}}_{m,n}$ can be regarded as the universal (in the Hopf algebra theoretical sense) counterpart of the elements $\bar{\Sha}_{m,n}$.
#### (c)
We describe an operation which transforms a collection ${\cal{T}}_{m,n}$ satisfying (\[ctshr\]) and (\[ico\]) into another, “dual”, collection $\check{\cal{T}}_{m,n}$ satisfying (\[ctshr\]) and (\[ico\]).
.2cm Keep the notation from the previous remark. Let $X\in {\mathrm{End}}(V^{\otimes m})$ and $Y\in {\mathrm{End}}(V^{\otimes n})$ be two operators. Then |\_[m,n]{}{ P}X\^[n]{}Y=XY\^[m]{}|\_[m,n]{}{ P} .Define $\check{\cal{T}}_{m,n}$ by \_[m,n]{}:=\_[n,m]{}|\_[m,n]{}{ P} \_[m,n]{}:=\_[n,m]{}|\_[m,n]{}{ P} .The equivalence of two definitions follows from |\_[m,n]{}{ P}\^[-1]{}=|\_[n,m]{}{ P} .
The relation (\[ico\]) is satisfied for the collection $\check{\cal{T}}_{m,n}$. The relation (\[ctshr\]) reads, by (\[pesha\]), \_[m,n+k]{}\_[n,k]{}|\_[n+k,m]{}{ P} |\_[k,n]{}\^[m]{}{ P}=\_[m+n,k]{} \_[m,n]{}\^[k]{}|\_[k,m+n]{}{ P}|\_[n,m]{}{ P} .Since |\_[n+k,m]{}{ P}|\_[k,n]{}\^[m]{}{ P} =|\_[k,m+n]{}{ P}|\_[n,m]{}{ P}it follows that the relation (\[ctshr\]) is as well satisfied for the collection $\check{\cal{T}}_{m,n}$.
.2cm With the help of the identity $ \bar{\Sigma}_m\{ P\}^2={\mathrm{Id}}$, it is straightforward to verify that the collection $\check{\cal{S}}_m$ for $\check{\cal{T}}_{m,n}$ is given by \_m=\_m|\_m{ P} .
Hecke and BMW algebras
======================
In the sequel we call the elements $\Sha_{m,n}$ additive shuffles and $^{^s}\!\Sha_{m,n}$ multiplicative shuffles. In this section we derive the sequences of the (anti-)symmetrizers for the Hecke and BMW algebras with the help of the multiplicative shuffles. We compare the multiplicative versions with known expressions for the (anti-)symmetrizers.
.2cm We derive a new expression for the (anti-)symmetrizers in terms of the highest multiplicative 1-shuffles alone and, for the Hecke algebras, in terms of the highest additive 1-shuffles alone.
.2cm In principle, the Hecke algebras can be considered as quotients of the BMW algebras and many formulas for the Hecke algebras can be obtained from this point of view. Because of importance of the Hecke algebras we prefer however to treat them separately.
Hecke algebras
--------------
#### 1.
The tower of the $A$-Type Hecke algebras $H_{M+1}(q)$ (see [*e.g.*]{} [@Jon1] and references therein) depends on a parameter $q\in {\mathfrak{k}}^*$; the algebra $H_{M+1}(q)$ is the quotient of the braid group ring ${\mathfrak{k}}B_{M+1}$ by \^2\_i=(q-q\^[-1]{})\_i+1 , i=1,…,M .
For $q^2\neq 1$, the baxterized elements have the form \_i(x):=(x\_i-x\^[-1]{}\_i\^[-1]{}) ;they are normalized, $\sigma_i(1)=1$, and satisfy the unitarity condition \_i(x)\_i(x\^[-1]{})=1- .
#### 2.
The symmetrizers $S_n$, $n=1,\dots,M+1$, ([@Jon0], [@W1], [@Gur]) are the non-zero elements, S\_1=1 , S\_nH\_n(q)H\_[M+1]{}(q) ,which satisfy \_iS\_n=S\_n\_i=qS\_n , i=1,…,n-1 ,which forces $S_n^2\sim S_n$; they are normalized by S\_n\^2=S\_n .The sequence $\{ S_n\}$ is defined by (\[desy\]), (\[desy’\]) and (\[desy2\]) uniquely (and it does exist for the Hecke quotients for generic $q$); the anti-symmetrizers are related to the symmetrizers by the isomorphisms $H_{M+1}(q)\rightarrow H_{M+1}(-q^{-1})$, $\sigma_i\mapsto\sigma_i$.
#### 3.
The symmetrizers can be quickly constructed with the help of the baxterized elements. Let $[n]_q:=(q^n-q^{-n})/(q-q^{-1})$, $[n]_q!:=[1]_q[2]_q\cdots [n]_q$ and $[n]_q^\$ :=[1]_q![2]_q!\cdots [n]_q!$. By (\[retr\]), $\sigma_i\, ^{^q}\!\Sigma_n=\, ^{^q}\!\Sigma_n\sigma_i=q\, ^{^q}\!\Sigma_n$, or, equivalently, $\sigma_i(q) ^{^q}\!\Sigma_n=[i+1]_q\, ^{^q}\!\Sigma_n$, so $(^{^q}\!\Sigma_n)^2=[n]_q^\$\, ^{^q}\!\Sigma_n$ and S\_n= \^[\^q]{}\_n .In particular, the symmetrizers satisfy the recurrent relation S\_n= \^[\^q]{}\_[1,n-1]{}S\_[n-1]{} .
#### 4.
We recall several other forms of the symmetrizers and compare them with (\[mshu\]) and (\[recm\]). A convenient recurrent relation for the symmetrizers is (see e.g. [@Jimb1], [@HIOPT]): S\_n=S\_[n-1]{} S\_[n-1]{} .This is checked either by verifying (\[desy\]), (\[desy’\]) and (\[desy2\]) and then by uniqueness or, using (\[recm\]), by the following calculation:
[rcl]{} \[n\]\_q!S\_n&=&\_1(q)…\_[n-2]{}(q\^[n-2]{}) \_[n-1]{}(q\^[n-1]{})S\_[n-1]{}=\_1(q)…\_[n-2]{}(q\^[n-2]{})\_[n-1]{}(q\^[n-1]{}) S\_[n-2]{}S\_[n-1]{}\
&=&\_1(q)…\_[n-2]{}(q\^[n-2]{})S\_[n-2]{}\_[n-1]{}(q\^[n-1]{})S\_[n-1]{}= S\_[n-1]{}\_[n-1]{}(q\^[n-1]{})S\_[n-1]{} .
Denote $\Sha_{1,n}\{ q\sigma\}$ (the additive shuffle built with the $q\sigma_1,\dots,
q\sigma_{n-1}$) by $\sha_{1,n}$. There is another recurrent relation for the symmetrizers in terms of the additive shuffles: S\_n= \_[1,n-1]{}S\_[n-1]{} .In other words, S\_n= \_n{ q} .This is checked again either by verifying (\[desy\]), (\[desy’\]) and (\[desy2\]) and then by uniqueness or, using (\[santis2\]), by induction:
[rcl]{} \[n\]\_qS\_n&=&S\_[n-1]{}\_[n-1]{}(q\^[n-1]{})S\_[n-1]{}= \_[1,n-2]{} S\_[n-2]{}\_[n-1]{}(q\^[n-1]{})S\_[n-1]{}\
&=&\_[1,n-2]{}\_[n-1]{}(q\^[n-1]{})S\_[n-1]{} =\_[1,n-2]{} ( \[n-1\]\_q\_[n-1]{}+q\^[1-n]{}) S\_[n-1]{}\
&=&( \[n-1\]\_q\_[1,n-2]{}\_[n-1]{}+ q\^[-1]{}\[n-1\]\_q) S\_[n-1]{}=q\^[1-n]{}\_[1,n-1]{}S\_[n-1]{} .
We stress that the factors $\frac{1}{[n]_q!}\ ^{^q}\!\Sha_{1,n-1}$ in (\[recm\]) and $\frac{q^{1-n}}{[n]_q}\ \sha_{1,n-1}$ in (\[reca\]) differ; the multiplicative and additive shuffles do not coincide although their products – the symmetrizers – do.
#### 5.
It turns out that the symmetrizer $S_n$ can be expressed in terms of the multiplicative 1-shuffle $^{^q}\!\Sha_{1,n-1}$ or in terms of the additive 1-shuffle $\sha_{1,n-1}$ only.
.2cm For the additive shuffle, we prove by induction that, for $k=1,\dots,n-1$, \_[j=0]{}\^[k-1]{}(\_[1,n-1]{}-q\^[j-1]{}\[j\]\_q) =q\^[k(k-1)]{}\_[1,n-1]{}\_[1,n-2]{}…\_[1,n-k]{} .For $k=1$ there is nothing to prove. Assume that (\[adle\]) holds for some $k<n-1$. By (\[lue\]), the right hand side is divisible, from the right, by $S_k^{\uparrow (n-k)}$. Multiply (\[adle\]) by the factor $(\sha_{1,n-1}-q^{k-1}[k]_q)$ from the right and substitute, in the right hand side, $$\sha_{1,n-1}=\sha_{1,k-1}^{\uparrow (n-k)}+q^k\sha_{1,n-1-k}\sigma_{n-k}\dots\sigma_{n-1}$$ in this factor. Since $S_k^{\uparrow (n-k)}\sha_{1,k-1}^{\uparrow (n-k)}=q^{k-1}[k]_q
S_k^{\uparrow (n-k)}$, the product in the right hand side simplifies,
[c]{}\_[1,n-1]{}\_[1,n-2]{}…\_[1,n-k]{}(-q\^[k-1]{}\[k\]\_q+ \_[1,k-1]{}\^[(n-k)]{}+q\^k\_[1,n-1-k]{}\_[n-k]{}…\_[n-1]{})\
=q\^k\_[1,n-1]{}\_[1,n-2]{}…\_[1,n-k]{}\_[1,n-1-k]{}\_[n-k]{}…\_[n-1]{} =q\^[2k]{}\_[1,n-1]{}\_[1,n-2]{}…\_[1,n-1-k]{}
(in the last equality we again used (\[lue\]) for $\sha_{1,n-1}\sha_{1,n-2}\dots\sha_{1,n-k}\sha_{1,n-1-k}$), establishing the induction step.
.2cm In particular, at $k=n-1$, we obtain, by (\[reca\]), the expression of $S_n$ in terms of $\sha_{1,n-1}$, S\_n= \_[j=0]{}\^[n-2]{}(\_[1,n-1]{}-q\^[j-1]{}\[j\]\_q) .
#### 6.
For the multiplicative shuffle, we prove by induction that, for $k=1,\dots,n$, ( \^[\^q]{}\_[1,n]{})\^k= \^[\^q]{}\_[1,n]{} \^[\^q]{}\_[1,n-1]{}… \^[\^q]{}\_[1,n+1-k]{} .For $k=1$ there is nothing to prove. Assume that (\[adle\]) holds for some $k<n$. The relations (\[symsh\]) and (\[ctsh\]) hold for $\Sigma_m=\, ^{^q}\!\Sigma_m$ and $\Sha_{m,n}=\, ^{^q}\!\Sha_{m,n}$. Therefore, (\[lue\]) holds as well and the product $\, ^{^q}\!\Sha_{1,n}\, ^{^q}\!\Sha_{1,n-1}\dots\, ^{^q}\!\Sha_{1,n-k}$ is divisible, from the right, by $\, ^{^q}\!\Sigma_{k+1}^{\uparrow (n-k)}$. The induction step is:
[rcl]{} \^[\^q]{}\_[1,n]{}… \^[\^q]{}\_[1,n+1-k]{} \^[\^q]{}\_[1,n]{}&=& \^[\^q]{}\_[1,n]{}… \^[\^q]{}\_[1,n+1-k]{} \^[\^q]{}\_[1,n-k]{}\_[n-k+1]{}(q\^[n-k+1]{})…\_n(q\^n)\
&=& \^[\^q]{}\_[1,n]{} \^[\^q]{}\_[1,n-1]{}… \^[\^q]{}\_[1,n+1-k]{} \^[\^q]{}\_[1,n-k]{} ,
since $S_{n+1}\sigma_k(q^k)=[k+1]_qS_{n+1}$, $k=1,\dots,n$.
.2cm In particular, at $k=n$, we obtain, by (\[mshu\]), the expression of $S_n$ in terms of $\, ^{^q}\!\Sha_{1,n}$, S\_[n+1]{}=( \^[\^q]{}\_[1,n]{})\^n .
#### 7. Remark.
Let $\hat{R}$ be a Hecke Yang–Baxter matrix and $\rho_{\hat{R}}$ the corresponding local representation of the tower of the Hecke algebras. The symmetrizers ${\cal{S}}_j$ built with ${\cal{T}}_{m,n}'=\rho_{\hat{R}}(\, ^{^s}\!\Sha_{m,n})$, at $s=q$, are the same as the symmetrizers built with ${\cal{T}}_{m,n}''=\rho_{\hat{R}}(\Sha_{m,n}\{ t\sigma\} )$, at $t=q$ (the symmetrizers coincide at $s^2=q^2$ and $t=q$ or $s^2=q^{-2}$ and $t=-q^{-1}$, these are the values for the anti-symmetrizers; the symmetrizers coincide trivially at $s^2=1$ and $t=0$; otherwise the symmetrizers for $\{{\cal{T}}_{m,n}'\}$ and $\{{\cal{T}}_{m,n}''\}$ differ). Therefore, for the Hecke algebras, the b-shuffle algebra on $\bigoplus_j {\mathrm{Im}}({\cal{S}}_j)$ coincides with the Nichols–Woronowicz algebra (or the symmetric algebra of the quantum space). Indeed, the composition law (\[shal\]) on $\bigoplus_j {\mathrm{Im}}({\cal{S}}_j)$ can be written in the following equivalent form: uv:=\_[m+n]{}(u’v’) ,where $u={\cal{S}}_mu'$ and $v={\cal{S}}_nv'$. Also, ${\mathrm{Im}}({\cal{S}}_j)\simeq V^{\otimes j}/{\mathrm{Ker}}({\cal{S}}_j)$, and the algebra on $\bigoplus_j {\mathrm{Im}}({\cal{S}}_j)$ can be defined alternatively as the algebra on $\bigoplus_j V^{\otimes j}/{\mathrm{Ker}}({\cal{S}}_j)$ with the composition law |[u]{}|[v]{}:=uv (\_[m+n]{}) ,where $\bar{u}\in V^{\otimes m}/{\mathrm{Ker}}({\cal{S}}_m)$ and $\bar{v}\in V^{\otimes n}/{\mathrm{Ker}}({\cal{S}}_n)$; $u\in V^{\otimes m}$ and $v\in V^{\otimes n}$ are representatives of $\bar{u}$ and $\bar{v}$, respectively. In the formulations (\[fe1\]) or (\[fe2\]), the algebra on $\bigoplus_j {\mathrm{Im}}({\cal{S}}_j)$ or $\bigoplus_j V^{\otimes j}/{\mathrm{Ker}}({\cal{S}}_j)$ depends only on the collection $\{ {\cal{S}}_j\}$; the composition laws (\[fe1\]) or (\[fe2\]) are well defined if ${\cal{S}}_{m+n}$ is divisible by ${\cal{S}}_m$ and ${\cal{S}}_n^{\uparrow m}$ from the right (which is, in general, weaker than ${\cal{S}}_{m+n}^{\phantom{\uparrow}}={\cal{T}}_{n,m}^{\phantom{\uparrow}}\,
{\cal{S}}_m^{\phantom{\uparrow}}\, {\cal{S}}_n^{\uparrow m}$); when, say, the representative $u$ of $\bar{u}$ changes, $u\sim u+\delta u$, ${\cal{S}}_m(\delta u)=0$, so $\delta u\otimes v\in {\mathrm{Ker}}({\cal{S}}_{m+n})$ and the product $\bar{u}\circ \bar{v}$ does not change, $u\otimes v\equiv (u+\delta u)\otimes v\ \ {\mathrm{mod}}
\ {\mathrm{Ker}}({\cal{S}}_{m+n})$.
.2cm However, the algebras on the space $\bigoplus_j V^{\otimes j}$, built with ${\cal{T}}_{m,n}'$ or ${\cal{T}}_{m,n}''$, are very different, as it is seen, for example from the comparison of the spectra of the multiplicative and additive 1-shuffles in Section \[secspec\]. The collections $\{ {\cal{T}}_{m,n}'\}$ and $\{ {\cal{T}}_{m,n}''\}$ seem to have different ranges of applicability (already for the BMW algebras, the symmetrizers for these two collections do not coincide).
BMW algebras
------------
The tower of the Birman-Murakami-Wenzl algebras $B\!M\!W_{M+1}(q,\nu)$ was introduced in [@M1] and [@BW]; it depends on two parameters, $q\in {\mathfrak{k}}^*$ and $\nu\in {\mathfrak{k}}\setminus \{0,q,-q^{-1}\}$. For $q^2\neq 1$, the algebra $B\!M\!W_{M+1}(q,\nu)$ is the quotient of the braid group ring ${\mathfrak{k}}B_{M+1}$ by \_i\_i=\_i\_i=\_i ,\_i\_[i-1]{}\_i=\^[-1]{}\_i, \_i\_[i-1]{}\^[-1]{}\_i=\_i ,where the elements $\kappa_i$ are defined by \_i-\_i\^[-1]{}=(q-q\^[-1]{})(1-\_i) .
The Hecke quotient is $\kappa_i=0$.
.2cm For $q^2\neq 1$, the baxterized elements have the form ([@Jon2], [@Mu], [@J], [@Is1]) \_i(x):=x\^[-1]{}( 1+\_i+ \_i) .Their classical counterparts (for the Brauer algebras) were found in [@Zam]. The elements (\[bmwbax\]) are normalized, $\sigma_i(1)=1$, and satisfy the same unitarity conditions (\[hunit\]). The spectral decomposition of the generator $\sigma_i$ contains three idempotents. The basic symmetrizer (the idempotent corresponding to the eigenvalue $q$) is proportional to $\sigma_i(q)$. However, $\sigma_i(q^{-1})$ is a mixture of two other idempotents. There are again isomorphisms $\iota:\, B\!M\!W_{M+1}(q,\nu) \simeq B\!M\!W_{M+1}(-q^{-1},\nu)$, $\sigma_i\mapsto\sigma_i$. The formula (\[bmwbax\]) is not invariant under $\iota$. The basic anti-symmetrizer (the idempotent corresponding to the eigenvalue $-q^{-1}$) is proportional to $\iota^{-1}(\sigma(x))$ at $x=q$.
.2cm Again, the symmetrizers $S_n$, $n=1,\dots,M+1$, are the non-zero elements, which satisfy S\_1=1 , S\_nBMW\_n(q)BMW\_[M+1]{}(q) ,(\[desy’\]) and (\[desy2\]); they exist and are defined uniquely by the conditions (\[desyb\]), (\[desy’\]) and (\[desy2\]).
.2cm Again, with the knowledge of the baxterized elements, one constructs the symmetrizers immediately: they are given by the same formula (\[mshu\]) and satisfy the same recurrence (\[recm\]); the anti-symmetrizers are related to the symmetrizers by the isomorphisms $\iota$.
.2cm The recurrency (\[santis2\]) holds for the BMW symmetrizers as well (it was used in [@WT],[@Fior]); it is derived from the baxterized form of the symmetrizers by the same calculation (\[dersan\]).
.2cm The recurrency relation (\[reca\]) does not hold for the BMW symmetrizers; the additive shuffles have to be modified. A version of such modification was suggested in [@HS] and can be derived by a calculation similar to (\[santider\]). For the Hecke algebras the expansions of the products $\sha_{1,n-1}\sha_{1,n-2}\dots\sha_{1,n-k}$ contain only reduced words; this is not any more so for the modified shuffles for the BMW algebras, the expansions contain similar terms (in a monomial basis, like the one suggested in [@K]) and the formulas are not as elegant as for the Hecke algebras.
.2cm The formula (\[syshh\]) holds, with the same derivation, for the BMW symmetrizers.
Spectrum of 1-shuffles
======================
Polynomial identities for the multiplicative (for the Hecke and BMW algebras) and additive (for the Hecke algebras) 1-shuffles follow, as a by-product, from (\[syshh\]) and (\[sysha\]). We establish the multiplicities of the eigenvalues in this section. The polynomial identity for the additive shuffle was discovered in [@Wal] for the symmetric groups and generalized to the Hecke algebras in [@Lus] with the help of the interpretation of the Hecke algebras in terms of flag manifolds over finite fields. The multiplicities of the eigenvalues of the additive shuffles were obtained in [@DFP] for the symmetric groups. We propose a different approach to the calculation of the multiplicities for the Hecke algebras; our method uses the traces of the operators of the left multiplication by the additive shuffles.
.2cm Let $u\in H_n(q)\subset H_m(q)$, $m\geq n$. Denote by $L_u$ the operator of the left multiplication by $u$, $L_u:H_m(q)\rightarrow H_m(q)$, $L_u(x):=ux$. We denote by ${\mathrm{Tr}}_{_{H_{m}}}(L_u)$ the trace of the operator $L_u$, considered as the operator on $H_m(q)$.
#### 1.
We start with the multiplicative shuffles. Since \^[\^q]{}\_[1,n]{}S\_[n+1]{}=\[n+1\]\_q!S\_[n+1]{} , we find, multiplying (\[syshh\]) by $(\, ^{^q}\!\Sha_{1,n}-[n+1]_q!)$, the following polynomial identity for the multiplicative shuffle (\^[\^q]{}\_[1,n]{})\^n(\^[\^q]{}\_[1,n]{}-\[n+1\]\_q!)=0 ,which holds for both Hecke and BMW algebras. This is the minimal polynomial, already for the Hecke algebras. It is seen without calculations in the Burau representation [@Bu] of $H_{n+1}$, \_j(q\^j)\_q\_[j-1]{}(
q\^[-j]{}&\[j\]\_q\
\[j\]\_q&q\^j
)\_q\_[n-j]{} .If the minimal polynomial is $t^i(t-[n+1]_q!)$ with $i<n$ (the eigenvalue $[n+1]_q!$ is present due to (\[ein\])) then $S_{n+1}$ is proportional to the smaller than $n$ power of $\, ^{^q}\!\Sha_{1,n}$. The matrix of the element $\sigma_j(q^j)$ in the Burau representation has only one non-zero entry under the main diagonal, on the intersection of line and $j$-th column. Therefore, the matrix of $^{^q}\!\Sha_{1,n}$ has only one sub-diagonal filled with (possibly) non-zeros. However, the matrix of $S_{n+1}$ is proportional to the Hankel type matrix $A^i_j:=q^{i+j}$; a smaller than $n$ power of the matrix of $^{^q}\!\Sha_{1,n}$ has zero in the very left entry of the bottom line and cannot be equal to the matrix of $S_{n+1}$.
.2cm Thus, the element $^{^q}\!\Sha_{1,n}$ is not semi-simple for $n>1$; the semi-simple part of $^{^q}\!\Sha_{1,n}$ is $[n+1]_q!S_{n+1}$ and the eigenvalue $[n+1]_q!$ is simple (the rank of the projector $L_{S_{n+1}}$ on $H_{n+1}(q)$ is one, because $S_{n+1}\sigma_j=qS_{n+1}$, $j=1,\dots,n$).
#### 2.
Similarly, multiplying (\[sysha\]) by $(\sha_{1,n-1}-q^{1-n}[n]_q)$, we find the following polynomial identity for the additive shuffle (\_[1,n-1]{}- q\^[1-n]{}\[n\]\_q) \_[j=0]{}\^[n-2]{}(\_[1,n-1]{}-q\^[1-j]{}\[j\]\_q)=0 . The $q$-numbers $q^{1-j}[j]_q\equiv 1+q^2+\dots+q^{2j-2}$, $j=1,2,\dots$, are polynomials in $q$, linearly independent over ${\mathbb{Z}}$. Therefore, there is a unique integer combination $\sum_{j\in\{ 1,2,\dots,n-2,n\} }n_jq^{1-j}[j]_q$, $n_j\in {\mathbb{Z}}$, of these $q$-numbers, which is equal to the trace of $L_{\sha_{1,n-1}}$; the coefficients $n_j$ in this combination are the multiplicities of the eigenvalues $q^{1-j}[j]_q$, $j>0$. The multiplicity $n_0$ of the eigenvalue $0$ is fixed by $\sum n_j={\mathrm{dim}}(H_n(q))\equiv n!$. Thus, the presence of the parameter $q$ gives a simple way to calculate the multiplicities.
.2cm[**Lemma 1.**]{} (i) If $u\in H_j$ then \_[\_[H\_[j+1]{}]{}]{}(L\_u)=\_[\_[H\_[j+1]{}]{}]{}(L\_[u\^[1]{}]{})= (j+1)\_[\_[H\_j]{}]{}(L\_u) .\_[\_[H\_[j+1]{}]{}]{}(L\_[\_1\_2…\_j]{}) =(q-q\^[-1]{})\_[\_[H\_j]{}]{}(L\_[\_1\_2…\_[j-1]{}]{}) , j>0 .\_[\_[H\_j]{}]{}(L\_[\_[k-l+1]{}…\_[k-1]{}\_[k]{}]{}) =(q-q\^[-1]{})\^l , j>kl .
.2cm[*Proof.*]{} Recall that, as a vector space, $H_{j+1}(q)\simeq\oplus_{k=-1}^{j-1}W_k$, where $W_k$ is the vector space consisting of elements $v\sigma_j\sigma_{j-1}\dots\sigma_{j-k}$ with $v\in H_j(q)$ (the word $\sigma_j\sigma_{j-1}\dots\sigma_{j-k}$ is, by definition, empty for $k=-1$); each $W_k$ is canonically isomorphic to $H_j(q)$ as a vector space, the isomorphism is $v\sigma_j\sigma_{j-1}\dots\sigma_{j-k}\mapsto v$. The Hecke versions of the automorphism $\mathfrak a$ and the anti-automorphism $\mathfrak b$, defined in (\[aau\]), transform the above decomposition of $H_{j+1}(q)$ into $H_{j+1}(q)\simeq\oplus_{k=-1}^{j-1}W_k'$, where $W_k'$ consists of elements $v\sigma_1\sigma_2\dots\sigma_{k+1}$ with $v\in H_j(q)^{\uparrow 1}$ and $H_{j+1}(q)\simeq\oplus_{k=-1}^{j-1}W_k''$, where $W_k''$ consists of elements $\sigma_{j-k}\dots\sigma_{j-1}\sigma_jv$ with $v\in H_j(q)$.
.2cm The operator $L_u$ (respectively, $L_{u^{\uparrow 1}}$) acts in each of the spaces $W_k$ (respectively, $W_k'$) separately and this action commutes with the isomorphisms $W_k\simeq H_j(q)$ (respectively, $W_k'\simeq H_j(q)^{\uparrow 1}$). This establishes (i).
.2cm In fact, it was enough to find the formula for ${\mathrm{Tr}}_{_{H_{j+1}}}(L_u)$; ${\mathrm{Tr}}_{_{H_{j+1}}}(L_u)={\mathrm{Tr}}_{_{H_{j+1}}}(L_{u^{\uparrow 1}})$ because $u^{\uparrow 1}$ is conjugate to $u$, $\sigma_1\dots\sigma_ju=u^{\uparrow 1}\sigma_1\dots\sigma_j$, for $u\in H_j(q)$.
.2cm (ii) Given a basis $\{ e_\alpha\}$ of a vector space $U$ we say that the vector $e_\alpha$ (from the basis) does not contribute to the trace of an operator $X:U\rightarrow U$ if $X_\alpha^\alpha=0$ (no summation), where $X_\alpha^\beta$ is the matrix of $X$ in the basis $\{ e_\alpha\}$.
.2cm We use the decomposition $H_{j+1}(q)\simeq\oplus_{k=-1}^{n-1}W_k''$. The operator $L_{\sigma_1\sigma_2\dots\sigma_j}$ maps $W_{-1}''$ to $W_{j-1}''$, so vectors from $W_{-1}''$ do not contribute to the trace of $L_{\sigma_1\sigma_2\dots\sigma_j}$. For $0\leq k<j$,
[l]{} (\_1…\_j)(\_[j-k]{}…\_[j-1]{}\_j)v=(\_[j-k+1]{}…\_j)(\_1 …\_j)\_jv\
=(\_[j-k+1]{}…\_j)(\_1…\_[j-1]{}) ((q-q\^[-1]{})\_j+1)v\
=(q-q\^[-1]{})(\_[j-k+1]{}…\_j)(\_1…\_j )v+(\_[j-k+1]{}…\_j)(\_1…\_[j-1]{})v\
=(q-q\^[-1]{})(\_1…\_j)(\_[j-k]{}…\_[j-1]{})v+(\_[j-k+1]{}…\_j)(\_1…\_[j-1]{} )v ,
the operator $L_{\sigma_1\sigma_2\dots\sigma_j}$ maps $W_k''$ to $W_{j-1}''\oplus W_{k-1}''$. Therefore, vectors from $W_k''$ do not contribute to the trace of $L_{\sigma_1\sigma_2\dots\sigma_j}$ for $k<j-1$. For $k=j-1$, the component $L_{\sigma_1\sigma_2\dots\sigma_j}^\diamond$ of the operator $L_{\sigma_1\sigma_2\dots\sigma_j}$, which maps $W_{j-1}''$ to $W_{j-1}''$, may have a non-zero trace. This component reads, by (\[cotr\]), $$L_{\sigma_1\sigma_2\dots\sigma_j}^\diamond(\sigma_1\dots\sigma_jv)=(q-q^{-1})\,
\sigma_1\dots\sigma_jL_{\sigma_1\sigma_2\dots\sigma_{j-1}}(v)\ ,$$ and the assertion (ii) follows.
.2cm (iii) Follows from (i) and (ii).$\Box$
.2cm By (\[dltrsh\]), the trace of the operator $L_{\sha_{1,n-1}}$ is \_[\_[H\_n]{}]{}(L\_[\_[1,n-1]{}]{})=\_[i=0]{}\^[n-1]{}(q\^2-1)\^i .
For the symmetric group ${\mathbb{S}}_n$, the multiplicity of the eigenvalue $j$ of $L_{\sha_{1,n-1}}$ is the number of permutations in ${\mathbb{S}}_n$ with exactly $j$ fixed points [@DFP]. Recall that the derangement number $d_n$ (the number of permutations in ${\mathbb{S}}_n$ without fixed points) is: d\_n=n!\_[i=0]{}\^nand the number $d_{n,j}$ of permutations in ${\mathbb{S}}_n$ with exactly $j$ fixed points is d\_[n,j]{}=
(
[c]{}[n]{}\
[j]{}
)
d\_[n-j]{}\_[i=0]{}\^[n-j]{} .For generic $q$, the multiplicities of the eigenvalues of $L_{\sha_{1,n-1}}$ are the same as for the symmetric group. By construction, $\sum_{j=0}^{n}d_{n,j}=n!$. Thus, to rederive the multiplicities we have only to check that $\sum_{j=0}^n d_{n,j}q^{1-j}[j]_q
={\mathrm{Tr}}_{_{H_n}}(L_{\sha_{1,n-1}})$, or, explicitly, \_[j=0]{}\^n\_[k=0]{}\^[n-j]{}q\^[1-j]{}\[j\]\_q =\_[i=0]{}\^[n-1]{}(q\^2-1)\^i .It is straightforward to verify that both left and right hand sides satisfy the same recurrency relation in $n$, f\_[n+1]{}=(n+1)f\_n+(q\^2-1)\^n ,with the same initial condition $f_0=0$, and thereby coincide.
.5cm [**Acknowledgements.**]{} The work of A. P. Isaev was partially supported by the grants RFBR 08-0100392-a, RFBR-CNRS 07-02-92166-a and RF President Grant N.Sh. 195.2008.2; the work of O. V. Ogievetsky was partially supported by the ANR project GIMP No.ANR-05-BLAN-0029-01.
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[^1]: Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; laboratoire affilié à la FRUMAM (FR 2291)
|
---
abstract: 'We show how using classical von Neumann index theory makes it possible a universal treatment of squeezing of arbitrary order. “Universal” means that the same approach applied to displacement (order $1$) and squeeze (order $2$) operators confirms toughly what is already known as well as provides rigorous arguments that the higher order squeezing can not be generalized in a “naive” way. We create an environment for answering all the emerging questions in positive ([**the ups**]{}) and negative ([**downs**]{}). In the latter case we suggest ways for further development.'
address: |
$^{1}$H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Division of Theoretical Physics, ul. Eliasza-Radzikowskiego 152, PL 31-342 Kraków, Poland\
$^{2}$Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL 30 348 Kraków, Poland
author:
- 'Katarzyna Górska$^{1}$, Andrzej Horzela$^{1}$ and Franciszek Hugon Szafraniec$^{2}$'
title: 'Squeezing of arbitrary order: the ups and downs'
---
In the eighties a tendency to generalize squeezed states, and squeeze operators in particular, to higher orders became present in the literature [@brau; @nieto]. The authors discussed, not expected by physicists, impossibility of exponentiating the operators $A_{\xi}^{(k)}=i{\xi}^{*}a_{-}^{k}- i{\xi}a_{+}^{k}$, $k\ge 3$ basing their arguments on showing non-analycity of the vacuum state. The latter is however not decisive for the lack of selfadjointness of those $A_{\xi}^{(k)}$s and creates a problem to be explained, cf. [@nagel]. The basic requirement therein, namely normalizability of squeezed states defined via the Bogolubov transform of $ a_{-}^{k}$, turns out to be misleading if $k\ge 3$. In the recent paper [@yzz] solutions to the Schrödinger equation for squeezed harmonic oscillators, considered in the Segal-Bargmann space, have been shown to be non-normalizable for $k\ge 3$. This not only remains in contradiction to what is in [@nagel] but also confirms earlier findings for similar $k$-photon Rabi model with the interaction $\sigma_{x}(g^{*}a_{-}^{k}+ga_{+}^{k})$ where $\sigma_{x}$ is the Pauli matrix. It has been known for several years [@cflo98] that this model suffers, for $k\ge 3$, from analogous pathologies as the generalized squeezing does. Discussion of the above, recently quite extensive [@yzz; @dajka1; @dajka2; @comments; @com2; @gar; @lo], does not get rid of difficulties arisen as the crucial question of selfadjointness of $A_{\xi}^{(k)}$, $k\ge 3$, is left untouched. An attempt at compensating the lack of selfadjointness with suitable modifications, like selfadjoint extensions, may lead to physically important consequences. The aim of our paper is to provide both communities, physicists and mathematicians, with adequate grounds for settling the appearing inconsistency. As a kind of surprise the main tool which works perfectly for this purpose turns out to be very classical and it is nothing but the von Neumann deficiency index approach. It makes the answers definite although reached after rather laborious calculations which we are presenting in detail so as to maintain mathematical rigor and to encourage others to follow.
We begin with preliminary notions to fix the language to be used. The main tool is to investigate essential selfadjointnesss of the operators $A_{\xi}^{(k)}$. We show that though the analytic vectors approach works well for $k=1,2$ is not sufficient to judge the problem for $k\Ge 3$. It is the von Neumann index theory which covers both cases giving definite answers: affirmative for $k=1,2$ and negative for $k\ge 3$. This streghtens universality of the apparatus we have chosen; all this is contained in Sections 2 and 3. Section 4 is devoted to analysis of possible subtleness’ appearing in the process of exponentiation of $A_{\xi}^{(k)}$ and related operators. In section 5 we go back to the case $k=1,2$ modeling them comprehensively in the Segal-Bargmann space. The paper is completed by concluding remarks in which we sum up its mathematical aspects as well as briefly discuss their physical consequences and the Appendix containing a substantial part of calculations needed in the Section 2.
Preliminaries
=============
Basic notions {#basic1}
-------------
Let $\hhc$ be a Hilbert space. For an operator $A$ in $\hhc$, $\dz A$ denotes its domain, $\ob A$ its range and $\jd A$ its null space (the kernel). $\sbar A$ stands always for the closure of a closable operator $A$ and $A^{*}$ for its Hilbert space adjoint.
If $\ddc\subset\dz A$ then the operator $A\res\ddc$ defined by $\dz{A\res\ddc}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\ddc$ and ${A\res\ddc}f\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}Af$, $f\in\ddc$, can be viewed as a [*restriction*]{} of $A$ to $\ddc$ and $A$ can be considered as an [*extension*]{} of $A\res\ddc$; this is a standard set theoretical notion. If $\ddc$ is dense in $\hhc$ then both $A$ and $A\res\ddc$ are a densely defined operators in $\hhc$ (that is $A\res\ddc$ acts within the same space $\hhc$ as $A$ does).
A linear subspace $\ddc$ of $\dz A$ is said to be a [*core*]{} of a closable operator $A$ if $\overline{A\res\ddc}=\sbar A$.
Furthermore, a subspace $\ddc\subset\dz A$ is said to be [*invariant*]{} for $A$ if $A\ddc\subset\ddc$. If this happens, $A\res\ddc$ can also be thought of as a densely defined operator acting in the Hilbert space $\sbar \ddc$, the closure of $\ddc$. Again if $\ddc$ is dense in $\hhc$ the only difference between this and the previous case is that in the latter $\ob{A}\subset\ddc$.
On the other hand, a subspace $\mathcal L$ of $\hhc$ is called [*invariant*]{} for $A$ if $A(\mathcal L\cap\dz A) \subset\llc$; then the [*restriction*]{} $A\rres{\mathcal L}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}A\res{\mathcal L\cap\dz A}$ is always considered as an operator in $\llc$. If $\ddc$ in the above is closed then $A\res\ddc=A\rres\ddc$. A step further, a closed subspace ${\mathcal L}$ [*reduces*]{} an operator $A$ if both $\mathcal L$ and $\mathcal L^\perp$ are invariant for $A$ as well as $P\dz A\subset\dz A$, where $P$ is the orthogonal projection of $\hhc$ onto $\mathcal L$; all this is the same as to require $P A\subset AP$. If this happens, the restriction $A\rres{\mathcal L}$ is called a [*part*]{} of $A$ in $\mathcal L$. If $\llc$ reduces $A$ and $A$ is densely defined then so is $A\rres\llc$.
Notice the word “invariant” has double meaning here but the circumstances we use it protect us from any confusion.
The operators {#s1.10.11}
-------------
Now let $\hhc$ be a separable Hilbert space (with the inner product to be linear in the first varaiable) and $(e_{n})_{n=0}^{\infty}$ be an orthonormal basis (i.e. an orthonormal complete set) in it[^1]. The (abstract) [*creation*]{} and [*annihilation*]{} operators (with respect to the orthonormal basis $(e_{n})_{n=0}^{\infty}$) are linearly extended from $$\begin{aligned}
\label{a}
\begin{gathered}\dz{a_{+}}=\dz{a_{-}}=\ddc\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\operatorname{lin}\big(e_{n}\big)_{n=0}^{\infty},
\\
a_{+}e_{n}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\sqrt{n+1}\,e_{n+1},\quad n=0,1,\ldots\qquad\text{(creation)}
\\
a_{-}e_{n}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\sqrt{n}\,e_{n-1},\quad n=1,\ldots, \quad a_{-}e_{0}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}0\qquad\text{(annihilation)}.
\end{gathered}
\end{aligned}$$
With the definitions we sort out selfadjointness of the operators $$A_{\xi}^{(k)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\operatorname{i}\xi^{*} a_{-}^{k}-\operatorname{i}\xi a_{+}^{k},\quad k=0,1, \ldots$$ with $\xi$ being a complex parameter. As our ultimate goal is to prove , $|\xi|$ has no impact on the problem and we drop it considering instead just the operators $$\begin{gathered}
A^{(k)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}{-}\operatorname{i}\big(\operatorname{e}^{\operatorname{i}\theta}a_{+}^{k}-\operatorname{e}^{-\operatorname{i}\theta}a_{-}^{k}\big), \quad\dz{A_{k}}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\ddc
\quad k=0,1, \ldots \end{gathered}$$ with $\theta$ being a (fixed) real parameter. Therefore $$\text{if $\xi=\operatorname{e}^{\operatorname{i}\theta}|\xi|$, $A^{(k)}_{\xi}=|\xi |A^{(k)}$ \;and\; $A^{(k)}=A^{(k)}_{\exp[\operatorname{i}\theta]}$.}$$ The operators $a_{+}$ and $a_{-}$ are adjoint each to the other, that is $$\is {a_{+}f}{g}=\is{f}{a_{-}g},\quad f,g\in\ddc;$$ in physical tradition this fact is nicknamed as “Hermitian adjoint” and symbolized by ${}^{\dag}$, which makes some sense as long as the $a_{+}$ and ${a_{-}}$ are formal algebraic objects and no domain is indicated. This means that $$a_{+}=(a_{-})^{*}\res\ddc,\quad a_{-}=(a_{+})^{*}\res\ddc$$ as $\ddc$ is invariant for both $(a_{+})^{*}$ and $(a_{-})^{*}$. Consequently, the operators $A^{(k)}$ are symmetric.
Moreover, it is a matter of direct calculation that $\ddc$ is a core of $(a_{+})^{*}$ and $(a_{-})^{*}$, and that for the closure one has $$\overline{a_{+}}=(a_{-})^{*},\quad \overline{a_{-}}=(a_{+})^{*}.$$ Notice that by means of the basis $(e_{n})_{n=0}^{\infty}$ $$\label{2.16.06}
A^{(k)}e_{n}= {-}\operatorname{i}\left\{\operatorname{e}^{\operatorname{i}\theta} \sqrt{\ulamek{\big(n+k\big)!}{n!}}\,e_{n+k}-\operatorname{e}^{-\operatorname{i}\theta} \sqrt{\ulamek{n!}{\big(n-k\big)!}}\,e_{n-k}\right\}$$ with notation $e_{-k}=e_{-k+1}=\ldots=e_{-1}=0$.
Defining$$\label{1.16.06}
\ddc^{(k,i)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\operatorname{lin}\big(e_{i+pk}\big)_{p=0}^{\infty},\quad
\hhc^{(k,i)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\operatorname{clolin}\big(e_{i+pk}\big)_{p=0}^{\infty},\;\quad i=0,\ldots,k-1$$ it is clear that $\ddc=\bigoplus_{i=0}^{k-1}\ddc^{(k,i)}$ and $\hhc=\bigoplus_{i=0}^{k-1}\hhc^{(k,i)}$. It is a kind of straightforward argument to verify the following.
\[1t.15.06\] Each $\hhc^{(k,i)}$ reduces $A^{(k)}$ and the domains $\dz{A^{(k,i)}}=\ddc_{i}$ are invariant for the parts $A^{(k,i)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\rres{\hhc^{(k,i)}}$ of $A^{(k)}$ in $\hhc^{(i)}$ as well as the operators $A^{(k,i)}$ are symmetric. $A_{k}$ is essentially selfadjoint[^2] if and only if so is each $A^{(k,i)}$.
Proposition \[1t.15.06\] allows to downgrade the search for essential selfadjointness of $A^{(k)}_{\xi}$ to that of any of $A^{(k,i)}$’s. With the notation $e^{( {k}, i)}_{p}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}e_{i+pk}$, $p=0,1,\ldots$, the formula reads as $$\label{1.18.06}
A^{(k,i)}e_{p}^{(k, i)}=-\operatorname{i}\left(\operatorname{e}^{\operatorname{i}\theta} \sqrt{\ulamek{[i+(p+1)k]!}{(i+pk)!}} \,e^{(k, i)}_{p+1} - \operatorname{e}^{-\operatorname{i}\theta}\sqrt{\ulamek{(i+pk)!}{[i+(p-1)k]!}}\,e^{(k, i)}_{p-1}\right).$$ Notice that $\big(e_{p}^{(k,i)}\big)_{p=0}^{\infty}$ is an orthonormal basis in $\hhc^{(k,i)}$. The operators $A^{(k,i)}$ act as Jacobi operators[^3] in $\hhc^{(k,i)}$ with zero diagonal. Therefore their deficiency indices are either $(0,0)$ or $(1,1)$, their representing measures are always symmetric with respect to $0$. Notice that each $A^{(k,i)}$ is a operator with a cyclic vector $e^{(k,i)}_{0}$, that is $\dz{A^{(k,i)}}=\operatorname{lin}\big(e^{(k,i)}_{p}\big)_{p=0}^{\infty}$. In conclusion,
\[t1.7.07\] $A^{(k)}$ becomes an orthogonal sum $\bigoplus_{i=0}^{k-1}A^{(k,i)}$ of Jacobi operators with respect to the bases $\big(e_{p}^{(k,i)}\big)_{p=0}^{\infty}$, $i=0,\ldots k-1$, which are in particular cyclic.
Essential selfadjointness of the operators $A^{(k,i)}$; the first attempt - biased {#s2.22.09}
==================================================================================
$\ccc^{\infty}$-vectors. A recollection {#s1.30.11}
---------------------------------------
Recall that, in general, $f\in\ddc^{\infty}(A)\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\bigcap_{n=0}^{\infty}\dz{A^{n}}$ is $$\begin{gathered}
\text{a {\em bounded} vector if there are $a>0$ and $b>0$ such that $\|A^{n}f\|\Le ab^{n}$ for $n=0,1,\dots$;}\notag
\\
\text{an {\em analytic} vector of $A$ if there is $t>0$ such that $\sum\nolimits_{n=0}^{\infty}\frac{t^{n}}{n!}\|A^{n}f\|<+\infty$;}\notag
\\ \text{
an {\em entire} vector if the convergence in the above holds for all $t>0$;}
\notag
\\
\text{a {\em quasianalytic} vector of $A$ if }\sum_{n=0}^{\infty}\|A^{n}f\|^{-1/n}=+\infty.\notag
\end{gathered}$$ All those vectors are customarily called $\ccc^{\infty}$-vectors of $A$. Let us introduce the self-evident notation $\bbc(A)$, $\aac(A)$, $\eec(A)$ and $\qqc(A)$ for the consecutive classes. The first three are always linear while the last may not be. Nevertheless the inclusions $ \bbc(A)\subset\eec(A)\subset\aac(A)\subset\qqc(A)$ are transparent. It may happen that even $\qqc(A)$ is a zero space. However when (essential) selfadjointness is around their nontriviality becomes essential.
\[t3.30.11\] $($a$)$ If $A$ is selfadjoint then $\bbc(A)$ $($consequently, $\aac(A)$, $\eec(A)$ and $\operatorname{lin}\qqc(A)$$)$ constitute a core of $A$.
$($b$)$ If $A$ is selfadjoint and $E$ is its spectral measure then $$\bbc(A)=\operatorname{lin}\zb{E(\sigma)f}{\sigma \text{ bounded, }f\in\hhc}.$$
$($c$)$ If $A$ is symmetric and any of $\bbc(A)$, $\aac(A)$, $\eec(A)$ and $\operatorname{lin}\qqc(A)$ is dense in $\hhc$ then $A$ is essentially selfadjoint.
Employing $\ccc^{\infty}$-vectors {#s1.12.09}
---------------------------------
Let us try to engage analytic and quasianalytic vectors in deciding for which $k$’s the operators $A^{(k,i)}$ are (essentially) selfadjoint. This is the first step towards answering the question of unitarity of so called higher order squeeze operators.
Let us collect first some formulae; the calculations are postponed to Appendix.
\[3Oct13-1\] For $k=1,2,3\ldots$ and with $j_{0} = n-r-1$ we have $$\label{3Oct13-1a}
\big(A^{(k,i)}\big)^{n} e^{(k,i)}_{p}
= \sum_{r=0}^{n} \prod_{s=1}^{r} \sum_{j_{s}=s-1}^{j_{s-1}+1} \frac{[i+(p-r+1+j_{s})k]!}{[i+(p-r+j_{s})k]!} \sqrt{\frac{[i+(p+n-2r)k]!}{(i+pk)!}} (-\operatorname{i}\operatorname{e}^{\operatorname{i}\theta})^{n-2r} e^{(k,i)}_{p+n-2r},$$ $$\begin{aligned}
\begin{split}\label{3Sep13-4}
\| \big(A^{(k, i)}\big)^n e^{(k, i)}_{p} \|^{2} = \sum_{r=0}^{n} \prod_{s=1}^{r} \sum_{j_{s}=s-1}^{j_{s-1}+1} \left\{\frac{[i+(p-r+1+j_{s})k]!}{[i+(p-r+j_{s})k]!}\right\}^{2} \frac{[i+(p+n-2r)k]!}{(i+pk)!},
\end{split}\end{aligned}$$ $$\label{1.29.01}
\sqrt{\frac{[i + (p+n)k]!}{(i+pk)!}}\Le\| \big(A^{(k, i)}\big)^n e^{(k, i)}_{p} \| \Le \sqrt{2}k^{\frac{n}{2}} \sqrt{\frac{[i + (p+n)k]!}{(i+pk)!}}.$$
\[9Sep13-5\] 1. The series $$\label{9Sep13-6}
\sum_{n=0}^{\infty} \frac{\|(A^{(k, i)})^{n} e^{(k, i)}_{p}\| t^{n}}{n!}$$ converges for $k=1$ with infinite radius of convergence (entire vectors) and for $k=2$ with $t < 1/(2\sqrt{2})$ (analytic vectors).
2\. The series diverges for $k = 3, 4, \ldots$ and furthermore the series $$\sum_{n=0}^{\infty} \| \big(A^{(k, i)}\big)^{n} e^{(k, i)}_{p} \|^{-\ulamek{1}{n}}$$ converges for $k {\Ge} 3$.
First we will proof the convergence of for $k=1, 2$. gives $$\label{9Sep13-7}
\sum_{n=0}^{\infty} \frac{\|\big(A^{(k, i)}\big)^{n} e^{(k, i)}_{p}\| t^{n}}{n!} \Le \frac{\sqrt{2}}{\sqrt{(i+kp)!}} \sum_{n=0}^{\infty} \frac{\sqrt{[i+(p+n)k]!}}{n!} k^{\ulamek{n}{2}} t^{n}.$$ Using d’Alembert’s test of convergence for the series on the right hand side of we get part 1 of the Proposition.
To prove the divergence of for $k=3, 4, \ldots$ we rewrite as $$\label{9Sep13-8}
\sum_{n=0}^{\infty} \frac{\|\big(A^{(k, i)}\big)^{n} e^{(k, i)}_{p}\| t^{n}}{n!} \geq \sum_{n=0}^{\infty} \frac{t^{n}}{n!} \sqrt{\frac{[i+(p+n)k]!}{(i+pk)!}}.$$ Employing d’Alembert’s test to the right hand side of we conclude the series in 2 is divergent.
Lemma \[3Oct13-1\] and Stirling’s formula give $$\begin{aligned}
\label{9Sep13-2}
&\sum_{n=0}^{\infty} \| \big(A^{(k, i)}\big)^{n} e^{(k, i)}_{p} \|^{-\ulamek{1}{n}} \Le \sum_{n=0}^{\infty} \left\{\frac{[i + (p+n)k]!}{(i+pk)!}\right\}^{-\ulamek{1}{2n}} \Le \sum_{n=0}^{\infty} [(i + (p+n)k)!]^{-\ulamek{1}{2n}}
\\&
\Le \sum_{n=0}^{\infty} [i + (p+n)k]^{-\ulamek{i + pk + 1/2}{2n} - \ulamek{k}{2}}
\exp\Big\{\ulamek{i + pk}{2n} + \ulamek{k}{2} - \ulamek{1}{2n[1 + 12i + 12(p+n)k]}\Big\} \\&
\qquad \Le \sum_{n=0}^{\infty} k^{-\ulamek{k}{2}} e^{\ulamek{k}{2}} n^{-\ulamek{k}{2}} = \left(\frac{e}{k}\right)^{\ulamek{k}{2}} \sum_{n=0}^{\infty} n^{-\ulamek{k}{2}}< +\infty,\quad k\Ge3.\qedhere\end{aligned}$$
\[t1.23.09\] $A_{\xi}^{(k)}$ is essentially selfadjoint for $k=1,2$ while any of $A^{(k,i)}$ $($hence $A_{\xi}$$)$ not be so if $k\Ge3$ (because 2 is a necessary, not a sufficient condition for essential slfadjointness).
The above confirms what is already recognized in this or another way for $k=1,2$; for $k\Ge 3$ it leaves the question unfastened for the time being.
Essential selfadjointness of the operators $A^{(k,i)}$. The second attempt - definite {#s2.22.09a}
=====================================================================================
The deficiency index approach to essential selfadjointeness {#def_{ind}}
-----------------------------------------------------------
The deficiency indices (sometimes called the defect numbers) $n_{+}$ and $n_{-}$ of a symmetric operator $A$ in a Hilbert space $\hhc$ are defined as follows $$n_{\pm} = \dim\,\ob{A \pm \operatorname{i}}^{\perp}.$$ It is included in the classical von Neumann theory of selfadjoint extensions of symmetric operators that $A$ is essentially selfadjoint (that is, its closure is selfadjoint), if and only if $$\label{1.15.02}
n_{+}=n_{-}=0.$$ Furthermore, the main part of the theory ensures the existence of selfadjoint extensions in the same space (that is in the sense described in the second paragraph of Subsection \[basic1\], Section \[s2.22.09\]) precisely when both deficiency indices are equal.
Towards determining the deficiency indices of $A^{(k,i)}$’s {#1s.16.06}
-----------------------------------------------------------
In order to determine the deficiency indices of $A^{(k,i)}$ take $f\in\hhc^{(i)}$ and check the cardinality of linearly independent $f$’s orthogonal to $\ob{A^{(k,i)} \pm \operatorname{i}}$ for both $\pm\operatorname{i}$, which reads as $$\label{2.15.02}
\langle ({A^{(k,i)}} \pm \operatorname{i})e^{{(k, i)}}_{p}, f \rangle = 0,\quad p=0,1,\ldots.$$ Notice that, due to the third of , $$\label{1.15.06}
{A^{(k,i)}}e^{{(k, i)}}_{p}= {-}\operatorname{i}\operatorname{e}^{\operatorname{i}\theta}a_{+}^{k}e^{{(k, i)}}_{p}\; \text{ for }\; p = -k, -k+1, \ldots, -1, {0}.$$
Develop $f$ as $f = \sum_{\alpha}f^{{(k, i)}}_{\alpha} e^{{(k, i)}}_{\alpha}$, $f^{{(k, i)}}_{\alpha} = \langle e^{{(k, i)}}_{\alpha}, f \rangle$ and write according to the left hand side of as follows $$\begin{gathered}
\langle (A^{(k,i)}\pm \operatorname{i})e^{(k, i)}_{p}, f \rangle = -\operatorname{i}\sum_{\alpha}\nolimits f^{(k, i)}_{\alpha} \langle (\operatorname{e}^{\operatorname{i}\theta}a_{+}^{k} -\operatorname{e}^{-\operatorname{i}\theta} a_{-}^{k} \mp 1)e^{(k, i)}_{p}, e^{(k, i)}_{\alpha} \rangle\\
-\sqrt{\ulamek{(i+pk)!}{[i+(p-1)k]!}} \operatorname{e}^{-\operatorname{i}\theta} \langle e^{(k, i)}_{p-1}, e^{(k, i)}_{\alpha}\rangle \mp
\langle e^{(k, i)}_{p}, e^{(k, i)}_{\alpha}\rangle\big\}
\\
= -\operatorname{i}\big\{\sum_{\alpha}\nolimits \operatorname{e}^{\operatorname{i}\theta}f^{(k, i)}_{\alpha} \sqrt{\ulamek{[i+(p+1)k]!}{(i+pk)!}} \delta_{p+1, \alpha} -\operatorname{e}^{-\operatorname{i}\theta} \sum_{\alpha}\nolimits f^{(k, i)}_{\alpha} \sqrt{\ulamek{(i+pk)!}{[i+(p-1)k]!}} \delta_{p-1, \alpha} \mp\sum_{\alpha}\nolimits f^{(k, i)}_{\alpha} \delta_{p, \alpha}\big\}
\\
= -\operatorname{i}\big\{\sqrt{\ulamek{[i+(p+1)k]!}{(i+pk)!}} \operatorname{e}^{\operatorname{i}\theta} f^{(k, i)}_{p + 1} - \sqrt{\ulamek{(i+pk)!}{[i+(p-1)k]!}} \operatorname{e}^{-\operatorname{i}\theta} f^{(k, i)}_{p-1} \mp f^{(k, i)}_{p}\big\}.\end{gathered}$$
Now now takes the form $$\label{08Feb2013-3}
\sqrt{\ulamek{[i+(p+1)k]!}{(i+pk)!}} \operatorname{e}^{\operatorname{i}\theta} f^{(k, i)}_{p + 1} - \sqrt{\ulamek{(i+pk)!}{[i+(p-1)k]!}} \operatorname{e}^{-\operatorname{i}\theta} f^{(k, i)}_{p-1} \mp f^{(k, i)}_{p} = 0,\;\; p=0,1,\ldots,$$ with $$\label{2.15.06}
f^{(k, i)}_{-q}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}f^{(k, i)}_{-q+1}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\ldots = f^{(k, i)}_{{-1}}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}0$$ which is in accordance with .
Let us treat the cases $k=1,2$ and $k\Ge 3$ separately.
the cases $k=1$ and $k=2$ {#s1.05.017}
---------------------------
It is well known that for the measure orthogonalizing polynomials $\pi_{n}$, $n=0,1\ldots$, to be unique (or, in other words, the corresponding moment problem to be determinate) it is necessary and sufficient $$\label{3.15.02}
\sum_{p=0}^{\infty}|\pi_{p}(z)|^{2}=+\infty$$ for any $z$ with $\operatorname{\mathfrak{Im}}z\neq 0$; cf. [@simon Theorem 3]. This means the would-be Fourier coefficients $f_{n}$ are not in $\ell^{2}$ which leaves the hypothetical $f$ out of the space $\hhc$ and is a counterpart of Proposition \[9Sep13-5\] part 1; both Hermite and Meixner-Pollaczek polynomials are determinate.
### Case $k=1$ {#exa1}
Here $i=0$ is the only possibility and the formulae and , after setting $g_{p}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\operatorname{e}^{-\operatorname{i}\!p\theta}\!f^{(1, 0)}_{p}$, take the form $$\label{08Feb2013-4}
\sqrt{p+1}g_{p+1} - \sqrt{p}g_{p-1} \mp g_{p} = 0,\quad p = 0,1\ldots,\; g_{-1}=0.$$ If [$g_{0}=0$]{} we get immediately that the only solution of is [$g_{n}=0$]{} for all $n=0,1,\ldots$ and both $\mp$. If not, then supposing [$g_{0}=1$]{} we can proceed as follows.
Normalizing the Hermite polynomials as $h_{p}(x) = \operatorname{i}^{p} H_{p}(x)/\sqrt{2^{p} p!}$ from the standard recurrence relation $$H_{p+1}(x) + 2p H_{p-1}(x) - 2x H_{p}(x) = 0,$$ one gets $$\label{08Feb2013-6}
\sqrt{p+1} h_{p+1}(x) - \sqrt{p}h_{p-1}(x) - x\!\sqrt{2} x h_{p}(x) = 0.$$ Comparing and and taking into account that $g_{0}=h_{0}=1$ and that the Hermite polynomials are the only solutions of we infer that $$g_{p} = h_{p}\Big(\!\pm \ulamek\operatorname{i}{\sqrt 2}\Big).$$ Consequently, due to the solution within $\hhc^{(1)}$ is $g_{p}=0$, $p=0,1\ldots$
### Case $k=2$ {#exa2}
Considering two parallel cases $i=0$ and $i=1$ we have to take into account Corollary \[t1.7.07\] which results in splitting . Thus the formula also splits in two, $i=0,1$, $$\label{08Feb2013-10}
\sqrt{(2p+i+2)(2p+i+1)}\operatorname{e}^{\operatorname{i}\theta} f^{(2, i)}_{p+1} - \sqrt{(2p+i)(2p+i-1)}\operatorname{e}^{-\operatorname{i}\theta} f^{(2, i)}_{p-1} \mp f^{(2, i)}_{p} = 0$$ with $p=0, 1, \ldots$ and $f^{(2, i)}_{-1} = 0$. The conditions $f^{(2, i)}_{0} = 0$ imply $f^{(2, i)}_{p} = 0$. Henceforth we take $f^{(2, i)}_{0} = 1$.
For $c^{(i)}_{p} = \operatorname{i}^{p} \operatorname{e}^{\operatorname{i}\! p\theta} f^{(2, i)}_{p}$, $i=0, 1$, reads $$\label{19Jun13-1}
\sqrt{(p+\ulamek{i+2}{2})(p + \ulamek{i+1}{2})} c^{(i)}_{p+1} \mp \ulamek{\operatorname{i}}{2} c^{(i)}_{p} + \sqrt{(p+\ulamek{i}{2})(p + \ulamek{i-1}{2})} c^{(i)}_{p-1} = 0.$$ Consider the Meixner-Pollaczek polynomials $P^{(\lambda)}_{n}(\;\cdot\;;\ulamek \pi 2)$, $n=0,1,\ldots$ and normalize them according to formula (9.7.2) in [@koe p. 213] $$\label{1.19.02}
p^{(\lambda)}_{n}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}{\sqrt{\frac{2^{2\lambda}n!}{2\pi \Gamma(n + 2\lambda)}}}\,P^{(\lambda)}_{n}(\,\cdot\,;\ulamek \pi 2)$$ which sends the recurrence relation $$(n+1)P^{(\lambda)}_{n+1}(x;\ulamek \pi 2) - 2 x P^{(\lambda)}_{n}(x;\ulamek \pi 2) + (n+2\lambda -1)P^{(\lambda)}_{n-1}(x;\ulamek \pi 2)=0$$ into $$\label{2.19.02}
\sqrt{(n+1)(n+2\lambda)}p^{(\lambda)}_{n+1}(x) - 2xp^{(\lambda)}_{n}(x) + \sqrt{n(n+2\lambda - 1)}p^{(\lambda)}_{n-1}(x)=0.$$ Comparing and for appropriate $i$ we get following two couples: $i=0$ corresponding to $\lambda = \ulamek{1}{4}$ and $i=1$ corresponding to $\lambda = \ulamek{3}{4}$. Moreover we have $$c^{(0)}_{n}=p^{(\tiny\frac 14)}_{n}{\Big(\!\pm\ulamek \operatorname{i}4\Big)} \quad \text{and} \quad c^{(1)}_{n}=p^{(\tiny\frac 34)}_{n}{\Big(\!\pm\ulamek \operatorname{i}4\Big)}.$$ For the same reason as above the series $$\sum_{n=0}^{\infty}|c^{(i)}_{n}|^{2}=\sum_{n=0}^{\infty}\Big|p^{(\lambda)}_{n}\Big(\!\pm\ulamek \operatorname{i}4\Big)\Big|^{2},$$ are divergent for both $\pm$. Because they are a subseries of $$\sum_{n=0}^{\infty}|f^{(2, i)}_{n}|^{2}, \quad i=0,1$$ the latter are divergent as well. The argument goes like in the case $k=1$ before.
: the case $k\Ge 3$ {#28.08}
-------------------
The recurrence , after fixing $i=0,1,\ldots,k-1$ and introducing $d^{(k, i)\, \pm}_{p} = \operatorname{e}^{\operatorname{i}\!p\theta}f^{(k, i)}_{p}$, takes the form $$\label{29Feb2013-3}
d^{(k,i)\, \pm}_{p+1} = \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}}\, d^{(k, i)\, \pm}_{p-1} \pm \sqrt{\frac{(i+pk)!}{(i+pk+k)!}}\, d^{(k, i)\, \pm}_{p}$$ with turning to $$\label{2.16.04}
d^{(k, i)\, \pm}_{-1}=0, \quad i=0,1, \ldots, k-1.$$
\[rem1\] With the zero sequence is the only solution of for each initial conditions $d^{(k, i)+}_{0}=0$ or $d^{(k, i)-}_{0}=0$. This implies that has at most one solution for each of the cases $+$ and $-$.
\[2.1\] If $d^{(k, i)+}_{0}>0$ then so are all the other entries of the sequence $(d^{(k, i)+}_{p})_{p}$. This can be inspected by induction applied to .
\[rem3\] If $d^{(k, i)+}_{0} \neq 0$ then $ d^{(k, i) +}_{p} \neq 0$ for all $p$.
Suppose the contrary, there exists $p$ such that $d^{(k, i) +}_{p+1} = 0$ and let it be the smallest such. By we have $$\sqrt{\frac{(i+kr)!}{(i + kr -k)!}}\, d^{(k, i) +}_{r-1} + d^{(k, i) +}_{r} = 0.$$ Consequently $d^{(k, i) +}_{p-1} $ and $d^{(k, i) +}_{p} $ are of different sings which contradicts Remark \[2.1\].
\[pro2\] $d^{(k, i) -}_{p} = (-1)^{p} d^{(k, i) +}_{p}$ for all $p$.
From for $d^{(k, i)\, -}_{p}$ we get $$\begin{aligned}
(-1)^{p+1} d^{(k, i)\, +}_{p+1} &= \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}}\, (-1)^{p-1} (-1)^{2}d^{(k, i)\, +}_{p-1} - \sqrt{\frac{(i+pk)!}{(i+pk+k)!}} (-1)^{p} d^{(k, i)\, +}_{p} \end{aligned}$$ which shortens to $$(-1)^{p+1} d^{(k, i)\, +}_{p+1} = \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}}\, (-1)^{p-1} d^{(k, i)\, +}_{p-1} - \sqrt{\frac{(i+pk)!}{(i+pk+k)!}} (-1)^{p} d^{(k, i)\, +}_{p}$$ Comparing this with for “-” and using the uniqueness in Remark \[rem1\] we get the conclusion.
\[1\] Proposition \[pro2\] implies $|d^{(k, i) -}_{p}| = |d^{(k, i) +}_{p}| = d^{(k, i) +}_{p}$ for all $p$. Denote this common number shortly by $d^{(k, i)}_{p}$. Henceforth, we can examine exclusively the equation $$\label{eq3}
d^{(k, i)}_{p+1} = \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}}\, d^{(k, i)}_{p-1} + \sqrt{\frac{(i+pk)!}{(i+pk+k)!}}\, d^{(k, i)}_{p}$$ for $d^{(k, i)}_{p}$’s.
[\[rem26.03.13-1\] With notation $\alpha^{(k)}_{p} \operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\sqrt{\ulamek{(i+pk)!}{(i+pk+k)!}}$ $$\label{1.10.04}
d^{(k, i)}_{p+1} - d^{(k, i)}_{p-1} < \alpha^{(k)}_{p} \alpha^{(k)}_{p-1} \ldots \alpha^{(k)}_{1} \alpha^{(k)}_{0} \big(d^{(k, i)}_{2} - d^{(k, i)}_{0}\big)$$]{}
Prove first $$d^{(k, i)}_{p+1} - d^{(k, i)}_{p-1} < \alpha^{(k)}_{p} \left(d^{(k, i)}_{p} - d^{(k, i)}_{p-2}\right)$$
Using , (notice $k\Ge3$) and we have $$\begin{aligned}
d^{(k, i)}_{p} &= \frac{(i+pk-k)!}{\sqrt{(i+pk)! (i+pk-2k)!}} d^{(k, i)}_{p-2} + \sqrt{\frac{(i+pk-k)!}{(i+kp)!}} d^{(k, i)}_{p-1} \\[0.5\baselineskip]
&< \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}} d^{(k, i)}_{p-2} + \left(1 - \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}}\right) d^{(k, i)}_{p-1}.\end{aligned}$$ That gives $$\frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}} d^{(k, i)}_{p-1} < \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}} d^{(k, i)}_{p-2} + d^{(k, i)}_{p-1} - d^{(k, i)}_{p}.$$ Inserting into , we get $$\begin{aligned}
d^{(k, i)}_{p+1} - d^{(k, i)}_{p-1} &< \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}} d^{(k, i)}_{p-2} + \left(\sqrt{\frac{(i+pk)!}{(i + pk + k)!}} - 1\right) d^{(k, i)}_{p} \\
& < \left(\sqrt{\frac{(i+pk)!}{(i + pk + k)!}} - 1\right) \left(d^{(k, i)}_{p} - d^{(k, i)}_{p-2}\right) < \sqrt{\frac{(i+pk)!}{(i + pk + k)!}} \left(d^{(k, i)}_{p} - d^{(k, i)}_{p-2}\right).\end{aligned}$$ Now the induction argument makes .
\[t2.10.02\] The sequence $(d^{(k, i)}_{p})_{p}$ is convergent.
It is clear that $$\label{25Apr13-2}
\sum_{r=p}^{\infty}\alpha^{(k)}_{r}\alpha^{(k)}_{r-1}\cdots\alpha^{(k)}_{0} = \sum_{r=p}^{\infty} \sqrt{\frac{i!}{(kr+k+i)!}}<+\infty.$$ Notice that $$\begin{aligned}
\label{1bis.16.04}
\begin{split}
|d^{(k, i)}_{p+m}-d^{(k, i)}_{p}&|\Le|d^{(k, i)}_{p+m}-d^{(k, i)}_{p+m-2}|+|d^{(k, i)}_{p+m-2}-d^{(k, i)}_{p+m-4}|+\cdots+|d^{(k, i)}_{p+2}-d^{(k, i)}_{p}|\\&
\Le\big(d^{(k, i)}_{2}-d^{(k, i)}_{0}\big)\sum_{r=p}^{m+p}\alpha^{(k)}_{r}\alpha^{(k)}_{r-1}\cdots\alpha^{(k)}_{0}.
\end{split}\end{aligned}$$ Because RHS is equal to $\big(d^{(k, i)}_{2}-d^{(k, i)}_{0}\big)$ multiplied a Cauchy fragment of a convergent series LHS tends to $0$ which says $(d^{(k, i)}_{p})_{p}$ is a Cauchy sequence, hence it is convergent.
\[t1.10.02\] None of the operators $A^{(k,i)}$, $k\Ge3$ and $i=0,\ldots,k-1$, is essentially selfadjoint.
As already experienced it is enough to show that the series $\sum_{p=0}^{\infty}(d^{(k, i)}_{p}){}^{2}$ is convergent. We already know, Corollary \[t2.10.02\], the sequence $(d^{(k, i)}_{p})_{p}$ is convergent. Then either $$\label{4.16.04}
\lim_{p\to+\infty}d^{(k, i)}_{p}\neq0.$$ or $$\label{3.16.04}
\lim_{p\to+\infty}d^{(k, i)}_{p}=0$$
Let us go on with as follows$$\begin{aligned}
\label{1ter.16.04}
\begin{split}
|d^{(k, i)}_{p+m}-d^{(k, i)}_{p}|&
\Le\big(d^{(k, i)}_{2}-d^{(k, i)}_{0}\big)\sum_{r=p}^{m+p}\alpha^{(k)}_{r}\alpha^{(k)}_{r-1}\cdots\alpha^{(k)}_{0}
={\rm const}\,\alpha^{(k)}_{p}\cdots\alpha^{(k)}_{0}
\\&\times\sum_{r=p+1}^{m+p}\alpha^{(k)}_{r}\alpha^{(k)}_{r-1}\cdots\alpha^{(k)}_{p+1}
\Le{\rm const}\,\alpha^{(k)}_{p}\cdots\alpha^{(k)}_{0}\sum_{r=p+1}^{\infty}\alpha^{(k)}_{r}\alpha^{(k)}_{r-1}\cdots\alpha^{(k)}_{p+1}\\&\stackrel{\eqref{25Apr13-2}}{=}{\rm const}\,\sum_{r=p+1}^{\infty} \sqrt{\frac{i!}{(kr+k+i)!}}
\Le{\rm const}\,\sum_{r=0}^{\infty} \sqrt{\frac{i!}{(kr+k+i)!}}.
\end{split}\end{aligned}$$
Suppose holds. Then for the sequence $$x^{(k, i)}_{p} \operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\frac{d^{(k, i)}_{p}}{d^{(k, i)}_{p+1}}.$$ we get immediately $x^{(k, i)}_{p}\to1$ as $p$ goes to $+\infty$.
From we have $d^{(k, i)}_{p} < d^{(k, i)}_{p -2}$ and $x^{(k, i)}_{p} < x^{(k, i)}_{p -2} x^{(k, i)}_{p-1} x^{(k, i)}_{p}$. Because $$\lim_{p\to+\infty} \sqrt{\frac{(i+pk-k)!}{(i+pk)!}} = 0 \quad \text{and} \quad \lim_{p\to+\infty} \frac{\sqrt{(i+pk-k)! (i+pk+k)!}}{(i+pk)!} = 1.$$ we get $$\begin{aligned}
\lim_{p\to+\infty} p \left(x^{(k, i)}_{p} - 1\right) &< \lim_{p\to+\infty} p \left(x^{(k, i)}_{p -2} x^{(k, i)}_{p-1} x^{(k, i)}_{p} - 1\right) = \lim_{p\to+\infty} p \left(\frac{\sqrt{(i+pk-k)! (i+pk+k)!}}{(i+pk)!}\, x^{(k, i)}_{p-2} \right.\nonumber \\
&\left.- \sqrt{\frac{(i+pk-k)!}{(i+pk)!}}\, x^{(k, i)}_{p-2} x^{(k, i)}_{p} - 1\right) = \frac{k}{2}\end{aligned}$$ and consequently $$\lim_{p\to+\infty} p \left[\big(x^{(k, i)}_{p}\big)^{2} - 1\right] = \lim_{p\to+\infty} p \left(x^{(k, i)}_{p} - 1\right) \lim_{p\to+\infty} \left(x^{(k, i)}_{p} +1\right) < 2 \frac{k}{2} = k.$$ Due to Raabe’s criterion we have $$\label{7}
\sum_{p=0}^{\infty}d^{(k,i)}_{p}$$ is convergent. This excludes the case to hold.
If happens, then passing in with $m$ to $+\infty$ we get $$\begin{aligned}
|d^{(k, i)}_{p}|&\Le{\rm const}\,\sum_{r=0}^{\infty} \sqrt{\frac{i!}{(kr+k+i)!}}\end{aligned}$$ Because in this case for $p$ sufficiently large $d^{(k, i)}_{p}{}^{2} \Le d^{(k, i)}_{p}$ we have $$\sum_{p=0}^{\infty}(d^{(k, i)}_{p}{})^{2}\Le {\rm{const}'}\sum_{p=0}^{\infty}d^{(k, i)}_{p}\Le{\rm const}''\sum_{p=0}^{\infty}\alpha^{(k)}_{r}\alpha^{(k)}_{r-1}\cdots\alpha^{(k)}_{0}\sum_{r=0}^{\infty} \sqrt{\frac{i!}{(kr+k+i)!}}.$$ Therefore convergence of the series has been proved.
Generating $\exp\{\operatorname{i}\!tA^{(k)}_{\xi}\}$. Squeeze operators of any order? {#s2.7.07}
======================================================================================
The groundwork thought over {#s3.7.07}
---------------------------
Suppose we are given a selfadjoint operator $B$, if $E$ stands for its spectral measure then the spectral integral $$\label{1.8.07}
\int_{\rrb}\operatorname{e}^{\operatorname{i}\!tx}E(\operatorname{d\!}x)$$ (understood as usually in the weak or strong operator topology) gives rise according to the rules of functional calculus to a one parameter ($t\in\rrb$) family of unitary operators which is customarily denoted as $\operatorname{e}^{\operatorname{i}\!tB}$. Due to the continuity property of spectral integral it is strongly continuous in $t$. If $B$ is essentially selfadjoint then its closure $\sbar B$ is selfadjoint so one can think of $\operatorname{e}^{\operatorname{i}\!t\sbar B}$.
On the other hand one has a definition: a family $\{U(t)\}_{t\in\rrb}$ of unitary operators in $\hhc$ is said to be a [*strongly continuous one-parameter unitary group*]{} if
1. $U(s+t)=U(s)U(t)$, $s,t\in\rrb$;
2. $\lim_{h\to 0} \big(U(t+h)-U(t)\big)f=f$ for $f\in\hhc$ and $t\in\rrb$.
It is clear that the exponential family $(\operatorname{e}^{\operatorname{i}\!tB})_{t\in\rrb}$ just defined is a strongly continuous one-parameter unitary group. The celebrated Stone theorem shows the way back: every strongly continuous one-parameter unitary group is of the form $(\operatorname{e}^{\operatorname{i}\!tB})_{t\in\rrb}$ with a uniquely determined selfadjoint operator $B$; it establishes a bijection between $(\operatorname{e}^{\operatorname{i}\!tB})_{t\in\rrb}$ and $(U(t))_{t\in\rrb}$ making them to be replaceable. In conclusion, the spectral integral definition of $(\operatorname{e}^{\operatorname{i}\!tB})_{t\in\rrb}$ is the primary way to defining the unitary group in question and this is made possible at least.
The operator $B$ is pretty often called the (infinitesimal) [*generator*]{} of the group $(U(t))_{t\in\rrb}$ and is defined by $$\begin{aligned}
\dz B\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\zb{f\in\hhc}{\poch{}t{}U(t)|_{t=0}f\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\lim_{h\to0}h^{-1}\big(U(h)-I\big)f\text{ exists}},\quad
\operatorname{i}\! Bf\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\poch{}t{}U(t)|_{t=0}f,\quad f\in\dz B.\end{aligned}$$ As a kind of extras attached to Stone’s theorem we have for $t\in\rrb$ $$\begin{gathered}
\label{3.8.07}
U(t)\dz B\subset \dz B,\quad
\poch{}t{}U(t)f=\operatorname{i}\!U(t)Bf=\operatorname{i}\!BU(t)f \text{ for }f\in\dz B.\end{gathered}$$
\[t1.11.01\] The repeated use of the second of leads to $$\poch{}t{n}U(t)f=\operatorname{i}^{n}U(t)B^{n}f=B^{n}U(t)f \text{ for }f\in\bigcap_{i=0}^{n}\dz {B^{i}}.\label{3a.8.07}$$ Therefore the question is for which $f$’s the Taylor series (the very left expression) $$U(t)f=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\poch{}t{n}U(t)\res{t=0}f=\sum_{n=0}^{\infty}\frac{(\operatorname{i}\!t)^{n}}{n!}U(t)B^{n}f=\sum_{n=0}^{\infty}\frac{(\operatorname{i}\!t)^{n}}{n!}B^{n}f.$$ converges and how.
More on the role of analytic vectors {#s2.8.07}
------------------------------------
The spectral integral allows to determine the unitary group once the spectral measure of its generator is known. A practical question is if one can avoid the spectral representation trying to promote suggestively a kind of Taylor series expansion by means of $\ccc^{\infty}$ vectors. More precisely, starting with an essentially selfadjoint operator $A$ the question is for which $f$’s the definition $$U(t)f\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\sum_{n=0}^{\infty}\frac{(\operatorname{i}\!t)^{n}}{n!}A^{n}f$$ makes sense. This has to be handled with some caution. An insight into the proof of Lemma 5.1 in [@nel] shows how to make this construction possible in the case when the set of analytic vectors $\aac{(A)}$ of a symmetric operator $A$ is dense.
The construction in [@nel Lemma 5.1] is local and can be reiterated resulting in the desired group. In the case when the set of entire vectors $\eec(A)$ is dense the construction can be made smoother, giving at once the group $(U(t))_{t\in\rrb}$ the operator $\sbar A$ generates. Therefore $\eec(A)$ is a subpace of $\hhc$ for which one can certainly replace integral with summation in the middle equality of $$\label{1.9.07}
\operatorname{e}^{\operatorname{i}\!t\sbar A}f\stackrel{\eqref{1.8.07}}=\int_{\rrb}\;\sum^{\infty}_{n=0}\frac{(\operatorname{i}\!t)^{n}}{n!}x^{n}E(\operatorname{d\!}x)f
=\sum^{\infty}_{n=0}\int_{\rrb}\frac{(\operatorname{i}\!t)^{n}}{n!}x^{n}E(\operatorname{d\!}x)f
=\sum^{\infty}_{n=0}\frac{(\operatorname{i}\!t)^{n}}{n!}A^{n}f,\; f\in\eec(A);$$ as a matter of fact the first equality holds for all $f\in\hhc$.
\[t1.19.01\] The role of $\ccc^{\infty}$ vectors in determining essential selfadjointness is described in some details in Fact \[t3.30.11\]. Though selfadjoint operators themselves have enough $\ccc^{\infty}$ vectors of any kind appearing there, an essentially selfadjoint ones (in particular the candidate for) may not have even quasianalytic vectors, they simply may not fit in with the domain of an operator which [*a priori*]{} is not closed. This makes the unseen at a first glance difference we want to put strong emphasis on. An acute awareness of this fact helps to monitor the situation we are in.
The displacement and squeeze operators {#s3.22.09}
--------------------------------------
Corollary \[t1.23.09\] or, alternatively, the results of Subsection \[s1.05.017\] lead directly to what the majority of mathematical physicist accept as granted (notice that what customarily emerges in the definitions is the complex parameter $\alpha=-\operatorname{i}\xi$ ).
\[t1.22.09\] The displacement $\exp\{\operatorname{i}tA_{\xi}^{(1)}\}$ and squeeze $\exp\{\operatorname{i}tA_{\xi}^{(2)}\}$ operators form a group of unitaries for $t\in\rrb$.
Keeping up with the notations of Subsection \[s1.10.11\] the important information launched in Corollary \[t1.7.07\] can be encapsulated now. It sheds more light on how the squeeze operator behaves.
\[t1.10.11\] The Hilbert space $\hhc$ decomposes as $\hhc=\hhc^{(2,0)}\bigoplus\hhc^{(2,1)}$ with $\hhc^{(2,0)}=\operatorname{clolin}\zb{e_{2n}}{n=0,1,\ldots}$ and $\hhc^{(2,1)}=\operatorname{clolin}\zb{e_{2n+1}}{n=0,1,\ldots}$. It forces the squeeze operators $\exp\{\operatorname{i}tA_{\xi}^{(2)}\}$ to decompose accordantly as $$\exp\{\operatorname{i}tA_{\xi}^{(2)}\}=\exp\{\operatorname{i}t|\xi|A^{(2,0)}\}\bigoplus\exp\{\operatorname{i}t|\xi|A^{(2,1)}\}$$ where $A^{(2,i)}$, $i=0,1$, are Jacobi operators acting as stated by in $\hhc^{(2,i)}$ respectively.
Moreover, the operators $\exp\{\operatorname{i}t|\xi|A^{(2,i)}\}$, $i=0,1$, can be retrieved from on the linear spaces $\ddc^{(2,i)}$ defined by as they are composed of analytic vectors of the operators $A^{(2,i)}$.
What happens if $B$ is not essentially selfadjoint - further developments
-------------------------------------------------------------------------
Due to Naĭmark a selfadjoint extension of a symmetric operator $A$ always exists (cf. [@kon Proposition 3.7]) if one allows it to be in a larger space, say $\kkc$, isometrically including $\hhc$. On the other hand, if $A$ has equal deficiency indices, the von Neumann theory provides with a plenty of selfadjoint extensions still within $\hhc$. Even if $A$ is a Jacobi operator having deficiency indices $(1,1)$, Naĭmark extensions are at least as much compelling as von Neumann ones, look at [@darek] for a stimulating example and its analytic background, and some whereabouts at [@lms].
Pick $B$ be either von Neumann’s or Naĭmark’s extension of $A$. Then $(\operatorname{e}^{\operatorname{i}\!tB})_{t\in\rrb}$ is well defined as described above; denote the group alternatively by $(U^{(B)}(t))_{t\in\rrb}$ stressing on its dependence on the choice of a selfadjoint extension of $A$.
Passing to the operator $A$ with domain, that is $A\dz A\subset \dz A$, which is our case we can still get something interesting. Because $U^{(B)}(t)Bf=BU^{(B)}(t)f$ for $f\in\dz B$ (the second part of ) and because $A\subset B$ we get from
$$\begin{gathered}
\label{3a.8.07}
\poch{}t{n}U^{(B)}(t)f=\operatorname{i}^{n}U^{(B)}(t)A^{n}f=\operatorname{i}^{n}B^{n}U^{(B)}(t)f \text{ for $n=0,1\ldots$ and }f\in\dz A\end{gathered}$$
regardless of the extension $B$.
Despite the fact that for $k\Ge3$ squeeze operators $\exp\{\operatorname{i}tA_{\xi}^{(k)}\}$ do not exist the situation is not completely hopeless. From the above we get a recipe which can be read as follows: taking $A=A^{(k,i)}$ for any $i$ with $k$ fixed we get a selfadjoint extension $B^{(k,i)}$ of $A^{(k,i)}$ in some $\kkc^{(k,i)}$ such that $$\poch{}t{n}U^{(B^{(k,i)})}(t)f=(\operatorname{i})^{n}U^{(B^{(k,i)})}(t)(A^{(k,i)})^{n}f=(\operatorname{i})^{n}(B^{(k,i)})^{n}U^{(B^{(k,i)})}(t)f \; \,\text{for } n=0,1\ldots \text{ and }f\in\dz {A^{(k,i)}}.$$ Summing up the above we come to the operator $B^{(k)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\bigoplus_{i=0}^{k-1}B^{(k,i)}$, selfadjoint in the space $\kkc^{(k)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\bigoplus_{i=0}^{k-1}\kkc^{(k,i)}$, such that $B^{(k)}$ extends $A^{(k)}\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\bigoplus_{i=0}^{k-1}A^{(k,i)}$ and $$\begin{aligned}
\poch{}t{n}U^{(B^{(k)})}(t)f=(\operatorname{i})^{n}U^{(B^{(k)})}(t)(A^{(k)})^{n}f=(\operatorname{i})^{n}(B^{(k)})^{n}U^{(B^{(k)})}(t)f\;\,\text{ for $n=0,1\ldots$ and }f\in\dz {A^{(k)}}.\end{aligned}$$ This opens a lot of possibilities which [we intend to explore in our future research.]{}
Back to $k=1$ and $k=2$. Models {#aaaa}
===============================
Because for $k=1,2$ the operator $A^{(k)}_{\xi}$ is essentially selfadjoint, $(\exp[\operatorname{i}tA^{(k)}_{\xi}])_{t\in\rrb}$ is a group of unitary operators (cf. Theorem \[t1.22.09\]).
With we have that the [*displacement*]{} operator $$D(z)=D(t,\xi)\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\exp[\operatorname{i}t A_{\xi}^{(1)}]=\exp[za_{+}-z^{*}a_{-}],\quad t\in\rrb,\;\xi\in\ccb$$ is unitary and $D(z^{*})=D(z)^{*}=D(-z)=D(z)^{-1}$; moreover, fixing $\xi\in\ccb$ we have $D(t,\xi)$ to be a group as $t\in\rrb$. The same refers to the [*squeeze*]{} operator $$S(z)=S(t,\xi)\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\exp[\operatorname{i}t A_{\xi}^{(2)}]=\exp[za_{+}^{2}-z^{*}a_{-}^{2}],\quad t\in\rrb,\;\xi\in\ccb.$$
$k=1$; the displacement operator
--------------------------------
### Reviving the models {#s1.1.02}
Because the splitting Corollary \[t1.7.07\] is not present when $k=1$ ($A^{(1)}_{\xi}=A^{(1,0)}_{\xi}$) the way of proposing notable expression for the displacement operator can be done just by “exponentiation” in the corresponding function space, so to speak. In particular we have at our disposal the following models
1. the $\llc^{2}(\rrb)$ representation (“configuration space”);
2. the Segal-Bargmann representation;
3. discrete representation by which we mean a one parameter family of harmonic oscillators acting on Charlier sequences considered in $\ell^{2}$, cf. [@yet];
4. the one parameter family of holomorphic oscillators as done in [@anal] (see also [@gaz]), which interpolates the models (a) and (b).
Let us take a brisk look at the case (b) as the most analytic one. The orthonormal basis in the Segal-Barmann space is $\big(\ulamek{z^{n}}{\sqrt{n!}}\big)_{n=0}^{\infty}$. Roughly here $D(t,\xi)=\exp[t(\xi z-\xi^{*}\partial_{z})]$ and the unitary equivalence between $\hhc$ and the Segal-Bargmann space is established by $$e_{n}\mapsto \ulamek{z^{n}}{\sqrt{n!}},\quad n
=0,1,\ldots,$$ and causes $D(t,\xi)$ to act as a Jacobi operator. Because $\ddc=\operatorname{lin}\big(\ulamek{z^{n}}{\sqrt{n!}}\big)_{n=0}^{\infty}$ is the set of entire vectors for $A^{(1)}_{\xi}$ formula applies $$D(t,\xi)f=\sum_{n=0}^{\infty}\frac{(\operatorname{i}t)^{n}}{n!}(\xi z-\xi^{*}\partial_{z})^{n}f,\quad f\in\operatorname{lin}\big(\ulamek{z^{n}}{\sqrt{n!}}\big)_{n=0}^{\infty}.$$ In the case of abstract Hilbert space the formula (which simplifies substantially as $k=1$ and $i=0$) combined with the the formula establishes a series representation of the displacement operator.
$k=2$; the squeeze operator {#s1.07.11}
---------------------------
The squeeze operator is defined as $$S(t,\xi)\operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\exp[\operatorname{i}t A_{\xi}^{(2)}]=\exp[t(\xi a_{+}^{2}-\xi^{*}a_{-}^{2})],\quad t\in\rrb,\;\xi\in\ccb$$ Benefitting from Corollary \[t1.7.07\] the orthogonal splitting[^4] $\hhc=\hhc^{(2,0)}\bigoplus\hhc^{(2,1)}$ which generates that of the operator $A^{(2)}$ as $A^{(2)}=A^{(2,0)}\bigoplus A^{(2,1)}$ and consequently $\exp[A^{(2)}]=\exp[A^{(2,0)}]\bigoplus\exp[A^{(2,1)}]$ clarifies the picture. However, instead of being in the space $\hhc$ the harmonic oscillator acts in we go a step further in modeling the action of the squeeze operator. More precisely, we duplicate the model in a way which is parallel to correspondence: “configuration space” $\mapsto$ an analogue of the Segal-Bargmann space. Here “$\mapsto$” has the appearance of a kind of Segal-Bargmann transform; another occasion when the Segal-Bargmann transform appears in connection of squeezed states is in a recent paper [@ali].
### $A^{(2,i)}$ as Jacobi operators in $\llc^{2}(\rrb)$
From the normalized polynomials $p^{(\lambda)}_{n}$ already given by we pass to the [*Meixner-Pollaczek functions* ]{} $$\mathfrak{p}^{(\lambda)}_{n}(x) \operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}\big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert p^{(\lambda)}_{n}(x),$$ which, due the formula (9.7.2) in [@koe], satisfy the following orthogonality relation $$\int_{-\infty}^{\infty} \mathfrak{p}^{(\lambda)}_{n}(x) \overline{\mathfrak{p}^{(\lambda)}_{m}(x)} \operatorname{d\!}x = \delta_{n, m}.$$ The sequence $(\mathfrak{p}^{(\lambda)}_{n})_{n=0}^{\infty}$ is therefore orthonormal in $\mathcal{L}^{2}(\mathbb{R})$, each of the operators $A^{(2,i)}_{\xi}$, $i=0,1$, acts in the Hilbert space $\operatorname{clolin}(\mathfrak{p}^{(\lambda)}_{n})_{n=0}^{\infty}$ which is a subspace of $\llc^{2}(\rrb)$.
### $A^{(2,i)}$ as multiplication operators in the space of the Segal-Bargmann type {#sb}
Let us introduce the Hilbert space $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$, with $$\label{4.11.13-6}
\nu(r) = \left[\frac{2^{\lambda}}{\pi\Gamma(2\lambda)}\right]^{2} r^{2\lambda-1} K_{2\lambda-1}(2 r),\quad r\in[0,+\infty).$$ The basis in $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ is formed by the monomials $$\label{4.11.13-3}
\varPhi_{\lambda, n}(z) \operatorname{{\stackrel{{\scriptscriptstyle{\mathsf{def}}}}{=}}}2^{-\lambda} \sqrt{2\pi}\, \Gamma(2\lambda) \frac{(-\operatorname{i}z)^{n}}{\sqrt{n! \Gamma(n + 2\lambda)}}, \quad z\in\mathbb{C}.$$
\[4.11.13-4\] $$\label{4.11.13-5}
\int_{\mathbb{C}} \varPhi_{\lambda, n}(z) \overline{\varPhi_{\lambda, m}(z)} \nu(|z|) \operatorname{d\!}z = \delta_{n, m}, \quad z\in\ccb,$$
Lemma \[4.11.13-4\] can be shown by substituting formulas and into and using [@APPrudnikov vol.2, formula (2.16.2.2), p. 343] [^5]
### From $\mathcal{L}^{2}(\mathbb{R})$ to $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ [*à la*]{} Segal-Bargmann {#from-mathcall2mathbbr-to-mathcalh_lambdamathbbc-nuz-operatornamedz-à-la-segal-bargmann .unnumbered}
$\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ is a reproducing kernel Hilbert space with the kernel calculated for $\varPhi_{\lambda, n}$ as $$\begin{aligned}
K^{(\lambda)}(t, \tau) & = \sum_{n=0}^{\infty} \varPhi_{\lambda, n}(t) \overline{\varPhi_{\lambda, n}(\tau)} = \frac{2\pi}{2^{2\lambda}} \Gamma^{2}(2\lambda) \sum_{n} \frac{(-\operatorname{i}t)^{n} (\operatorname{i}\bar{\tau})^{n}}{n! \Gamma(n + 2\lambda)} \\
& = \frac{2\pi}{2^{2\lambda}} \Gamma^{2}(2\lambda) (t\bar{\tau})^{-\lambda+1/2} I_{2\lambda-1}(2\sqrt{t\bar{\tau}}),\quad t,\tau\in\ccb.\end{aligned}$$ where $I_{\alpha}$ is the modified Bessel function of the first kind. This is one more kernel from which, applying the procedure developed in [@ho1; @ho2], one may get a new class of coherent states.
Let us find the unitary mapping of $\mathcal{L}^{2}(\mathbb{R})$ onto $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$. We start with the formula[^6] for the generating function for the Meixner-Pollaczek polynomials, see [@koe formula (9.7.12)]: $$\begin{aligned}
\sum_{n=0}^{\infty} \frac{(-\operatorname{i}z)^{n}}{(2\lambda)_{n}} P^{(\lambda)}_{n}(x; \ulamek{\pi}{2}) &= \operatorname{e}^{z} {_{1}F_{1}}(\lambda+\operatorname{i}x; 2\lambda; -2z) = z^{-\lambda} M_{-\operatorname{i}x, \lambda -1/2}(-2 z), \quad z\in\mathbb{C},\end{aligned}$$ where $M_{\sigma, \nu}$ is the Whittaker function\[wit\]. Expressing $P^{(\lambda)}_{n}(x, \ulamek{\pi}{2})$ in terms of $\mathfrak{p}^{(\lambda)}_{n}(x)$ we get $$\begin{aligned}
\sum_{n=0}^{\infty} \varPhi_{\lambda, n}(z) \overline{\mathfrak{p}^{(\lambda)}_{n}(x)} &= \operatorname{e}^{z} \big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert {_{1}F_{1}}(\lambda+\operatorname{i}x; 2\lambda; -2z) = (2 z)^{-\lambda} \big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert M_{-\operatorname{i}x, \lambda-1/2}(-2z)\end{aligned}$$ and determine the transformation $\ggf_{\lambda}$ which sends $ \mathfrak{p}^{(\lambda)}_{n}$ to $\varPhi_{\lambda, n}$ as an integral one $$\label{4.11.13-10}
\varPhi_{\lambda, n}(z) = (\ggf_{\lambda} \mathfrak{p}^{(\lambda)}_{n})(z) = \int_{-\infty}^{\infty} G_{\lambda}(\bar{x}, z) \mathfrak{p}^{(\lambda)}_{n}(x) \operatorname{d\!}x$$ with the kernel $$G_{\lambda}(\bar{x}, z) = (2 z)^{-\lambda} \big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert M_{-\operatorname{i}x, \lambda-1/2}(-2z), \quad x\in\mathbb{R}, \quad z\in\mathbb{C}.$$
\[4.11.13-13\] $$\label{4.11.13-14}
\int_{-\infty}^{\infty} G_{\lambda}(\bar{x}, z) \overline{G_{\lambda}(\bar{x}, z)} \operatorname{d\!}x = \frac{2\pi}{2^{2\lambda}} \Gamma^{2}(2\lambda) |z|^{-2\lambda + 1} I_{2\lambda-1}(2|z|) = K^{(\lambda)}(z, \bar{z}).$$
Using we get $$\begin{aligned}
\begin{split}
\int_{-\infty}^{\infty} G_{\lambda}(\bar{x}, z) \overline{G_{\lambda}(\bar{x}, z)} \operatorname{d\!}x &= (2 r)^{-2 \lambda} \int_{-\infty}^{\infty} \big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert^{2} M_{-\operatorname{i}x, \lambda-1/2}\, (-2z) M_{\operatorname{i}x, \lambda-1/2}(-2\bar{z}) \operatorname{d\!}x \\
& = (2 r)^{-2 \lambda} \operatorname{e}^{\operatorname{i}\pi\lambda}\int_{-\infty}^{\infty} \big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert^{2} M_{\operatorname{i}x, \lambda-1/2}(2 z) M_{\operatorname{i}x, \lambda-1/2}(-2 \bar{z}) \operatorname{d\!}x \\
& = \pi (2 r)^{-2 \lambda + 1} \Gamma^{2}(2\lambda) \lim_{\rho\to 0} \frac{\exp[-(z-\bar{z})\tanh(\rho)]}{\cosh(\rho)} \operatorname{i}^{2\lambda-1}J_{2\lambda-1}\big(\ulamek{2\operatorname{i}r}{\cosh(\rho)}\big) \\
& = \frac{2\pi}{2^{2\lambda}} \Gamma^{2}(2\lambda) r^{-2\lambda + 1} \operatorname{i}^{2\lambda-1}J_{2\lambda-1}(2\operatorname{i}r) = \frac{2\pi}{2^{2\lambda}} \Gamma^{2}(2\lambda) r^{-2\lambda + 1} I_{2\lambda-1}(2r).\qedhere \end{split}\end{aligned}$$
### From $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ back to $\mathcal{L}^{2}(\mathbb{R})$; unitarity of $\ggf_{\lambda}$. {#from-mathcalh_lambdamathbbc-nuz-operatornamedz-back-to-mathcall2mathbbr-unitarity-of-ggf_lambda. .unnumbered}
This can be proved by showing that range of $\ggf_{\lambda}$ is dense in $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$. Let us take the function $f_{q}\in\mathcal{L}^{2}(\mathbb{R})$, of the form $f_{\bar{q}}(w) = \overline{G_{\lambda}(\bar{q}, w)}$. Then and gives $$\begin{aligned}
\int_{-\infty}^{\infty} G_{\lambda}(\bar{q}, z) f_{\bar{q}}(w) \operatorname{d\!}q = \int_{-\infty}^{\infty} G_{\lambda}(\bar{q}, z) \overline{G_{\lambda}(\bar{q}, w)} \operatorname{d\!}q = K^{(\lambda)}_{\bar{w}}(z).\end{aligned}$$ The functions $K^{(\lambda)}_{\bar{w}}(z)$ are complete in $\mathcal{H}_{\lambda}$, this finishes the proof of unitarity of $G_{\lambda}$.
Define the operator $\funk W{ \mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z}{\mathcal{L}^{2}(\mathbb{R})}$ by $$(Wf)(x) = \int_{\mathbb{C}} \overline{G_{\lambda}(\bar{x}, w)} f(w) \nu(|w|) dw, \quad w\in\ccb.$$ For $g = \ggf_{\lambda}(W f) \in\mathcal{H}_{\lambda}$, we get $$\begin{aligned}
g(z) &= \int_{-\infty}^{\infty} G_{\lambda}(x, \bar{z}) \left[\int_{\mathbb{C}} \overline{G_{\lambda}(\bar{x}, w)} f(w) \nu(|w|) \operatorname{d\!}w\right] \operatorname{d\!}x
= \int_{\mathbb{C}} f(w) \left[\int_{-\infty}^{\infty} G_{\lambda}(x, \bar{z}) \overline{G_{\lambda}(\bar{x}, w)} \operatorname{d\!}x\right] \nu(|w|) \operatorname{d\!}w \\
& = \int_{\mathbb{C}} f(w) K_{\lambda}(\bar{z}, w) \nu(|w|) \operatorname{d\!}w = f(z)\end{aligned}$$ which means $W$ is the inverse of $\ggf$.
The integral kernel corresponding to $G^{-1}_{\lambda}$ is given by $$\begin{aligned}
G^{-1}_{\lambda}(\bar{x}, z) = \overline{G_{\lambda}(\bar{x}, z)} &= (2\bar{z})^{-\lambda} \big\vert\Gamma(\lambda-\operatorname{i}x)\big\vert M_{\operatorname{i}x, \lambda-1/2}(-2\bar{z})
= (2\bar{z})^{-\lambda} \big\vert\Gamma(\lambda+\operatorname{i}x)\big\vert M_{\operatorname{i}x, \lambda-1/2}(-2\bar{z}) .\end{aligned}$$
### The image of the Jacobi operator $A^{{(2)}}_{\xi}$ in $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ {#the-image-of-the-jacobi-operator-a2_xi-in-mathcalh_lambdamathbbc-nuz-operatornamedz .unnumbered}
\[4.11.13-18\] $$\label{mult}
G_{\lambda} \left(x\mathfrak{p}^{(\lambda)}_{n}(x)\right)(z) = \operatorname{i}\frac{1-z^{2}}{2 z} \varPhi_{\lambda, n}(z),\quad z\in\ccb.$$
Using the recurrence relation and , we get $$\begin{aligned}
G_{\lambda} \left(x\mathfrak{p}^{(\lambda)}_{n}(x)\right)(z) & = \int_{-\infty}^{\infty} G_{\lambda}(\bar{x}, z) x\mathfrak{p}^{(\lambda)}_{n}(x) \operatorname{d\!}x \\
& = \int_{-\infty}^{\infty} G_{\lambda}(\bar{x}, z) \left[\ulamek{1}{2}\sqrt{(n+1)(n+2\lambda)} \mathfrak{p}^{(\lambda)}_{n+1}(x) + \ulamek{1}{2}\sqrt{n(n+2\lambda-1)} \mathfrak{p}^{(\lambda)}_{n-1}(x)\right] \operatorname{d\!}x \\
& = \ulamek{1}{2}\sqrt{(n+1)(n+2\lambda)} \varPhi_{\lambda, n+1}(z) + \ulamek{1}{2}\sqrt{n(n+2\lambda-1)} \varPhi_{\lambda, n-1}(z) \\
& = \ulamek{-\operatorname{i}z}{2} \varPhi_{\lambda, n}(z) + \ulamek{\operatorname{i}}{2 z} \varPhi_{\lambda, n}(z) = \ulamek{\operatorname{i}}{2} (\ulamek{1}{z} -z) \varPhi_{\lambda, n}(z).\qedhere\end{aligned}$$
The $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ parallels of the other members of the harmonic oscillator family can be derived as follows. From it can be proved that “creation” and “annihilation” operators with respect to the basis $(\varPhi_{\lambda, n})_{n=0}^{\infty}$ act as $$\label{5.11.13-1}
-\operatorname{i}z \varPhi_{\lambda, n}(z) = \sqrt{(n+1)(n+2\lambda)} \varPhi_{\lambda, n+1}(z), \quad \operatorname{i}\frac{\operatorname{d\!}}{\operatorname{d\!}z} \varPhi_{\lambda, n}(z) = \sqrt{\frac{n}{n+2\lambda-1}} \varPhi_{\lambda, n-1}(z).$$ Formulas imply (“number” operator) $$z \frac{\operatorname{d\!}}{\operatorname{d\!}z} \varPhi_{\lambda, n}(z) = n \varPhi_{\lambda, n}(z)$$ which shares the appearance with that of the classical Segal-Bargmann space.
The counterparts of in $\llc^{2}(\rrb)$ can be get from unitarity of $G_{\lambda}$ and formulas and $$\begin{aligned}
G^{-1}_{\lambda}\left[-\operatorname{i}z\varPhi_{\lambda, n}(z)\right] =\sqrt{(n+1)(n+2\lambda)} \mathfrak{p}^{(\lambda)}_{n+1}(x),\quad G^{-1}_{\lambda}\left[\operatorname{i}\frac{\operatorname{d\!}}{\operatorname{d\!}z} \varPhi_{\lambda, n}(z)\right] = \sqrt{\frac{n}{n+2\lambda-1}} \mathfrak{p}^{(\lambda)}_{n-1}(x).\end{aligned}$$
### Symmetricity of the multiplication operator of {#symmetricity-of-the-multiplication-operator-of .unnumbered}
The operator $M_{\lambda}$ of multiplication[^7] by $ \operatorname{i}\frac{1-z^{2}}{2 z}$ in $\mathcal{H}_{\lambda}[\mathbb{C}; \nu(|z|) \operatorname{d\!}z]$ is, according to , an image of the symmetric operator (the Jacobi one) it must be necessary symmetric too. The multiplication by a rational function with a pole at $0$ acting in a space of entire functions may look strange at a first glance though our reasoning does not leave any doubt. However, just for disbelievers we add an alternative, direct argument for this a little bit amazing fact.
Symmetricity of $M_{\lambda}$ means $$\int_{\mathbb{C}} \ulamek{\operatorname{i}}{2}\Big(\ulamek{1}{z} - z\Big) \Phi_{\lambda, n}(z) \overline{\Phi_{\lambda, m}(z)} \nu(|z|) \operatorname{d\!}z = \int_{\mathbb{C}} \Phi_{\lambda, n}(z)\, \overline{\ulamek{\operatorname{i}}{2}\Big(\ulamek{1}{z} - z\Big) \Phi_{\lambda, m}(z)} \nu(|z|) \operatorname{d\!}z.$$ Passing to polar coordinates under the integral and using explicitly and gives the above equality.
Concluding remarks {#fhsz:s1}
==================
We have proposed a precise solution of an intriguing problem of possible generalization of higher order squeezing. As we have already pointed out in the very introduction the existing so far attempts do not explain satisfactorily why there is a disparity between the case $k=1,2$ and that of $k\Ge 3$. What is hidden behind is the fact that a Hilbert space operator can not live without its domain being explicitly manifested. The example we have in mind is a symmetric operator and selfadjoint as well, in which case the domain makes the difference (cf. [@kon] or for much more particularities also [@fabio]). This is invisible when the notion of a Hermitian operator is the only in use, with a consequence of an automatic transplantation of the ${^{\dagger}}$ operation from finite matrices together with its algebraic properties to would-be Hilbert space operators. The typical argument: $U=\exp(\pm\operatorname{i}H)$ is unitary if $H$ is Hermitian, i.e. $H=H^{\dagger}$, is far from being correct as long as $H$ is not (essentially) selfadjoint - the case of $A_{\xi}^{(3)}$ makes a strong warning here. Therefore some caution even for the trivially looking cases of $A_{\xi}^{(1)}$ and $A_{\xi}^{(2)}$ has to be undertaken - this is a message our universal approach conveys. Not taking into account behaviour of domains may result in serious, troublesome problems as the paper [@gal] inquires into.
Although we have focused ourselves on mathematical aspects of higher order squeezing the paper sends also a clear message to physicists: generalizing naively squeezing operators to higher order fails because the out-coming operators do not obey fundamental quantum mechanical requirements postulated by von Neumann - they are “ill-defined” in the physical jargon. Moreover, this fact is by no means restricted to squeezing circumstances. As we have already emphasized the same situation one faces if studying the $k$-photon Rabi model. The $3$-photon Rabi model has been recently suggested [@mondloch] to explain the mechanism of phase locking through the spontaneous three-photon scattering which, if confirmed experimentally, allows us to conjecture that “ill-defined” phenomenological description may be cured in a mathematically rigorous way and to achieve this one should look for selfadjoint extensions of the $3$-photon Rabi interaction.
Summing up, though our main goal has been to prove rigorously impossibility of generalizing squeezing to higher orders in a naive way, one of the benefits of our investigations is to call reader’s attention to a need of being aware how important and helpful a domain is for studying properties of a specific operator; for the thorough discussion of the issue the chapter [@fabio] highly recommended.
aknowledgment
=============
[The authors are extremely grateful to the referees for their deep insight into the paper which benefited in its final version. The third author was supported by the MNiSzW grant NN201 546438.]{}
Appendix. Laborious though indispensable calculations {#appendix.-laborious-though-indispensable-calculations .unnumbered}
=====================================================
[Proof of Lemma \[3Oct13-1\].]{} For , Pochhammer symbol appears again, cf. footnote [@xdefthefnmark[\[poh\]]{}footnotemark]{}, use induction as follows$$\begin{aligned}
& (A^{(k,i)})^{n+1} e^{(k,i)}_{p} = \sum_{r=0}^{n} \prod_{s=1}^{r} \sum_{j_{s}=s-1}^{j_{s-1}+1} \frac{(1+i+(p-r+j_{s}))_{k}}{[(1+i+pk)_{(n-2r)k}]^{-1/2}} (-\operatorname{i}\operatorname{e}^{\operatorname{i}\theta})^{n-2r} A^{(k,i)} e^{(k,i)}_{p+n-2r} \\
&\quad = \sum_{r=0}^{n} \prod_{s=1}^{r} \sum_{j_{s}=s-1}^{j_{s-1}+1} \frac{(1+i+(p-r+j_{s}))_{k}}{[(1+i+pk)_{(n+1-2r)k}]^{-1/2}} (-\operatorname{i}\operatorname{e}^{\operatorname{i}\theta})^{n+1-2r} e^{(k,i)}_{p+n+1-2r} \\
&\quad + \sum_{r=0}^{n} \prod_{s=1}^{r} \sum_{j_{s}=s-1}^{j_{s-1}+1} \frac{(1+i+(p-r+j_{s}))_{k} (1+i+pk)_{(n-2r)k}}{\sqrt{(1+i+pk)_{(n-1-2r)k}}} (-\operatorname{i}\operatorname{e}^{\operatorname{i}\theta})^{n-1-2r} e^{(k,i)}_{p+n-1-2r}.
\end{aligned}$$ For apply the Pythagorean law to .
To prove the right hand side of the inequality it is enough to show $\| (A^{(k, i)})^n e^{(k, i)}_{p} \|^{2} \Le k^{n+1} [i + (p+n)k]!/(i+pk)!$. Indeed, $$\begin{gathered}
\| (A^{(k, i)})^n e^{(k, i)}_{p} \|^{2} \Le (1+i+pk)_{nk} + \sum_{r=1}^{n} \frac{[(1+i+(p+n-2r)k)_{k}]^{2}}{(n-r+1)^{-1}} \sum_{j_{2}=1}^{n-r+1} [(1+i+(p-r+j_{2})k)_{k}]^{2} \\
\times \prod_{s=3}^{r} \sum_{j_{s}=s-1}^{j_{s-1}+1} [(1+i+(p-r+j_{2})k)_{k}]^{2} (1+i+pk)_{(n-2r)k} \Le \ldots \Le (1+i+pk)_{nk}\\ + \sum_{r=1}^{n} \left[\frac{(1+i+pk)_{(n-r-1)k}}{(1+i+pk)_{(n-2r)k}}\right]^{2}
(n-r+1)^{r-1} \sum_{j_{s}=r-1}^{n-1} \left[(1+i+(p-r+j_{s})k)_{k}\right]^{2}
\Le (1+i+pk)_{nk}\\ + k^{2} \left[\frac{(1+i+pk)_{(n-1)k}}
{(1+i+pk)_{(n-2)k}}\right]^{2} (1+i+pk)_{(n-2)k}
= (1+i+pk)_{nk} \left[1 + k^{n} - \frac{k^{n+1}}{i+(p+n)k}\right]\\ \Le (1+k^{n}) (1+i+pk)_{nk} \Le 2k^{n} (1+i+pk)_{nk}.\end{gathered}$$ The left hand side of the inequality can be automatically get from , because $$\| (A^{(k, i)})^n e^{(k, i)}_{p}\|^{2}\geq (1+i+pk)_{nk}.$$
### Further inequalities {#further-inequalities .unnumbered}
For $p=0, 1, \ldots$ and $k\Ge 3$ we have $$\label{1.4.2}
\frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}} + \sqrt{\frac{(i+pk)!}{(i+pk+k)!}} - 1 < 0, \quad i=0, 1, \ldots.$$
[Proof of ]{}. Indeed, going with the left hand side of on $$\begin{aligned}
&\prod_{j=0}^{k-1} \left(\frac{kp-j+i}{kp+j+1+i}\right)^{\frac{1}{2}} + \prod_{j=0}^{k-1} \frac{1}{\sqrt{kp+j+1+i}} - 1 < \left(\frac{kp+i}{kp+1+i}\right)^{k/2} + \left(\frac{1}{kp+1+i}\right)^{k/2} - 1 \\
& < 1 - \frac{k}{2(kp+1+i)} + \frac{1}{kp+1+i} - 1 < - \frac{k-2}{2(kp+1+i)} \end{aligned}$$ which makes ; here the assumption $k\Ge 3$ is .
For $p=0, 1, \ldots$, $k=1, 2, \ldots$, and $i=0, 1, \ldots, k-1$ $$\label{2.4.2}
\frac{(i+pk-k)!}{\sqrt{(i+pk)! (i+pk-2k)!}} - \frac{(i+pk)!}{\sqrt{(i+pk-k)! (i+pk+k)!}} < 0.$$
[Proof of ]{}. Proceeding as in the proof of we get $$\begin{aligned}
&\prod_{j=0}^{k-1} \left(\frac{kp-k-j+i}{kp-k+j+1+i}\right)^{\frac{1}{2}} - \prod_{j=0}^{k-1} \left(\frac{kp-j+i}{kp+j+1+i}\right)^{\frac{1}{2}} < \left(\frac{kp-k+i}{kp-k+1+i}\right)^{k/2} - \left(\frac{kp+i}{kp+1+i}\right)^{k/2} \\
& < \left(1 - \frac{1}{kp-k+1+i}\right)^{k/2} - \left(1 - \frac{1}{kp+1+i}\right)^{k/2}\
\\
&=\left[\left(1 - \frac{1}{kp-k+1+i}\right)^{k/2} + \left(1 - \frac{1}{kp+1+i}\right)^{k/2}\right]^{-1}\left[ \left(1 - \frac{1}{kp-k+1+i}\right)^{k} - \left(1 - \frac{1}{kp+1+i}\right)^{k}\right]
\\
&\Le\frac1{\sqrt 2}\left[ \left(1 - \frac{1}{kp-k+1+i}\right) - \left(1 - \frac{1}{kp+1+i}\right)\right]\,\sum_{l=0}^{k-1}\left(1 - \frac{1}{kp-k+1+i}\right)^{l} \left(1 - \frac{1}{kp+1+i}\right)^{k-l-1}<0.\end{aligned}$$ which implies .
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[^1]: We follow the customary mathematical notation. In order to make our mathematical reasoning more clear we intentionally do not use the Fock space notation. We believe it does not cause any problem for the physicists.
[^2]: An operator is called [*essentially selfadjoint*]{} if its closure is selfadjoint.
[^3]: A Jacobi operator in a separable Hilbert space acts according to the same (tridiagonal) pattern with respect to a chosen orthonormal basis as Jacobi matrix does (when considered in $\ell^{2}$ with respect to the zero-one basis; in this case it has to be considered in principle as an unbounded operator defined on “finite” vectors).
[^4]: It corresponds somehow to what is in [@man] where odd and even coherent states are considered.
[^5]: $\int_{0}^{\infty} x^{\alpha-1}K_{\nu}(cx) \operatorname{d\!}x = 2^{\alpha-2} c^{-\alpha} \Gamma(\ulamek{\alpha+\nu}{2}) \Gamma(\ulamek{\alpha-\nu}{2})$. By the way, the formula (3.26) in [@Barut] is incorrect which is irrelevant for the rest of that paper.
[^6]: \[poh\] called sometimes [*shifted factorial*]{} (see [@koe p. 4]) employed here.
[^7]: Notice that the formula for $M_{\lambda}$ is the same regardless the space it acts in; though both the space as well as the domain depend on $\lambda$.
|
---
author:
- 'Yang-Yu Liu'
- 'Albert-László Barabási'
title: Control Principles of Complex Networks
---
=1
|
---
abstract: 'Most problems in natural language processing can be approximated as inverse problems such as analysis and generation at variety of levels from morphological *(e.g., cat+Plural$\leftrightarrow$cats)* to semantic *(e.g., (call + 1 2)$\leftrightarrow$“Calculate one plus two.”)*. Although the tasks in both directions are closely related, general approach in the field has been to design separate models specific for each task. However, having one shared model for both tasks, would help the researchers exploit the common knowledge among these problems with reduced time and memory requirements. We investigate a specific class of neural networks, called Invertible Neural Networks (INNs) [@ardizzone2018analyzing] that enable simultaneous optimization in both directions, hence allow addressing of inverse problems via a single model. In this study, we investigate INNs on morphological problems casted as inverse problems. We apply INNs to various morphological tasks with varying ambiguity and show that they provide competitive performance in both directions. We show that they are able to recover the morphological input parameters, i.e., predicting the lemma (e.g., cat) or the morphological tags (e.g., Plural) when run in the reverse direction, without any significant performance drop in the forward direction, i.e., predicting the surface form (e.g., cats).'
author:
- |
Gözde Gül Şahin, Iryna Gurevych\
Research Training Group AIPHES and UKP Lab,\
Department of Computer Science, Technische Universität Darmstadt\
www.ukp.tu-darmstadt.de\
www.aiphes.tu-darmstadt.de\
bibliography:
- 'emnlp-ijcnlp-2019.bib'
title: 'Two Birds with One Stone: Investigating Invertible Neural Networks for Inverse Problems in Morphology'
---
Introduction
============
Inverse problem is a general term used in natural sciences and mathematics to describe the process of recovering the hidden model parameters, $\mathbf{x}$, from a set of observations, $\mathbf{y}$. In general, the forward problem, i.e., generating observations/outputs from parameters, is well-defined; while the inverse problem is generally ill-posed, i.e., no (unique) solution exists. For instance, inferring seismic properties of the Earth’s interior from surface observations is a typical inverse problem in geophysics [@snieder1999inverse]; since forward problem is well-defined and can be simulated by a forward model, recovering the seismic properties that lead to a specific surface value is ill-posed. Although inverse problems have been tackled in many fields such as imaging [@bertero1998introduction; @adler2019uncertainty], astronomy [@osborne2019radynversion; @ardizzone2018analyzing] and geophysics [@snieder1999inverse], natural language processing (NLP) has not yet witnessed an explicit exploration, mostly due to the discrete nature of human language. However NLP contains many tasks that resemble inverse problems such as semantic parsing $\leftrightarrow$ text generation, morphological analysis $\leftrightarrow$ morphological generation, text-to-data (e.g., database records) $\leftrightarrow$ data-to-text generation and many more. Recently, the NLP field has replaced traditional discrete representations of text with dense and low-dimensional continuous ones, referred to as word/sentence vectors. This has enabled us to reformulate some of the classical morphological tasks as inverse problems within an existing framework that has been previously employed for such problems in other fields.
Inverse problems always exist together with their forward problem. Even so, traditional methods in NLP, optimize the inverse problems in an isolated fashion via a supervised loss for direct posterior probability learning, $p(\mathbf{x} \,|\, \mathbf{y})$, causing challenges for ill-posed problems, i.e., multiple possible $\mathbf{x}$ values in $\mathbf{y}\rightarrow\mathbf{x}$ [^1]. In addition, the direct formulation ignores the connection to its forward problem, losing the potential to exploit the shared knowledge between those two. Furthermore, direct optimization of those problems requires two dedicated models to be trained separately, increasing the computational requirements.
Recently, a new class of neural networks, called Invertible Neural Networks (INNs), have been introduced by @ardizzone2018analyzing following the research line of @dinh2014nice [@DinhSB17; @kingma2018glow]. INNs offer three key functionalities: (i) modeling inverse problems within a single network (shared parameters), (ii) having an invertible architecture that allows getting the inverse mapping $\mathbf{y}\rightarrow\mathbf{x}$ for free during prediction; and enables efficient bi-directional training *(i.e., training the network at both ends)*; and (iii) addressing the ill-posed problems via an additional latent output variable $\mathbf{z}$ to have a one-to-one input and output mapping as $\mathbf{x}\leftrightarrow[\mathbf{y}, \mathbf{z}]$.
{width=".95\columnwidth"}
In this work, we model two well-known morphological tasks: *morphological inflection* and *lemmatization* together with their approximated inverses via modified invertible neural networks. *Morphological inflection* is defined as inflecting a lemma with a set of inflectional morphemes to generate a surface form. Some examples of morphological inflection for Turkish, Tagalog and French are given in the left side of Fig. \[fig:inn\_tasks\]. It is considered as a well-defined problem, since (almost always) only one surface form can be generated given the lemma and the tag combination. Its inverse problem is *morphological analysis*, that aims to extract the lemma and the tags given the surface form. Unlike inflection, it may be ambiguous, i.e. there may be multiple possible analysis of a single surface form, depending on the linguistic properties of the language. Note that the examples in Fig. \[fig:inn\_tasks\]-left have only one possible analysis. *Lemmatization* is the task of finding the lemma of a given surface form. Although it is possible to have multiple lemmas per surface form, it is treated as a well-defined task in literature [^2]. Its inverse problem is surface form generation, similar to morphological inflection, however without the guidance of morphological tags. We name it as surface form sampling, since one can sample a $z$ variable and generate surface forms given the lemma as shown in Fig. \[fig:inn\_tasks\]-right.
In this work, we recognize, for the first time, that NLP contains problems similar to “inverse problems” that are found in many other fields such as imaging, geophysics, medical and astronomy. This recognition would provide new ways to approach traditional NLP problems by adapting the existing solutions developed for inverse problems in other fields. We experiment with several languages from diverse language families and morphological typologies and show that:
- INNs have the ability to optimize for both problems simultaneously providing strong results for both, however generally 2%-6% less then a strong method designed specifically for the forward task under our experimental settings,
- Bi-directional training is *crucial* to provide competitive scores for both sides of the network, e.g., adding reverse training to morphological inflection, boosts the lemma prediction performance (inverse problem), while causing only a slight drop in morphological inflection (forward process),
- Introduction of additional categorical latent variables provide improvements in lemmatization for *all languages*, even surpassing a strong model for some languages,
- INNs implicitly learn simple morphological tag distributions even without a dedicated loss function, however words needed direct supervision.
We believe that this initial exploration of INNs for inverse problems in morphology would encourage research in that direction for more complex NLP tasks.
Background: Invertible Neural Networks (INNs) {#sec:INN}
=============================================
INNs are originally proposed to address ill-posed inverse problems that have a well-defined forward process, i.e., a unique $x \rightarrow y$, whereas the inverse problem is ambiguous, i.e., multiple $y \rightarrow x$ mappings. It is addressed by introducing an additional latent variable, as illustrated in Fig. \[fig:inn\_idea\], that turns $\mathbf{x} \leftrightarrow [\mathbf{y}, \mathbf{z}]$ into a bijective mapping. In other words, it enables the reparametrization of $p(\mathbf{x} \,|\, \mathbf{y})$ into a deterministic function $\mathbf{x} = f(\mathbf{y}, \mathbf{z})$. Therefore we can define an inverse function $x = f^{-1} (\mathbf{y}, \mathbf{z}) = g(\mathbf{y}, \mathbf{z})$ to estimate the full posterior distribution, $p(\mathbf{x} \,|\, \mathbf{y})$. Recently proposed neural network components that are invertible such as coupling layers can now be employed as described in more details later in this section. Such networks are unique since they can be run backwards to get the inverse $\mathbf{y} \rightarrow \mathbf{x}$ without any additional cost at prediction time. Furthmermore, INNs offer efficient training procedure that can optimize both ends of the network simultaneously, which is referred to as bi-directional training.
![Additional latent variable $\mathbf{z}$ for one-to-one mapping[]{data-label="fig:inn_idea"}](INN_idea_smaller.pdf){width=".65\columnwidth"}
Invertible Component {#ssec:coupling}
--------------------
INNs consist of layers of “reversible block”s shown with Fig. \[fig:inn\_arch\] that contain complemantary affine coupling layers proposed by @DinhSB17. First, the input vector $u$ is split into two halves $\mathbf{u}_1$ and $\mathbf{u}_2$ (in a fixed way ), which are then transformed by the learned affine functions $s_i$ and $t_i$, where $i \in \{ 1,2 \}$ using the following equations: $$\begin{aligned}
\mathbf{v}_1 &= \mathbf{u}_1 \odot \exp\!\big(s_2(\mathbf{u}_2)\big) + t_2(\mathbf{u}_2) \\ \mathbf{v}_2 &= \mathbf{u}_2 \odot \exp\!\big(s_1(\mathbf{v}_1)\big) + t_1(\mathbf{v}_1), \end{aligned}$$ where $\odot$ denotes element-wise multiplication. The output $\mathbf{v}$ is then simply calculated by concatenating $\mathbf{v}_1;\mathbf{v}_2$. The input $\mathbf{u}$ can then be recovered with: $$\begin{aligned}
\mathbf{u}_2 &= (\mathbf{v}_2 - t_1(\mathbf{v}_1)) \odot \exp\!\big(-s_1(\mathbf{v}_1)\big) \\ \mathbf{u}_1 &= (\mathbf{v}_1 - t_2(\mathbf{u}_2)) \odot \exp\!\big(-s_2(\mathbf{u}_2)\big). \end{aligned}$$ $s_i$ and $t_i$ need not be invertible and generally represented by multi-layered feed-forward neural networks with nonlinear activations, which can then be trained by standard backpropogation algorithm. Similar to previous methods, in order to vary the splits $[\mathbf{u}_1, \mathbf{u}_2]$ among different layers, we employ a permutation layer between the reversible blocks which shuffle the elements in a randomized but a fixed way.
Bi-directional Training {#ssec:training}
-----------------------
As discussed by @GroverDE18, networks that are invertible offer a unique oppurtunity to optimize for both the input and output domains *simultaneously*, i.e., apply the loss functions $\mathcal{L}_x$, $\mathcal{L}_y$ and $\mathcal{L}_z$ given in Fig. \[fig:inn\_arch\] for the forward and the inverse passes at the same time.
![Demonstration of one layer INN. Revertible block is shown inside the dashed box. The operations (e.g., $+, \odot$) are only shown for the forward direction for convenience. They are inverted to subtraction and element-wise division in reverse direction. $\mathcal{L}$ denotes loss function.[]{data-label="fig:inn_arch"}](inn_arch.pdf){width=".70\columnwidth"}
This is achieved by performing forward and inverse iterations in an alternating fashion, and then accumulating the gradients to update the network parameters. This training procedure has been shown to be highly beneficial in auto-encoders [@teng2019invertible] and our own experiments.
Method {#sec:method}
======
Different applications of INNs [@osborne2019radynversion; @ardizzone2018analyzing; @adler2019uncertainty] have commonly used continuous latent variables with Gaussian priors as $\mathbf{z} \sim \mathcal{N}(\mu,\,\sigma^{2})$. Unlike in previously addressed ill-posed inverse problems, most NLP tasks contain categorical and discrete values. For instance in lemmatization the $\mathbf{z}$ variable is expected to capture the morphological tags which are categorical by definition. Therefore, we have employed categorical latent variables [@DBLP:conf/iclr/JangGP17] that is based on the idea of approximating a categorical distribution via a differentiable distribution called *Gumbel-Softmax (GS)*. This distribution can be smoothly annealed into a categorical distribution via a temperature parameter, $\tau$. As $\tau \rightarrow 0$, the GS distributions are identical to samples from a categorical distribution (one-hot), where as $\tau \rightarrow \infty$ GS samples become uniform. Then the method can be formally defined as: Given the input vector $\mathbf{x} \in \mathbb{R}^n$, the output vector $\mathbf{y} \in \mathbb{R}^m$, we assume that the forward process $\mathbf{y}=\mathbf{s}(\mathbf{x})$, *i.e., morphological inflection and lemmatization*, is well-defined by an arbitrary transformation function $\mathbf{s}$. Our goal is to approximate the posterior $p(\mathbf{x} \,|\, \mathbf{y})$ by a tractable distribution $q(\mathbf{x} \,|\, \mathbf{y})$. This is then reparameterized by a deterministic function $g$, represented by the neural network with parameters $\theta$. Categorical output latent variable $\mathbf{z} \in \mathbb{R}^{d \times cat}$ is drawn from a *Gumbel-Softmax (GS)* distribution, where $d$ is latent variable dimension and $cat$ is the number of categories for each dimension. Then $g$ is defined as: $$\begin{aligned}
\mathbf{x} = g (\mathbf{y},\mathbf{z};\theta) \quad \textrm{where} \quad \mathbf{z} \approx p(z) = GS(z;\tau) \end{aligned}$$ We learn the inverse model $g (\mathbf{y},\mathbf{z};\theta)$ jointly with the forward model $f (\mathbf{x}; \theta)$ that approximates the $s(\mathbf{x})$ function: $$\begin{aligned}
[\mathbf{y},\mathbf{z}] = f (\mathbf{x};\theta) = [f_y(\mathbf{x};\theta), f_z(\mathbf{x};\theta)] = g^{-1}(\mathbf{x};\theta) \\ \textrm{and} \quad f_y(\mathbf{x};\theta) \approx s(\mathbf{x}) \end{aligned}$$ It should be noted that both functions $f$ and $g$ share the same parameters $\theta$, therefore can be implemented by a **single** model. The invertibility property, $f = g^{-1}$ is ensured by the architecture that consists of invertible affine coupling layers defined in previous background section. One of the restrictions of using an invertible architecture is the equality constraint on the dimensions of the input $\mathbf{x}$ and the output $[\mathbf{y},\mathbf{z}]$. To satisfy this constraint, we pad the lower-dimensional side (i.e., input or output) with zeros [^3]. Finally the posterior distribution is calculated as the following as shown in [@ardizzone2018analyzing]: $$\begin{aligned}
q(\mathbf{x}=g(\mathbf{y},\mathbf{z};\theta)|\mathbf{y}) = p(\mathbf{z}) |J_x|^{-1}, \textrm{where} \\J_x = det \left( \frac{\partial {g(\mathbf{y},\mathbf{z}|\theta)}}{\partial {[\mathbf{y},\mathbf{z}]}}\bigg|_{\mathbf{y},f_z(\mathbf{x})} \right) \end{aligned}$$ where $J_x$ is the Jacobian determinant and can be easily calculated due to its triangular structure. We refer the reader to the original study [@ardizzone2018analyzing] for details of the above derivations.
For all morphological tasks, the words (lemmas and surface forms) are divided into smaller units, i.e., subword units, via the Byte-Pair-Encoding (BPE) algorithm [@SennrichHB16a]. We then used the 100 dimensional pretrained multilingual embeddings released by @heinzerling2018bpemb that provide vector representations for the BPE subword units. Finally each word is represented as sum of its subword unit vectors that we denote as $\vec{lemma} \in \mathbb{R}^{100}$ and $\vec{surface} \in \mathbb{R}^{100}$. For *morphological inflection*, we define the vector $\vec{t} \in [0,1]^N$ to represent the morphological tagset, where $N$ is the number of distinct morphological tags in the training set. Then $t_i$ is set to $1$, if the feature $i$ is among the tagset of the surface form.
#### Variables:
For *morphological inflection*, as given in Fig. \[fig:inn\_tasks\]-left, $\mathbf{x}$ is defined as $\mathbf{x} = [\vec{lemma};\vec{t}]$ where $;$ denotes concatentation operation and $\mathbf{y}$ is simply equal to $\vec{surface}$. According to Fig. \[fig:inn\_tasks\]-right, $\mathbf{x}$ and $\mathbf{y}$ are equal to $\vec{surface}$ and $\vec{lemma}$ accordingly for *lemmatization*.
#### Loss Functions:
We use a supervised loss, implemented via cosine distance function, to minimize the error between the network prediction $\vec{surface'}$ and the gold value $\vec{surface}$ defined as: $$\begin{aligned}
\mathcal{L}_{surface} = 1 - \frac{\vec{surface} \cdot \vec{surface'}}{\|\vec{surface}\| \|\vec{surface'}\|}\end{aligned}$$ for both tasks. The loss between the predicted $\vec{lemma'}$ and the gold $\vec{lemma}$ is again calculated via cosine distance. For morphological inflection where $\vec{t'}$ is the predicted and $\vec{t}$ is the gold tag vector, we minimize the error using binary cross entropy loss defined as below: $$\begin{aligned}
\mathcal{L}_t = - \frac{1}{N} \sum_{i=0}^{N} (t_i.log(t'_i) + (1-t_i) log(1-t'_i)) \end{aligned}$$ $\mathcal{L}_x$ in Fig. \[fig:inn\_arch\] is defined as $\mathcal{L}_{lemma} + \mathcal{L}_t$ for *morphological inflection*, where it is equal to $\mathcal{L}_{surface}$ for *lemmatization* task. Similarly $\mathcal{L}_y$ is defined as $\mathcal{L}_{surface}$ and $\mathcal{L}_{lemma}$ respectively for *morphological inflection* and *lemmatization*. Finally, we use KL divergence loss for $\mathbf{z}$. The final losses are calculated as following: $$\begin{aligned}
\mathcal{L}_{inflection} = \alpha_x (\mathcal{L}_{lemma}+\mathcal{L}_{t})+ \alpha_y \mathcal{L}_{y} + \alpha_z \mathcal{L}_{z} \\ \mathcal{L}_{lemmatization} = \alpha_x \mathcal{L}_{x}+ \alpha_y \mathcal{L}_{y} + \alpha_z \mathcal{L}_{z} \end{aligned}$$ The relative weights of the losses denoted with $\alpha$ are adjusted as hyperparameters.
#### Training Procedure:
Bi-directional training is performed iteratively to calculate forward and backward losses. The pseudocode for training *morphological inflection* task is given in Algorithm \[fig:pseudo\].
$\mathbf{x} \gets [\vec{lemma};\vec{t}]$ $\mathbf{y} \gets \vec{surface}$ $\mathbf{y'}, \mathbf{z'} \gets INN(\mathbf{x})$ $\mathbf{z} \gets GumbelSoftmax (\mathbf{z'})$ $\mathbf{x'} \gets INN(\mathbf{y,z}, \textbf{reverse}=\texttt{True})$ $\mathcal{L}_{total} \gets \alpha_x \mathcal{L}_{x}+ \alpha_y \mathcal{L}_{y} + \alpha_z \mathcal{L}_{z} $ $\mathcal{L}_{total}.backward()$
\[fig:pseudo\]
#### Testing Procedure:
In both models, we choose the word with the highest cosine similarity to the predicted vector during testing. This nearest neighbor search is very efficiently implemented in the gensim framework [@rehurek_lrec]. To predict morphological tagset, we use sigmoid activation function on $\vec{t'}$ and choose the features with activations above $0.5$.
Experiments
===========
We first describe the dataset used in the paper, then detail our experimental design, training settings and evaluation measures.
Dataset {#ssec:datasets}
-------
**Wicentowski** is the most commonly used dataset for lemmatization, which is one of the most interesting ill-posed inverse problems investigated in this paper. Previous studies [@Dreyer2011; @rastogi2016weighting] use a subset of the dataset created by @Wicentowski:2003:MLM:936459 [^4]. However, since previous works evaluate on the lemmatization task only, this subset does not contain any morphological tags but only the unique lemma-surface pairs. Therefore we have used the original dataset that additionally contains morphological tags. In this dataset, each language has its own set of morphological tags. Furthermore, they are arbitrarily written (e.g., 1P and Past instead of person=“1P” and tense=“Past”) without providing the morphological category like *person* and *tense*. The statistics for the dataset is given in Table \[tab:data-stat\].
Experimental Design {#ssec:exp_des}
-------------------
We perform morphological experiments with the proposed framework on 6 different languages from diverse language families and morphology types, namely Turkish (Turkic, agglutinative), Romanian (Romance, fusional), Finnish (Uralic, agglutinative), French (Romance, fusional), Irish (Celtic, fusional) and Tagalog (Austronesian, agglutinative). Agglutinative and fusional languages have different morphological properties. Typical properties of agglutinative languages include (1) the ability to generate/derive words by attaching morphemes like beads on a string; and (2) associating each morpheme with one certain semantic unit only, e.g., using the Turkish “lar” morpheme only for plurality. This leads to having a high ratio of out-of-vocabulary words, however the meaning of words are generally predictable. Unlike agglutinative languages, fusional languages typically have smaller ratio of morphemes per word, i.e., not as productive as agglutinative languages. However one morpheme is generally associated with multiple meaning units, e.g., tense and person information conveyed with one morpheme. This generally results in lower out-of-vocabulary ratios with more ambiguous words, especially when the context is unknown.
Training and Evaluation {#ssec:train_eval}
-----------------------
For all models, we have used 100-dim vectors extracted from pretrained Byte-Pair-Encoding (BPE) model with vocabulary size of 1K, provided by @heinzerling2018bpemb. We have initialized the weight parameters orthogonally. For all INN models, we have used 3 invertible blocks, with 3 fully connected linear layers with hidden dimension of 128 (with ReLU activations in the intermediate layers) as affine coefficient function. We used gradient clipping and early stopping to prevent overfitting. Models are optimized with Adam optimizer with the initial learning rate of 0.001, decreased by 0.3 if scores on development set do not improve for 5 epochs. Unless otherwise stated, we used the weights $\alpha_x=20$, $\alpha_t=10$, $\alpha_y=80$ and $\alpha_z=1$ respectively for $\mathcal{L}_x$, $\mathcal{L}_t$, $\mathcal{L}_y$ and $\mathcal{L}_z$. These weights are intuitively chosen following the previous studies that has employed INNs. We have trained all models for 30 epochs, and did not perform a comprehensive hyperparameter search since the goal of this study is not delivering state-of-the-art results, rather providing and investigate a different perspective.
For the baseline models, we used 3 fully connected linear layers with hidden dimension of 128, with ReLU activations in intermediate layers. We kept the other settings the same as the INNs (except from zero-padding, since it is not necessary for feed forward networks). Our method is not directly comparable to previous methods for three main reasons: (a) none of the previous morphological inflection or works report two (three when tag scores are included) scores with the same model, (b) none of the previous lemmatization studies offer sampling surface forms with the given lemma, (c) most of the previous works are able to generate unseen words, while we treat the generation as “a selection from a large vocabulary” problem that simplifies the problem space. Nevertheless, we have chosen one of the recent state-of-the-art models by [@AharoniG17] as a reference for comparison, to provide an insight about the proposed model’s performance. For evaluation we use the percentage of exact match score for lemma and surface form predictions, and F1 score for morphological tag predictions due to having multiple labels per form.
Since a nearest neighbor search is performed between the network output and the existing word embedding space using cosine similarity, words that have not been encountered in the training data, would not be predicted. In order to address this, we have extended the vector space with 500K most common words, which can be considered quite many, along with the words encountered in the test data during prediction time.
As discussed previously, *morphological analysis*, i.e., inverse of the *morphological inflection* task, may be ill-posed depending on the linguistic properties of the language. For instance, agglutinative languages have one-to-one morpheme-to-tag mapping, while for fusional languages one morpheme may stand for multiple tags. This means that, *morphological analysis* of fusional languages is ill-defined while (mostly) the opposite is true for agglutinative ones. In case of ill-posedness, continuous sampling from $\mathbf{z}$ is necessary to find distributions of the morphological tags, which are hard to score. To simplify scoring of predicted tags, we ignore $\mathbf{z}$ and only consider the most likely tagset in a current setting. Since the ambiguity greatly varies with the language families, we focus on the tag scores of languages with less ambiguous input.
Results and Analysis
====================
We present the results of lemmatization and the morphological inflection tasks for 6 languages in Table \[tab:wicentowski\_res\]. First, we observe that all models in all experimental settings outperform the baseline.
#### Lemmatization:
Inroducing latent variables help increasing the performance of lemmatization *in all languages*, surpassing the [@AharoniG17] by a small margin for Turkish and Irish, and providing similar results for other languages. More specifically, relative performance to [@AharoniG17] ranges between \[-2%, +3%\]. In addition, we observe that larger the $z$ dimension gets, better the scores become in all languages. It may be due to simply providing the network with a larger representation space, allowing for a more flexible learning process. We have then used the trained model to sample surface forms for the given lemma. We have observed that, the model almost always generates a valid surface form when run backwards. However we have also noticed the diversity of the generated surface forms being quite low. This may be due to the statistical properties of the dataset, i.e., observing one dominant morphological tag combination throughout the trainig set. Second reason, is the problem similar to the one described in @BowmanVVDJB16. This study uses Variational Autoencoders (VAE) to generate diverse sentences via the help of latent variable, $z$, that aims to encode useful global information hence enable generation of diverse sentences. However, they observe *KL-divergence* loss of zero that causes the model to ignore $z$.
#### Morphological Inflection:
One of the most important findings for this task is the evidence of network’s ability to provide remarkably strong results for both directions when $\mathcal{L}_y$+$\mathcal{L}_x$ are used. This suggests that, optimizing the same network parameters ($\theta$) with loss functions of dual problems, is feasible and necessary for dual strong results. Although majority of the time, adding $\mathcal{L}_t$, slightly decreased the lemma recovery scores, in Irish and Turkish, we observe improvement on both scores, which may be due to relatively small number of training data. We see mixed results for tag F1 scores, due to varying morphological ambiguity in languages. For agglutinative languages, Finnish, Tagalog and Turkish, where each morpheme is associated with a tag, the ambiguity is lower, therefore F1 scores are higher; in contrast to fusional languages, Romanian and French. Interestingly, even when the network is only optimized for $\mathcal{L}_y$, INNs started to implicitly learn the morphological tags *for all languages*, demonstrated by the tag F1 scores ranging between $11.84$-$36.29$. However no improvement had been observed for the lemma, suggesting that the distribution of the lemma is too complicated to learn implicitly, hence an explicit supervision is necessary. Finally, for all agglutinative languages, lemma exact match scores in *lemmatization* task are better than or very similar to the scores in the inverse task of *morphological analysis*; which can be explained by one-to-one morpheme to tag mapping, that is implicitly learned without any guidance.
Related Work
============
#### Morphological Tasks
Most morphological problems dealt in this paper are also known as string-to-string transduction problems. @DreyerSE08 introduce a general modeling framework based on weighted finite state transducers (WFST) that employ n-gram features and latent variables. They perform experiments on morphological form generation and lemmatization and show that incorporating latent variables improves the results dramatically. Although they use the same framework for both problems, the models are trained separately for each problem. Furthermore, the framework learns the best alignment between the lemma and the inflected form, *separately* for each task, which may not be known beforehand in a realistic scenario. @SchnoberEDG16 compare more recent encoder-decoder architectures such as seq2seq with hard monotonic attention [@AharoniG17] with traditional transduction techniques based on conditional random fields and WFSTs on classical string transduction tasks including lemmatization. They find that although traditional models have similar performance to neural models in most cases, their performances fall behind of the neural models for lemmatization task. This suggests that the lemmatization can be considered as a more complex/ambiguous problem compared to other tasks. @RibeiroNCC18 introduce a method to reduce the transduction to sequence labeling problem, and show that although neural methods perform on par or worse (e.g., on Finnish OCR) in most cases, it is the opposite for the morphological inflection task.
#### Invertibility in NLP
@He_emnlp_18 employ invertible transformations (coupling layers that are very similar to the ones used in this study), to perform unsupervised learning of syntactic structure. They demonstrate the efficiency of the invertibility property on POS tagging and dependency parsing. Recently, @ziegler2019latent proposed a flow-based model for discrete sequences such as text, and show that their proposed autoregressive model performs on par with traditional sequential models on character-level language modeling task.
#### Joint models
A few decades ago, the researchers have attempted to design a unified framework, i.e., a reversible, single grammar, for parsing and generation [@DBLP:conf/coling/Shieber88; @wintner2000amalia]. However, the grammar development has been replaced by advanced statistical tools; therefore those works have not been explored recently. Another set of models that are conceptually similar to ours, perform paired training [@DBLP:conf/acl/KonstasIYCZ17; @DBLP:conf/icml/HuYLSX17; @CaoZLLY19], however still optimize separate models and generally have a more complicated training procedure.
Conclusions
===========
We have proposed modeling several inverse problems in morphology together with their dual problem, such as morphological analysis $\leftrightarrow$ inflection, with recently proposed invertible neural networks that uses a single model for both problems, offer efficient bi-directional training with the help of invertibility layers and provide free inverse mapping. We showed that they are capable of simultaneous optimization of such dual problems providing strong results on both; and lemmatization benefits from additional categorical latent variables. We demonstrated that simple distributions are implicitly learned while complex, multimodal distributions needed supervision. We hope that these initial encouraging results for inverse problems in morphology would inspire the researchers to explore other inverse problems of NLP.
Acknowledgments
===============
This work has been supported by the German Research Foundation through the German-Israeli Project Cooperation (DIP, grant DA 1600/1-1 and grant GU 798/17-1). We would like to thank anonymous reviewers, Jonas Pfeiffer, Mohsen Mesgar, Ilia Kuznetsov and Yevgeniy Puzikov for their constructive feedback. We are grateful to Lynton Ardizzone, Jakob Kruse and Ullrich Köthe for the initial discussions and kindly sharing the INN implementation before the official release with us.
[^1]: In case of a unique solution, we refer to it as a well-posed inverse problem, otherwise it is named as ill-posed following @kabanikhin2008definitions
[^2]: For instance, the Turkish word “dolar” can have the lemma “dol” (to fill), “dola” (to wrap) or “dolar” (dollar), which can only be determined when the surface form “dolar” is given within a sentence. However this task is considered different and named contextual lemmatization. Therefore such cases are not handled in scope of the lemmatization task.
[^3]: One could implement an additional loss to keep padding values closer to zero throughout the training. However we have not seen any significant change in performances when implemented.
[^4]: We thank the author for sharing the dataset with us.
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abstract: 'We present a new causal quantum mechanics in one and two dimensions developed recently at TIFR by this author and V. Singh. In this theory both position and momentum for a system point have Hamiltonian evolution in such a way that the ensemble of system points leads to position and momentum probability densities agreeing exactly with ordinary quantum mechanics.'
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by 1.5truecm by 1truecm
TIFR/TH/98-45\
November 1998\
$^\star$\
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[*Tata Institute of Fundamental Research,*]{}\
[*Homi Bhabha Road, Mumbai 400 005, INDIA*]{}
5ex =10000
PACS: 03.65.Bz
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$^\star$ Based on lecture presented at the Golden Jubilee Workshop on Foundations of Quantum Theory, TIFR, 9–12 September, 1996.
[**1.**]{} . At the 1927 Solvay conference Einstein discussed the example of a particle passing through a narrow hole on to a hemispherical fluorescent screen which records the arrival of the particle. Suppose that a scintillation is seen at a point $P$ at time $t = T$, and suppose that the hole is so narrow that the wave packet corresponding to the particle is uniformly spread all over the screen at $t$ slightly less than $T$. Was the particle somewhere near $P$ at $t = T - \epsilon$ ($\epsilon$ small)? Ordinary quantum mechanics says that the probabilities at $t = T - \epsilon$ for the particle being anywhere on the screen are uniform (and not particularly large in the vicinity of $P$). Thus the naive history corresponding to the idea of a particle with a trajectory ( trajectory) is denied.
There have been recent attempts to define ‘consistent histories’ in quantum mechanics of open systems$^1$. Apart from detailed features found unattractive by some$^2$, there is the basic proposition by the authors themselves that only very special sets of histories are ‘consistent’, and only these can be assigned probabilities. For example, in a double slit interference experiment we cannot assign a probability that the particle reached a region of the screen having earlier passed through slit 1 (except in the case of vanishing interference).
[**2.**]{} . One of the definitions of causality (e.g. that advocated by Jauch) is that “Different results should have different causes”. Consider a quantum superposition $\alpha|+\rangle +
\beta|-\rangle$ for a spin-1/2 particle, where $|+\rangle$ and $|-\rangle$ are eigenstates of $\sigma_z$ with eigenvalues +1 and -1 respectively. When this state is prepared repeatedly and passed through a Stern-Gerlach apparatus to measure $\sigma_z$, a fraction $|\alpha|^2$ of the particles ends up at the detector corresponding to $\sigma_z = +1$, and a fraction $|\beta|^2$ goes to the detector corresponding to $\sigma_z = -1$. Thus different results (going to one detector or the other) arise from exactly the same cause (the same initial state). Of course this lack of causality might be restored in a theory in which the wave function is not a complete description of the state of the system.
[**3.**]{} . In quantum mechanics, $$|\psi(\vec x,t)|^2 d\vec x$$ is the probability of ‘observing’ position to be in $d\vec x$ if position were measured. It is not the probability of position ‘being’ in $d\vec x$ independent of observation. In fact, the same state vector also yields $$|\tilde \psi (\vec p,t)|^2 d\vec p$$ which is the probability of observing momentum to be in the interval $d\vec p$ if momentum were to be measured. The standard dogma is that a simultaneous measurement of position and momentum is not possible. For, if it were possible, it would collapse the state vector into a simultaneous eigenstate of position and momentum which does not exist. Thus, quantum mechanics does not give probabilities for position and momentum in the same experimental situation or ‘’. Moreover, quantum mechanics cannot be embedded in a stochastic hidden variable theory in which ‘reality’ is context independent$^3$. Consider for example Bell’s theorem$^3$ in the context of Einstein-Podolsky-Rosen type measurements of $\vec \sigma_1
\cdot \vec a \ \vec \sigma_2 \cdot \vec b$ for a system of two spin 1/2 particles in the singlet state. It shows that even if the measurements of $\vec \sigma_1 \cdot \vec a$ and $\vec \sigma_2 \cdot
\vec b$ are made at spacelike separation, statistical predictions of quantum mechanics are inconsistent with the assumption that the measured value of $\vec \sigma_1 \cdot \vec a$ has a reality that is independent of whether $\vec \sigma_2 \cdot \vec b$ or $\vec \sigma_2
\cdot {\vec b}'$ is measured together with it. In this example context dependence takes the form of violation of Einstein locality.
[**4.**]{} . De Broglie and Bohm$^4$ (dBB) proposed a theory with position as a ‘hidden variable’ so that $\{\vec
x,|\psi\rangle\}$, i.e., the state vector supplemented by the instantaneous position is the complete description of the state of the system. Here $\vec x = (\vec x_1, \cdots, \vec x_N)$ denotes the configuration space co-ordinate which evolves according to $${d\vec x_i \over dt} = {1 \over m_i} \vec \nabla_i S(\vec x(t),t),$$ where $m_i$ denotes the mass of particle $i$, and the Schrödinger wave function is given by, $$\langle \vec x|\psi(t)\rangle \equiv R \exp i S,$$ with $R$ and $S$ real functions of $(\vec x,t)$. DBB show that if we start at $t=0$ with an ensemble of particles whose position density coincides with $|\psi(\vec x,0)|^2$ at $t = 0$, and evolves with time according to (1), then the position density coincides with $|\psi (\vec
x , t) |^2$ at any arbitrary time $t$. Thus, the phase space density is $$\rho_{dBB} (\vec x , \vec p , t) = |\psi (\vec x, t)|^2 \delta \left(\vec p
- \vec \nabla S (\vec x, t)\right)$$ whose marginal at arbitrary time reproduces the position probability density, $$\int \rho_{dBB} (\vec x, \vec p , t) \ d \vec p = | \psi (\vec x, t)
|^2 .$$ Further, the time evolution (1) corresponds to evolution according to a $c$-number causal Hamiltonian $H_c (\vec x, \vec p, t)$ in which the potential has an added term “the quantum potential” which depends on the wave function.
As far as the position variable is concerned the $dBB$ theory restores history and causality without altering the statistical predictions of quantum mechanics. The lack of Einstein locality is however an essential feature of quantum mechanics. It is in sharp focus in Eq. (1): the velocity of the $i$th particle depends on the instantaneous position of all the particles however far they may be.
The momentum and other variables besides position do not have the same favoured status as position however. As Takabayasi$^5$ pointed out the $dBB$ phase space density does not yield the correct quantum momentum density, i.e., $$\int \rho_{dBB} (\vec x, \vec p, t) d \vec x \neq |\tilde\psi (\vec p,
t)|^2 .$$ To overcome this problem $dBB$ introduce a measurement interaction whose purpose is to convert the preexisting momentum prior to measurement into one whose distribution agrees with the quantum distribution. In contrast, for position, the value observed is the same as the preexisting value. ‘Momentum’ therefore has not the same reality as ‘Position’.
[**5.**]{} . We asked the question$^{6,7}$, is it possible to remove this asymmetrical treatment of position and momentum and build a new causal quantum mechanics in which momentum and position can have simultaneous reality? In Ref. 6, 7 we spelt out an affirmative answer in one dimensional configuration space. here we recall the one dimensional construction and also give the two dimensional generalization.
The point of departure is to seek a phase space density of the form $$\rho (x, p, t) = |\psi (x, t)|^2 \delta (p - \hat p (x, t)) ,$$ where $\hat p (x, t)$ is not given by the $dBB$ formula. Rather, $\hat
p (x, t)$ is to be determined by the requirement $$\int \rho (x, p, t) dx = | \tilde \psi (p, t) |^2 .$$ If we assume that $\hat p (x, t)$ is a monotonic function of $x$ (non-decreasing or non-increasing), $$\delta (p - \hat p (x, t)) = \frac{\delta (x - \hat x (p, t))} {\left|
\frac{\partial \hat p (x, t)} {\partial x}\right|}$$ and $$\rho (x, p, t) = \frac{|\psi (x, t)|^2}{\left| \frac{\partial \hat p
(x, t)} {\partial x} \right|} \delta (p - \hat p (x, t))$$ If we determine $\hat p (x, t)$ such that $$|\psi (x, t)|^2 = \left| \frac{\partial \hat p (x, t)}{\partial
x}\right| |\tilde\psi (p, t)|^2 ,$$ we obtain $$\rho (x, p, t) = | \tilde\psi (p, t) |^2 \ \delta (p - \hat p (x, t))$$ which obeys the desired Eq. (6). Two explicit solutions to Eq. (9), corresponding to non-decreasing $(\epsilon = 1)$ and non-increasing $(\epsilon = -1)$ functions $\hat p (x, t)$ are given by, $$\int^{\hat p (x, t)}_{-\infty} dp' |\tilde\psi (p', t)|^2 =
\int^{\epsilon x}_{-\infty} dx' |\psi (\epsilon x', t)|^2 .$$ Instead of Eq. (5) or Eq. (10), the phase space density may now be written in the symmetric form, $$\rho (x, p, t) = |\psi (x,t)|^2 |\tilde\psi (p, t)|^2 \delta \left(
\int^p_{-\infty} dp' |\tilde\psi (p',t)|^2 - \int^{\epsilon
x}_{-\infty} dx' |\psi (\epsilon x', t)|^2 \right) .$$ We have shown$^{6,7}$ that this phase space density corresponds to evolution of position and momentum according to a $c$-number causal Hamiltonian of the form $$H_c (x,p,t) = \frac{1}{2m} (p - A (x,t))^2 + V (x,t) ,$$ with both $A(x,t)$ and $V(x,t)$ having parts which depend on the wave function. Thus we have two quantum potentials instead of just one in the $dBB$ theory. The existence of $H_c$ ensures that the phase space density obeys the Liouville condition. In one dimension it turns out that $$\frac{dx}{dt} = \left(\frac{dx}{dt}\right)_{dBB},$$ but $$\hat p (x,t) = m \frac{dx}{dt} - A (x,t) \neq \left( m
\frac{dx}{dt}\right)_{dBB} .$$
It may be recalled that like the phase space distribution (12), the Wigner distribution$^8$ also reproduces $|\psi (x,t)|^2$ and $|\tilde
\psi (p, t)|^2$ as marginals. However the Wigner distribution is not positive definite and cannot therefore have a probability interpretation; further the condition of Hamiltonian evolution of $x,
p$ and the consequent Liouville property are valid for the phase space density (12) but not for the Wigner distribution.
In higher dimensions there is a surprise. E.g. for $n=2$, we can construct a causal quantum mechanics in which the quantum probability densities corresponding to three different complete commuting sets (CCS) of observables, e.g. $(X_1, X_2), (P_1, X_2), (P_1, P_2)$ is simultaneously realized. Explicitly, the positive definite phase space density $$\rho (\vec x, \vec p, t) = |\psi (x_1, x_2, t)|^2 |\psi (p_1, x_2,
t)|^2 |\psi (p_1, p_2, t)|^2 \delta (A_1) \delta (A_2)$$ where $$\begin{aligned}
A_1 &\equiv& \int^{p_1}_{-\infty} |\psi (p'_1, x_2, t)|^2 dp'_1 -
\int^{x_1}_{-\infty} |\psi (x'_1, x_2, t)|^2 dx'_1 , \\
A_2 &\equiv& \int^{p_2}_{-\infty} |\psi (p_1, p'_2, t)|^2 dp'_2 -
\int^{x_2}_{-\infty} |\psi (p_1, x'_2, t)|^2 dx'_2 ,\end{aligned}$$ reproduces as marginals the correct quantum probability densities $|\psi (x_1, x_2, t)|^2 , |\psi (p_1, x_2, t)|^2$ and $|\psi (p_1,
p_2, t)|^2$. (For notational simplicity we have omitted the tildas denoting Fourier transforms). The corresponding velocities are however in general different from the $dBB$ velocities. The calculation of the velocities, and the explicit causal Hamiltonian for $n \geq 2$ will be given in detail in a later publication$^9$ where we show that the quantum probability densities corresponding to $n+1$ CCS can be simultaneously realized (i.e., with one phase space density). A recent quantum phase space contextuality theorem proves that$^{10}$ it is impossible to simultaneously realize quantum probability densities corresponding to all possible CCS. We conjecture that the causal quantum mechanics we have constructed is maximally realistic.
[**References**]{}\
1. R.B. Griffiths, [*J. Stat. Phys.*]{} , 219 (1984); [*Phys. Rev. Lett.*]{} , 2201 (1993); M. Gell-Mann and J.B. Hartle, in Proc. 25th Int. Conf. on High Energy Physics, Singapore 1990, eds. K.K. Phua and Y. Yamaguchi (World Scientific, 1991); R. Omnés, [*Rev. Mod. Phys.*]{} , 339 (1992) and “” (Princeton Univ. Press 1994).
2. F. Dowker and A. Kent, [*J. Stat. Phys.*]{} , 1575 (1996); [*Phys. Rev. Lett.*]{} , 3038 (1995).
3. A.M. Gleason, [*J. Math. & Mech.*]{} , 885 (1957); S. Kochen and E.P. Specker, [*J. Math. & Mech.*]{} , 59 (1967); J.S. Bell, [*Physics*]{} , 195 (1964); A. Martin and S.M. Roy, [*Phys. Lett.*]{} , 66 (1995).
4. L. de Broglie, “Nonlinear Wave Mechanics, A Causal Interpretation”, (Elsevier 1960); D. Bohm, [*Phys. Rev.*]{} , 166; 180 (1952); D. Bohm and J.P. Vigier, [*Phys. Rev.*]{} , 208 (1954).
5. T. Takabayasi, [*Prog. Theor. Phys.*]{} , 143 (1952).
6. S.M. Roy and V. Singh, ‘Deterministic Quantum Mechanics in One Dimension’, p. 434, Proceedings of International Conference on Non-accelerator Particle Physics, 2-9 January, 1994, Bangalore, Ed. R. Cowsik (World Scientific, 1995).
7. S.M. Roy and V. Singh [*Mod. Phys. Lett.*]{} , 709 (1995).
8. E. Wigner, [*Phys. Rev.*]{} , 749 (1932).
9. S.M. Roy and V. Singh, preprint TIFR/TH/98-42.
10. A. Martin and S.M. Roy, [*Phys. Lett.*]{} , 66 (1995).
|
---
abstract: 'Swarms of coupled mobile agents subject to inter-agent wireless communication delays are known to exhibit multiple dynamic patterns in space that depend on the strength of the interactions and the magnitude of the communication delays. We experimentally demonstrate communication delay-induced bifurcations in the spatio-temporal patterns of robot swarms using two distinct hardware platforms in a mixed reality framework. Additionally, we make steps toward experimentally validating theoretically predicted parameter regions where transitions between swarm patterns occur. We show that multiple rotation patterns persist even when collision-avoidance strategies are incorporated, and we show the existence of multi-stable, co-existing rotational patterns not predicted by usual mean field dynamics. Our experiments are the first significant steps towards validating existing theory and the existence and robustness of the delay-induced patterns in real robotic swarms.'
author:
- Victoria Edwards
- Philip deZonia
- 'M. Ani Hsieh'
- Jason Hindes
- Ioana Triandaf
- Ira B Schwartz
title: Delay Induced Swarm Pattern Bifurcations in Mixed Reality Experiments
---
> There is interest in deploying swarms of autonomously interacting robots for cooperative tasks such as search and rescue, exploration and mapping, distributed sensing and estimation, and more. However, accounting for the uncertainties of the physical world is challenging when validating with simulation alone, and experimental validation using large numbers of real robots requires overcoming significant logistical and resource constraints; e.g., space, cost, manpower. A useful intermediate approach for systematic experimental validation is to couple simulation with real robots in a mixed reality framework. Mixed reality retains the key features of physical experiments that are hard to capture through simulation alone. In particular, realistic inter-agent wireless communication is extremely difficult to capture in simulation given the computational complexities of faithfully modeling the physics of RF propagation. The mixed reality framework enables the study of the impact of wireless communication effects, such as delays in communication, on swarms with physical robots. Existing theory has shown that when inter-agent communication delays are introduced into an ideal swarm with simple interaction rules, collective rotation patterns can emerge. In this work, we take steps toward testing such patterns with swarms of real robots, and characterize when transitions occur between patterns in a mixed reality setting.
Introduction
============
Natural swarms have inspired researchers to understand how simple organisms produce complex emergent patterns and behaviors. Swarms in nature are composed of individual agents with relatively simple behaviors that interact locally to synergistically yield complex collective behavior. Examples of such swarms in nature include: schools of fish [@Couzin2013; @Calovi2014], flocks of starlings [@Leonard2013; @Cavagna2015] and jackdaws [@Ouellette19], colonies of bees [@Li_Sayed_2012] and ants [@Theraulaz2002], aggregation of locust [@Topaz12], and crowds of people [@Rio_Warren_2014].
When examining biological swarms, recent analysis has shown that there exists a delay in reaction time between agents. In other words, as agents move, they react in response to the past positions of their neighbors rather than their instantaneously detected positions. For example, delays have been measured in schooling fish [@JL_fish], bats [@Luca_bats], birds [@Nagy_pigeons] and crowds of people [@JF_people]. Since natural agents move in an almost continuous manner, it is natural to model swarms based on biological ideas as continuous systems with communication delays.
Swarms can be modeled by differential delay equations where the delay is included in the communication network between agents. Such delays can act as a destabilizing parameter, which means for a swarm with communication delay different spatio-temporal patterns will emerge. Pattern emergence was shown by analyzing the mean field of a globally coupled swarm in the presence of communication delays where a Hopf bifurcation can be observed [@Romero2012]. Recent work has generalized this result with an exact stability analysis and the inclusion of range dependent delay [@Hindes2020; @Schwartz2020]. In Szwaykowska et al. [@Szwaykowska2016], theoretical analysis showed that the delay-induced bifurcation picture is robust to random link removal in a swarm’s communication network and to agent heterogeneity. Preliminary mixed reality experiments tested one of the theoretically predicted swarming behaviors for a restricted parameter set, but did not show transitions between swarm behaviors based on parameter changes.
In general, existing works in the design and control of artificial swarms have focused on the synthesis of local interaction rules that give rise to global swarm behavior. These works focus on bottom-up strategies where the objective is to develop provably correct single robot strategies that yield some desired swarm behavior [@Tanner03a; @Tanner03b; @Tanner07; @Jadbabaie03; @Viraghn14; @Gazi05; @Desai01]. Unsurprisingly, these works are based on a strict set of assumptions, which are necessary to ensure the desired emergent behavior. Nevertheless, while the strategies have been tested in simulation, validation on physical systems is often problematic since the necessary assumptions do not always hold in the real-world. Thus, the extensive body of work has provided a strong theoretical foundation for the design of swarming strategies but few have been experimentally validated on actual systems.
Experimental validation with physical robots invariably introduces uncertainty in the form of actuation, sensing, and robot-robot and robot-environment interaction noise. Since swarms are complex nonlinear dynamical systems, they can typically exhibit multiple steady-state patterns [@Romero2014]. In the presence of noise it is possible to transition from one steady-state to another [@Freidlin1998; @Szwaykowska2018]. Additionally, changes in system parameters can result in changes to the stability nature of steady-states [@Wells15; @Hartle17; @Ansmann16]. In this work we conduct experiments with robots, which requires dealing with uncertainties inherent in physical experiments as well as changes in system parameters.
The dramatic reduction in the price to performance ratio of embedded processors, sensors, and computers and the ubiquity of wireless communications technologies have made experiments with ever larger number of robots more viable. Examples of these experiments include the Kilobots [@Rubenstein12] which consists of 1000 robots interacting in a limited environment and the Crazyswarm [@Preiss17] which consists of 50 unmanned aerial vehicles executing planned trajectories through the environment. Ultimately, the logistics of dealing with a large number of physical entities in a confined workspace require trade-offs: either the simplification of the environment or the use of open-loop, [*e.g.*]{}, pre-computed, strategies that are not adaptive or reactive to changes in the environment or internal state of the swarm. Given these logistical challenges, experimental validation of swarming strategies are more often conducted using small numbers of robots which do not account for large number effects [@Li17; @Pickem17; @Dorigo2013; @Ipparthi17; @Warkentin18]. Unsurprisingly, experiments on small numbers of robots in controlled laboratory settings limit the types and frequency of robot-robot and robot-environment interactions. Furthermore, such methodologies cannot suitably evaluate the performance of the coordination strategies for arbitrarily large team sizes. Thus, the results may not fully account for the many factors that affect the dynamics of emergent swarming behavior.
In this work, we addressed these experimental challenges in engineered swarm systems by proposing a mixed reality experimental framework as a first significant step towards full experimental validation. Mixed reality is the use of both virtual robots and real robots in both the simulated and real world [@Honig15], that retains critical features of physical robots while enabling scaling to larger numbers of agents, or larger workspaces, without being subject to the physical limitations of resources. The benefits that come from mixed reality include: ability to work with large numbers of robots [@Edwards18] and ensuring safety in human robot interactions [@Quinlan10; @Huang19; @Rosen19]. Experiments using mixed reality come at a lower cost due to the reduced number of robots needed, while still introducing complex dynamics of the real world from a few real robots. Mixed reality is a significant first step towards full scale experimental validation of theoretical findings. Furthermore, the mixed reality framework provides opportunities to gain additional insights into the theory and improving experiment design.
Our current research uses mixed reality as a way to further study the controller proposed in Szwaykowska et al. [@Szwaykowska2016], and to map out experimentally a complete bifurcation picture in terms of physical parameters. In addition to uncovering the bifurcation structure of the swarm dynamics, we will focus on understanding transitions between behaviors and the impacts of adding collision avoidance. The new experiments are done using two different platforms of interest: one uses an Unmanned Aerial Vehicle (UAV), and the other uses an Autonomous Surface Vehicles (ASV), both within a mixed reality framework. The use of two different platforms has several advantages. First, it tests the universal bifurcation structure of delay coupled swarms across different platforms and vastly different time scales. Second, it allows for different numbers of robots and constraints to be tested safely during experimentation.
The layout of the paper is as follows. In section \[sec:method\] we describe in detail the model used to control the swarm. In section \[sec:experiments\] we outline the experimental setup for both UAV and ASV experiments, and in section \[sec:results\] we explain the results observed. In section \[sec:discussion\] we discuss different phenomena that arose during experimentation, [*[e.g.]{}*]{}, platform differences, obstacles encountered, and multi-stability. In section \[sec:conclusion\] we conclude.
Methodology {#sec:method}
===========
Consider a swarm composed of $N$ robots positioned in the plane, $r_i \in \mathbb{R}^2$, where $i \in {1 ... N}$. We begin by detailing the development of our single agent and swarm ensemble models.
Single agent model
------------------
The dynamics for each agent in the system consist of a self propulsion term, an attraction term, and a repulsion term. This can be mathematically represented as follows using the original equation proposed by Mier-y-Teran-Romero [@Romero2012] for the $i^{th}$ agent:
$$\dot{r_i} = v_i,$$
and
$$\begin{aligned}
\ddot{r_i} = (1 - ||\dot{r_i}||^2)\dot{r_i} - \sum_{j\in N} \nabla_{r_i} U[r_i(t),r^{\tau}_j(t)],
\label{eq:original}\end{aligned}$$
To model the communication topology between agents, we consider a fully connected graph model, $G = (\mathcal{E}, \mathcal{V})$, where $\mathcal{E}, \mathcal{V}$ are the set of edges and vertices, or nodes, respectively. We improve upon the validity of this model by having the robots communicate a delayed position $r^{\tau}_j(t) = r_j(t - \tau)$. Note that the robots interact with one another with a fixed time delay, $\tau$, which captures realistic finite-time effects of robot to robot communication.
We assume a harmonic interaction potential defines the attraction term
$$\begin{aligned}
U(r_i, r_j^{\tau}) = f(r_i, r_j) + \frac{a}{2N}(r_i - r_j^\tau)^2,
\label{harmonic_equation}\end{aligned}$$
where $a$ is a constant, and $f(r_i, r_j)$ is a repulsion term.
In previous theoretical work the repulsion force was added to only a fraction of the agents in the experiments [@Szwaykowska2016]. However, for real systems to interact safely in the world repulsion forces for all agent interactions are necessary, making it important to extend this work to consider the addition of repulsion to all agents in the swarm. As long as the repulsion force selected is an anti-symmetric function in the neighboring robot’s states, the analysis performed in section \[sec:ensemble\_swarm\_model\] of the global swarm behavior will be preserved.
For the experimentation done in this paper, two anti-symmetric functions were selected. The original repulsion force presented in Szwaykowska et al.[@Szwaykowska2016] is:
$$\begin{aligned}
f(r_i, r_j) = c_re^{\frac{||r_i - r_j||}{l_r}},
\label{eq:exponential_rep}\end{aligned}$$
where $c_r$ is the strength of the repulsion, and $l_r$ is the radius of repulsion considered between agents.
However, Equation \[eq:exponential\_rep\] does not account for limitations of physical platforms, [*[e.g.]{}*]{}, max speeds or acceleration capacity. As such a sigmoid repulsion function is used:
$$\begin{aligned}
f(r_i, r_j) = \Big(c_{r} - \frac{c_{r}}{1+e^{-k(|r_i - r_j|-R_{rep})}}\Big)\frac{r_i - r_j}{|r_i - r_j|}.
\label{eq:sigmoid_repulsion}\end{aligned}$$
where $c_{r}$ is the maximum repulsion strength, $R_{rep}$ is the inter agent distance at which the repulsion force is at half strength, and $k$ represents how quickly the magnitude of the repulsion force switches from maximum strength to zero. Note that the repulsion term is independent of the delay since the interactions for repulsion are local in space.
Ensemble Swarm Model {#sec:ensemble_swarm_model}
--------------------
The mean field of a swarm is computed by taking $R=\frac{1}{N}\sum_{i=1}^{N}r_i$ to denote the center of mass, and consider the limit as $N \rightarrow \infty$. From a mean field analysis of Equation \[eq:original\], analytical expressions can be derived for different swarm states[@Romero2012]. The swarm state is the global representation of the entire swarm evaluated by observing the center of mass of the swarm, $R$, in lieu of the position of all agents. The state of the robot, $r_i$, is the individual dynamics for the local behavior of the robot.
The mean field of the original controller for the swarm behavior was shown theoretically to have several bifurcating regions in parameter space. Each region implies the stability of certain swarm behaviors including: flocking, ring, and rotating swarm states.
Figure \[fig:dim\_bifurcation\] is the converted dimensional version of the original dimensionaless bifurcation structure proposed in Szwaykowska et al [@Szwaykowska2016]. The transition between region II and III was theoretically predicted by a Hopf bifurcation curve of the mean field, and a pitchfork bifurcation curve is predicted to separate regions I and II. Figure \[fig:sim\_example\] illustrates the three basic modal patterns of the swarm behavior as a function of the coupling strength, $\alpha$, and communication delay, $\tau$.
For the application of the dynamical model to the real world, we consider the dimensionalized equation for each agent as follows:
$$\begin{aligned}
\ddot{r_i} &= \beta(v_g^2 - ||\dot{r_i}||^2)\dot{r}_i - \frac{\alpha}{N}\sum_{j=1,j \neq i}^N(r_i - r_j^\tau) + \sum_{j=1, j\neq i}^N\nabla_r f(r_i, r_j),
\label{eq:dimensional}\end{aligned}$$
where $v_g$m/s is the asymptotic velocity of the agent in the absence of coupling, $\alpha$1/s$^2$ is the coupling strength, $\beta$s/m$^2$ is a dimensional factor, and $\nabla_{r_i} f(r_i, r_j)$ is the repulsion force.
In order to further study the theoretically predicted swarm states, it was necessary to compute a new dimensional bifurcation diagram, which considers the physical parameters used in Equation \[eq:dimensional\]. To achieve Figure \[fig:dim\_bifurcation\] the following conversions were used:
$$\begin{aligned}
t' = \beta v_g^2 t, \\
r'_i = \beta v_g r_i, \\
a = \frac{\alpha}{\beta^2v_g^4},
\label{eq:parameter}\end{aligned}$$
where $a, t',$ and $r'_i$ are dimensionless.
The conversion of the bifurcation diagram allows for regions of interest to be isolated, specifically around the regions of uncertainty along the Hopf bifurcation. The original mean-field analysis does not change through conversion, which is demonstrated in the appendix where the mean field was re-derived for Equation \[eq:dimensional\].
The three desired swarm states are highlighted in Figure \[fig:dim\_bifurcation\]. In region I the swarm is in a translating state, where all agents are in alignment going in one direction. In region II the swarm is in a ring state, where all agents move about a stationary center of mass, and in region III the swarm is in a rotating state, where all agents cluster and move as a collective on a circular orbit around the origin. Along the boundaries of each region the swarm state will transition from one swarm state to another. This is referred to as the transition between swarm states. The possibility of multi-stable co-existing rotational patterns along these regions makes experimental verification important to further understand the limitations of the mean field analysis.
Experiments {#sec:experiments}
===========
The objective of our experiments is to test the mean field predictions described in Section \[sec:method\] in swarms composed of a few real robots. Discrepancies can then be used to build more accurate theories and analysis for future experiments with larger numbers of real robots. Mixed reality experiments were conducted using both the Ascending Technologies Inc. Pelican quadrotor shown in Figure \[fig:UAV\] and custom built ASVs shown in Figure \[fig:ASV\]. We show experimentally all three swarm states (translating, ring, and rotating), along with the transition between swarm states as predicted by the bifurcation diagram in Figure \[fig:dim\_bifurcation\] within the mixed-reality framework. We describe the details of our mixed reality architecture, experimental platforms, and experimental methodology in the following sections.
Mixed reality system architecture
---------------------------------
An outline of a mixed reality system is in Figure \[fig:mixed\_reality\], and an example of a mixed reality experiment is depicted in Figure \[fig:mr\_robot\_example\]. In building a mixed reality, global positions of the real robots are necessary, [*e.g.*]{}, GPS, infrared camera system, SLAM. The ground truth position of the robots are provided to the simulator, which maintains all simulated agents. The resulting response outputs for the robots are computed and sent to the robot from the simulator, considering interactions with all simulated agents.
The mixed reality system is controlled by the simulator, which maintains the positions of the simulated robots and updates the positions of the real robots based on ground truth information, as seen in Figure \[fig:mixed\_reality\]. The simulator workspace is defined as an unbounded region with no obstacles. The origin of the simulated workspace corresponds to the origin of the physical workspace. Delayed information is stored in a fixed length list for each agent which holds previous positions. The length of the list corresponds to the amount of delay specified: larger $\tau$ is a longer list, and smaller $\tau$ is a shorter list. The last entry in the list represents the delayed information received by an agent. All agents leveraged the global knowledge of the simulator to compute the distances between agents. Sensing between agents was abstracted away, which allowed for the focus of the results to be on the swarm states. From the simulator, new control commands were sent to the UAV/ASVs based on delayed information of simulated agents, as if they were in the world. Mixed reality allowed the UAV/ASVs to express the swarm behavior without the risk of multiple real robots interacting in unpredictable ways or the cost of running a multi-robot experiment.
Experimental Platforms
----------------------
In this work we employed two experimental platforms: unmanned aerial vehicles (UAVs) and autonomous surface vehicles (ASVs). The validation using two separate platforms shows the applicability to any vehicle platforms whose dynamics can be abstracted into Equation \[eq:original\]. The UAV, Figure \[fig:UAV\], is a quadrotor vehicle, which is equipped with an Odriod for onboard computing, WiFi communication, and an inertial guidance system (AscTec Autopilot). The vehicle is approximately $65.1$cm $\times$ $65.1$cm and $18$cm tall and weighs $1.65$kg. The workspace is a $15$m $\times$ $10$m $\times$ $8$m room. The ASVs, Figure \[fig:ASV\], are differential drive surface vehicles equipped with a micro-controller board, XBee radio module, and an inertial measurement unit (IMU). The vehicles are approximately $12$ cm long and have a mass of about $45$g each. The ASVs are deployed in a $3$ m $\times$ $5$ m $\times$ $1$m oval tank. Both workspaces include an infrared (IR) camera system to provide robot localization information.
Experimental Methodology
------------------------
The experiments with the UAVs consisted of 1 physical and 49 simulated robots. The motions of simulated agents are updated using a double-point-integrator with Equation \[eq:dimensional\]. For the translating swarm state, and the transition swarm state experiments the original configurations consisted of all agents facing in the same direction in a fixed pattern, with the same initial input velocity of 0.2m/s. For the ring and rotating swarm state experiments, the UAV was placed at $[0, 0, 0]$ with the simulated robots placed around a rough ring shape with initial velocities in x and y selected between $[-0.3, 0.3]$m/s. To achieve transition between swarm states $\Delta \tau$ was added every 10s to $\tau$. Details of experimental parameters are in Table \[table:UAV\_params\]. Parameters were chosen for Equations (\[eq:parameter\]) to satisfy maximum speed constraints for the real robots and finite workspace size. Using the theoretical equations predicting the ring and rotating state radii, parameters were selected which provided the desired radii size ($\approx 0.75$m). The parameters were tested in simulation before being tested in experiments.
A small amount of repulsion was introduced to the experiment, Equation \[eq:exponential\_rep\], where $c_r = 1.2$ and $l_r = 0.01$. To accommodate the low repulsion forces, the experiments were constrained to a two dimensional slice such that each real and simulated agent were on a unique plane. The interactions between agents were achieved by projecting all agents onto the same two dimensional plane, and velocities for the robot were a two dimensional velocity with an additional altitude component [@Mellinger11].
Swarm State Duration $\alpha$ 1/s$^2$ $\tau$s $\beta$s/m$^2$ $v_g$m/s $\Delta \tau$
------------------- ---------- ------------------ --------- ---------------- ---------- --------------- --
Translating 30 0.01 0.01 20.0 0.2 0.0
Ring 90 0.09 2.5 20.0 0.2 0.0
Rotating 90 1.0 4.0 20.0 0.2 0.0
Tran - Ring - Rot 200 1.5 0.01 20.0 0.2 0.3
: Experimental parameters for UAV mixed reality experiments[]{data-label="table:UAV_params"}
Next we considered the impacts of adding more robots to the swarm, thus requiring stronger repulsive forces. To do this we used a team of ASVs shown in Figure \[fig:ASV\]. The experiments consisted of 3 physical and 12 simulated robots. These experiments were specifically designed to observe transitions between swarm states by modifying the delay provided to the system. These were untested properties of the swarm model, and mixed reality allowed for testing theoretical predictions.
The parameters for the ASV mixed reality experiments are in Table \[table:ASV\_params\]. For all experiments starting in the translating swarm state, the ASVs were initialized in a formation based on relative positions and moving forward at the desired speed $v_g$. For the experiment starting in the ring swarm state the ASVs were initialized at three equidistant points on a circle of radius 0.3m pointing counter clockwise and tangent to the circle, and the virtual agents were initialized with random heading at points along the circumference of the circle with a normally distributed amount of noise added to their positions. Due to the physical constraints of the ASVs, sigmoid repulsion[@repulsion_details] (Equation \[eq:sigmoid\_repulsion\]) was used in Equation \[eq:dimensional\].
Swarm State Duration $\alpha$ 1/s$^2$ $\tau_0$s $\beta$s/m$^2$ $v_g$m/s $\tau_1$s
--------------- ---------- ------------------ ----------- ---------------- ---------- ----------- --
Tran-Ring-Rot 660 0.01 0.0 18.0 0.0471 10, 35
Tran-Rot-Tran 400 0.01 0.0 18.0 0.0471 35
Tran-Ring 300 0.01 0.0 18.0 0.0471 10
Ring-Rot 380 0.01 10.0 18.0 0.0471 35
: Experimental parameters for ASV mixed reality experiments[]{data-label="table:ASV_params"}
Experimental Evaluation
-----------------------
Evaluation of all experiments was done using swarm polarization, which is a measure of alignment between agents in a swarm. Swarm polarization is computed as follows: $$\begin{aligned}
sp = \frac{||\sum_i r_i||}{\sum_i ||r_i||}.\end{aligned}$$ This metric evaluates the swarm state, for example; in the ring state all the individual positions of the agents will cancel out resulting in a swarm polarization of $\approx 0$, while in the rotating and translating states the swarm is aligned resulting in a swarm polarization of $\approx 1$. We note that swarm polarization has been proposed in earlier works, [@Armbruster16; @Tunstrom13], and is a measure of alignment, but can be computed in different ways.
In addition to swarm polarization, the velocity of the center of mass and the acceleration of the center of mass were computed. These metrics aided in determining the difference between the translating and rotating swarm states, which have the same swarm polarization value. The rotating swarm state has a high velocity and acceleration of the center of mass, while in the translating swarm state the center of mass velocity is high but the acceleration is low. Velocity of the center of mass is computed as follows: $$\begin{aligned}
c_{vel} = ||\frac{\sum_i^N v_i}{N}||,\end{aligned}$$
and the acceleration of the center of mass is:
$$\begin{aligned}
c_{acc} = ||\frac{\sum_i^N \dot{v_i}}{N}||.
\end{aligned}$$
Additional comparisons were done using theoretically predicted results from equations in Szwaykowska et al. [@Szwaykowska2016], and experimental results for the ring swarm state radius and period along with the rotating swarm state radius and period are presented in Figure \[theory\_v\_experiment\_figure\]. These equations are listed in the appendix.
Results {#sec:results}
=======
UAV Simulation of Parameters in Different Bifurcation Regions
-------------------------------------------------------------
![ The resulting swarm polarization from 240 simulation trials with 50 simulated agents. Each trial was 100s and used UAV parameters, $\alpha \in [ 0.0, 4.75]$1/s$^2$, $\tau \in [0.01, 5.51]$s, $\beta = 20.0$s/m$^2$, and $v_g = 0.2$m/s. For each $\alpha$, $\tau$ was updated by 0.5s. At the end of the range for $\tau$, $\alpha$ was updated by 0.251/s$^2$. Color is a representation of the final swarm polarization at the end of an experiment. The trends of the contours while slightly off of the theoretically predicted bifurcation curve still meet the general trend of the plot. A transition from white to black to white in the plot, corresponds to the transition from the translating to ring to rotating swarm states. []{data-label="bifurcation_data_figure"}](Figure4.pdf){width="\linewidth"}
Simulation trials using UAV parameters were executed with different combinations of $\alpha$ and $\tau$, testing the theoretically predicted bifurcation regions in Figure \[fig:dim\_bifurcation\]. The simulation trials used parameters corresponding to the UAV experiments. The trials ran for 100s, with $\beta = 20.0$s/m$^2$ and $v_g = 0.2$m/s. The initial $\alpha = 0.0$1/s$^2$ and $\tau = 0.01$ were the parameters used for the first trial. At the completion of a trial $\tau$ was updated by $0.5$. The range of $\tau$ was $\tau \in [0.01, 5.51]$s for each value of alpha. When a trial for each value in the range of $\tau$ was complete $\alpha$ was updated by 0.25. The range of $\alpha$ was $\alpha \in [0.0, 4.75]$1/s$^2$. In total, two hundred and forty simulation trials were run with 50 simulated agents. Figure \[bifurcation\_data\_figure\] shows the resulting plot of the final swarm polarization at the end of each simulation, for each set of parameters, which shows fair qualitative agreement with mean field predictions.
Regions of multi-stability exist along the transition between swarm states, meaning that for certain initial conditions the swarm will transition to another swarm state or it may stay in the same swarm state [@Romero2014]. This is clearly seen by the jagged edge along the bifurcation in Figure \[bifurcation\_data\_figure\]. This result showed that more experimentation using physical robots needed to be done, exploring each of the three swarm states along with transitions between the swarm patterns.
UAV Mixed Reality Results {#sec:uav_mr}
-------------------------
The first set of experimental results used a UAV mixed reality framework. Experiments of the swarm in the translating swarm state were achieved using parameters from region I in Figure \[fig:dim\_bifurcation\], where all agents move with the same average velocity. The swarm would quickly start moving out of the translating swarm state and into the ring swarm state in the last seconds of the experiment. This was due to the instability of the swarm when translating, from the introduction of any noise in the system, which can switch the system out of the translating swarm state and into the ring swarm state.
Experiments where the ring swarm state was expressed used parameters from region II in Figure \[fig:dim\_bifurcation\], where all agents move, clockwise or counter-clockwise, around a stationary center of mass. To further test that the dynamic model matched experimental results, comparisons were made between the theoretically predicted radius and the period of the ring swarm state to the experimental values. Figures \[fig:ring\_period\] and \[fig:ring\_radius\] show the average swarm ring period and radius experimental result (blue points) and the theoretically predicted value (red line). These experiments show that at steady-state the average swarm ring radius converges to approximately 0.68m and the average swarm ring period converged to approximately 21.22s, these values were both close to the theoretically predicted values of 0.66m and 20.94s. To achieve the experimental ring radius and ring period, parameters were selected which put the swarm clearly in the ring swarm state. The theoretical values were computed based on the parameters selected for the experiments. The proximity of the measured ring radius and ring period comes from ensuring well calibrated experiments which expressed the desired behavior. This result qualitatively supports the comparison between the swarm theory and the experimental results.
The rotating swarm state experiments used parameters from Region III in Figure \[fig:dim\_bifurcation\]. During the rotating swarm state experiment all agents clustered together and moved in a collective around a stationary point. Figures \[fig:rot\_period\] and \[fig:rot\_radius\] show the average radius and period of the swarm in the rotating state, comparing the theoretically predicted values (red line) to the experimental results (blue points). The theoretical period was 6.55s and the theoretical radius was 0.32m, compared with the converged average of the experimental period was 5.68s and the experimental radius was 0.26m. Similarly to the experiments with the swarm in the ring state, parameters were specifically selected to put the swarm clearly in the rotating swarm state. The measured rotating radius and rotating period are close to theoretical predictions because of well tuned experiments resulting in the desired behavior. The plots highlight that the swarm does converge to the theoretically predicted swarm behavior even with the addition of a real robot.
![ Swarm polarization, center of mass acceleration, center of mass speed for an experiment with 1 UAV and 49 simulated robots. A 200s experiment transitioning between translating, ring, and rotating swarm state, where the experiment started with the following parameters $\alpha = 1.5$1/s$^2$, $\tau_0 = 0.01$s, $\beta = 20.0$s/m$^2$, $v_g = 0.2$m/s, and every 10 seconds $\tau$ was updated by $\Delta \tau = 0.3$s. The increment in tau caused the transition between the translating state to the ring state to occur between $\tau = [1.81, 2.41]$s and the transition from the ring to the rotating state to occur at $\tau= [3.61, 4.21]$s, labeled on the plot by red lines. []{data-label="switching_exp_data_figure"}](Figure6.pdf){width="\linewidth"}
Finally, Figure \[switching\_exp\_data\_figure\] depicts the experimental results from an experiment transitioning through all three swarm states. The translating swarm state had swarm polarization of $\approx 1$, a low center of mass acceleration, and a high center of mass speed. The speed of the center of mass decreased rapidly, with the addition of greater values of $\tau$ causing the swarm to begin to switch into the ring swarm state. The transition to the ring swarm state occurred between $\tau = [1.8, 2.41]$s, where the swarm had a stationary center of mass resulting in 0 swarm polarization, 0m/s center of mass speed, and 0 m/s$^2$ center of mass acceleration. The final transition to the rotating swarm state occurred between $\tau = [3.61, 4.21]$s, resulting in a swarm polarization of $\approx 1$, a high center of mass acceleration, and a high center of mass speed. Although qualitatively accurate, the existence of small discrepancies between predicted and measured dynamics for a swarm with a single real UAV suggests the need for a more accurate description of the UAV dynamics.
ASV Mixed Reality Results
-------------------------
ASV mixed reality experiments were performed to further investigate the transition between swarm states, and safely increase the number of robots. These experiments consisted of 3 ASVs and 12 simulated robots. Each experiment had multiple trials of different types of sigmoid repulsion [@repulsion_details]. The first ASV mixed reality experiments tested the transitions between all three swarm states. Additional experiments tested the transitions from translating swarm state to rotating swarm state, from the translating swarm state to the ring swarm state, and finally from the ring swarm state to the rotating swarm state.
The results from the experiments are presented in the following figures. Figure \[upenn\_exp\_type3rep\_swarm\_stats\] depicts results from the translating to rotating swarm transition. The behavior of the swarm during the translating to ring swarm state transition is observed in Figure \[upenn\_exp\_type2rep\_swarm\_stats\]. Finally, results for the transition between the ring to rotating swarm state are presented in Figure \[upenn\_exp\_type1rep\_swarm\_stats\], with Figure \[fig:upenn\_experiments\_8\_frames\] depicting the time series snapshots for $3$ ASV and 12 virtual agents transitioning from the ring to rotating swarm state.
From the experimental results we observed and measured the different theoretically predicted swarm states. The ring swarm state can be identified by the low swarm polarization, center of mass acceleration, and center of mass speed indicative of an unaligned swarm that is stationary. The rotating swarm state can be identified by the high polarization, center of mass acceleration, and center of mass speed indicative of an aligned swarm that is constantly moving in a circle. All of these traits can be seen in Figure \[upenn\_exp\_type1rep\_swarm\_stats\]. The oscillations in the center of mass speed and acceleration, in Figure \[upenn\_exp\_type1rep\_swarm\_stats\], were the result of the swarm moving on an ellipsoidal trajectory, slowing down near the loci and speeding up near the semi-minor axis. The dips in the polarization were the result of the existence of two groups of agents in the swarm that turn in different directions when the whole swarm reverses direction near the loci of the ellipse. The ellipsoidal motion gradually relaxed to a circle with all of the agents turning together in the same direction, with center of mass acceleration and velocity reaching steady state values and the polarization approaching a steady state value of 1.
Likewise, in Figure \[upenn\_exp\_type2rep\_swarm\_stats\] there is a spike in polarization around the 50s mark. This was due to the swarm reversing direction coherently before scattering into the ring swarm state. In the translating and rotating swarm state, the ASVs adopted a hexagonal grid formation in the rough shape of a disk that remained rigid even during transitions between the rotating and translating state.
The experimental results successfully reproduced and extended the results obtained from the mixed reality UAV experiments described in section \[sec:uav\_mr\]. The ASV mixed reality experiments showed the persistence of the swarm states and the transition between the states even when collision avoidance routines were executed on all robots and virtual agents. These results are a step towards experimental validation because full scale robot experiments will require collision avoidance for safe robot-robot interactions.
![Swarm polarization, center of mass acceleration, and center of mass speed for the ASV swarm using local sigmoidal repulsion[@repulsion_details] during a mixed reality experiment. A delay of 10s was introduced at t = 25s causing a translating to ring transition. The spike in polarization at t = 50s is caused by the agents momentarily reversing direction and translating in an aligned formation before breaking up.[]{data-label="upenn_exp_type2rep_swarm_stats"}](Figure7.jpg){width="\linewidth"}
![ Swarm polarization, center of mass acceleration, and center of mass speed for ASV swarm using sensed sigmoidal repulsion[@repulsion_details] during a mixed reality experiments. A delay of 35s was introduced at t = 40s causing a translating to rotating transition. The dips in swarm polarization were caused by the agents disagreeing on which way to turn in the early stages of the transition, where the agents were in a degenerate rotating state (one characterized by the swarm moving back and forth along a line). As the eccentricity of the ellipse lessened, more agents agreed on which direction to turn at the vertices of the ellipse.[]{data-label="upenn_exp_type3rep_swarm_stats"}](Figure8.jpg){width="\linewidth"}
![ Swarm polarization, center of mass acceleration, and center of mass speed for the ASV swarm using global sigmoidal repulsion[@repulsion_details] during a mixed reality experiment. A delay of 40s was introduced at t = 60s causing a ring to rotating swarm state transition. The oscillation in all measures was the result of the motion of the agents moving on a gradually widening ellipse. When the eccentricity of the ellipse was high, the agents slowed down as they changed direction (not necessarily the same direction) at the ends of the ellipse, and sped up near the center of the ellipse. This behavior gradually smoothed out as the motion approached that of a circle for large t.[]{data-label="upenn_exp_type1rep_swarm_stats"}](Figure9.jpg){width="\linewidth"}
Discussion {#sec:discussion}
==========
Mixed reality provided a general framework to test the theoretical predicted behaviors with two different robotic platforms. Our results cover a large swath of the parameter space for the theoretically predicted swarming patterns. Our experimental results were obtained using vehicles with distinct dynamics, communication delays, and coupling strengths, nevertheless the experiments all exhibited the global patterns and pattern switches predicted by theory.
Different bifurcation diagrams were built for each combination of $\beta$ and $v_g$, which helped inform where the transition points between swarm states might occur in a physical experiment. The robotic platform differences contributed to a wide range of parameters that were necessary to achieve the different swarm states. Additionally, different initial conditions for the swarms were outlined in the experimental section to compensate for platform differences.
In general, the UAV mixed reality experiments had faster dynamics and limited battery life, which restricted flight time. Shorter UAV experiments tested individual behaviors at higher speeds, as well as select long experiments to observe transitions through all the swarm states. Mixed reality allowed for the use of only one physical robot to capture complex interactions with the real world, while still maintaining a swarm of simulated agents [@new_results]. Likewise, the 3 dimensional experiments with the dynamics of each robot constrained to the 2 dimensional slices demonstrate that the behaviors are possible for a UAV and a direction of future work is to consider what unconstrained 3 dimensional dynamics looks like with the addition of multiple real UAVs. In the ASV mixed reality experiments, the dynamics were slower and the workspace constrained to 2D. Nevertheless, these experiments captured the full effects of on-board collision avoidance routines on swarm pattern formation and pattern transitions and proved the persistence of the swarm states and the robustness of the communication delay induced switches in the global pattern formation. These experiments are a first attempt at validating the theoretically derived results, Figure \[fig:dim\_bifurcation\], in the presence of realistic real-world robot-robot and robot-environment interactions.
During experimentation, there were problems that each robotic platform exhibited. In mixed reality experiments with the UAV for conditions testing the rotating swarm state, the UAV lagged behind the center of mass. This issue stems from internal safety features on the UAV to prevent high speeds. In the rotating swarm state the swarms acceleration is high, meaning that the simulated robots exceed the set point velocity $v_g$. However, the real robot has internal mechanisms onboard which prevent rapid increases in speed. We attempted to use different $\alpha$ values for the simulated robots and the real robot to try and pull the swarm together, but the real robot did not merge with the rest of the virtual cluster. This behavior was not expressed in the ASV experiments.
The ASV mixed reality experiments introduced collision avoidance to the swarm dynamics. The addition of collision avoidance risks the complete destruction of the theoretically predicted swarming patterns. Since collision avoidance presents a hard constraint, pairs of vehicles that are in immediate danger of collision would end up being forcefully pushed apart by their collision avoidance routines. The result is either rings of larger thickness or agents that swing towards the center of the ring, Figure \[upenn\_exp\_loose\_ring\].
It is hard to analytically describe the multi-stability present in Equation \[eq:dimensional\]. In addition to multi-stability observed in Figure \[bifurcation\_data\_figure\], multi-stability was also observed in Figure \[multi\_stable\_figure\], where given three similar initial conditions the amount of time the system used a specific amount of delay impacted the transition between the ring and rotating swarm state. For example, in Figure \[fig:mult\_stable\_fast\] the transition to the rotating swarm state happened for $\tau = 4.81$s, in Figure \[fig:mult\_stable\_medium\] the transition happened for $\tau = 3.5$s and in Figure \[fig:mult\_stable\_slow\] the transition happened at $\tau = 4.0$s. These variances in transition points come from different initial configurations of the swarm, and the evolution of the system.
Conclusion and Future Work {#sec:conclusion}
==========================
Through manipulating communication delay and coupling strength we took the first significant steps to test different emergent swarm states using mixed reality. The mixed reality framework allowed for the study of emergent swarm behavior through the use of simulation to increase the number of agents while maintaining critical real world interactions, which are hard to model, with physical platforms. Because mixed reality can handle many different levels of abstraction we were able to use two distinct robotic platforms to validate the swarm behavior. We selected the level of abstraction necessary for the behavior to exhibit, and in our case it was focused on agents that exhibited simple dynamics and used delayed information as would be done in the real world. Both the ASVs and the UAV tested all three of the theoretically predicted behavior along with transitions between the predicted swarm states. This emphasizes that the proposed swarm model has the potential to be applicable across platforms, and highlights the impacts of communication delay on systems behavior.
There is a range of theory that supports the proposed model, and the presented results are significant steps toward showing theory is valid, as well as also demonstrating that there are areas of interest that the theory is not capturing. Understanding the multi-stability in this system is difficult analytically, but with the use of simulation and mixed reality we were able to observe multi-stability and the impacts it has on the proposed theoretical model when paired with real vehicles.
The next steps for this work include investigating how the addition of more real world assumptions change the predicted emergent patterns. For example, our communication model of global coupling is not practical with all real robots due to network limitations. Next steps, are to study the impacts of range based communication. While often swarms are studied for homogeneous agents it is also possible to consider the impacts of heterogeneity as was done in Szwaykowska et al. [@Szwaykowska2016], this may require different types of collision avoidance to be used to account for different hardware limitations. Finally, there exist different dynamic models, where delay can be added. This presents new potential patterns to be studied using the described forms of analysis. These changes will continue to add to the understanding of our current models and the use of this type of emergent behavior in the physical world.
Acknowledgment
==============
Edwards, Triandaf, Hindes, and Schwartz gratefully acknowledge the Office of Naval Research (ONR) for their support under N0001412WX20083, support of the NRL Base Research Program N0001412WX30002, and the NRL Karles Fellowship Program. DeZonia and Hsieh gratefully acknowledge the support of ONR Award No. N00014-18-1-2580 and ARL DCIST CRA W911NF-17-2-0181.
Appendix
========
Mean Field Analysis
-------------------
We will consider the mean field analysis when the collision avoidance term is anti-symmetric to the robot’s neighbors, resulting in negligible impact to the remaining analysis. We will consider the mean of the swarm as $R(t) = \frac{1}{N} \sum_{i=0}^N r_i(t)$, which is the average center of mass, for all agents in the swarm. $R(t)$ has units meters. Likewise consider that $\delta r_i(t) = r_i(t) - R(t)$ is the distance of each particle $r_i$ in meters from the center of mass in meters, thus $\delta r_i(t)$ has units meters.
We know that the $\sum_{i=0}^N \delta r_i = \sum_{i=0}^N \delta \dot{r_i} = \sum_{i=0}^N \delta \ddot{r_i} = 0$. We will now consider the substitution into Equation \[eq:dimensional\] using the above relationships:
$$\begin{aligned}
\ddot{R} + \ddot{\delta r_i} = \beta(v_g^2 - || \dot{R} + \delta \dot{r_i}||^2) (\dot{R} + \delta \dot{r_i}) \nonumber \\
- \frac{\alpha}{N} \sum_{j=1}^N (R + \delta r_i - R^{\tau} - \delta r_j^{\tau}), \end{aligned}$$
where $R^{\tau} = R(t - \tau)R$ and the units of $\ddot{R}$ is m/sec$^2$. Multiplying out all the relationships and condensing like terms we get the following:
$$\begin{aligned}
\ddot{R} = -\ddot{\delta r_i} + \beta \dot{R} (v_g^2 - ||\dot{R}||^2) + \beta (v_g^2 - || \dot{R}||^2) \delta \dot{r_i}
\nonumber \\
- \beta ( || \delta \dot{r_i}||^2 + 2<\dot{R}, \delta \dot{r_i}>)(\dot{R} + \delta \dot{r_i})
\nonumber \\
- \frac{\alpha}{N} \sum_{j=1}^N (R - \delta r_i - R^{\tau} - \delta r_j^{\tau}),\end{aligned}$$
where $<.,.>$ is the dot product. Taking the sum over all $r_i$ in the swarm and dividing by $N$ you get the following equation:
$$\begin{aligned}
\ddot{R} = -\frac{1}{N}\sum_{i=0}^N \delta \ddot{r_i} + \beta \dot{R} (v_g^2 - ||\dot{R}||^2)
+ \frac{\beta}{N} (v_g^2 - || \dot{R}||^2)\sum_{i=0}^N\delta \dot{r_i}
\nonumber \\
- \frac{\beta}{N} (|| \delta \dot{r_i}||^2 + 2 < \dot{R}, \delta \dot{r_i}>)(\dot{R} + \delta \dot{r_i})
\nonumber \\
- \frac{\alpha}{N}(R - R^{\tau}) - \sum_{i=0}^N\sum_{j=0, i \neq j}^N (\delta r_i - \delta r_j^{\tau}).\end{aligned}$$
The observations were made in previous work that the tendency is for particles to stay close to the center of mass, which means the impacts of deviations from the center of mass play less and less importance as N increases [@Szwaykowska2016]. If we consider as $N \rightarrow \infty$ then we can ignore the individual impacts of $\delta r_i$. We know that $\sum_{i=0}^N \delta \ddot{r_i} = 0$, eliminating the first term in the equation.
Thus giving us the following equation in m/s$^2$:
$$\begin{aligned}
\ddot{R} = \beta(v_g^2 - ||\dot{R}||^2) \dot{R} - \frac{\alpha}{N}(R - R^{\tau}).\end{aligned}$$
Evaluation Metrics
------------------
The mathematics behind the prediction of the radius of the ring and rotating states along with the derivation of the angular velocity, was originally presented in Szwaykowska et al. [@Szwaykowska2016]. $\rho_{ring}$ is the radius of the ring and $\omega_{ring}$ is the angular velocity of the swarm. The equations are:
$$\begin{aligned}
\rho_{ring} = \frac{\sqrt{ 1 / a }}{v_g \beta},
\\
\omega_{ring} = \frac{\sqrt{a}}{v_g^2 \beta}. \end{aligned}$$
Likewise, the radius of rotation and angular velocity of the swarm in the rotating state are represented by $\rho_{rot}$ and $\omega_{rot}$. The equations are: $$\begin{aligned}
\rho_{rot} = \frac{1}{|\omega_{rot}|v_g \beta}\sqrt{1 - \frac{a\sin(\omega_{rot} \tau)}{\omega_{rot}} },
\\
\omega_{rot}^2 = \frac{a}{v_g^2 \beta} [ 1 - \cos(\omega_{rot} \tau)]. \end{aligned}$$
Note that the $\alpha$ used in the experiment needs to be converted to $a$ using the following conversion: $ a = \frac{\alpha}{\beta^2 v_g^4}$. Due to noise related to measuring $\omega_{ring}$ and $\omega_{rot}$ values from experimental data, we instead compute the period of the swarm in the ring and rotating swarm state through the following conversion $T_{ring} = \frac{2\pi }{\omega_{ring}}$ and $T_{rot} = \frac{2\pi}{\omega_{rot}}$.
![ Example of ring state with occupied middle region due to the slow acceleration of the ASVs. The inability of the ASVs to accelerate back up to full speed after a run-in with another agent resulted in them traveling on a highly eccentric path that frequently passed through the center.[]{data-label="upenn_exp_loose_ring"}](Figure11.jpg){width="0.8\linewidth"}
![Behavior of swarm with global sigmoidal repulsion during a ring to rotating transition. A delay of 40s was added at t = 60s.[]{data-label="fig:upenn_experiments_8_frames"}](Figure12.jpg){width="0.8\linewidth"}
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---
abstract: 'It has been known for some time that General Relativity can be regarded as a Yang-Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an $SO(1,4)$ or $SO(2,3)$ gauge field $A^{a}_{{\phantom}{a}b\mu}$ and Higgs field $V^{a}$ which acts to break the symmetry down to that of the Lorentz group $SO(1,3)$. This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field $\Phi$ of electroweak theory is taken as a genuine dynamical field satisfying a Klein-Gordon equation, the gauge independent component $V^2$ of the Higgs-type field $V^a$ is typically regarded as non-dynamical. Instead, in many treatments $V^a$ does not appear explicitly in the formalism or is required to satisfy $V^2\equiv \eta_{ab}V^{a}V^{b}=const.$ by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection $A^{a}_{{\phantom}{a}b\mu}$ and Higgs field $V^a$ as genuine dynamical fields. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar-tensor model of gravity. A specific choice based on the symmetry group $SO(1,4)$ yields a positive cosmological constant and an effective mass $M$ of the gravitational Higgs field ensuring the constancy of $V^2$ at low energies and agreement with empirical data if $M$ is sufficiently large. More general actions are discussed corresponding to variants of Chern-Simons modified gravity and scalar-Euler form gravity.'
author:
- 'H.F. Westman$^{1}$'
- 'T.G. Zlosnik$^{2}$'
bibliography:
- 'references.bib'
title: Gravity from dynamical symmetry breaking
---
Introduction
============
The question of whether the gravitational field is essentially similar to other fields in nature or not has been a long running theme in physics. In particular, how far do the commonalities between gravitation and the gauge theories of particle physics extend? The basic ingredients of these gauge theories are gauge fields which are one-forms valued in the Lie algebra of of some group $G$. In gauge theories with a spontaneously broken symmetry there are in addition to gauge fields Higgs fields which are scalar fields valued in some representation of $G$ which may break the symmetry $G$ down to a residual symmetry $H\subset G$ at the level of the equations of motion via the attainment of non-vanishing vacuum expectation values. An example of this in the standard model of particle physics is the electroweak theory. In that model the gauge fields are a $U(1)$ gauge field $B\equiv B_{\mu}dx^{\mu}$ coupling to hypercharge and an $SU(2)$ gauge field $C^{{\cal A}}_{{\phantom}{{\cal A}}{\cal B}} \equiv C^{{\cal A}}_{{\phantom}{{\cal A}}{\cal B}\mu}dx^{\mu}$ coupling to isospin, where ${\cal A},{\cal B},..$ are $SU(2)$ indices. These fields are accompanied by a $U(1)\times SU(2)$-valued field $\Phi^{\cal A}$, called the electroweak Higgs field. When $\Phi^{\dagger}\Phi \equiv \left<\Phi^{\dagger}\right>\left<\Phi\right>\neq 0$ the $SU(2)\times U(1)$ gauge symmetry is broken leaving a remnant $U(1)$ symmetry preserving the form of $\Phi^{\cal A}$. In this example then the group $G$ is identified with $U(1)\times SU(2)$ and $H$ is identified with the $U(1)$ symmetry of electromagnetism.
We now consider the gravitational field. This field is typically described entirely by a rank-2 space-time tensor $g_{\mu\nu}$ referred to as the metric tensor. We will refer to this as the second-order formulation of gravity. In the sense of the above definitions, this field is neither a gauge field nor a Higgs field [^1]. However, the metric formulation of gravity may be indeed be recovered from ingredients that have more, but not everything, in common with gauge theory. This is possible via the first-order Einstein-Cartan formulation of gravity [@Kibble:1961ba] wherein gravity is described by an $SO(1,3)$ gauge field $\omega^{I}_{{\phantom}{I}J}\equiv \omega^{I}_{{\phantom}{I}J\mu}dx^{\mu}$, referred to as the spin-connection, and a Lorentz valued one-form $e^{I}\equiv e^{I}_{\mu}dx^{\mu}$, referred to as the co-tetrad. From the Einstein-Cartan perspective, gravitation is described by gauge invariant actions with the Lorentz group $SO(1,3)$ as the local symmetry group $G$. The metric tensor is identified with $\eta_{IJ}e^{I}_{\mu}e^{J}_{\nu}$ where $\eta_{IJ}=\mathrm{diag}(-1,1,1,1)$ is the invariant $SO(1,3)$ matrix.
While the connection $\omega^{I}_{{\phantom}{I}J}$ has all the mathematical characteristics of a standard gauge field the co-tetrad $e^I$ does not appear to have a counterpart in gauge theory [^2]. It is a $SO(1,3)$-valued vector but also possesses a space-time index $\mu$, a combination not found in any other known fundamental field in physics. This dissimilarity with gauge theory may ultimately be superficial if it turns out that the co-tetrad is a composite object formed of more familiar objects. For example, an approach which has attracted recent interest is to retain the $SO(1,3)$ gauge field and regard $e^{I}$ as just such a composite object. For instance [@Akama:1978pg; @Diakonov:2011im; @Obukhov:2012je] have considered models involving sets of $spin(1,3)$ spinor fields $\{\Psi_{(i)}\}$ where $e^{I}_{\mu}\equiv i\sum_{(i)}\left(\bar{\Psi}_{(i)}\gamma^{I}D_{\mu}\Psi_{(i)}-D_{\mu}\bar{\Psi}_{(i)}\gamma^{I}\Psi_{(i)}\right)$.
An alternative to the above spinorial approach is to regard the $SO(1,3)$ symmetry of the Einstein-Cartan theory as the remnant symmetry $H\subset G$ after spontaneous symmetry breaking, i.e. we begin with a symmetry group larger than $SO(1,3)$ and via symmetry breaking retain only local Lorentz symmetry. The ingredients necessary for this will thus be standard quantities from gauge theory: a gauge field $A^{a}_{{\phantom}{a}b}\equiv A^{a}_{{\phantom}{a}b\mu}dx^{\mu}$ and a Higgs field $V^{a}$ where $a,b,..$ are gauge indices of the larger group. The larger group can be taken to be the de Sitter $SO(1,4)$ group, the anti-de Sitter $SO(2,3)$, or the Poincare group $ISO(1,3)$. This approach is that of Cartan-geometry [@Wise:2009fu] where the gauge field and Higgs field admit a simple geometrical interpretation in terms of ‘idealized waywisers’ [@Westman:2012xk]. The gauge connection $A^{a}_{{\phantom}{a}b }$ dictates how much a symmetric space (either the de-Sitter, anti-de Sitter, or Minowksi spacetime) is rotated when rolling without slipping along some path on the spacetime manifold. The Higgs field $V^a$ corresponds to the ‘point of contact’ between the symmetric space and the manifold.
If we are interesting in eliminating differences between gravity and the gauge theories of particle physics, this approach is promising as one may use the above ingredients to construct an object which, like the co-tetrad, possesses a space-time index and a gauge index. This is simply the covariant derivative of $V^{a}$: $$\begin{aligned}
D_{\mu}V^{a} \equiv \partial_{\mu}V^{a}+A^{a}_{{\phantom}{a}b \mu}V^{b}. \label{dvo}\end{aligned}$$ By construction, this quantity transforms as a one-form under coordinate transformations and as a vector under $SO(1,4)/SO(2,3)$ transformations. For concreteness we restrict attention to cases where the larger group is either of the 10-parameter groups $SO(1,4)$ and $SO(2,3)$. Consider a field $V^{a}$, where $a,b,c,..$ now specifically refer to $SO(1,4)$ or $SO(2,3)$ indices. If $V^{2}\equiv \eta_{ab}V^{a}V^{b} = - \ell^{2}$ for $SO(2,3)$ and $V^{2}=\ell^{2}$ for $SO(1,4)$ then a gauge may be found where $V^{a}\overset{*}{=}\ell \delta^{a}_{4}$ (the notation $\overset{*}{=}$ denotes an equation holding in a particular preferred gauge), where $\ell$ is some real non-zero constant. One may check that the generators of $SO(1,4)/SO(2,3)$ which leave the above form of $V^{a}$ invariant are those of the Lorentz group $SO(1,3)$. Therefore for the symmetry breaking ansatz $V^{2}= \mp \ell^{2}$ we have $G=SO(1,4)/SO(2,3)$ and $H=SO(1,3)$. We will refer to any indices projected along $V^{a}$ as $4$ and the remaining $SO(1,3)$ indices as $I,J..$. Given the ansatz for $V^{2}$ we have from (\[dvo\]) that: $$\begin{aligned}
D_{\mu}V^{a} &\overset{*}{=}& A^{a}_{{\phantom}{a}4\mu}\ell \label{dv}\end{aligned}$$ If can be seen that if $V^{2}=\mathrm{const.}$ then $D_{\mu}V^{a}$ only contains non-vanishing components orthogonal to $V^{a}$ and transforms homogeneously under the $SO(1,3)$ transformations defined by $V^{a}$. Therefore $D_{\mu}V^{a}$ behaves like a co-tetrad. It additionally follows that $A^{I}_{{\phantom}{I}J}$ can play the role of the spin-connection $\omega^{I}_{{\phantom}{I}J}$. In these senses then, the familiar objects of Einstein-Cartan theory may be recovered in the symmetry broken phase of a $SO(1,4)/SO(2,3)$ gauge theory. No objects beyond a gauge field and a Higgs field have been introduced.
Though this construction seems promising, it must be shown that $SO(1,4)/SO(2,3)$ invariant actions built from the pair $\{A^{a}_{{\phantom}{a}b},V^{a}\}$ exist that can reduce to Einstein-Cartan gravity, and thus General Relativity, after symmetry breaking. This was first shown to be possible by Stelle and West [@Stelle:1979va; @Stelle:1979aj] and later by Pagels [@Pagels:1983pq]. Due to the relationship between the variables $\{A^{a}_{{\phantom}{a}b},V^{a}\}$ and Cartan geometry [@SharpeCartan; @Wise:2006sm], we refer to gravity seen through these variables as Cartan Gravity.
In these papers, the recovery of the co-tetrad via (\[dv\]) was aided by actually enforcing the symmetry breaking ansatz $V^{2}=\mp \ell^{2}$ by adding a Lagrange multiplier to the gravitational action. By analogy, in the electroweak theory this would be akin to fixing $\Phi^{\dagger}\Phi =\mathrm{const.}$ via a similar constraint. It appears to be the case however that fluctuations of the scalar $\Phi^{\dagger}\Phi$, in the form of the electroweak Higgs particle, have been detected. Therefore in electroweak theory, variations in the action $\Phi^{{\cal A}}$ are not restricted. If indeed Cartan gravity represents a step closer to the remaining parts of physics, it would seem that fluctuations in $\eta_{ab}V^{a}V^{b}$ should similarly be allowed.
This implies that symmetry breaking solutions to a Cartan gravity model should follow from free variation of fields. We note that the authors of [@Stelle:1979va] indeed considered actions where $V^{a}$ was allowed to vary. However, we note that these actions are non-polynomial in the fields $\{A^{a}_{{\phantom}{a}b},V^{a}\}$. As a simplifying principle we shall seek to consider only actions that are polynomial in these fields. Therefore our restrictions represent a point of departure from prior work [@Wilczek:1998ea; @Leclerc:2005qc; @Tresguerres:2008jf].
The plan of this paper is as follows: In Section \[poly\] we exhibit the most general polynomial Cartan gravity action. In Section \[decomp\] we discuss a choice of variables that enables the general action of \[poly\] to be written in a particularly transparent form and facilitates reformulating the first-order theory in second-order form. In \[spece\] we consider a subset of this general action and show that these types of theories are equivalent to a class of scalar-tensor theories. As may be expected, the scalar degree of freedom is encoded in the norm $V^{2}$. In Section \[general\] we consider the physical interpretation of more general actions, and in Section \[vnull\] we discuss behaviour in the event that $V^{2}=0$. In Section \[discussion\] we discuss the results of the previous sections.
Throughout the paper we work in units in which $c$ and $\hbar$ are dimensionless and numerically equal to one. Duration is measured in meters $m$ and mass in inverse meters $m^{-1}$.
Polynomial actions for gravity {#poly}
==============================
In accordance with the discussion in the previous section we will look to construct gravitational actions from the pair $\{A^{a}_{{\phantom}{a}b},V^{a}\}$ which are each allowed to vary freely up to the restriction that $\delta A^{a}_{{\phantom}{a}b}=\delta V^{a}=0$ on the boundary. As a further guide, we will additionally look to construct only actions that are invariant under $SO(1,4)/SO(2,3)$ and polynomial in the basic fields $\{A^{a}_{{\phantom}{a}b},V^{a}\}$. We shall see that these restrictions still allow a non-trivial collection of possible terms in the action and we shall here consider the most general combination of them.
Given our restrictions, what terms may be considered? In general these terms will be actions (defined as integrals over space-time four-forms) built from the invariants $\eta_{ab},\epsilon_{abcde}$, the field $V^{a}$, and the curvature two-form of $A^{a}_{{\phantom}{a}b}$ defined as $F^{a}_{{\phantom}{a}b}=\frac{1}{2}F^{a}_{{\phantom}{a}b\mu\nu}dx^{\mu}\wedge dx^{\nu}$ where $F^{a}_{{\phantom}{a}b\mu\nu}\equiv 2\partial_{[\mu|}A^{a}_{{\phantom}{a}b|\nu]}+2A^{a}_{{\phantom}{a}c[\mu|}A^{c}_{{\phantom}{c}b|\nu]}$. The $SO(1,4)$/$SO(2,3)$ indices may be raised with $\eta^{ab}$ and lowered with $\eta_{ab}$. For notational compactness we denote the wedge product $y\wedge z$ between differential forms $y$ and $z$ simply as $yz$. For example, if $y$ is a three-form and $z$ is a one-form then we have:
$$\begin{aligned}
\int yz &=& \int y \wedge z \\
&=& \frac{1}{3!} \int y_{\mu}z_{\nu\sigma\delta} dx^{\mu}\wedge dx^{\nu}\wedge dx^{\sigma}\wedge dx^{\delta} \\
&=& \frac{1}{3!}\int \varepsilon^{\mu\nu\sigma\delta}y_{\mu}z_{\nu\sigma\delta} d^{4}x\end{aligned}$$
where $\varepsilon^{\mu\nu\sigma\delta}$ is the Levi-Civita density. For an exposition of the calculus of variations using differential forms we refer to [@Westman:2012xk]. The most general action consistent with our requirements is as follows:
$$\begin{aligned}
{\nonumber}S[V^{a},A^{ab}]&=&\int \left(\alpha_{1} \epsilon_{abcde}V^{e}+\alpha_{2}V_{a}V_{c}\eta_{bd}+\alpha_{3}\eta_{ac}\eta_{bd}\right)F^{ab}F^{cd}+\left(\beta_{1} \epsilon_{abcde}V^{e}+\beta_{2}V_{a}V_{c}\eta_{bd}+\beta_{3}\eta_{ac}\eta_{bd}\right)DV^{a}DV^{b}F^{cd}\\
&&+\gamma_{1} \epsilon_{abcde}V^{e}DV^{a}DV^{b}DV^{c}DV^{d} \label{act1}\end{aligned}$$
The $\alpha$,$\beta$, and $\gamma$ may themselves be polynomial functions of $V^{2}$. The action (\[act1\]) may look unfamiliar. It contains terms quadratic in the curvature $F^{ab}$ but terms contributing to the equations of motion necessarily couple to the gravitational Higgs field $V^{a}$; the only such term which may be independent of $V^{a}$ is the $\alpha_{3}$ term. However, if $\alpha_{3}$ is constant then the term is simply proportional to $\int F_{ab}F^{ab}= \int d (A^{ab}F_{ab}+\frac{1}{3}A^{ac}A_{a}^{{\phantom}{a}d}A_{cd})$ and so may be neglected as a boundary term.
Decomposing the connection {#decomp}
==========================
The equations of motion deduced from a variational principle of polynomial actions written in terms of forms and gauge covariant derivatives can be shown to invariably be first-order partial differential equations [@Westman:2012zk]. Thus, the above polynomial action will lead to first-order partial differential equations in the dynamical variables $V^a$ and $A^{a}_{{\phantom}{a}b}$ which do not look very familiar. In order to see how one may recover more familiar looking second-order differential equations we shall in this section introduce a ‘covariant’ decomposition of the connection $A^{a}_{{\phantom}{a}b}$. We first recall a simplifying strategy from the Einstein-Cartan theory of gravity where gravity is described by the pair $\{\omega^{I}_{{\phantom}{I}J},e^{I}\}$. However, in doing variations of actions it is convenient to make the following decomposition: $$\begin{aligned}
\omega^{IJ}= \bar{\omega}^{IJ}+{\cal C}^{IJ} \label{vij}\end{aligned}$$ where $\bar{\omega}^{IJ}$ is determined *entirely* by $e^{I}$ and its partial derivatives via the following equation:
$$\begin{aligned}
D^{(\bar{\omega})}e^{I}=0. \label{toro}\end{aligned}$$
which can be solved yielding $$\begin{aligned}
\bar{\omega}^{IJ}_{{\phantom}{ab}\mu} &=& 2(e^{-1})^{\nu[I}\partial_{[\mu}e^{J]}_{\nu]}+e_{\mu K}(e^{-1})^{\nu I}(e^{-1})^{\alpha J}
\partial_{[\alpha}e^{K}_{\nu]}\end{aligned}$$ where $(e^{-1})^{\mu}_{I}$ is the inverse of the matrix $e^{I}_{\mu}$ referred to as the tetrad. By inspection $\bar{\omega}^{IJ}$ transforms as a gauge field and thus alone becomes responsible for the inhomogeneous transformation law of $\omega^{IJ}$; the one-form ${\cal C}^{IJ}$, referred to as the contorsion, transforms homogeneously under a gauge transformation [@Leclerc:2005ig; @Hehl:2007bn]. The two-form $D^{(\omega)}e^{I}$ is referred to as the torsion and hence, as a solution to (\[toro\]), $\bar{\omega}^{IJ}$ is often referred to as the torsion-free spin-connection. The equations of motion for the Einstein-Cartan theory may be obtained by varying $e^{I}$ and ${\cal C}^{IJ}$ independently. A reason for why this is useful is that the equations of motion for ${\cal C}^{IJ}$ take a particularly simple form in Einstein-Cartan theory in the presence of the matter fields of the standard model of particle physics. Indeed, of these fields it is only spinorial matter that is expected to couple to the contorsion [@Hehl:1971qi]; the coupling is such that one can solve algebraically for the contorsion in terms of spinor current. This allows the contorsion to be reinserted into the action and eliminated from the variational principle. The resultant variational principle, dependent upon $e^{I}$ and $\bar{\omega}^{IJ}(e^{K})$, corresponds to the Einstein-Hilbert action and after the addition of the Gibbons-Hawking boundary term yields the Einstein equations for the metric tensor $g_{\mu\nu}$ following variation with respect to $e^{I}$.
We would like to make a similar decomposition in the Cartan gravity case. The idea again will be to encode the inhomogeneous transformation property of the connection in a quantity defined analogously to (\[toro\]). The additional, homogeneously transforming terms in the connection will hopefully simplify analysis if some of them may be eliminated from the variational principle.
Consider the following decomposition of $A^{ab}$:
$$\begin{aligned}
A^{ab} &=& \bar{A}^{ab} + B^{ab} \label{split1}\end{aligned}$$
As for the case of $\bar{\omega}^{IJ}$ in the Einstein-Cartan model, the field $\bar{A}^{ab}$ encodes the inhomogeneous transformation properties of the connection $A^{ab}$. We propose the following analogue of equation (\[toro\]):
$$\begin{aligned}
D^{(\bar{A})}D^{(A)}V^{a}=0 \label{cotro}\end{aligned}$$
We will return to the interpretation of this equation shortly. The remaining part of the connection is the field $B^{ab}$, and by construction it transforms homogeneously with respect to $SO(1,4)/SO(2,3)$ gauge transformations. As follows, to simplify analysis we will assume that $V^{a}\neq 0$. From a dynamical perspective this restriction is not natural but we will see that the recovery of a four-dimensional metric in the theory is dependent the norm of $V^{a}$ being non-zero. Given this assumption, we may define the quantity $E^{a} \equiv B^{ab}V_{b}$. Furthermore, we may define the field $C^{ab}\equiv B^{ab}- (2/V^{2})E^{[a}V^{b]}$. We shall see that after symmetry breaking, $E^{a}$ plays the role of the co-tetrad $e^{I}$ whilst $C^{ab}$ plays the role of the contorsion ${\cal C}^{IJ}$. In this sense, the form $B^{ab}$ unifies co-tetrad and contorsion into a single object. For the null case where $V^{a}\neq 0$, $V^{2}=0$ we shall see that though the field $E^{a}$ is still well-defined, it corresponds to a metric tensor $\eta_{ab}E^{a}_{\mu}E^{b}_{\nu}$ which is three dimensional and therefore care is required.
In the event that $V^{2}\neq 0$ it may be seen that (\[cotro\]) is satisfied if two independent conditions hold:
$$\begin{aligned}
D^{(\bar{A})}V^{a} &=& \frac{1}{2V^{2}}dV^{2}V^{a} \label{c1}\\
D^{(\bar{A})}E^{a} &=& 0 \label{c2}\end{aligned}$$
This implies the following solution:
$$\begin{aligned}
\label{wab}
\bar{A}^{ab}= -\frac{2}{V^{2}}dV^{[a}V^{b]}+\bar{\omega}^{ab}\end{aligned}$$
where
$$\begin{aligned}
\bar{\omega}^{ab}_{{\phantom}{ab}\mu} &=& 2(E^{-1})^{\nu[a}\partial_{[\mu}E^{b]}_{\nu]}+E_{\mu c}(E^{-1})^{\nu a}(E^{-1})^{\alpha b}
\partial_{[\alpha}E^{c}_{\nu]}\end{aligned}$$
Where we have used the field $(E^{-1})^{\mu a}$, the ‘inverse’ of $E^{a}$, which satisfies: $V_{a}(E^{-1})^{a\mu}=0$, $E^{a}_{\mu}(E^{-1})^{\mu}_{b}= P^{a}_{{\phantom}{a}c}P^{c}_{{\phantom}{c}b}$, $E_{a \mu}(E^{-1})^{a\nu}=\delta^{\nu}_{\mu}$. The matrix $P^{a}_{{\phantom}{a}b} \equiv \delta^{a}_{b}-V^{a}V_{b}/V^{2}$ acts to project along internal ‘directions’ orthogonal to $V^{a}$. By inspection the solutions for $\bar{\omega}^{IJ}$ and $\bar{\omega}^{ab}$ are identical up to interchange of $\{e^{I}_{\mu},(e^{-1})^{\mu}_{I}\}$ with $\{E^{a}_{\mu},(E^{-1})^{\mu}_{I}\}$. The notational coincidence is deliberate. As noted, after appropriate symmetry breaking induced by $V^{a}$ the field $E^{a}$ will play the roll of the co-tetrad $e^{I}$ and $\bar{\omega}^{ab}$ will play the roll of the torsion-free spin-connection $\bar{\omega}^{IJ}$. We will assume that $\bar{\omega}^{ab}$ is well defined and so the applicability of this variable will be restricted by the conditions this places upon $E^{a}$. The additional $V^{a}$-dependent contribution to $\bar{A}^{ab}$ encodes the inhomogeneously transforming part of $A^{ab}V_{b}$. By way of interpretation one may think of the five ‘degrees of freedom’ of $V^{a}$ as being comprised of a norm $V^{2}$ and an orientation $U^{a}$ unit-vector (i.e. $U^{a}\equiv V^{a}/\sqrt{V^{2}}$, $|\eta_{ab}U^{a}U^{b}| = 1$). If $V^{2}\neq 0$ over a region of the manifold then one can choose a gauge where $U^{a}\overset{*}{=} \delta^{a}_{{\phantom}{a}4}$. This fixes $\bar{A}_{ab}-\bar{\omega}_{ab}=0$ implying that $A^{ab}V_{b}\overset{*}{=} E^{a}$.
In summary then, we adopt the following decomposition of $A^{ab}$:
$$\begin{aligned}
A^{ab} &=& {\bar A}^{ab}(E^{c},V^{d})+ C^{ab} + \frac{2}{V^{2}}E^{[a}V^{b]} \label{aab}\end{aligned}$$
Consequently if $V^{2}\neq0$ and $\bar{\omega}^{a}_{{\phantom}{a}b}$ is well defined we may consider the action (\[act1\]) as a functional of of $\{V^{a},C^{ab},E^{a}\}$ complemented by the constraints on the projections of $C_{ab}V^{a}=E_{a}V^{a}=0$. Given the decomposition (\[aab\]), we have
$$\begin{aligned}
F^{ab} &=& \bar{R}^{ab}-\frac{1}{V^{2}}E^{a}E^{b}+C^{a}_{{\phantom}{a}c}C^{cb}+D^{(W)}C^{ab} {\nonumber}\\
&&+ \frac{2}{V^{2}}V^{[b}C^{a]}_{{\phantom}{a]}c}E^{c}+\frac{1}{V^{4}}E^{[a}V^{b]}dV^{2}{\nonumber}\\
DV^{a} &=& \frac{1}{2V^{2}}dV^{2}V^{a}+E^{a}{\nonumber}\end{aligned}$$
where $\bar{R}^{ab}=d\bar{A}^{ab}+\bar{A}^{a}_{{\phantom}{a}c}\bar{A}^{cb}$. We can now look to write the individual contributions to the action $S[V^{a},A^{ab}]$ in terms of the variables $\{V^{a},C^{ab},E^{a}\}$. It will prove useful to group the contributions in terms of the manner in which $C^{ab}$ appears in the action. For some contributions terms with derivatives of $C^{ab}$ can be eliminated by partial integration resulting in an action with $C^{ab}$ entering only algebraically, whereas other terms are quadratic in derivatives of $C^{ab}$ so that the derivatives $D^{(\bar{A})}C^{ab}$ cannot be removed by partial integration. The former contributions (after partial integration) are as follows:
$$\begin{aligned}
S_{\alpha_{2}}[V^{a},E^{a},C^{ab}] &=& \int \left(\alpha_{2}E^{a}E^{b}C_{am}C^{m}_{{\phantom}{m}b}+\frac{\alpha_{2}}{V^{2}}dV^{2}C_{ab}E^{a}E^{b}\right)\\
S_{\beta_{1}}[V^{a},E^{a},C^{ab}] &=& \int \beta_{1}\epsilon_{abcde}V^{e}E^{a}E^{b}\left(\bar{R}^{cd}-\frac{1}{V^{2}}E^{c}E^{d}+C^{c}_{{\phantom}{c}m}C^{md}\right){\nonumber}\\
{\nonumber}&&-\left(\frac{\partial\beta_{1}}{\partial V^{2}}+\frac{\beta_{1}}{2V^{2}}\right)\epsilon_{abcde}V^{e}dV^{2}C^{ab}E^{c}E^{d}\\
S_{\beta_{2}}[V^{a},E^{a},C^{ab}] &=& \int \frac{\beta_{2}}{2} dV^{2}C_{ab}E^{a}E^{b}\\
S_{\beta_{3}}[V^{a},E^{a},C^{ab}] &=& \int\left(\beta_{3}E^{a}E^{b}C_{am}C^{m}_{{\phantom}{m}b}+\frac{\beta_{3}}{V^{2}}\left(1-\frac{V^{2}}{\beta_{3}}\frac{\partial\beta_{3}}{\partial V^{2}}\right)dV^{2}C_{ab}E^{a}E^{b}\right)\\
S_{\gamma_{1}}[V^{a},E^{a}] &=& \int \gamma_{1}\epsilon_{abcde}V^{e}E^{a}E^{b}E^{c}E^{d}\end{aligned}$$
The remaining contributions, which involve derivatives of $C^{ab}$ are:
$$\begin{aligned}
S_{\alpha_{1}}[V^{a},E^{a},C^{ab}] &=& \int \alpha_{1}\epsilon_{abcde}V^{e}\left(\bar{R}^{ab}\bar{R}^{cd}-\frac{2}{V^{2}}\left(\bar{R}^{ab}+C^{a}_{{\phantom}{a}m}C^{mb}\right)E^{c}E^{d}+\frac{1}{V^{4}}E^{a}E^{b}E^{c}E^{d}\right) \\
&& -2\left(\frac{\partial\alpha_{1}}{\partial V^{2}}+\frac{\alpha_{1}}{2V^{2}}\right)\epsilon_{abcde}V^{e}dV^{2}C^{ab}\left(\bar{R}^{cd}-\frac{1}{V^{2}}E^{c}E^{d}+\frac{1}{3}C^{c}_{{\phantom}{c}m}C^{md}+\frac{1}{2}D^{(W)}C^{cd}\right)\\
S_{\alpha_{3}}[V^{a},E^{a},C^{ab}]&=& \int \alpha_{3}\bar{R}^{ab}\bar{R}_{ab} -2\frac{\partial\alpha_{3}}{\partial V^{2}}dV^{2}C_{cd}\left(\bar{R}^{cd}-\frac{1}{V^{2}}E^{c}E^{d}+\frac{1}{3} C^{c}_{{\phantom}{a}m}C^{md}+\frac{1}{2} D^{(W)}C^{cd}\right)\end{aligned}$$
This distinction between the appearance of $C^{ab}$ is important. For any combination of the actions for which $C^{ab}$ appears algebraically (after a partial integration), the equation of motion for $C^{ab}$ may be used to solve for $C^{ab}$ itself. Consequently, this solution can be re-inserted back into the actions, yielding a variational principle that depends solely on $V^{a}$ and $E^{a}$. In this case the contorsion field does not have independent degrees of freedom and is instead completely fixed by other fields. On the other hand, whenever the action contains contributions from the $\alpha_1$ and $\alpha_3$ terms the contorsion may not be solved for algebraically and thus should be regarded as a genuine dynamical field with independent degrees of freedom.
The above seven actions remain completely general up to the requirement that $V^{2}\neq 0$ and that $\bar{\omega}^{ab}$ is well-defined. We will now concentrate on a sub-case which illustrates the consistency of the use of variables that assume $V^{2}\neq 0$. Though the approach of using variables which require a non-zero $V^{2}$ is clearly restrictive we will see that the variables enable in a simple way the recovery of more familiar looking geometric objects such as $g_{\mu\nu}$ and the torsion-free Ricci scalar ${\cal R}$. Indeed, this may be seen as a generalisation of the relationship between Einstein-Cartan gravity and the second-order formulation of gravity. The second-order formulation follows from the adoption of the ansatz (\[vij\]) and does not admit more general solutions such as $e^{I}=0,\omega^{IJ}\overset{*}{=}0$ that vacuum Einstein-Cartan theory does. Clearly though, the description of gravity using the Einstein-Hilbert action and its variable $g_{\mu\nu}$ is justified if solutions to the equation of motion are consistent with its assumed invertibility, even if it is regarded as only sometimes recoverable from the Palatini action of Einstein-Cartan theory.
We postpone discussion of the possibility that $V^{2}=0$ to Section \[vnull\] but we stress that from the perspective wherein $V^a$ is a genuine dynamical field, [*ad hoc*]{} restrictions on $V^2$ (e.g. $V^2>0$, $V^2\neq0$, etc.) are rather unnatural. Instead, from the dynamical perspective we should in principle allow for possible solutions in which $V^2$ changes sign and can become zero on hyper surfaces. Such changes in the sign of $V^2$ will be accompanied with a change in metric signature.
Let us examine this a bit closer and also establish some notational conventions that will be used in the following sections. We will find that a metric formalism for gravity may be recovered and that the signature of the metric tensor will depend on the sign of $V^{2}$ for a given group. For instance for $SO(1,4)$ if $V^{2}>0$ then the subgroup that leaves $V^{a}$ invariant is $SO(1,3)$ (corresponding to metric signature $(-,+,+,+)$). If $V^{2}<0$ then the subgroup is $SO(4)$ (metric signature $(+,+,+,+)$). For $SO(2,3)$ if $V^{2}>0$ then the subgroup is $SO(2,2)$ (metric signature $(-,-,+,+)$ and for $V^{2}<0$ the subgroup is $SO(1,3)$ (again corresponding to metric signature $(-,+,+,+)$). It is useful then to define a *positive definite* scalar field $\phi\equiv |\sqrt{V^{2}}|$ such that
$$\begin{aligned}
\eta_{ab}V^{a}V^{b} = \sigma \phi^{2}\end{aligned}$$
The constant $\sigma=1$ for $SO(1,4)$ with $(-,+,+,+)$ metric signature and for $SO(2,3)$ with $(-,-,+,+)$ signature, whilst $\sigma=-1$ for $SO(1,4)$ with $(+,+,+,+)$ signature and for $SO(2,3)$ with $(-,+,+,+)$ signature. Therefore in addition to keeping note of the norm of $V^{2}$ we must also keep track of the group being used as that will say what the metric signature is for a given $V^{2}$. Therefore we introduce the symbol $\theta$ which takes the value $-1$ if the metric has an even number of timelike dimensions (inclusive of the case that there are $0$) and the value $+1$ if the metric $g_{\mu\nu}$ has an odd number of timelike dimensions.
A Specific Example {#spece}
==================
We are now ready to work out the physics of specific choices of actions and cast it in a more familiar second-order form in terms of the metric tensor $g_{\mu\nu}$. Consider then the case where, of the ‘$\alpha$,$\beta$,$\gamma$’, only $\beta_{1}$,$\beta_{2}$ are non-vanishing. Furthermore we will assume that these two quantities are constant i.e. do not depend on $V^{2}$. As such $\beta_1$ and $\beta_2$ become constants with dimensions of $m^{-3}$ and $m^{-4}$ respectively. In the following $V^a$ and $E^a$ have dimensions $m$ and the remaining objects such as $A^{ab},C^{ab}, \bar{\cal R}^{ab},\dots$ dimensionless.
The combined action is then: $$\begin{aligned}
S_{1} &=& \int \beta_{1}\epsilon_{abcde}V^{e}E^{a}E^{b}\left(\bar{R}^{cd}-\frac{1}{V^{2}}E^{c}E^{d}+C^{c}_{{\phantom}{c}m}C^{md}\right){\nonumber}\\
&&-\left(\frac{\beta_{1}}{2V^{2}}\epsilon_{abcde}V^{e}-\frac{\beta_{2}}{2}\eta_{ac}\eta_{bd}\right)dV^{2}C^{ab}E^{c}E^{d} \label{simact}\end{aligned}$$ If $V^{a}$ is non-vanishing then it selects a subgroup which $V^{a}$ is invariant under. If we again utilize the orientation unit-vector $U^{a}\equiv V^{a}/|\sqrt{V^{2}}|$ then $\epsilon_{abcd} \equiv \epsilon_{abcde}U^{e}$ is invariant under the remnant transformations. It can be seen from (\[simact\]) that variation of the action with respect to $C^{ab}$ will yield an equation linear in $C^{ab}$, which may be used to obtain the following solution for the field:
$$\begin{aligned}
C_{ab} &=& \frac{1}{2V^{2}}E_{[a}\partial_{b]}V^{2}+\frac{\beta_{2}\theta}{8\beta_{1}\phi}\epsilon_{abcd}\partial^{c}V^{2}E^{d}\end{aligned}$$
where $\partial_{a}V^{2}\equiv (E^{-1})^{\mu}_{a}\partial_{\mu}V^{2}$. Substitution of $C^{ab}$ back into (\[simact\]) yields:
$$\begin{aligned}
{\nonumber}S_{1}[V^{2},E^{a}] &=& \int \beta_{1}\epsilon_{abcd}\phi E^{a}E^{b}\bar{R}^{cd} -\beta_{1}\frac{1}{V^{2}}\epsilon_{abcd}\phi E^{a}E^{b}E^{c}E^{d}\\
&& +\epsilon_{abcd}\frac{\beta_{1}\phi}{16V^{4}}\left(1-\frac{\beta_{2}^{2}\theta\phi^{2}}{4\beta_{1}^{2}}\right)\left(6\partial_{m}V^{2}\partial^{b}V^{2}E^{a}E^{m}-\partial^{m}V^{2}\partial_{m}V^{2}E^{a}E^{b}\right)E^{c}E^{d} \label{s1dev}\end{aligned}$$
We are now in a position to write $S_{1}$ in terms of more familiar variables. From $E^{a}$ we may define a metric tensor $\bar{g}_{\mu\nu} \equiv \eta_{ab}E^{a}_{\mu}E^{b}_{\nu}$ and inverse metric $\bar{g}^{\mu\nu}\equiv\eta_{ab}(E^{-1})^{a\mu}(E^{-1})^{b\nu}$. This compels us to identify $\bar{R}_{\mu\nu\alpha\beta}= E^{a}_{\mu}E^{b}_{\nu}{\bar R}_{ab\alpha\beta}$ with the torsionless Riemann curvature tensor. Furthermore we define the Ricci tensor $\bar{{\cal R}}^{\mu}_{{\phantom}{\mu}\nu} \equiv \bar{R}^{\mu\alpha}_{{\phantom}{\mu\alpha}\nu\alpha}$ and Ricci scalar $\bar{{\cal R}} \equiv \bar{{\cal R}}^{\mu}_{{\phantom}{\mu}\mu}$. By standard methods, the action (\[s1dev\]) may be rewritten as an integration over a scalar density:
$$\begin{aligned}
\label{beforeredef}
S_{1}[\phi,\bar{g}_{\mu\nu}] &=& \int \beta_{1}\left(2\bar{{\cal R}}+\frac{3}{ \phi^{2}}\left(1-\frac{\beta_{2}^{2}\theta\phi^{2}}{4\beta_{1}^{2}}\right)\bar{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi- \frac{24\sigma}{\phi^{2}}\right)\phi\sqrt{-\bar{g}}d^{4}x\end{aligned}$$
We may cast the action in a more familiar form by considering the conformal transformation of the metric: $g_{\mu\nu}=\Omega^2 \bar g_{\mu\nu}$. The Ricci scalar $\cal R$ corresponding to $g_{\mu\nu}$ is related to $\bar {\cal R}$ according to [@Wald:1984rg] $\bar{\cal R}= (\Omega^{2}{\cal R}+6g^{\mu\nu}\nabla_\mu\log\Omega\nabla_\nu\log\Omega+\bar{g}^{\mu\nu}\bar{\nabla}_\mu\bar{\nabla}_\nu\log\Omega)$. If we choose $\Omega^2=\phi/\phi_0$, for some constant $\phi_0$, the action in the new variables $(\phi,g_{\mu\nu})$ becomes
$$\begin{aligned}
\label{s1norm}
S_{1}[\phi,g_{\mu\nu}] &=& \int \left(2\beta_1\phi_0{\cal R}-\frac{3\phi_0\beta_{2}^{2}\theta}{4\beta_{1}}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi- 24\beta_1\sigma\frac{\phi_0^2}{\phi^{3}}\right)\sqrt{-g}d^{4}x\end{aligned}$$
where the boundary term $\bar{\nabla}_\mu (6\beta_1\phi_0\bar{\nabla}^\mu\log\frac{\phi}{\phi_0})\sqrt{-\bar{g}}$ has been removed. We can now identify the gravitational constant as $\kappa=2\beta_1\phi_0=\frac{1}{16\pi G}$ where $G$ is Newton’s constant. Furthermore, perhaps surprisingly, after this conformal transformation the ‘wrong-signed’ kinetic contribution of equation $ $is cancelled and the scalar field $\phi^2=|V^2|$ is recognized as a scalar field satisfying a Klein-Gordon equation with potential $U(\phi)=24\beta_1\sigma\frac{\phi_0^2}{\phi^3}$. This potential does not have a minimum and does not allow for a stable vacuum expectation value of $\phi$. In such a case there is no privileged choice of the value of $\phi_0$ which was introduced merely to make the conformal factor $\Omega$ dimensionless. However, if the potential could be modified by adding other polynomial terms to the action enabling a stable vacuum expectation value, then $\phi_0$ could be identified with that vacuum expectation value. How to achieve this will be considered below in section \[stabil\].
$SO(1,4)$ leading to $(-,+,+,+)$ metric signature
-------------------------------------------------
Recall that for the group $SO(1,4)$ then if $V^{2} >0$ then the invariant subgroup is $SO(1,3)$ and with our conventions on $\eta_{ab}$ the metric $g_{\mu\nu}$ is of signature $(-,+,+,+)$. Therefore we have $\sigma=+1$ and $\theta=+1$. We can see from (\[s1norm\]) that the action (\[simact\]) corresponds to General Relativity coupled to a scalar field $\phi \equiv |\sqrt{V^{2}}|$ with right-sign kinetic term but with a potential with sign that depends upon the original gauge group. For $SO(1,4)$ we have a potential $1/\phi^{3}$. Therefore the field will tend to ‘roll down’ the potential to increasing values of $\phi$.
$SO(2,3)$ leading to $(-,+,+,+)$ metric signature
-------------------------------------------------
For the group $SO(2,3)$ if $V^{2}<0$ then the invariant subgroup is $SO(1,3)$ and with our conventions on $\eta_{ab}$ the metric $g_{\mu\nu}$ is of signature $(-,+,+,+)$, corresponding now to $\sigma=-1$ and $\theta=+1$. Now due to the different sign of we have the potential $-1/\phi^{3}$. This potential will tend to lead to runaway evolution toward $\phi=0$. Therefore upon beginning from the action (\[simact\]), we see that symmetry breaking is dynamically favoured for the case $SO(1,4)$.
Stabilizing $\phi$ {#stabil}
------------------
We have seen that the action (\[simact\]) is a model of dynamical symmetry breaking in the $SO(1,4)$ case. However, the potential for the field $\phi$ does not have a minimum, and the model in the absence of matter corresponds to a Peebles-Ratra rolling quintessence model [@Ratra:1987rm; @Peebles:1987ek]. We now consider additional polynomial contributions to the action that would alter the potential as to create local maxima and minima. By inspection, aside from the $\alpha_{1}$ action it is only the $\gamma_{1}$ action that can provide additional contributions to the potential. All other terms involve the field $C^{ab}$ which generically will depend upon derivatives of $|V^{2}|=\phi^2$. It is straightforward to rewrite the $\gamma_{1}$ action in terms of the variables of the previous section:
$$\begin{aligned}
\label{s1stab}
S_{1}[\phi,g_{\mu\nu}] &=& \int \left(2\beta_1\phi_0{\cal R}-\frac{3\phi_0\beta_{2}^{2}\theta}{4\beta_{1}} g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-U(\phi)\right)\sqrt{-g}d^{4}x\end{aligned}$$
where we have defined the scalar field potential $$\begin{aligned}
\label{potential}
U(\phi) \equiv 24\phi_0^2\left(\frac{\beta_{1}\sigma}{\phi^{3}}-\frac{\gamma_{1}(\phi^{2})}{\phi}\right).\end{aligned}$$ A local minimum in the potential corresponds to a real, positive solution to the equations $(dU/d\phi)_{\phi=\phi_{0}}=0$, $(d^{2}U/d\phi^{2})_{\phi=\phi_{0}} >0$.
Note that a peculiarity of this approach is that appears impossible to add a term to $U(\phi)$ corresponding to a cosmological constant (i.e. a constant part of $U(\phi)$). This would, for instance, correspond to a contribution to $\gamma_{1}$ of the form $\sqrt{ \sigma V^{2}}$ which is excluded by the requirement that the action (\[act1\]) is polynomial in $A^{ab}$ and $V^{a}$. However, if $\phi$ settles to a stationary point $\phi_{0}$ then it creates an effective cosmological constant. Furthermore the quadratic term in an expansion around $\phi_{0}$ defines a mass $M$ of the scalar field.
An empirically viable model
---------------------------
It is now interesting to examine whether there exist a specific choice of action which would lead to a modification of General Relativity but yet not ruled out by empirical fact. In particular, such model should imply a stable positive cosmological constant at sufficiently low energies. Of the six specific models displayed in Fig. 1 and 2 only one model seems free of potential or immediate problems.
Clearly, the model corresponding to the blue line of Fig. 1 is ruled out because of its stable negative value of the cosmological constant emulated by $U(\phi_{min})$. We also note that all models based on $SO(2,3)$ seem potentially problematic as none of the potentials in Fig. 2 are bounded from below. This can potentially cause instability problems, especially in a quantum context with the possibility of quantum tunnelling. For this reason we will put them aside and regard them in absence of further analysis as unsuitable. Furthermore, the dashed green line in Fig. 1 corresponds to an asymptotically zero cosmological constant ($U(\phi)\rightarrow 0$ as $\phi\rightarrow\infty$) and it is therefore not clear without further analysis that this model would be empirically viable.
This leaves us with the model corresponding to the bright green line in Fig. 1 which is characterized by having a global stable minimum of the potential $U(\phi)$. This model is based on $SO(1,4)$, a spacelike contact vector $V^a$, with $\beta_1$ and $\beta_2$ as constants, and with $\gamma=-CV^2=-C\phi^2$. We shall now see how the parameters $\beta_1,\beta_2$, and $C$ are related to more familiar constants such as the gravitational and cosmological constants $\kappa$ and $\Lambda$ and an effective mass $M$ for the gravitational Higgs field $V^a$.
The model is then defined by the action with the scalar field potential $$\begin{aligned}
U(\phi)=24\phi_0^2\left(\frac{\beta_1}{\phi^3}+C\phi\right)\end{aligned}$$ corresponding to a choice of $\gamma(\phi^2)=-C\phi^2$ for some constant $C$ of dimensions $m^{-7}$.
As mentioned above, the constant $\phi_0$ was introduced to make the conformal factor $\Omega=\frac{\phi}{\phi_0}$ dimensionless. As such it is arbitrary. However, it is natural to fix it by requiring it to coincide with the value of $\phi$ at the potential minimum, i.e. we impose $$\begin{aligned}
0=\left.\frac{\partial U}{\partial \phi}\right|_{\phi=\phi_0}=24\phi_0^2\left(-\frac{3\beta_1}{\phi_0^4}+C\right)\end{aligned}$$ which implies $$\begin{aligned}
\label{VEV}
\phi_0^4=\frac{3\beta_1}{C}.\end{aligned}$$ Furthermore, whenever the scalar field has settled to its vacuum expectation value $\phi_0$ it is constant everywhere and the corresponding value of the potential $U(\phi_0)$ will then emulate a cosmological constant $\Lambda$ defined by $\int \kappa({\cal R}-2\Lambda)\sqrt{-g}$. The gravitational constant must be identified as $\kappa=2\beta_1\phi_0$ and together with equation we obtain $\Lambda=\frac{U(\phi_0)}{2\kappa}=\frac{48\beta_1}{\kappa\phi_0}=\frac{24}{\phi_0^2}$. We can now express $\beta_1$ and $C$ in terms of $\Lambda$ and $\kappa$ $$\begin{aligned}
\beta_1=\frac{\kappa\sqrt{\Lambda}}{\sqrt{96}}\qquad C=\frac{\kappa\Lambda^{5/2}}{768\sqrt{6}}.\end{aligned}$$ Thus, if the constants $\beta_1$ and $C$ are chosen properly we reproduce the predictions of General Relativity plus a cosmological constant whenever the scalar field is in its ground state. Note that in the absence of matter, a ‘genuine’ cosmological constant truly independent of $\phi$ cannot be constructed- such a term could only arise from non-polynomial terms in the action (\[act1\]) such as a contribution to $\gamma_{1}$ of the form $\sqrt{V^{2}}$.
At sufficiently high energies scalar field $V^2$ may get excited from its lowest energy state, the vacuum expectation value $\phi_0$. At which energy that happens is dictated by the effective mass $M$ of the scalar field.
In order to determine the effective mass we need to put the scalar field kinetic term in a standard form. This is achieved by a rescaling $\phi=a\varphi$ with $a^{-2}=\frac{3\phi_0\beta_2^2}{4\beta_1}$. The action now takes on the standard form $$\begin{aligned}
\bar S_{1}[\varphi,g_{\mu\nu}] &=& \int \left(\kappa{\cal R}- g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi- U(\varphi)\right)\sqrt{-g}d^{4}x\end{aligned}$$ with $U(\varphi)=24\phi_0^2(\frac{\beta_1}{a^3\varphi^3}+Ca\varphi)$. The effective mass $M$ of the scalar field $\varphi$ is defined as ($\phi_0\equiv a\varphi_0$) : $$\begin{aligned}
M^2\equiv\left.\frac{1}{2}\frac{\partial^2 U}{\partial \varphi^2}\right|_{\varphi=\varphi_0}=\frac{1}{2}24\phi_0^2 \frac{12\beta_1}{a^3\varphi_0^5}=\frac{\kappa^2\Lambda^3}{288\beta_2^2}\end{aligned}$$ and we see that for given values of $\kappa$ and $\Lambda$ it is the parameter $\beta_2$ that determines the effective mass of the scalar field $\varphi$.
We have now seen how General Relativity is recovered in the low energy limit from a gauge theory with dynamical symmetry breaking defined by the action $$\begin{aligned}
S[A^{a}_{{\phantom}{a}b\mu},V^{a}] &=& \int \left(\beta_{1}\epsilon_{abcde}V^{e}+\beta_{2}V_{a}V_{c}\eta_{bd}\right)DV^{a}DV^{b}F^{cd} \\
&& - CV^2\epsilon_{abcde}V^{e}DV^{a}DV^{b}DV^{c}DV^{d}.\end{aligned}$$ containing the two basic ingredients of such a theory: a $SO(1,4)$-valued gauge field $A^{a}_{{\phantom}{a}b\mu}$ and a Higgs field $V^a$ with norm $V^2$ subject to a Klein-Gordon equation. In particular, at the fundamental level neither metric nor co-tetrad are necessary for the formulation of this theory.
More General Cases {#general}
==================
It is natural to ask how general the results of the previous section are. In Appendix \[gencas\] an identical calculation is performed for the case where all $\alpha,\beta,\gamma$ except $\alpha_{1}$ and $\alpha_{3}$ are non-zero and allowed to have a polynomial dependence upon $V^{2}$. As for the case considered in Section \[stabil\], if $V^{2}\neq 0$ then the action is equivalent to a metric theory of gravity coupled to a scalar field. By inspection one may again make a conformal transformation of the metric to recover the familiar Einstein-Hilbert Lagrangian. To recover a canonical kinetic term for the scalar field, it is necessary to perform a redefinition of the scalar field.
Finally we consider the $\alpha_{1}$ and $\alpha_{3}$ terms. As mentioned, variation of these actions with respect to $C^{ab}$ will yield field equations containing derivatives of $C^{ab}$. Therefore the presence of these terms in an action generally precludes the possibility of eliminating $C^{ab}$ from the variational principle. In this sense these actions involve ‘propagating contorsion’. We now consider the impact of these terms.
Effect of the $\alpha_{1}$ term
-------------------------------
We first consider the limiting case where $V^{2}$ is constant and non-vanishing. The action in this limit is commonly referred to as the Macdowell-Mansouri action [@MacDowell:1977jt]. Here it may be checked that the term $\epsilon_{abcde}V^{e}\bar{R}^{ab}\bar{R}^{cd}$ reduces to a boundary term whereas the remaining terms (i.e. those that do not vanish when $dV^{2}=0$) amount to Einstein-Cartan gravity in the presence of a cosmological constant. The Macdowell-Mansouri action has been studied in a wide variety of contexts [@Wise:2006sm; @Wise:2011ab; @Gibbons:2009af; @Randono:2010cq; @Gryb:2012qt; @Obregon:2012zz; @Durka:2012wd; @Freidel:2012np]. When $V^{2}\neq 0$ and varying, a number of terms in $S_{\alpha_{1}}$ that are proportional to $dV^{2}$ may now contribute. This action has been considered briefly in [@Banados:1996gi] but this model, with the remnant $SO(1,3)$ symmetry of the symmetry broken phrase, seems largely unexplored. Cosmological solutions for a very closely related action to $S_{\alpha_{1}}$ have been recently considered [@TolozaT:2013wi].
Effect of the $\alpha_{3}$ term
-------------------------------
The structure of the $\alpha_{3}$ term is rather more familiar. Recall that in the event that $V^{2}$ is constant then $\alpha_{3}(V^{2})$ is necessarily constant and the term is simply a boundary term. If $V^{2}\neq 0$ and varying and $\alpha_{3}$ carries a dependence on $V^{2}$ then the $\alpha_{3}$ term corresponds to an additional term in gravitation in the Chern-Simons modified gravity theory [@Alexander:2008wi; @Cantcheff:2008qn; @Ertem:2009ur; @Alexander:2009tp; @Garfinkle:2010zx; @Furtado:2010qv; @Canizares:2012ji], with $\alpha_{3}$ playing the role of the Chern-Simons scalar field $\Phi_{cs}$. Typically also present in these theories are an Einstein-Hilbert (or Palatini) action as well as an action for the dynamics of $\Phi_{cs}$. A direct comparison between Chern-Simons modified gravity a subset of the general actions (\[act1\]) requires some care. We have seen that the $\beta_{1}$ action may reduce to the Einstein-Hilbert action only after conformal transformation of $E^{a}$, therefore a more accurate comparison follows after an additional conformal transformation of the $\alpha_{3}$ action. A persistent commonality though is the presence of derivatives of the contorsion, if allowed for, in the action of the theory. As in the case of the $\alpha_{1}$ term, the presence of a non-vanishing $\alpha_{3}(V^{2})$ may preclude the existence of a second-order formulation in terms of a metric and scalar field.
Null or vanishing $V^{a}$ {#vnull}
=========================
It has become apparent in the previous sections that the tensor $\bar{g}_{\mu\nu}=\eta_{ac}V_{b}V_{d}B^{ab}_{{\phantom}{ab}\mu}B^{cd}_{{\phantom}{cd}\nu}= E^{a}_{\mu}E_{a \nu}$ is to be identified, up to a conformal factor, with the familiar metric tensor in situations where $V^{a}\neq 0$. However, dynamically it may be the case that $V^{a}$ becomes null (i.e. $\eta_{ab}V^{a}V^{b}=0$). This condition may be satisfied with $V^{a}=0$ or $V^{a}\neq 0$. When $V^{a}=0$ then $\bar{g}_{\mu\nu}=0$ and so it is unclear whether a metric description exists in regions where this is satisfied. The null case with $V^{a}\neq 0$ is a bit more subtle. In this case we may choose a gauge where $V^{a}=\psi(x^{\mu})(1,0,0,0,\pm 1)$, where first and last components refer to internal time and spatial components. The remaining components will be labelled by lowercase middle-alphabet Latin indices $i,j..$.
$$\begin{aligned}
\bar{g}_{\mu\nu}= \psi^{2}\eta_{ij}\left(B^{i}_{{\phantom}{i}0} \pm B^{i}_{{\phantom}{i}4}\right)_{\mu}\left(B^{j}_{{\phantom}{j}0}\pm B^{j}_{{\phantom}{j}4}\right)_{\nu}\end{aligned}$$
Therefore in the null case the signature of the metric will be that of $\eta_{ij}$ i.e. the signature will be three dimensional rather than four dimensional. For $SO(1,4)$ then the metric has signature $(+,+,+)$ and for $SO(2,3)$ the metric will have signature $(-,+,+)$. The $\alpha_{1}$ action in isolation was studied for the former possibility in [@Westman:2012np], wherein it was indeed found that a three dimensional geometric description was appropriate. Recall that for $SO(1,4)$ when $V^{2} >0$ then the metric is of signature $(-,+,+,+)$ whereas when $V^{2}<0$ the metric is of signature $(+,+,+,+)$. It is tempting to wonder whether situations may occur where the transition of $V^{2}$ through $0$ dynamically facilitates a change of metric signature. Such a possibility has been examined in the $(3+1)$ formulation of General Relativity [@Ellis:1991st]. Consideration of explicit models in Cartan Gravity, however, is beyond the scope of this current work.
Discussion and outlook {#discussion}
======================
A common view in physics has been that gravity should ultimately be thought of just another force field on par with the strong and electroweak fields of the standard model. From a mathematical perspective such a view is limited by the difference in mathematical machinery used by the force fields of particle physics on the one hand and the gravitational field on the other. More specifically, while the force fields of particle physics are mathematically represented by gauge connections valued in some suitable Lie algebra, the gravitational field is typically described by a symmetric second rank metric tensor $g_{\mu\nu}$. The mathematical difference is still present within the Einstein-Cartan formulation of gravity in terms of a $SO(1,3)$-valued gauge connection $\omega^{IJ}$ and a co-tetrad $e^I$. In particular, the co-tetrad regarded as a fundamental object appears to have no analogue within ordinary gauge theory [^3].
However, it has been know for some time that gravity can be formulated exclusively in terms of objects of gauge theories with spontaneously broken symmetry. Within this Cartan-geometric formulation of gravity the descriptor of gravity consist of the pair $\{A^{a}_{{\phantom}{a}b},V^{a}\}$. These objects admit a simple geometrical interpretation in terms of ‘idealized-waywisers’ [@Westman:2012xk], but from a particle physics point of view we recognize $A^{a}_{{\phantom}{a}b \mu}$ as a standard gauge connection and $V^a$ as a symmetry breaking Higgs field. In this paper we have shown that there exist actions which are polynomial in these fields and their exterior derivatives which allow for a dynamical symmetry breaking mechanism closely mirroring that of the electroweak theory of particle physics. It is noteworthy that this is achieved without any metric tensor or co-tetrad appearing in the action. Instead, these objects are regarded as compound objects which can be constructed from the more fundamental variables $\{A^{a}_{{\phantom}{a}b},V^{a}\}$ should it be convenient to do so. In relation to this, we note that the constrained-norm form of Cartan gravity has been recovered from a perspective where $V^{A}$ itself is a composite object, regarded as a fermionic condensate constructed from either $spin(1,4)$ or $spin(2,3)$ valued spinor fields [@Randono:2010ym].
To our knowledge the recovery of such symmetry breaking solutions following free variation of $\{A^{a}_{{\phantom}{a}b},V^{a}\}$ in a polynomial action principle is new. By way of contrast, the constancy of $V^2$ is typically imposed by hand within Cartan gravity without any dynamical mechanism proposed. From a pure mathematical point of view we can of course regard $V^a$ as a mathematical redundant representation of a preferred section on the gauge fibre bundle that is necessary for the mathematical construction of Cartan geometry [@Petti:2006ue]. On that view the norm $V^2$ does not play any role in the construction and we might as well impose $V^2=\mathrm{const.}$. However, from a particle physics point of view, and in particular with the electroweak Higgs field in mind, the restriction $V^2=\mathrm{const.}$ does not appear natural. Instead, it suggests a natural modification of Cartan gravity in which the Higgs field $V^a$ is treated as a genuine dynamical field subject to equations of motion. This was the basic motivation for this paper.
Although the class of actions (\[act1\]), which always lead to first-order partial differential equations, look rather unfamiliar they may (excluding $\alpha_1$- and $\alpha_3$-terms) be recast into a more familiar second-order form where the actions reduce to particular examples of scalar-tensor theories. Furthermore, we showed how a specific subset of (\[act1\]) lead to a scalar potential with a global minimum playing the role of a vacuum expectation value. Agreement with General Relativity is guaranteed as long as the effective mass $M$ calculated from the potential is large enough to suppress the excitation of the Higgs field. Interestingly the requirement that (\[act1\]) is polynomial in basic variables precludes the possibility of a genuine ‘bare’ cosmological constant term according to the volume form $\sqrt{-g}d^{4}x$. For high energies deviations from General Relativity are expected due to the presence of gravitational Higgs bosons. A particularly stringent constraint on the mass of $M$ may come due to the coupling of $V^{a}$ to matter fields [@Westman:2012zk].
One perhaps surprising result of this paper is that $\phi\equiv |\sqrt{V^{2}}|$ appears as a standard scalar field when the action is rewritten in a second-order form. From a differential forms point of view this shows that the Hodge dual pops up naturally. The natural appearance of the Hodge dual when rewriting polynomial first-order field equations in a second-order form was also noted in [@Westman:2012zk]. This seems to indicate that the use of auxiliary fields, see e.g. [@Smolin:2007rx; @Lisi:2010td], may not be necessary in order to reproduce equations with such a Hodge structure, e.g. the Klein-Gordon and Yang-Mills equations.
One central simplifying assumption in the analysis of this paper was that $V^2$ be everywhere non-zero. We have argued that this is akin to the use of an invertible metric $g_{\mu\nu}$ in the variational principle of second-order General Relativity, the use of which is consistent whenever solutions to the resultant equations of motion do not threaten the assumption of invertibility. However, if we take the dynamical perspective seriously such a restriction on $V^{2}$ must be regarded as [*ad hoc*]{}. Instead, one should take the view that the field equations should dictate what possible solutions of $V^a$ we can have. As such we cannot exclude the possibility of $V^2$ changing sign or of $V^a$ even becoming zero on hypersurfaces. Under such conditions the scalar-tensor formalism breaks down but perhaps not the applicability of the basic variables $\{A^{a}_{{\phantom}{a}b},V^{a}\}$. As an interesting dynamical possibility we might have metric signature change. This would occur whenever $V^2$ changes sign, selecting out a different subgroup of $SO(1,4)/SO(2,3)$. Within a model based on $SO(1,4)$, $V^2$ may be spacelike in some region leading to the usual Lorentzian signature $(-,+,+,+)$ of the metric (based on subgroup $SO(1,3)$). However, it might be possible that the first-order field equations allow solutions where the sign of $V^2$ varies from region to region. In particular, in regions where $V^2$ is timelike, the ‘spacetime’ becomes Euclidean with signature $(+,+,+,+)$ (based on subgroup $SO(4)$). It would therefore be interesting to investigate whether there are solutions to the first-order field equations that admit a signature change and thus a dynamical and classical realization of Hartle and Hawking’s no-boundary proposal [@Hartle:1983ai]. Indeed this represents a subtlety in this approach to viewing gravity as a gauge theory: changes in the character of the symmetry breaking or even symmetry restoration inevitably coincide with the loss of familiar notions of space and time.
The present work may be taken to suggest a more radical view on the form of field equations in Nature. Typically second-order field equations are regarded as the ones chosen in Nature. In particular, gravity, Yang-Mills fields, and scalar fields are all described by second-order partial differential equations. However, a notable exception are fermionic fields described by Dirac equations which are naturally on first-order. Further, the Cartan-geometric formulation of gravity and the coupling to matter fields through the gauge prescription [@Westman:2012zk] puts all fields equations on first-order form. Of course, this may be regarded as a mere reformulation. However, as we have noted in this paper, the inclusion of terms like $\alpha_1$ and $\alpha_3$ will not allow for a second-order equivalent. With such contributions to the action the contorsion field becomes a genuinely dynamical field which cannot be solved for algebraically in terms of other fields. However, whenever the scalar $V^2$ becomes constant the contorsion field can be solved for algebraically and a second-order formulation becomes possible. From this perspective, we may perhaps regard the ubiquitousness of second-order field equations in nature as a contingent feature of a universe where the Higgs vield $V^a$ has settled to its vacuum expectation value. However, at a more fundamental level the theory would be governed by first-order equations that cannot be cast into second-order ones.
As a further departure from the standard picture of gravity, it is conceivable that the $SO(1,4)/SO(2,3)$ invariance is itself part of a symmetry broken phase of a model with a larger symmetry. For instance, it may be interesting to construct an equivalent of the actions (\[act1\]) for a gauge theory of the conformal group $SO(2,4)$; conceivably the dynamics of symmetry breaking may allow for a General Relativity limit, and more comprehensive differences at higher energies.
General Case {#gencas}
============
It may be shown after rather lengthy calculation that the case where only $\alpha_{1}$ and $\alpha_{3}$ are assuming to be zero and all other variables are allowed to depend polynomially upon $V^{2}$ leads to the following second-order action upon elimination of $C^{ab}$:
$$\begin{aligned}
S_{2}[\phi,\bar{g}_{\mu\nu}] &=& \int \left(2\beta_{1}\bar{{\cal R}}-\xi(\phi)\bar{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi {\phantom}{\frac{\sigma\beta_{1})}{\phi^{2}}}\right. \\
&& \left.
-\frac{24\left(\sigma\beta_{1}-\gamma_{1}\phi^{2}\right)}{\phi^{2}}\right)\phi\sqrt{-\bar{g}}d^{4}x\end{aligned}$$
where
$$\begin{aligned}
\xi(\phi) &=& 3\frac{\left(\theta Q^{2}-4\left(\frac{\partial(\beta_{1}\phi)}{\partial \phi^{2}}\right)^{2}+\frac{2\sigma (\alpha_{2}+\beta_{3})\theta}{\beta_{1}\phi}\left(\frac{\partial(\beta_{1}\phi)}{\partial \phi^{2}}\right)Q\right)}{\beta_{1}\left(1+\frac{\theta}{\phi^{2}}\left(\frac{\alpha_{2}+\beta_{3}}{2\beta_{1}}\right)^{2}\right)} \\
Q &\equiv & \frac{\sigma\alpha_{2}}{\phi^{2}}+\frac{\beta_{2}}{2}+\frac{\sigma\beta_{3}}{\phi^{2}}
\left(1-\frac{\phi^{2}}{\beta_{3}}\frac{\partial\beta_{3}}{\partial \phi^{2}}\right)\end{aligned}$$
As before we may make a conformal transformation. If $\beta_{1}$ itself has a dependence upon $V^{2}$ we define we make the following conformal transformation $\bar{g}_{\mu\nu}=
(\kappa/2\beta_{1}\phi)g_{\mu\nu}$, in terms of which the action becomes:
$$\begin{aligned}
S_{2}[\phi,g_{\mu\nu}] &=& \int \left( \kappa{\cal R} -\left(\frac{\kappa}{2\beta_{1}}\right)\left(\frac{12}{\beta_{1}}\left(\frac{\partial (\beta_{1}\phi)}{\partial \phi^{2}}\right)^{2}+\xi(\phi)\right)g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi
-24\left(\frac{\kappa}{2\beta_{1}}\right)^{2}\left(\frac{\sigma\beta_{1}}{\phi^{3}}-\frac{\gamma_{1}}{\phi}\right)\right)\sqrt{-g}d^{4}x \end{aligned}$$
where recall that the $\alpha_{i},\beta_{i},\gamma_{i}$ may depend polynomially on $\phi^{2}$ and the constant $\kappa \equiv 1/16\pi G$.
[^1]: In the second-order formulation of gravity, the Christoffel connection is indeed a $GL(4)$-valued connection but it is constructed from $g_{\mu\nu}$ and therefore not usually thought of as the fundamental variable.
[^2]: A similar remark applies to the Palatini first-order formulation in which the dynamical variables are the $GL(4)$-valued gauge field $\Gamma^\rho_{\mu\nu}$, the affine connection, and the metric $g_{\mu\nu}$, which again does not appear to have a natural counterpart in gauge theory.
[^3]: An interesting alternative is in schemes that unify the $SO(1,3)$ gauge field with gauge fields of grand unified theories into a connection ${\cal A}^{\Omega}_{\,\,\,\,\Gamma\mu}$ for a larger Lie group ${\cal G}$ [@Percacci:1984ai; @Percacci:2009ij; @Nesti:2009kk]. In these theories there is also taken to exist an object $\theta^{\Omega}_{\mu}$, which is a ${\cal G}$ vector and space-time one-form. It is this object which may display a symmetry breaking solution to play the role of the $SO(1,3)$-valued co-tetrad.
|
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abstract: 'Recently a new class of asymptotically AdS ultra-spinning black holes has been constructed with a noncompact horizon of finite area [@HennigarKubiznakMann:2014], in which the asymptotic rotation is effectively boosted to the speed of light. We employ this technique for four-dimensional $U(1)^4$ and five-dimensional $U(1)^3$ gauged supergravity black holes. The obtained new exact black hole solutions for both cases possess a noncompact horizon; their topologies are a sphere with two punctures. We then demonstrate that the ultra-spinning limit commutes with the extremality condition as well as the near horizon limit for both black holes. We also show that the near horizon extremal geometries of the resulting ultra-spinning gauged supergravity black holes lead to the well-known result which contains an AdS$_2$ throat. We then obtain the $[(d-1)/2]$ central charges of the dual CFTs. By assuming the Cardy formula, we show that despite the noncompactness of the horizon, microscopic entropy of the dual CFT is precisely equivalent to the Bekenstein-Hawking entropy.'
---
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Introduction
============
The first solution of Einstein field equations which is their only $4$D spherically symmetric vacuum solution, describing the first black hole was given by the Schwarzschild metric. Moreover many other black hole solutions have been found. Black holes possess a special null surface called the event horizon that is generated by a null Killing vector field, where no object behind it can escape to infinity. Bekenstein discovered that to prevent the violation of the second law of thermodynamics in the presence of a black hole, it must be viewed as a thermodynamic object with entropy. Hawking showed that the black hole acts as a black body with finite temperature that can radiate away its mass. The thermodynamic behavior of black holes points to the existence of an underlying black hole microstate structure. One of the main curiosities is How these microstates can explain the macroscopic Bekenstein-Hawking entropy from a statistical viewpoints. A complete answer has not been given yet, but in string theory, for a large class of extremal supersymmetric black holes, this question has been answered [@StromingerVafa]: Bekenstein-Hawking entropy can indeed be reproduced from a statistical entropy as the logarithm of the degeneracy of BPS states indeed [@Sen]. Recently the horizon fluffs proposal was demonstrated in [@AfsharGrumillerSheikhJabbari] to identify the microstates of three-dimensional Bañados–Teitelboim–Zanelli (BTZ) black holes as states that are marked by the conserved charges related to the nontrivial diffeomorphisms on the near horizon region. Also this proposal evaluated in [@SheikhJabbariYavartanoo] for the general AdS$_3$ black holes in the class of Bañados geometries.
The Kerr/CFT correspondence [@GuicaHartmanStrominger-KerrCFT] provides a rich setup, supporting the idea that whatever the states of quantum gravity are in the near horizon region of an extremal Kerr-AdS black hole, they are holographically dual to quantum states of a two-dimensional chiral (left-moving part) CFT. Since then many further examples [@HartmanStrominger:2009CFTDual; @ChowCvetic-CFT:2008] have been investigated for a large class of black hole solutions. In all cases the statistical microscopic entropy of the dual CFT using the Cardy formula [@Cardy] precisely agrees with the Bekenstein-Hawking entropy of the black holes.
There is a classification of black hole solutions based on their horizon topology. In particular, the famous Hawking theorem for stationary, asymptotically flat vacuum Einstein black hole solutions in four dimensions asserts that their horizon topologies necessarily are $S^2$. By relaxing some of the Hawking’s theorem assumptions, one may comes up with the some classes of black hole solutions with different horizon topology, such as $S^3$ and $S^2 \times S^1$ (black ring solutions) horizon topologies in five-dimension. Also in asymptotically AdS spacetimes, the horizon of a black hole may have a compact Riemann surface of any genus $g$ instead of spherical horizons [@Non-sphericalHorizon], as well as black ring solutions with horizon topology $S^1 \times S^{d-3}$. In the case of adding a rotation to them, the horizon would then be noncompact and the obtaining geometries would describe a rotating black membranes with horizon topology $\mathbb{H} \times S ^{d-4}$. It was shown in four-dimension in [@GneccchiHristovKlemm:2014] and elaborated upon in [@Klemm:2014] that in Einstein-Maxwell-Lambda theory or more generally, in the presence of a scalar potential for $\mathcal{N}=2$ gauged supergravity, one can obtain black hole solutions with non-compact horizon topology with finite area, which can be topologically viewed as spheres with two punctures. Other examples of this kind of topology were constructed in [@HennigarKubiznakMann:2014; @HennigarKubiznakMann:2015] by performing an ultra-spinning (super-entropic) limit to Kerr-AdS and $d$-dimensional multi-spinning Myers-Perry black holes.
The first effort to study ultra-spinning black holes was made by Emparan and Myers [@EmparanMyers:2003] for exploring the stability of Myers-Perry black holes at any arbitrarily large angular momentum with keeping mass fixed, in which the rotation parameter $a \rightarrow \infty$. One result is that area decreases as angular momentum increases, and in higher dimensional $d$, area shrinks to zero. By capturing the null geodesics in the plane of rotation one can realize that the horizon of this kind of ultra-spinning black hole is highly spread out in the plane of rotation while it shrinks in the perpendicular direction. Also it was argued that a Gregory-Laflamme-type instability can be seen for these solutions. The analogue limit can be used in the case of rotating AdS black holes in $d \ge 6$ by taking the rotation parameter approaches the AdS radius $\ell$, which gives the geometry of a black membrane by keeping the mass of the black hole fixed [@CaldarelliEmparan:2008]. In [@ArmasObers] another technique was proposed by which one can also perform $a \rightarrow \infty$ while restricting the ratio $a/\ell$ to remain fixed. Furthermore, Caldarelli et al. constructed a new type of solution with horizon topology $\mathbb{H}^2 \times S^{d-4}$ by keeping the horizon radius $r_+$ fixed while zooming in to the pole and taking $a \rightarrow \ell$, which is called hyperboloid membrane limit [@CaldarelliEmparan:2008; @CaldarelliLeigh:2011]. Recently a simple ultra-spinning (super-entropic) limit was introduced in [@HennigarKubiznakMann:2014; @HennigarKubiznakMann:2015] for Kerr-Newman-AdS and $d$-dimensional multi-spinning Myers-Perry black holes. This technique begins with the Kerr-AdS black hole in an asymptotically rotating frame, and then boosts this rotation to the maximum value $a \rightarrow\ell$, while keeping the metric finite by introducing a change to corresponding azimuthal coordinate, and finally one can compactify this new coordinate to generate a new rapid black hole solution. The resulting black hole has a non-compact horizon but finite area, which is an interesting feature. Also it was shown in the context of an extended thermodynamic phase space where the cosmological constant can vary [@KastorRayTraschen:2009]that the entropy of some of these new solutions violates the reverse isoperimetric inequality, so such black holes are called “super-entropic” [@HennigarKubiznakMann:2014; @HennigarKubiznakMann:2015].
The aim of this work is to find new rotating black hole solutions by taking the recent ultra-spinning (super-entropic) limit [@HennigarKubiznakMann:2014] on some existing black holes. The interesting geometry of resulting ultra-spinning black holes motivated us to further explore the applicability of this limit for some gauged supergravity black holes in four and five dimensions to generate a new type of black hole solutions. Since Supergravity solutions correspond to consistent string theory backgrounds, microscopic degeneracy of states can be investigated using the AdS/CFT correspondence, providing more motivation for their study in ultra-spinning limit. We will focus specifically on two gauged supergravity solutions in $\mathcal{N}=2$ and $\mathcal{N}=4$. After that the Kerr/CFT correspondence for obtaining gauged supergravity ultra-spinning black holes are investigated. It was shown in [@Mann-SEBH-CFT:2015] that despite the noncompactness of event horizons there exists this correspondence for the ultra-spinning Kerr-Newmann-AdS black hole as well as the ultra-spinning limit of minimal gauged supergravity black holes in five dimension.
This paper is organized as follows. In Sec. II, we consider $U(1)^4$ gauged supergravity black hole in four dimensions and we discuss the geometry of the obtained ultra-spinning limit version. We then explore extremality conditions and the near horizon limit under the ultra-spinning limit for this black hole, indicating that both of them commute with the ultra-spinning limit. Next we provide a brief review of Kerr/CFT correspondence, followed by finding microscopic entropy via the Cardy formula, which is exactly equal to the black hole entropy. In Sec. III we have presented a similar analysis for multispinning $U(1)^3$ gauged supergravity black hole in five dimensions. Our conclusions with some remarks are given in Section IV.
Four-Dimensional $U(1)^4$ Gauged Supergravity With Pairwise Equal Charge {#section-4d}
========================================================================
Here, we consider a four-dimensional rotating gauged supergravity black hole with pairwise-equal charged, which was constructed first in [@ChongCveticLuPope-4dgaugedBH] as a $U(1)\times U(1)$ Abelian subgroup of the $SO(4)$ gauged $\cal {N}$$=4$ supergravity. This solution can be consistently embedded in $\cal{N}$$=8$ gauged supergravity, in which two independent electromagnetic charges can be carried by fields in $U(1)$ subgroups of two $SU(2)$ sectors in $SO(4) \sim SU(2) \times SU(2)$. Its relevant bosonic Lagrangian as a truncation of the $\mathcal{N}=4$ is given by $$\begin{aligned}
\label{Action4d}\nonumber
\mathcal{L}_4&=& R * \mathds{1} - \frac{1}{2} * d \varphi \wedge d \varphi - \frac{1}{2} e^{2 \varphi} * d \chi \wedge d \chi - \frac{1}{2} e^{- \varphi} * F_{_{(2)2}} \wedge F_{_{(2)2}} - \frac{1}{2} \chi F_{_{(2)2}} \wedge F_{_{_{(2)2}}} \\ \nonumber
&-& \frac{1}{2(1+\chi^2 e ^{2\varphi})} \big(e^{\varphi} * F_{_{(2)1}}\wedge F_{_{(2)1}} - e^ {2 \varphi} \chi \, F_{_{(2)1}} \wedge F_{_{(2)1}} \big) \\
&-&g^2\big(4+ 2 \cosh \varphi + e^{\varphi} \chi^2\big)* \mathds{1},\end{aligned}$$ where $\varphi$ is the dilaton and $\chi$ is axion. Indeed this special solution is a $U(1)^2$ subset of $U(1)^4$ which arises by setting two electric charges equal $(\delta_2=\delta_4)$, and taking the two magnetic charges equal $(\delta_1=\delta_2)$ as well. For more details we refer the reader to [@ChongCveticLuPope-4dgaugedBH].
Four field strengths can be written in terms of potentials as $$\begin{aligned}
F_{_{(2)1}} &=& d A_{_{(1)1}}, \quad \quad F_{_{(2)2}} = d A_{_{(1)2}},\end{aligned}$$ and $g$ denotes the gauge-coupling constant, which is related to the AdS radius $\ell$ by $g = \ell^{-1}$. Its non-extremal black hole solution in asymptotic rotating frame (ARF) is given by [@ChongCveticLuPope-4dgaugedBH] $$\begin{aligned}
\label{Metric4d)}\nonumber
ds^2&=&-\frac{\Delta_r}{W}\big(dt - \frac{a \sin^2\theta}{\Xi} d \phi \big)^2+ W\big(\frac{dr^2}{\Delta_r}+\frac{d\theta^2}{\Delta_\theta}\big)+\frac{\Delta_\theta \sin^2\theta}{W}\big(a dt-\frac{r_1r_2+a^2}{\Xi}d\phi\big)^2,\\\end{aligned}$$ where $$\begin{aligned}
\label{Metric4d-2)}\nonumber
\Delta_r&=&r^2+a^2 -2m r+\frac{1}{\ell^2}r_1r_2(r_1r_2+a^2),\\
\Delta_{\theta}&=&1-\frac{a^2}{\ell^2}\cos^2\theta, \quad \quad W=r_1r_2 + a^2 \cos^2\theta,\\\nonumber
r_i&=&r+2ms_i^2=r+q_i, \quad \quad \Xi=1-\frac{a^2}{\ell^2},\\ \nonumber
s_i&=&\sinh \delta_i, \quad \quad c_i=\cosh \delta_i.\end{aligned}$$ Also the axion, dilaton and gauge potentials read $$\begin{aligned}
e^{\varphi}&=&\frac{r_1^2+a^2\cos^2\theta}{W}, \quad \quad \quad \quad \quad \quad \qquad \quad \quad \quad \chi=\frac{a(r_2-r_1)\cos^2\theta}{r_1^2+a^2\cos^2\theta},\\ \nonumber
A_{_{(1)1}}&=&\frac{2\sqrt{2}m (dt-a \sin^2\theta \Xi^{-1} d \phi)}{W} s_1 c_1 r_2, \quad \quad \quad A_{_{(1)2}}=\frac{2\sqrt{2}m (dt-a \sin^2\theta \Xi^{-1} d \phi)}{W} s_2 c_2 r_1.\nonumber\end{aligned}$$ The coordinate change $\tilde{\phi}=\phi+a g^2 t$ yields an asymptotically static frame (ASF). The Hawking temperature, entropy, angular velocity and electrostatic potentials on the horizon (in the asymptotically rotating frame) are $$\begin{aligned}
\label{Thermo4d}\nonumber
T_H&=&\frac{r_+^2-a^2+a^2/\ell^2(r_+^2-q_1q_2)+(r_++q_1)(r_++q_2)(3r_+^2+q_1r_+ + q_2 r_+ - q_1q_2)/\ell^2}{4 \pi r_+ [(r_++q_1)(r_++q_2)+a^2]},\\ \nonumber
S&=&\frac{\pi[(r_++q_1)(r_++q_2)+a^2]}{\Xi}, \quad \quad \quad \quad \quad \quad \Omega=\frac{\Xi\, a}{(r_++q_1)(r_++q_2)+a^2}, \\
\Phi_1&=&\Phi_2=\frac{2ms_1 c_1(r_++q_2)}{(r_++q_1)(r_++q_2)+a^2}, \quad\qquad\qquad \Phi_3=\Phi_4=\frac{2ms_2 c_2(r_++q_1)}{(r_++q_1)(r_++q_2)+a^2},\end{aligned}$$ where $r_+$ is the largest root of $\Delta_r=0$ as the outer horizon.
The charge quantities including mass, angular momentum and pairwise electric potentials were constructed in [@CvetiGibbons:2005] and are $$\begin{aligned}
\label{Charges4d}\nonumber
E&=&\frac{m}{\Xi^2}(1+s_1^2+s_2^2)=\frac{2m+q_1+q_2}{2 \, \Xi^2}, \\ \nonumber
J&=&\frac{m a}{\Xi^2}(1+s_1^2+s_2^2)=\frac{a(2m+q_1+q_2)}{2 \, \Xi^2},\\
Q_1&=&Q_2=\frac{ms_1 c_1}{2\Xi}, \quad\quad\quad \quad Q_3=Q_4=\frac{m s_2 c_2}{2 \Xi}.\end{aligned}$$ In the next subsection we use the ultra-spinning (super-entropic) limit upon the metric (\[Metric4d)\]) in order to obtain a new charged-AdS black hole solution in gauged supergravity.
This novel ultra-spinning (super-entropic) limit can be interpreted as a simple method to generate a new black hole solution in which the rotation parameter $a$ reaches its maximum amount, equal to the AdS radius $\ell$. This procedure limit consists of three steps. i) Transforming metric to an asymptotic rotating frame to avoid a singular metric in the ultra-spinning limit. We then need only define a new azimuthal coordinate $\varphi=\phi/{\Xi}$ afterward. ii) This rotation has to be boosted effectively to the speed of light, namely, by taking the $a \rightarrow \ell$ limit. iii) In final step, we compactify the new azimuthal direction $\varphi$ [@HennigarKubiznakMann:2014].
We note that employing an asymptotically rotating coordinate should be assumed as a crucial point in the ultra-spinning limit technique. The uniqueness of the choice of ARF to have a nonsingular black hole solution has been discussed in [@HennigarKubiznakMann:2015]. In fact, avoiding to start from an ARF leads us to a singular limit.
Ultra-spinning limit
--------------------
Since the metric (\[Metric4d)\]) is already written in an asymptotically rotating frame, we therefore need only to introduce a new azimuthal coordinate $\varphi=\phi/\Xi$, followed by taking the limit $a \rightarrow \ell$. Hence we straightforwardly obtain the following solution $$\label{UsMetric4d}
ds^2=-\frac{\tilde{\Delta}_r}{\tilde{W}}\big(dt - \ell \sin^2\theta d \varphi \big)^2+ \tilde{W}\big(\frac{dr^2}{\tilde{ \Delta}_r}+\frac{d\theta^2}{\sin^2\theta}\big)+\frac{\sin^4\theta}{\tilde{W}}\big[\ell dt-(r_1r_2+\ell^2 )d\varphi\big]^2,$$ where $\tilde{\Delta}_r$ and $\tilde{W}$ are given by (\[Metric4d-2)\]) as $a \rightarrow l$. To exclude a conical singularity in $\varphi$ direction, one can identify it with period $2\pi/\Xi$. Since the new azimuthal coordinate $\varphi$ is non-compact, we now compactify it by requiring $$\varphi \sim \varphi + \mu,$$ where the parameter $\mu$ is dimensionless. Note that there is still an axial Killing vector $\partial_\varphi$ in the new coordinate direction $\varphi$. Therefore, it is straightforward to show that the obtained metric (\[UsMetric4d\]) appears as a new exact solution. Also we can easily find the dilaton, axion and gauge fields in this limit as $$\begin{aligned}
\label{fieldsUS4d}
e^{\tilde{\varphi}}&=&\frac{r_1^2+\ell^2\cos^2\theta}{\tilde{W}}, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \tilde{\chi}=\frac{\ell(r_2-r_1)\cos\theta}{r_1^2+\ell^2\cos^2\theta},\\ \nonumber
\tilde{A}_{_{(1)1}}&=&\frac{2\sqrt{2}m s_1 c_1 r_2 (dt-\ell \sin^2\theta d \varphi)}{\tilde{W}}, \quad \quad \quad \tilde{A}_{_{(1)2}}=\frac{2\sqrt{2}m s_2 c_2 r_1 (dt-\ell \sin^2\theta d \varphi)}{\tilde{W}}.
\end{aligned}$$
**Horizon geometry:** To ensure that the new solution (\[UsMetric4d\]) is describing a black hole, we examine the largest root of $\tilde{\Delta}_r$ in (\[UsMetric4d\]), which is supposed to demonstrate the location of the horizon as $r_+$. It is required now to check $\tilde{\Delta}^{\prime}_r \geq 0 $. We therefore find following mass bound $$\begin{aligned}
\label{mass4d}
m &\geq& m_0\equiv \frac{\sqrt{3}}{18 \ell^2}\big[(q_1-q_2)^2-4\ell^2\big]^{\frac{3}{2}}-\frac{q_1+q_2}{2}, \quad \quad |q_2-q_1| \geq 2 \ell.
\end{aligned}$$ For $m > m_0$ a horizon exists, while for $m < m_0$ there exists a naked singularity. Also to ensure that our obtained geometry would be free of any closed timelike curves (CTC) we examine $g_{\varphi\varphi} \ge 0$ $$\begin{aligned}
g_{\varphi\varphi}=\frac{\ell^2\sin^4\theta\big[(2 m+ r_+ (q_1+q_2)+q_1q_2\big] }{ (r_+ +q_1)(r_++q_2) + \ell^2 \cos^2\theta}.
\end{aligned}$$ Apparently $g_{\varphi\varphi}$ is positive in the entire spacetime.
Now, let us take a deeper look at the geometry of the horizon. The induced metric on a constant $(t,r)$ surface yields, $$\begin{aligned}
\label{ds2h-1}
ds^2_h=\frac{\tilde{W}_+}{\sin^2\theta}d\theta^2+\sin^4\theta\frac{\big((r_+ +q_1)(r_++q_2)+\ell^2\big)^2}{\tilde{W}_+}d\varphi^2.
\end{aligned}$$ where $\tilde{W}_+=\tilde{W}|_{r_+}$. However, this geometry seems to be singular in $\theta=0$ and $\pi$, so we will show that the symmetry axis $\theta=0,\pi$ is actually not part of the spacetime. For a precise study about these poles, one can probe the metric in the small $\theta=0$ limit by introducing the change of variables similar to [@HennigarKubiznakMann:2015], $$\label{definek}
k=\ell(1-\cos\theta),$$s Horizon metric (\[ds2h-1\]) for small $k$ becomes $$\begin{aligned}
\label{ds2h-2}
ds^2_h=(r_+ +q_1)(r_++q_2)\bigg[\frac{dk^2}{4k^2}+\frac{4k^2}{\ell^2}d\varphi^2\bigg],
\end{aligned}$$ which shows clearly a metric of constant negative curvature on a quotient of the hyperbolic space $\mathbb{H}$$^2$. Not that, due to the symmetry, the $\theta=\pi$ limit gets the same result. Thus, there is no true singularity at these two points, but some sort of boundaries. Therefore topologically, the event horizon is a sphere with two punctures, which implies that our obtained black hole enjoys a finite area but non-compact horizon.
In order to visualize the geometry of the horizon (\[ds2h-1\]) we can embed it in Euclidean $3$-space as a surface of revolution [@GneccchiHristovKlemm:2014]. We then identify the induced metric on the horizon (\[ds2h-1\]) by using a flat metric in cylindrical coordinates as $$\label{CylindricalCo1}
ds_3^2=dz^2+dR^2+R^2d \Phi^2,$$ and consider $z = z(\theta)$, $R = R(\theta)$. Setting $\Phi=\frac{2\pi}{\mu} \varphi$, one gets $$\label{CylindricalCo2}
R^2(\theta)=\bigg(\frac{\mu}{2\pi}\bigg)\frac{[(r_++q_1)(r_++q_2)+\ell^2]^2}{(r_++q_1)(r_++q_2)+\ell^2 \cos^2\theta}\sin^4\theta,$$ $$\label{CylindricalCo3}
\bigg(\frac{dz(\theta)}{d\theta}\bigg)^2 + \bigg(\frac{dR(\theta)}{d\theta}\bigg)^2=\frac{(r_++q_1)(r_++q_2)+\ell^2 \cos^2\theta}{\sin^2\theta},$$ which is a differential equation for $dz/d\theta$. We integrated (\[CylindricalCo3\]) numerically for the values $\ell=1$, $\mu=2\pi$, and $r_+=0.8$, by choosing different amounts of $q_1$ and $q_2$. The resulting surfaces of revolution are shown in Fig. 1.
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![[**Fig. 1. **Horizon embeddings in 4d****]{}. Plots show two-dimensional nonacompact horizons embedded in $\mathbb{R}$$^3$ as a surface of revolution, displaying the topology of a sphere with two punctures. We have set $\mu = 2\pi$, $r_+ = 0.8$, and $\ell=1$ for all diagrams, with $q_1=10$, $q_2=0$ (left); $q_1=10$, $q_2=5 $ (middle); and $q_1=3$, $q_2=6$ (right). []{data-label="HorizonEmbedding4d"}](Hd4rp=8q1=10q2=0.pdf "fig:"){width="30.00000%" height="0.3\textheight"} ![[**Fig. 1. **Horizon embeddings in 4d****]{}. Plots show two-dimensional nonacompact horizons embedded in $\mathbb{R}$$^3$ as a surface of revolution, displaying the topology of a sphere with two punctures. We have set $\mu = 2\pi$, $r_+ = 0.8$, and $\ell=1$ for all diagrams, with $q_1=10$, $q_2=0$ (left); $q_1=10$, $q_2=5 $ (middle); and $q_1=3$, $q_2=6$ (right). []{data-label="HorizonEmbedding4d"}](Hd4rp=8q1=10q2=5.pdf "fig:"){width="30.00000%" height="0.3\textheight"} ![[**Fig. 1. **Horizon embeddings in 4d****]{}. Plots show two-dimensional nonacompact horizons embedded in $\mathbb{R}$$^3$ as a surface of revolution, displaying the topology of a sphere with two punctures. We have set $\mu = 2\pi$, $r_+ = 0.8$, and $\ell=1$ for all diagrams, with $q_1=10$, $q_2=0$ (left); $q_1=10$, $q_2=5 $ (middle); and $q_1=3$, $q_2=6$ (right). []{data-label="HorizonEmbedding4d"}](Hd4rp=8q1=3q2=6.pdf "fig:"){width="30.00000%" height="0.3\textheight"}
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\[Fig1-4dH\]
**Conformal Boundary :** Let us here, find the conformal boundary of our obtained new black hole solution (\[UsMetric4d\]) (the conformal factor is $\ell^2/r^2$) $$\label{ConformalbdryUS4d}
ds^2_{bdry}=-dt^2+\ell\sin^2\theta dt d\varphi +\frac{1}{\sin^4\theta}d\theta^2.$$ It appears that the new coordinate $\varphi$ would be a null one on the conformal boundary. Now we study this metric near the pole $\theta=0$ using (\[definek\]). For small $k$ we have then $$\label{ConformalbdryUS4d-k}
ds^2_{bdry}=-dt^2 +2 k dt d\varphi+\frac{1}{4 k^2}dk^2.$$ This metric can be interpreted as an AdS$_3$ written in Hopf-like fibration over $\mathbb{H}^2$. It means again that the poles $\theta=0, \, \pi$ are indeed not parts of the spacetime, and they are being removed from the boundary. However, to answer what happens precisely at $\theta=0$ and $\pi$, we should study the behavior of the geodesics in the entire spacetime. To do that, it should be shown that no outgoing null geodesics from inside the horizon would not be able to reach to the symmetry axis $\theta=0$, in a finite affine parameter, similar to the strategy taken in [@HennigarKubiznakMann:2015]. This study of course, is not in the scope of the current paper, so we postpone studying it to another further work.
As we recall, the AdS/CFT viewpoint hints at exploring a dual $3$-dimensional CFT which exists on the AdS$_3$ conformal boundary (\[ConformalbdryUS4d-k\]). We note, depending on the choice of the AdS coordinate slicing, different manifolds may be achieved such that one can found a dual CFT that lives on the correspondence slicing. Each coordinate indeed covers the whole or a part of AdS boundary; for instance, in global coordinate the dual CFT lives on $R \times S^p$ geometry, while in the Poincar$\acute{e}$ patch the CFT exists on $R^{p,1}$ manifold. Therefore, in the slicing (\[ConformalbdryUS4d-k\]) one may expect the dual CFT to reside on an AdS$_3$ geometry. These CFTs can be viewed as possible deformations around the conformal fixed point, which may be related to each other by the Wilsonian renormalization group flow equation.
**Thermodynamic quantities :** Although, the event horizon of the obtained ultra-spinning black hole (\[UsMetric4d\]) is noncompact, it has a finite area and entropy as $$\label{EntropyUS4d}
S =\frac{\mu}{2} \big[ (r_+ +q_1)(r_+ + q_2)+\ell^2 \big].$$ For solution (\[UsMetric4d\]) we extract the Hawking temperature, electrostatic potentials and angular velocity on the horizon. Our results are $$\begin{aligned}
\label{thermoUS4d}\nonumber
T_H&=& \frac{2r_+^2-\ell^2-q_1q_2+(r_+ +q_1)(r_+ +q_2)(3r_+^2+q_1r_+ + q_2r_+-q_1q_2)/\ell^2}{4\pi r_+
\big[(r_++q_1)(r_++q_2)+\ell^2\big]},\\ \nonumber
\Phi_1&=& \Phi_2 = \frac{\sqrt{q_1(2m+q_1)}(r_++q_2)}{(r_++q_1)(r_++q_2)+\ell^2}, \quad \quad \quad \Phi_3= \Phi_4 = \frac{\sqrt{q_2(2m+q_2)} (r_++q_1)}{(r_++q_1)(r_++q_2)+\ell^2}, \\
\Omega&=&\frac{\ell}{(r_+ +q_1)(r_+ + q_2)+\ell^2}.
\end{aligned}$$ The electric potential is computed using $\Phi^I=K^\mu A_\mu^I$, where $K^\mu=\partial_t+\Omega_s \partial_\varphi$ is the null Killing vector generating the horizon. We then calculated the following conserved charges as $$\begin{aligned}
\label{ChargesUS4d}
E&=&\frac{2m+q_1+q_2}{2}, \quad\quad\quad \quad \quad\quad\quad \quad \quad\quad \quad \quad J=\frac{\ell(2m+q_1+q_2)}{2},\\ \nonumber
Q_1&=&Q_2=\frac{\sqrt{q_1(2m+q_1)}}{4}, \quad\quad \quad\quad \quad\quad\quad Q_3=Q_4=\frac{\sqrt{q_2(2m+q_2)}}{4}.
\end{aligned}$$ Angular momentum can be obtained by using the Komar integral based on the Killing vector $\partial_\varphi$, also the electric charges are calculated by using the Gaussian integrals. For computing mass we use the conformal approach of Ashtekar, Magnon and Das (AMD) [@Ashtekar]. In the AMD method the electric part of the Weyl tensor plays an essential role: its spatial conformal boundary integral gives us the mass. It was shown that for a large variety of gauged supergravity black holes this mass exactly agrees with the first law of thermodynamics calculation [@Chen:2006:MassAMD] . **Extremality under ultra-spinning limit:** For an extremal black hole the inner and outer horizons coincide to a single horizon at $r_0$, with vanishing Hawking temperature and generically non-zero entropy.
Here, in order to answer whether extremality condition will be preserved under the ultra-spinning limit or not, we consider two different approaches: i) Finding the extremality condition of a given black hole and then applying the ultra-spinning limit.ii) Performing the ultra-spinning limit to a given geometry, and then finding the extremality condition of the obtained ultra-spinning black hole afterwards.
Firstly, let us to start from situation (i), the extremality conditions of metric (\[Metric4d)\]) can be found by using (\[Thermo4d\]) and (\[Metric4d-2)\]), imposing $T_H|_{r=r_0}=0$ and $\Delta_r=0$. Hence the degenerate horizon $r_0$ reads $$\begin{aligned}
\label{ExCondition4d-1}
r_0=\frac{1}{6}\bigg[\big(3\mathcal{B}+3\sqrt{\mathcal{A}^3+\mathcal{B}}\big)^{1/3}-\frac{\mathcal{A}}{\big(3\mathcal{B}+3\sqrt{\mathcal{A}^3+\mathcal{B}}\big)^{1/3}}-3(q_1+q_2)\bigg],
\end{aligned}$$ where $\mathcal{A}=3^{2/3}\big[2(a^2+\ell^2)-(q_1-q_2)^2\big]$, and $\mathcal{B}=9\ell^2(2m+q_1+q_2)$. Also the following constraints between parameters of solution and $r_0$ are achieved $$\begin{aligned}
\label{ExCondition4d-2}
q_1&=&\frac{a^2 q_2 -4 r_0^2(r_0+q_2)+\sqrt{4(r_0^2-q_2^2)[-a^2\ell^2 +r_0^2(a^2+\ell^2+q_2^2+4q_2r_0+3r_0^2) ]}}{2(r_0^2-q_2^2)}.
\\ \nonumber
m&=&\frac{a^4 q_2 + 4 \ell^2 q_2(r_0-q_2)+2 (r_0 + q_2)^2\big[2r_0(r_0+q_2)^2+\chi)\big]-a^2\big[4\ell^2(q_2-r_0)+4r_0(r_0+q_2)^2+\chi\big]}{4 \ell^2 (r_0-q_2)^2},
\end{aligned}$$ where $\chi=\sqrt{a^4q_2^2+4(r_0+q_2)\big(r_0^2 (q_2 + r_0) ((q_2 + r_0)^2 - a^2) + 4 l^2 (q_2 - r_0) (r_0^2 - a^2 ) \big)}$.
Now, upon taking the limit $a \rightarrow \ell$, one easily gets $$\begin{aligned}
\label{ExConditionUS4d-1}
q_1&=&\frac{\ell^2 q_2 - 4 q_2 r^2 - 4 r^3+\sqrt{ 4 r_0^2 (q_2 + r_0)^4 + \ell^4 (-3 q_2^2 + 4 r_0^2)-8 \ell^2 r_0^3 (q_2 + r_0)}}{2(r_0^2-q_2^2)}, \\ \nonumber
m&=&\frac{\ell^4 (-3 q_2 + 4 r_0) + 2 (q_2 + r_0)^2\big[2r_0(q_2+r_0)^2+\chi_s\big]-\ell^2\big[4r_0^2(3q_2+r_0)+\chi_s\big]}{4 \ell^2 (r_0-q_2)^2},
\end{aligned}$$ where $\chi_s=\sqrt{4r_0^2(r_0+q_2)^4+\ell^4(4r_0^2-3q_2^2)-8\ell^2r_0^3(r_0+q_2)}$. The horizon also obtained as $$\begin{aligned}
\label{ExConditionUS4d-2}
r_0=\frac{1}{6}\bigg[\big(3\mathcal{B}+3\sqrt{\tilde{\mathcal{A}}^3+\mathcal{B}}\big)^{1/3}-\frac{\tilde{\mathcal{A}}}{\big(3\mathcal{B}+3\sqrt{\tilde{\mathcal{A}}^3+\mathcal{B}}\big)^{1/3}}-3(q_1+q_2)\bigg],
\end{aligned}$$ where $\tilde{\mathcal{A}}=3^{2/3}\big[2\ell^2-(q_1-q_2)^2\big]$.
Now, we examine approach (ii). In this way we only need to find the extremality condition of our obtained ultra-spinning black hole (\[UsMetric4d\]) by using (\[thermoUS4d\]). Our resulting conditions are exactly same as (\[ExConditionUS4d-1\]) and (\[ExConditionUS4d-2\]). It means that the extremality conditions commute with the ultra-spinning limit for a four-dimensional $U(1)^4$ gauged supergravity (\[Metric4d)\]) black hole.
Near horizon geometry
---------------------
Extremal black holes have several interesting characteristics. One of them is that the near horizon region of extremal black holes intuitively can be interpreted as an isolated geometry and isolated thermodynamical system. It has been discussed in [@BardeenHorowitz; @KunduriLucietti] that focusing on the near horizon geometry of extremal black holes leads to a new class of solutions to the same theory of gravity, which their conserved charges are the same as the ones of the original black hole. Near horizon extremal geometries (NHEG) have no horizon and no singularity unlike black holes, with enhanced $SL(2,\mathbb{R}) \times U(1)^N$ symmetry, and their asymptotic behaviours are different. Analogues to black hole thermodynamics, the laws of NHEG dynamics have been derived in [@Sheikh-JabbariHajianSeraj:2013], in which despite the absence of a event horizon in NHEG there is an entropy associated to them as a Noether charge. However in this case system cannot be excited while it keeps $SL(2,R)$ isometry [@Reall; @JohnstoneSheikh-Jabbari]. It was provided a proof in that $AdS_2$ sector appears in the near horizon geometry of any regular stationary extremal black hole.
In this section we try to find the near horizon geometry of the obtained ultra-spinning black hole (\[UsMetric4d\]) in the extremal limit. Now we can write the near horizon expansion as $$\begin{aligned}
\nonumber
\tilde{\Delta}_r=X(r-r_0)^2+\mathcal{O}(r-r_0)^3,
\end{aligned}$$ where $$\begin{aligned}
X=2+\frac{1}{\ell^2}(6r_0^2+6q_1r_0+6q_2r_0+q_1^2+q_2^2+4q_1q_2).
\end{aligned}$$ Now, in order to find the near horizon geometry in the extremal limit, we use following dimensionless coordinate changes $(\hat{t},\hat{r},\hat{\theta},\hat{\varphi})$ to extract the NHEG as an exact solution. $$\begin{aligned}
\label{NHG-Coordinate4d}
r&=&r_0(1 +\lambda \hat{r}),\quad \quad \quad \varphi=\hat{\varphi}+\Omega_H^0 \hat{t}, \quad \quad \quad
t=\frac{\hat{t}}{2\pi T^{\prime}_H r_0 \lambda },\quad \quad \quad \hat{\theta}=\theta.
\end{aligned}$$ Here, the quantity $\Omega_H^0$ is the angular velocity on the horizon in extremal limit, which is a shift for the coordinate $\varphi$ to attain the comoving coordinate of the horizon. In these new coordinates the killing vector $\xi=\lambda/r_0 \partial_t$ is the horizon generator. Also the quantity $T ^{\prime \, 0}_H$ is defined as $$T^{\prime \, 0}_H=\left.\frac{\partial T_H}{\partial \hat{r}_+ }\right\vert_{\hat{r}_+=r_0}.$$
Now upon taking the limit $\lambda \rightarrow 0$, the near-horizon metric reads, $$\begin{aligned}
\label{NH4d}
ds^2&=&\frac{\tilde{W}_0}{X}\big(-\hat{r}^2 d\hat{t}^2 + \frac{d \hat{r}^2}{\hat{r}^2}\big) + \frac{\tilde{W}_0}{\sin^2\theta}d\hat{\theta}^2,\\ \nonumber
&+& \frac{\sin^4 \theta}{\ell^2 \tilde{W}_0}\big[(r_0+q_1)(r_0+q_2)+\ell^2\big]^2 \left( d\hat{\varphi} + k \hat{r} d\hat{t} \right)^2,
\end{aligned}$$ where $$\label{k4d}
k=\frac{\ell (2r_0+q_1+q_2)}{X\big[(r_0+q_1)(r_0+q_2)+\ell^2\big]},$$ and $\tilde{W}_0=\tilde{W}|_{\hat{r}=r_0}$. This metric can be viewed as the direct product of AdS$_2 \times S^2$, in which the AdS sector is written here by Poincar$\acute{e}$-type coordinates$(\hat{t}, \hat{r})$. It appears that the NHEG of the ultra-spinning black hole (\[UsMetric4d\]) gives the well-known result which contains an AdS$_2$ sector. The metric (\[NH4d\]) similar to the horizon geometry, seems to be singular in $\theta=0, \pi$. We showed in previous subsections that these points are not truly singularities but they are some kinds of boundaries. Therefore the S$^2$ sector of NHEG (\[NH4d\]) inherits the noncompactness characteristic and then it topologically would be a sphere with two punctures.
Metric (\[NH4d\]) can be cast into the general form of the NHEG constructed in [@ChowCvetic-CFT:2008] which is calculated for the most general gauged extremal and stationary supergravity black holes as follows $$\begin{aligned}
\label{NHGeneralform}
ds^2&=&\Gamma(\theta)\bigg[-\rho^2 dt^2 + \frac{d\rho^2}{\rho^2} + \alpha(\theta) d\theta^2\bigg]+\gamma(\theta)(d\phi+k \rho dt)^2,\\ \nonumber
A^I&=& f(\theta)(d\phi+k \rho dt)+e^p r dt.
\end{aligned}$$ Where $\Gamma(\theta), \gamma(\theta)$ and $\alpha(\theta)$ are those extracted from (\[NH4d\]). Additionally, to explore the behaviour of the gauge fields in the near horizon limit (\[NHG-Coordinate4d\]) we should carry out a new gauge transformation $A^I \rightarrow A^I + d\Lambda^I$ on parameter $\Lambda$ [@CompereKerrCFT]. $$\label{NHG-gauge2}
\Lambda=\frac{\Phi_e^{I \, ext}}{\lambda}r_0 \hat{t},$$ where $\Phi_e^{I \, ext}$ are the electrostatic potentials on the horizon in the extremal limit. This gauge transformation can be realized as a simple embeddings of a $U(1)$ gauge field in a higher-dimensional auxiliary spacetime [@CompereKerrCFT].
A quick review on Kerr/CFT
--------------------------
Here, we test the conjecture Kerr/CFT correspondence and its extensions for our noncompactness horizon black hole. Kerr/CFT explains that there exists a duality between the near horizon states of a $4$d extremal kerr black hole and a certain $d=2$ chiral conformal field theory [@GuicaHartmanStrominger-KerrCFT]. The general NHEG (\[NHGeneralform\]) by imposing a set of consistent boundary conditions admits an enhanced $SL(2, R)_R \times U(1)^n_L$ isometry group [@GuicaHartmanStrominger-KerrCFT; @Barnich]. Indeed it includes all the symmetries of AdS$_2$ plus translations in $\varphi^i$ coordinates. The symmetry generators are given by the following Killing vectors fields $$\label{symGenSL2U1}
\xi_1=\partial_t, \quad \xi_2=t\partial_t - r \partial_r, \quad \xi_3=\frac{1}{2}\big(\frac{1}{r^2}+t^2\big)\partial_t - t r \partial_r -\sum_{i=1}^{n}\frac{k_i}{r}\partial_\phi^i, \quad \quad \bar{\xi}_i=\partial_\phi^i,$$ One can write a relation between the $SL(2, R)$ and $U(1)^n$ symmetry generators [@Sheikh-JabbariHajianSeraj:2013] $$\label{SU2U1}
n^i\xi_i=k^i\bar{\xi}_i,$$ where $n_i$ is the unit vector normal to AdS$_2$.
Therefore, because of the exact similarity between NHEG of our solutions (\[NHG-Coordinate4d\]) and (\[NHGeneralform\]), one can deduce that the metric (\[NH4d\]) is invariant under diffeomorphisms generated by $\bar{\xi}_1$ and $\xi_{1,2,3}$. Therefore an asymptotic symmetry group $(ASG)$ associated to every consistent set of boundary conditions can be found. Asymptotic symmetries of (\[NH4d\]) may contain diffeomorphisms $\zeta$ and a $U(1)$ gauge transformation such that [@HartmanStrominger:2009CFTDual] $$\begin{aligned}
\label{ASG1}
\delta_\zeta g_{\mu\nu}&=& \mathcal{L}_\zeta g_{\mu\nu}, \quad\quad \delta_\zeta A_\mu= \mathcal{L}_\zeta A_\mu, \quad\quad \delta_\Lambda A=d\Lambda.
\end{aligned}$$ The infinitesimal field variations are defined by $a_\mu=\delta A_\mu$ and $h_{\mu\nu}=\delta g_{\mu\nu}$. The combined transformation $(\zeta,\Lambda)$ has an associated charge $Q_{\zeta,\Lambda}$ defined in [@Barnich], which generates the symmetry $(\zeta,\Lambda)$ under Dirac brackets.
The charge $Q_{\zeta,\Lambda}$ must be finite for all field variations, required to satisfy a consistent boundary condition. Now we choose the same boundary conditions as in [@GuicaHartmanStrominger-KerrCFT]. The method used in [@GuicaHartmanStrominger-KerrCFT] assumed that $\partial_\phi$ proportional to the zero mode of a nontrivial Virasoro algebra, as well as indicating the boundary conditions in terms of power law falloff of the components of the metric fluctuations $h_{\mu\nu}$ as $$\begin{aligned}
\label{ASG-Boundary}
\begin{pmatrix}
h_{tt}=\mathcal{O}(r^2)& h_{t\varphi}=\mathcal{O}(1)& h_{t\theta}=\mathcal{O}(1/r)&h_{tr}=\mathcal{O}(1/r^2)\\
& h_{\varphi\varphi}=\mathcal{O}(1)& h_{\varphi\theta}=\mathcal{O}(1/r)&h_{\varphi r}=\mathcal{O}(1/r)\\
& & h_{\theta\theta}=\mathcal{O}(1/r)&h_{\theta r}=\mathcal{O}(1/r^2)\\
& & &h_{r r}=\mathcal{O}(1/r^3)\\
\end{pmatrix}.
\end{aligned}$$ Also, the boundary condition for the gauge field reads $$\label{GaugeBoundary}
a_\mu=\mathcal{O}(r,\frac{1}{r},1,1/r^2).$$ The most general diffeomorphisms $\zeta_m$ that preserve the above boundary conditions are given by $$\begin{aligned}
\label{GeneralDiffeo}
\zeta_\epsilon=\epsilon(\phi) \partial_\psi - r \epsilon^\prime(\phi) \partial_r, \quad \quad \quad \bar{\zeta}=\partial_t,
\end{aligned}$$ and the Virasoro algebra can be extracted as $$\label{Virasoro1}
i[\zeta_m,\zeta_n]=(m-n)\zeta_{m+n}, \quad \quad \epsilon_n(\phi)=-e^{-in \phi}.$$ The gauge field transformation $A$ under $\zeta_\epsilon$ does not satisfy the boundary condition (\[GaugeBoundary\]). So to restore $\delta A_\varphi=\mathcal{O}(1/r)$ we must impose a suitable compensating $U(1)$ gauge transformation as $\Lambda=-f(\theta)\epsilon(\phi)$ [@HartmanStrominger:2009CFTDual]. The Virasoro algebra with vanishing central charge of ASG reads $$\begin{aligned}
\label{Virasoro3}\nonumber
[\Lambda_m,\Lambda_n]_\zeta&=&\zeta_m^\mu\partial_\mu\Lambda_n-\zeta_n^\mu\partial_\mu\Lambda_m,\\
i[(\zeta_m,\Lambda_m),(\zeta_n,\Lambda_n)]_\zeta&=&(m-n)(\zeta_{m+n},\Lambda_{m+n}).
\end{aligned}$$ The gauge transformation $(\zeta_n,\Lambda_n)$ and charges $Q_n$ associated to them satisfy a similar algebra up to a central extension. Using (\[NHGeneralform\]), one can derive the Dirac brackets algebra of $Q_n$ as [@Barnich; @HartmanStrominger:2009CFTDual] $$\begin{aligned}
\label{ASG3} \nonumber
i \big\{Q_{\zeta_\epsilon,\Lambda},Q_{\zeta_{\bar{\epsilon}},\tilde{\Lambda}}\big\}_{DB}&=&iQ_{(\zeta_\epsilon,\Lambda),(\zeta_{\bar{\epsilon}},\tilde{\Lambda})}-\frac{i k}{16 \pi}\int{d\theta d\varphi} \sqrt{\frac{\alpha(\theta)\gamma(\theta)}{\Gamma(\theta)}}\bigg(f(\theta) \Lambda \tilde{\epsilon}^\prime+\Gamma(\theta)\epsilon^\prime\tilde{\epsilon}^{\prime\prime}\\
&+&[f(\theta)^2+\gamma(\theta)]\epsilon\tilde{\epsilon}^\prime-(\epsilon,\Lambda \leftrightarrow \tilde{\epsilon},\tilde{\Lambda})\bigg).
\end{aligned}$$ The algebra of the charge $Q_n$ associated to ASG generators $(\zeta_n,\Lambda_n)$ from (\[ASG3\]) is $$\label{Virasoro4}
i\big\{Q_m,Q_n\big\}_{DB}=(m-n)Q_{m+n} + \frac{c}{12}(m^3-\alpha m) \delta_{m+n,0}.$$ We note that, $\alpha$ is a constant and can be absorbed to $Q_0$. The central charge $c_L$ has combinations of $k^{grav}$ and $k^{gauge}$ as $c=c_{grav}+c_{gauge}$. They can be nicely calculated in the manner described in [@HartmanStrominger:2009CFTDual; @ChowCvetic-CFT:2008] as $$\label{CentarlCharge}
c_{grav}=\frac{3k_i}{2\pi}\int_0^{\pi}d\theta \sqrt{\Gamma(\theta)\alpha(\theta)\gamma(\theta)}, \quad \quad c_{gauge}=0,$$ where the constants $k_i$’s are given by corresponding Frolov-Thorne temperatures as below $$\label{Frolov1}
k_i=\frac{1}{2\pi T_i}.$$ Here, $T_i$’s referred to the left- and right-moving temperatures of the quantum field theory coming from the Frolov-Thorne vacuum, are restricted to an extreme Kerr black hole [@FrolovThorne; @GuicaHartmanStrominger-KerrCFT]. In order to determine these temperatures, one can use a scalar field expansion in terms of eigenstates of the asymptotic energy $E$ and angular momentum $J$ $$\label{FrolovVacume}
\Phi=\sum_{E,J,l}^{} \phi_{E J l} \, e^{-i E \hat{t} + i J \hat{\varphi}} f_l(r,\theta).$$ In the near horizon region (\[NHG-Coordinate4d\]) we have $$\label{FrolovVacumeNH}
e^{-i E \hat{t} + i J \hat{\varphi}} = e^{-i (E-\Omega^0_H J)\hat{t} r_0/\lambda + i J \hat{\varphi}} =e^{-i n_R \hat{t} + n_L \hat{\varphi}},$$ where $n_R= (E-\Omega^0_H J) r_0/\lambda$ and $n_L=J$ are the left and right charges associated to $\partial_\phi$ and $\partial_t$ in the near horizon metric. One can find a diagonal density matrix associated to the vacuum by tracing (\[FrolovVacume\]) over the region inside the horizon. The Boltzmann weighting factor in the energy-angular momentum eigenbasis reads as $$\label{BoltzmanWeighting}
e^{-\frac{(E-\Omega_H J)}{T_H}}$$ In the nonrotating $(\Omega_H=0)$ case, this vacuum reduces to the Hartle-Hawking vacuum. One can hence write the Boltzmann factor in terms of $n_L$, $n_R$ and $T_i$ as $$\label{FrolovVacumeBoltzman}
e^{- \frac{(E-\Omega_H J)}{T_H}}=e^{-\frac{n_L}{T_L}-\frac{n_R}{T_R}}.$$ Dimensionless Frolov-Thorne temperatures $T_i$ where defined firstly for higher-dimensional Kerr-AdS black holes in [@LuMeiPope-KerrCFT] and are $$\label{FrolovTRTL}
T_L=\lim\limits_{r_+\rightarrow r_0}\frac{T_H^0}{\Omega^0_H-\Omega_H}=-\frac{\partial T_H / \partial r_+}{\partial \Omega_H / \partial r_+}\vert_{r_+=r_0}, \quad \quad \quad T_R=\frac{r_0}{\lambda}T_H\vert_{r_+=r_0}.$$ For an extremal solution, The right-temperature vanishes, while there are $[(d-1)/2]$ left-temperatures associated to the $2$d CFTs for each azimuthal $\varphi$ coordinate. However, the extreme Kerr black hole has vanishing Hawking temperature, but quantum fields states outside the horizon live in a thermal state.
We recall that (\[FrolovVacume\]) can also be extended using the first thermodynamics law for a charged rotating black hole $$\label{FirstLaw}
TdS=dM-(\Omega_H dJ + \Phi dQ).$$ Now by imposing the extremality constraint $T^{ex} dS=0$, it gets finally $$dS=\frac{d J}{T_L}+\frac{dQ}{T_e}.$$ Also the Boltzmann factor can be extended to $$\label{BoltzmanGeneral}
e^{-n_R/T_R-n_L/T_L-Q/T_e}.$$
Ultra-spinning/CFT description
------------------------------
In this section we shall study Kerr/CFT correspondence for our obtained extremal ultra-spinning $U(1)^4$ gauged supergravity black hole (\[UsMetric4d\]). Our strategy besides confirming the existence of the CFT dual, will be to verify that the ultra-spinning limit and the Kerr-CFT limit commute with each other. It has been shown in [@HennigarKubiznakMann:2015] that the ultra-spinning limit commutes with the near horizon limit for $5$d Kerr-AdS and minimal gauged supergravity black holes.
Here, we consider two different procedures to explore Kerr/CFT for our ultra-spinning black hole. i) Beginning with an extremal ultra-spinning black hole and then taking near horizon limit into account. ii) Finding the near horizon limit of an extremal black hole and then applying the ultra-spinning limit afterwards. These two different ways are clearly shown in Fig 2.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- --
![[**Fig. 2.**]{} This diagram illustrates two different order limits for a general rotating AdS black hole(R-AdS BH). Horizontal arrows (blue) represent the near horizon (NH) limit, to provide Kerr-CFT limit. Also the vertical ones (red) show the ultra-spinning (US) limit. We show that in both paths the resulting limit (US-KCFT) are exactly the same. ](fig2.pdf "fig:"){width=".6\textwidth" height=".21\textheight"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- --
\[Fig2\] Let us here, start the lower path in Fig. 2. The resulting geometry is already obtained in (\[NH4d\]). To find the central charge associated to metric (\[NH4d\]), we need to write the extended version of the first law as $$\label{FirstLawUS4d}
TdS=dM-\Omega_H dJ - \sum_{i}^{}{\Phi_i dQ_i} - K d\mu.$$ Here, we consider $\mu$ as a thermodynamic parameter shown as a chemical potential [@HennigarKubiznakMann:2014]. It was explained in [@Herzog] that the compactified null length can be signified as a chemical potential. Since the new coordinate $\varphi$ is a compact null coordinate on the conformal boundary that becomes compactified by $\mu$, then $\mu$ is assumed to be a chemical potential, and its thermodynamic conjugate is denoted by $K$. Note that $\mu$ is a dimensionless quantity, so in Smarr formula there is no $K\mu$. Considering the extremality constraint on (\[FirstLawUS4d\]) gives $$\label{FirstLawUS4d2}
TdS=-\big[ (\Omega_H-\Omega_H^{ex}) dJ + \sum_{i=1}^{4}{(\Phi_i-\Phi_i^{ex}) dQ_i} + (K-K^{ex}) d\mu \big].$$
Then the Boltzmann factor (\[FrolovVacumeBoltzman\]) in the extremal limit takes the following form $$\label{BoltzmanUS4d2}
e^{-n_R/T_R-n_L/T_L-\sum_{i=1}^{4} Q_i/T_{i,e}-\mu/T_\mu},$$ where $n_R=(E-\Omega_H^{ex} J - \sum_{i}^{}\Phi_i^{ex} Q_i -K^{ex}\mu) r_0/ \lambda$ and $n_L=J$. Also $T_R$ and $T_L$ are given by (\[FrolovTRTL\]). The Frolov-Thorne temperatures $T_e$ and $T\mu$ are defined as $$\label{FroloVUS4d}
T_{e,i}=- \frac{\partial T_H/ \partial r_+ }{\partial \Phi_i/ \partial r_+}|_{r_+=r_0},\quad\quad T_{\mu}=-\frac{\partial T_H/ \partial r_+ }{\partial K/ \partial r_+}|_{r_+=r_0}.$$ Now, using (\[FrolovTRTL\]) and (\[FroloVUS4d\]) we obtain following left- and right-moving temperatures $$\begin{aligned}
\label{TL4d}
T_R&=&0,\quad \quad \quad T_L=\frac{1}{2\pi k}=\frac{X\big[(r_0+q_1)(r_0+q_2)+\ell^2\big]}{2 \pi \ell (2r_0+q_1+q_2)},
\end{aligned}$$ and finally using (\[CentarlCharge\]) we find central charge as $$\label{cUS4d}
c=3\frac{\mu}{\pi} \frac{\ell (2r_0+q_1+q_2)}{X}.$$
We emphasis that the left-moving part of CFT identifies the quantum field states on the near horizon region (\[NH4d\]). Namely, in the extremal limit the vacuum state in the bulk reduces to a mixed density matrix on the CFT side. The Boltzmann weighting factor reads $$\label{density matrixUS4d}
\rho=e^{-\frac{J}{T_L}-\sum_{i}^{}{\frac{Q}{T_{e,i}}}-\frac{\mu}{T_\mu}}.$$
Indeed, the CFT dual of the generalised Hartel-Hawking vacuum has temperature $T_L$. Now we apply the thermodynamic Cardy formula that counts microstates of an unitary and modular invariance CFT at large $T$, and gives the entropy of CFT relating to its temperature and central charge [@Cardy]. $$\label{Cardy}
S=\frac{\pi^2}{3}c_L T_L.$$ This relation confirms that $c_L$ can be viewed as a measure of degrees of freedom and it determines the asymptotic density of states.
Now, using (\[TL4d\]) and (\[cUS4d\]) we obtain the microscopic entropy of the dual CFT $$S_{CFT}=\frac{\mu}{2} \big[ (r_+ +q_1)(r_+ + q_2)+\ell^2 \big],$$ which agrees precisely with the macroscopic Bekenstein-Hawking entropy (\[EntropyUS4d\]). So this remarkable result confirms the Kerr/CFT conjecture again.
Now, we turn to the upper path in Fig. 2. In the first step we have to find the near horizon geometry of the metric (\[Metric4d)\]) in the extremal limit, and then take the ultra-spinning limit on the NHEG. In [@ChowCvetic-CFT:2008] the NHEG of the metric (\[Metric4d)\]) has been constructed. Its ultra-spinning limit can easily be calculated by replacing new coordinate $\varphi=\frac{\phi}{\Xi}$ and then setting $a \rightarrow \ell$ and finally compactifying the new coordinate $\varphi$ with period $\mu$. One can easily check that the final result is exactly the same as (\[NH4d\]). So we can conclude that their CFT dual and central charge are the same too.
Hence these two results coming from procedures i) and ii) confirm that the ultra-spinning and the near horizon limits commute with each other for a $4$D gauged supergravity black hole solution. Another issue that can also be discussed is that, in black hole (\[UsMetric4d\]), the angular momentum $J$ is not independent of the mass $M$. As we see in Eq. (\[thermoUS4d\]), they are related by the chirality-type condition $M = J/ l$. We follow the thermodynamic interpretation as that in [@Klemm:2014]; thus we can define $L_0$ and $\tilde{L}_0$ in terms of $M$ and $J$ as $$\label{L0}
L_0=\frac{(M+J/l)}{2}, \quad \quad \quad \tilde{L}_0=\frac{(M-J/l)}{2}.$$ So in order to derive the Smarr formula in terms of $L_0$ and $\tilde{L}_0$, we should consider $M=M(L_0, \tilde{L}_0, Q , \mu) $. Since in our case $\tilde{L}_0$ vanishes, the first law becomes $$\label{FirstLawL0}
TdS=(1-\Omega)dL_+-\sum_i \Phi dQ - K d\mu,$$ and the chirality condition $M=J/l$ states that the black hole microstates can be explained by chiral excitations of a $3$D conformal field theory.
$5$-dimensional $U(1)^3$ gauged supergravity black holes {#section-5d}
========================================================
In this section we consider a charged rotating black hole which is constructed as a solutions of $SO(6)$ gauged five-dimensional supergarvity, whose relevant part of bosonic action is given by [@Gunaydin-5dSupergravity] $$\label{Action5d}
S_{5d}=\frac{1}{16\pi}\int d^5x \sqrt{-g} \bigg( R-\frac{1}{2}\partial\overrightarrow{\phi}^2-\frac{1}{4}\sum_{i=1}^{3}X_i^{-2}(F^i)^2+\frac{4}{\ell^2}\sum_{i=1}^{3}X_i^{-1}+\frac{1}{24}
\epsilon_{ijk}\epsilon^{\mu\nu\rho\sigma\lambda}F^i_{\mu\nu}F^j_{\rho\sigma}A^k_\lambda,
\bigg),$$ where $\overrightarrow{\phi}=(\phi_1,\phi_2)$, and $$\label{E3}
X_1=e^{-\frac{1}{\sqrt{6}}\phi_1-\frac{1}{\sqrt{2}}\phi_2}, \quad\quad\quad\quad X_2=e^{-\frac{1}{\sqrt{6}}\phi_1+\frac{1}{\sqrt{2}}\phi_2}, \quad\quad\quad\quad X_3=e^{\frac{2}{\sqrt{6}}\phi_1}.$$ A six parameter family of solutions including three electric charges, two angular momenta and mass, which is the most general asymptotically AdS$_{5}$ black hole solution to this theory was constructed in [@Wu-General5GaugedSupergravity]. Here, we consider the $U(1)^3$ Cartan subgroup of $SO(6)$ with three charge parameters $\delta_I$ that satisfy $\delta_1=\delta_2:=\delta$ and $\delta_3=0$, as well as two independent rotation parameters. These four-parameters family of solutions was derived in [@ChongCveticLuPope-5dGaugedSupergravity]. Their metric is given by $$\begin{aligned}
\label{Metirc5d}\nonumber
ds^2&=&H^{-\frac{4}{3}}\bigg[-\frac{\Delta}{\rho^2}\big(dt-a\sin^2\theta\frac{d\phi}{\Xi_a}-b\cos^2\theta\frac{d\psi}{\Xi_b}\big)^2\\
&+&\frac{C}{\rho^2}\big(\frac{ab}{f_3}dt-\frac{b}{f_2}\sin^2\theta\frac{d\phi}{\Xi_a}-\frac{a}{f_1}\cos^2\theta\frac{d\psi}{\Xi_b}\big)^2 +\frac{Z\sin^2\theta}{\rho^2}\big(\frac{a}{f_3}dt-\frac{1}{f_2}\frac{d\phi}{\Xi_a}\big)^2\\\nonumber
&+&\frac{W\cos^2\theta}{\rho^2}
(\frac{b}{f_3}dt-\frac{1}{f_1}\frac{d\psi}{\Xi_b})^2\bigg] + H^{\frac{2}{3}}\big(\frac{\rho^2}{\Delta}dr^2+\frac{\rho^2}{\Delta_\theta}d\theta^2\big),
\end{aligned}$$ where $$\begin{aligned}
\label{Metirc5d2}\nonumber
H&=&1+q/\rho^2,\quad \quad \quad \quad \quad \rho^2=r^2+a^2\cos^2\theta+b^2\sin^2\theta, \\ \nonumber
f_1&=&a^2+r^2, \quad \quad \quad f_2=b^2+r^2, \quad \quad \quad \quad f_3=f_1 f_2+qr^2, \\\nonumber
\Delta&=&\frac{1}{r^2}(a^2+r^2)(b^2+r^2)-2m+(a^2+r^2+q)(b^2+r^2+q)/\ell^2,\\
\Delta_\theta &=& 1-\frac{a^2}{\ell^2}\cos^2\theta-\frac{b^2}{\ell^2}\sin^2\theta, \quad \quad \quad C=f_1f_2(\Delta+2m-q^2/\rho^2),\\\nonumber
Z&=&-b^2C+\frac{f_2f_3}{r^2}[f_3-\frac{r^2}{\ell^2}(a^2-b^2)(a^2+r^2+q)\cos^2\theta],\\\nonumber
W&=&-a^2C+\frac{f_1f_3}{r^2}[f_3+\frac{r^2}{\ell^2}(a^2-b^2)(b^2+r^2+q)\sin^2\theta],\\\nonumber
\Xi_a&=&1-\frac{a^2}{\ell^2},\quad \quad \quad \Xi_b=1-\frac{b^2}{\ell^2}.
\end{aligned}$$ The gauge and scalar fields are $$\begin{aligned}
\label{Metirc5d-gauge}\nonumber
A^1&=&A^2=\frac{\sqrt{q^2+2mq}}{\rho^2}(dt-a\sin^2\theta\frac{d\phi}{\Xi_a}-\cos^2\theta\frac{d\psi}{\Xi_b}),\\
A^3&=&\frac{q}{\rho^2}(b\sin^2\theta\frac{d\phi}{\Xi_a}+a\cos^2\theta\frac{d\psi}{\Xi_b}).\\ \nonumber
X_1&=&X_2=H^{-\frac{1}{3}}, \qquad X_3=H^{\frac{2}{3}}
\end{aligned}$$ The metric (\[Metirc5d\]) is written in an asymptotic rotating frame. One can use a sort of coordinates to be asymptotically static frame (ASF) by taking $\phi=\tilde{\phi}+\frac{a}{\ell^2}t$ and $\psi=\tilde{\psi}+\frac{b}{\ell^2}t.$
The Hawking temperature, entropy and the electrostatic potentials on the horizon in the asymptotically static frame are $$\begin{aligned}
\label{TemEnt5d} \nonumber
T_H&=&\frac{2r_+^6+r_+^4(\ell^2+a^2+b^2+2q)-a^2b^2\ell^2}{2\pi r_+\ell^2[(r_+^2+a^2)(r_+^2+b^2)+qr_+^2]}, \quad \quad
S_{BH}=\frac{\pi^2[(r_+^2+a^2)(r_+^2+b^2)+qr_+^2]}{2r_+\Xi_a\Xi_b}, \\
\Phi_1&=&\Phi_2=\frac{\sqrt{q^2+2mq} \, r_+^2}{(a^2+r_+^2)(b^2+r_+^2)+qr_+^2}, \quad \quad \quad \qquad \qquad \Phi_3=\frac{a q b}{(a^2+r_+^2)(b^2+r_+^2)+qr_+^2}.
\end{aligned}$$ The Killing vector field that generates the Killing horizon is $K=\partial_t+\Omega_a^S \partial_\phi+\Omega_b^S \partial_\psi$, where $\Omega_a^S$ and $\Omega_b^S$ are the angular velocities on the horizon in (ASF). In ARF, the angular velocities on the horizon are written as $$\label{OmegaARF5d}
\Omega_a^R=\Omega_a^S-\frac{a}{\ell^2}=\frac{\Xi_a a (r_+^2+b^2)}{(a^2+r_+^2)(b^2+r_+^2)+ q r_+^2}, \quad \quad \quad
\Omega_b^R=\Omega_b^S-\frac{b}{\ell^2}=\frac{\Xi_b b (r_+^2+a^2)}{(a^2+r_+^2)(b^2+r_+^2)+ q r_+^2}.$$
Conserved charges including two angular momenta, two electric charges and mass as mentioned in [@ChongCveticLuPope-5dGaugedSupergravity] are $$\begin{aligned}
\label{Charges5d} \nonumber
J_a&=&\frac{\pi a (2m+q \Xi_b)}{4 \Xi_b \Xi_a^2}, \quad \quad \quad \qquad \qquad J_b=\frac{\pi b (2m+q \Xi_a)}{4 \Xi_a \Xi_b^2}, \\
Q_1&=&Q_2=\frac{\pi \sqrt{q^2+2mq}}{4 \Xi_a \Xi_b}, \qquad \qquad \qquad Q_3=-\frac{\pi a b q}{4 \ell^2 \Xi_a \Xi_b},\\ \nonumber
E&=&\frac{\pi \big[2m(2\Xi_a+2\Xi_b-\Xi_a\Xi_b)+q(2\Xi_a^2+2\Xi_b^2+2\Xi_a\Xi_b-\Xi_a^2\Xi_b-\Xi_b^2\Xi_a)\big]}{8 \Xi_a^2 \Xi_b^2}.
\end{aligned}$$
Ultra-spinning limit
--------------------
We are now ready to perform the ultra-spinning limit on the metric (\[Metirc5d\]), by following the same steps as in the previous section. This black hole rotates in two different directions $\phi$ and $\psi$, corresponding to the rotation parameters $a$ and $b$. But we are only allowed to take the ultra-spinning limit for one azimuthal direction which is selected as $\phi$ for us. Since the metric is already written in ARF, we begin by introducing a new azimuthal coordinate $\varphi=\phi / \Xi_a$. Then upon taking the $a \rightarrow \ell$ limit while we keep the parameter $b$ fixed, we will have $$\label{Delta5d}
\Delta_{\theta}=\Xi_b \, \sin^2\theta.$$ Hence we obtain the following new black hole solution as $$\begin{aligned}
\label{MetricUS5d}\nonumber
ds^2&=&\tilde{H}^{\frac{2}{3}}\bigg[-\frac{\tilde{\Delta}}{\rho^2}(dt-\ell\sin^2\theta d\varphi-b\cos^2\theta\frac{d\psi}{\Xi_b})^2\\ \nonumber
&+&\frac{\tilde{C}}{\tilde{\rho}^2}(\frac{\ell b}{\tilde{f_3}}dt-\frac{b}{f_2}\sin^2\theta d\varphi-\frac{\ell}{\tilde{f_1}}\cos^2\theta\frac{d\psi}{\Xi_b})^2 +\frac{\tilde{Z}\sin^2\theta}{\tilde{\rho}^2}(\frac{\ell}{\tilde{f_3}}dt-\frac{1}{f_2} d\varphi)^2\\
&+&\frac{\tilde{W}\cos^2\theta}{\tilde{\rho}^2}
(\frac{b}{\tilde{f_3}}dt-\frac{1}{\tilde{f_1}}\frac{d\psi}{\Xi_b})^2 \bigg] + \tilde{H}^{\frac{2}{3}}(\frac{\tilde{\rho}^2}{\tilde{\Delta}}dr^2+\frac{\tilde{\rho}^2}{\Xi_b \sin^2\theta}d\theta^2),
\end{aligned}$$ where the $\tilde{H}$ , $\tilde{Z}$ , $\tilde{W}, \tilde{\Delta}$ and $\tilde{\rho}$ are given by (\[Metirc5d2\]) in which $a \rightarrow \ell$. Also the new coordinate $\varphi$ can be compactified by $\varphi \sim \varphi + \mu$. The gauge potentials and scalar fields in this limit become $$\begin{aligned}
\label{USMetirc5d-gauge}\nonumber
A^1&=&A^2=\frac{\sqrt{q^2+2mq}}{\tilde{\rho}^2}(dt-\ell\sin^2\theta d\varphi-\cos^2\theta\frac{d\psi}{\Xi_b}),\\
A^3&=&\frac{q}{\tilde{\rho}^2}(b\sin^2\theta d\varphi+\ell\cos^2\theta\frac{d\psi}{\Xi_b}),\\ \nonumber
X_1&=&X_2=\tilde{H}^{-1/3}, \quad\quad\quad\quad X_3=\tilde{H}^{2/3}.
\end{aligned}$$ Now, it is easy to check that the metric (\[MetricUS5d\]) and fields (\[USMetirc5d-gauge\]) satisfy the equation of motion, and we can call (\[MetricUS5d\]) as a new exact gauged supergravity black hole solution.
It is worth mentioning that taking the ultra-spinning limit in the $\psi$ direction (instead of $\phi$) is also carried out in the same way. But it is impossible to take the ultra-spinning limit in both directions simultaneously, because the $g_{\theta \theta}$ component in the metric (\[MetricUS5d\]) will diverge in the $b \rightarrow \ell$ limit, and $1/\Xi_b$ divergence may not be absorbed into a new coordinate.
**Horizon geometry:** Here, we find the induced metric on the horizon $$\begin{aligned}
\label{ds2h-5d}
ds^2_h&=&\tilde{H}^{-\frac{4}{3}}\bigg[\frac{\tilde{C}}{\tilde{\rho}^2}(\frac{b}{f_2}\sin^2\theta d\varphi+\frac{\ell}{\tilde{f_1}}\cos^2\theta\frac{d\psi}{\Xi_b})^2 +
\frac{\tilde{Z}\sin^2\theta}{\tilde{\rho}^2}\frac{1}{f_2^2} d\varphi^2\\ \nonumber
&+&\frac{\tilde{W}\cos^2\theta}{\tilde{\rho}^2}
(\frac{1}{\tilde{f_1}}\frac{d\psi}{\Xi_b})^2 \bigg] +\tilde{H}^{\frac{2}{3}}\frac{\tilde{\rho}^2}{\Xi_b \sin^2\theta}d\theta^2.
\end{aligned}$$
As in the four-dimensional case, this metric seems to be ill defined in $\theta=0$ $( 0 \leq \theta \leq \pi/2)$. But to show that there is not problem near this point, one can perform the change of coordinates $k=l(1-\cos \theta)$, for small $k$, by which the horizon metric reads, $$\label{eq12}
ds^2_h=\frac{\rho_+}{\Xi_b}\left[\frac{dk^2}{4k^2}+4k^2\frac{(\ell^2+r_+^2)^3b^2\Xi_b}{r_+^2 \ell^2\rho_+^4}d\varphi^2+\frac{4bkm}{\rho_+ ^4}d\varphi d\psi \right] + \frac{2m}{\rho_+^2 \Xi_b} d \psi^2.$$ For $\psi=$constant slices, it reduces to a metric of constant negative curvature on a quotient of the hyperbolic space $\mathbb{H}^2$, indicating that the horizon is non-compact. In Fig. 3 the embedding topology for constant $\psi$ slices of the horizon are displayed for $\mu=2\pi$.
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**Conformal Boundary :** To gain a deeper understanding of the obtained geometry (\[MetricUS5d\]), let us here take a look at its conformal boundary, with the conformal factor $\ell^2/r^2$. $$\begin{aligned}
\label{eq13}
ds^2_{bdry}&=&-dt^2+\ell\sin^2\theta dt d\varphi+\frac{b\cos^2\theta}{\Xi_b}dt d\psi+\frac{\ell^2\cos^2\theta}{\Xi_b}d\psi^2\\ \nonumber
&+&\frac{b\ell\sin^2\theta\cos^2\theta}{\Xi_b}d\varphi d\psi +\frac{\ell^2}{\Xi_b\sin^2\theta}d\theta.
\end{aligned}$$ The coordinate $\varphi$ is a null coordinate on the conformal boundary. Moreover, as before we analyze this metric near the pole $\theta=0$. Then for small $k$ we have $$\label{eq14}
ds^2_{bdry}=-dt^2+\frac{1}{\Xi_b}\big[\frac{dk^2}{4k^2} + 2k dt d \varphi+ b dt d\psi + \ell^2 d \psi^2 + 2b k d\varphi d\psi \big]$$ which for $\psi=$constant slices represents an AdS$_3$ written Hopf-like fibration over $\mathbb{H}^2$. It means that the pole $\theta=0$ is removed from the boundary and is indeed not part of the space-time indeed.
**Thermodynamic quantities:** For new obtained ultra-spinning black hole (\[MetricUS5d\]), we precisely derived the thermodynamic quantities including temperature, entropy and angular velocities on the horizon as well as electrostatic potentials on the horizon as $$\begin{aligned}
\label{ThermoUS5d} \nonumber
T_H&=&\frac{2r_+^6+r_+^4(2\ell^2+b^2+2q)-b^2l^4}{2\pi r_+\ell^2[(r_+^2+\ell^2)(r_+^2+b^2)+qr_+^2]}, \quad \quad \quad
S_{BH}=\frac{\mu\pi[(r_+^2+\ell^2)(r_+^2+b^2)+qr_+^2]}{4r_+\Xi_b}, \\ \nonumber
\Omega_a&=&\frac{\ell(b^2+r_+^2)}{(r_+^2+\ell^2)(r_+^2+b^2)+qr_+^2}r,\qquad \qquad \qquad \Omega_b=\frac{b(r_+^4+2r_+^2\ell^2+r_+^2q+\ell^4)}{\ell^2(r_+^2+\ell^2)(r_+^2+b^2)+q\ell^2r_+^2},\\
\Phi_1&=&\Phi_2=\frac{r_+^2\sqrt{q^2+2mq}}{(\ell^2+r_+^2)(b^2+r_+^2)+qr_+^2}, \quad \quad \quad \quad \Phi_3=\frac{q \ell b}{(\ell^2+r_+^2)(b^2+r_+^2)+qr_+^2}.
\end{aligned}$$
**Extremality under ultra-spinning limit:** To ensure that the extremality condition is preserved under the ultra-spinning limit for this $5$d supergravity black hole, we similarly explore two different ways as mentioned in Fig. 2. So, we first obtain the extremality conditions of main metric (\[Metirc5d\]) with a horizon at $r=r_0$ using Eq. (\[TemEnt5d\]) by imposing $\Delta=0$ and $T_H|_{r=r_0}=0$. We find two following conditions $$\begin{aligned}
\label{ExCondition5d}
q&=&- a b + \frac{r_0^2\sqrt{a^2+b^2+\ell^2+2r_0^2}}{\ell}, \\ \nonumber
m&=&\frac{1}{2}\frac{\ell^4(a^2+r_0^2)(b^2+r_0^2)+r_0^2(Y- a^2 \ell)(Y - b^2 \ell)}{r_0^2 \ell^4},\end{aligned}$$ where $Y=a b \ell - r_0^2\sqrt{a^2+b^2\ell^2+2r_0^2}$.
Now, we find the extremality conditions of our obtained ultra-spinning black hole using (\[ThermoUS5d\]) $$\begin{aligned}
\label{ExtUS5d}
q&=&-b \ell + \frac{r_0^2\sqrt{b^2+2(\ell^2+r_0^2)}}{\ell} \\ \nonumber
m&=&\frac{1}{2}\frac{\ell^4(b^2+r_0^2)(\ell^2+r_0^2)+r_0^2(Y_s-\ell^2(b-\ell))(Y_s + b \ell(b-\ell))}{r_0^2 \ell^4}\end{aligned}$$ where $Y_s=r_0^2(\ell+\sqrt{b^2+2(\ell^2+r_0^2)})$. It is easy to check that Eq. (\[ExtUS5d\]) will be achieved by taking $a \rightarrow l$ limit on (\[ExCondition5d\]). Hence as the previous case, the extremality condition and the ultra-spinning limit commute with each other for five-dimensional $U(1)^3$ gauged supergavity class of black holes.
Near horizon geometry
---------------------
For an extremal solution of an ultra-spinning black hole (\[MetricUS5d\]) with a horizon at $r=r_0$, we have found the extremality conditions (\[ExtUS5d\]). Then one can write the near horizon expansion as $$\Delta=\tilde{X}(r-r_0)^2+\mathcal{O} (r-r_0)^3,$$ where $$\label{V5d}
\tilde{X}=2+\frac{3 \ell^2 b^2}{r_0^4}+\frac{1}{\ell^2}(6 r_0^2+b^2+2 q).$$ To obtain the near horizon geometry, we use the coordinate changes similar to (\[NHG-Coordinate4d\]) as $$\label{NHG-Coordinate5d}
r=r_0(1+\lambda \hat{r}), \quad \quad \quad \varphi=\hat{\varphi}+\Omega_a^0 \hat{t}, \quad \quad \psi=\hat{\psi}+\Omega_b^0\hat{t}, \quad \quad t=\frac{\hat{t}}{2\pi T^{\prime \, 0}_H r_0 \lambda}, \quad\quad \theta=\hat{\theta}.$$ After applying the scaling parameter $\lambda \rightarrow 0$, the near horizon geometry in terms of the vielbeins takes the form of $$\label{NHG-Metric5d}
ds^2=H_0^{2/3}\frac{\rho_0^2}{\tilde{X}}\big(-\hat{r}^2 d\hat{t}^2 + \frac{d\hat{r}^2}{\hat{r}^2}\big) + F(\theta)d\hat{\theta}^2 + \sum_{i=1}^{2}\hat{e}^i\hat{e}^i,$$ where $F(\theta)=H_0^{2/3}\frac{\rho_0^2}{\sin^2\theta \, \Xi_b}$ and vielbeins are $$\begin{aligned}
\label{Vielbeins1} \nonumber
\hat{e}^1=\alpha_1 \hat{e}_1 + \alpha_2 \hat{e}_2, \quad \quad \hat{e}^2=\beta_1 \hat{e}_1 + \beta_2 \hat{e}_2, \\
\hat{e}_1=d\hat{\varphi} + k_\varphi \hat{r} dt, \quad \quad \hat{e}_2=d\hat{\psi} + k_\psi \hat{r} dt,
\end{aligned}$$ and $$\begin{aligned}
\label{Vielbeins2} \nonumber
k_\varphi&=&\frac{2 \ell [(r_0^2+b^2)^2+qb^2]}{\tilde{X}[(r_0^2+\ell^2)(r_0^2+b^2)+qr_0^2] r_0}, \qquad \quad k_\psi=\frac{2b \, \Xi_b[(r_0^2+\ell^2)^2+q \ell^2]}{\tilde{X} [(r_0^2+\ell^2)(r_0^2+b^2)+qr_0^2] r_0},\\ \nonumber
\alpha_1&=&H_0^{-2/3}\frac{(r_0^2+\ell^2+q)\sin\theta }{\sqrt{\rho_0^2(1-\Xi_b \sin^2\theta)}}, \qquad \qquad \quad \alpha_2=-H_0^{-2/3}\frac{(r_0^2+b^2+q)\,\ell\,b\sin\theta }{\sqrt{\rho_0^2(1-\Xi_b \sin^2\theta)}}, \\\nonumber
\beta_1&=&H_0^{-2/3}\frac{ b [(r_0^2+l^2)\rho_0^2+q r_0^2]\sin^2\theta }{r_0\,\rho_0^2 \sqrt{1-\Xi_b \sin^2\theta}}, \qquad \quad \beta_2=H_0^{-2/3}\frac{ \ell [(r_0^2+l^2)\rho_0^2+q r_0^2]\cos^2\theta }{r_0\,\rho_0^2 \, \Xi_b \sqrt{1-\Xi_b \sin^2\theta}}. \\ \nonumber
\end{aligned}$$
Ultra-spinning/CFT description
------------------------------
In order to find the CFT dual of obtained $5$d ultra-spinning black hole (\[MetricUS5d\]), as it was done through the previous section, we can follow two different paths of Fig. 2. We firstly start from the lower path. In (\[NHG-Metric5d\]) we represented the NHEG of ultra-spinning solution. Also for the current case we choose the same boundary conditions as in [@GuicaHartmanStrominger-KerrCFT], $$\begin{aligned}
\label{ASG-Boundary5d}
\begin{pmatrix}
\mathcal{O}(r^2)& \mathcal{O}(1)& \mathcal{O}(1)& \mathcal{O}(1/r)& \mathcal{O}(1/r^2)\\
& \mathcal{O}(1) & \mathcal{O}(1) &\mathcal{O}(1/r) & \mathcal{O}(1/r)\\
& & \mathcal{O}(1) &\mathcal{O}(1/r) & \mathcal{O}(1/r)\\
& & & \mathcal{O}(1/r)& \mathcal{O}(1/r^2)\\
& & & & \mathcal{O}(1/r^3)\\
\end{pmatrix},
\end{aligned}$$ using the basis $(\hat{t},\hat{\varphi},\hat{\psi},\hat{\theta},\hat{r})$. One can then show that the five-dimensional near horizon geometry (\[NHG-Metric5d\]) provides two copies of commuting Virasoro algebras [@LuMeiPope-KerrCFT], generated by a pair of commuting diffeomorphisms $$\begin{aligned}
\label{diffeo}
\zeta_\varphi=-e^{in \varphi} \partial_\varphi- i n r e^{-in \varphi} \partial_r, \qquad
\zeta_\psi=-e^{in \psi} \partial_\psi- i n r e^{-in \psi} \partial_r.
\end{aligned}$$ These diffeomorphisms leading us to an asymptotic symmetry algebra of transformations that satisfy above boundary conditions. In [@ChowCvetic-CFT:2008] a general form for NHEG of rotating black holes in $d=2n+1$ dimensions was constructed $$\begin{aligned}
\label{NHGgeneral}
ds^2&=&\Gamma(y)\big(-\rho^2 dt^2 + \frac{d\rho^2}{\rho^2}\big)+\sum_{\alpha=1}^{n-1}F_\alpha dy_\alpha^2 + \sum_{i,j=1}^{n-1}\tilde{g}_{ij} \tilde{e}_i \tilde{e}_j, \\ \nonumber
\tilde{e}_i&=&d\phi_i+k_i \rho dt, \quad \quad \quad k_i=\frac{1}{2 \pi T_i}. \quad \quad \quad
\end{aligned}$$ The NHEG of our new solution is exactly recast in this form. The metric (\[NHGgeneral\]) has $(n-1)$ copies of the Virasoro algebra. The central charges in this form are given by [@ChowCvetic-CFT:2008] $$\begin{aligned}
\label{Centralgeneral}
c_i=\frac{3}{2\pi}k_i \int{d^{n-1}y_\alpha \bigg(det \tilde{g}_{ij} \prod_{\alpha=1}^{n-1} F_\alpha\bigg)^{1/2} } \int {d \phi_1 \dots d \phi_{n-1}}.
\end{aligned}$$ Then, for (\[NHG-Metric5d\]) we obtain two central charges $c_1$ and $c_2$ associated to diffeomorphisms $\partial_{\hat{\varphi}}$ and $\partial_{\hat{\psi}}$ respectively $$\begin{aligned}
\label{CentralgeneralUS5d}
c_1&=&\frac{3 k_\varphi}{8 \pi} \int \sqrt{F(\theta) (\alpha_2\beta_1-\alpha_1\beta_2)} d\theta d\varphi d\psi=\frac{3 \mu \ell [(r_0^2+b^2)^2+qb^2]}{\tilde{X} \Xi_b r_0^2}, \\
c_2&=&\frac{3 k_\psi}{8 \pi} \int \sqrt{F(\theta) (\alpha_2\beta_1-\alpha_1\beta_2)} d\theta d\varphi d\psi=\frac{3 \mu b [(r_0^2+\ell^2)^2+q \ell^2]}{\tilde{X} r_0^2}. \nonumber
\end{aligned}$$ We note that the central charges contain the chemical potential $\mu$, which comes from compactification of the new azimuthal coordinate. Then similar to the $U(1)^4$ case we can write the first law of thermodynamics in its extremality limit, appearing as $$TdS=-[(\Omega_\varphi-\Omega_\varphi^{ex})dJ_\varphi+(\Omega_\psi-\Omega_\psi^{ex})dJ_\psi+\sum_{i=1}^{3} (\Phi_i-\Phi_i^{ex}dQ_i)+(K-K^{ex})d\mu],$$ and the Boltzmann weighting factor for this case reads $$e^{-n_R/T_R-n_\varphi/T_\varphi-n_\psi/T_\psi-\sum_{i=1}^{4} Q_i/T_{t,\, i}},$$ where $n_R=(E-\Omega_\varphi^{ex}J_{Ex}-\Omega_\psi^{ex}J_{\psi}-\sum \Phi^{ex}_i Q_i)r_0/\lambda$, $n_\varphi=J_\varphi$ and $n_\psi=J_\psi$. We extract again the Frolov-Thorne temperatures $$\begin{aligned}
T_R&\equiv& \frac{T_H r_0}{\lambda}=0,\\ \nonumber
T\varphi&=&-\frac{\partial T_H/ \partial_{r_+}}{\partial \Omega_\varphi/\partial_{r_+}}|_{ex}=\frac{1}{2 k_\varphi}, \quad \quad \quad T_\psi=-\frac{\partial T_H/\partial_{r_+} }{\partial \Omega_\psi/\partial_{r_+}}|_{ex}=\frac{1}{2 k_\psi}, \nonumber
\end{aligned}$$ where $K_\varphi$ and $K_\psi$ are given by (\[Vielbeins2\]). Ultimately the CFT entropy using the Cardy formula is computed as $$S_{CFT}=\frac{\pi^2}{3}c_1 T_\varphi+\frac{\pi^2}{3}c_2 T_\psi=\frac{\mu\pi[(r_+^2+\ell^2)(r_+^2+b^2)+qr_+^2]}{4r_+\Xi_b},$$ which is exactly the same as the Bekenstein-Hawking entropy (\[ThermoUS5d\]).
Now, we examine CFT duality via the upper path of Fig 2. The near horizon geometry of metric (\[Metirc5d\]) in extremal limit was studied in [@ChowCvetic-CFT:2008]. One can easily check that by performing the ultra-spinning technique based on their result, our calculated NHEG (\[NHG-Metric5d\]) will exactly be held. It means that we can again confirm that the near horizon limit commutes with the ultra-spinning limit for this $5$D black hole. Hence the CFT dual of $U(1)^3$ gauged supergravity black hole (\[Metirc5d\]) gives us the same result for both the upper and lower paths of Fig. 2.
Discussion {#Discussion}
==========
For better understanding of the physics of gauged supergravity black holes in large angular momentum, we have employed the novel ultra-spinning (super-entropic) limit proposed in [@HennigarKubiznakMann:2014] to generate new exact supergravity black hole solutions. In particular, we use this simple ultra-spinning technique for four-dimensional singly spinning $U(1)^4$ and five-dimensional doubly spinning $U(1)^3$ gauged supergravity black holes. Our obtained black holes for both cases have an unusual horizon that is noncompact but with finite area. However in the first glance we see singularities in the coordinate angles $\theta=0,\pi$. We have shown that these poles are not parts of the spacetime and can be viewed as a sort of boundary that introduces punctures to the spacetime, providing a non-compact horizon. In [@HennigarKubiznakMann:2014; @HennigarKubiznakMann:2015] it was shown that a relation between this kind of ultra-spinning limit and super-entropic black holes can be found by exploring the thermodynamics behavior of the obtained black holes in the context of the extended phase space thermodynamics. In our upcoming work we shall study the properties of these obtained ultra-spinning black holes in the extended phase space to find the range of parameter space, giving us super-entropic black holes.
Also we have shown that the extremality conditions and the near horizon limit are preserved under ultra-spinning limit, demonstrating that they commute with the ultra-spinning limit. We have also presented the NHEG of both ultra-spinning gauged black holes despite the noncompactness of their horizons, possessing an AdS$_2$ sector and a $S^{d-2}$ with two punctures. The appearance of the AdS$_2$ factor in the NHEG prompts us to explore whether these unusual new solutions exhibit the well-defined Kerr/CFT correspondence. We have also investigated kerr/CFT for both cases. Assuming the Cardy formula, we have shown that the microscopic entropy of the dual CFT for both new ultra-spinning gauged supergravity black holes in four and five dimensions precisely agrees with its Bekenstein-Hawking entropy. Recently a formula as a higher-dimensional generalization of the Cardy formula for the large energy limit of generic CFT has been presented in [@Shaghoulian], which relates the entropy of a CFT to the vacuum energy on $S^1 \times \mathbb{R}^d$. Now, there is a curiosity to explore, whether this formula can reproduce the entropy of the ultra-spinning black holes at a high temperature.
A further direction for our next research will be an investigation of applying a hyperboloid membrane limit [@CaldarelliEmparan:2008] to these supergravity black holes and their ultra-spinning versions to generate other new exact solutions.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank M. M. Sheikh-Jabbari for fruitful discussions, insights, and helpful comments on the draft. Also we like to thank D. Klemm and especially M. H. Vahidinia for their useful discussions. S. M. N. wishes to acknowledge Saeedeh Rostami for many comments, and the organizers of the “Recent Trends in String Theory and Related Topics” workshop, held in May 2016 in Tehran, at which some preliminary results of this work were presented. S. M. N. would also like to thank the members of the quantum gravity group at the Institute for Research in Fundamental Sciences (IPM) for useful discussions, as well as the IPM for hospitality while this project was completed.
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|
---
abstract: 'We present a new algorithm for approximate inference in probabilistic programs, based on a stochastic gradient for variational programs. This method is efficient without restrictions on the probabilistic program; it is particularly practical for distributions which are not analytically tractable, including highly structured distributions that arise in probabilistic programs. We show how to automatically derive mean-field probabilistic programs and optimize them, and demonstrate that our perspective improves inference efficiency over other algorithms.'
author:
- 'David Wingate, Theo Weber'
bibliography:
- 'avi.bib'
nocite: '[@herbst10; @wainwright2008graphical; @beal2003variational; @winn2006variational; @peters05; @honkela2008natural; @amari98; @kakade02; @welling2008hybrid]'
title: |
Automated Variational Inference\
in Probabilistic Programming
---
Introduction
============
Automated Variational Inference for Probabilistic Programming
-------------------------------------------------------------
Probabilistic programming languages simplify the development of probabilistic models by allowing programmers to specify a stochastic process using syntax that resembles modern programming languages. These languages allow programmers to freely mix deterministic and stochastic elements, resulting in tremendous modeling flexibility. The resulting programs define prior distributions: running the (unconditional) program forward many times results in a distribution over execution traces, with each trace generating a sample of data from the prior. The goal of inference in such programs is to reason about the posterior distribution over execution traces conditioned on a particular program output. Examples include BLOG [@milch05], PRISM [@sato97], Bayesian Logic Programs [@kersting07], Stochastic Logic Programs [@Muggleton96stochasticlogic], Independent Choice Logic [@poole08], IBAL [@pfeffer01], Probabilistic Scheme [@radul07], $\Lambda_\circ$ [@park08], Church [@goodman08], Stochastic Matlab [@wingate11], and HANSEI [@kiselyovS09].
It is easy to sample from the prior $p(x)$ defined by a probabilistic program: simply run the program. But inference in such languages is hard: given a known value of a subset $y$ of the variables, inference must essentially run the program ‘backwards’ to sample from $p(x|y)$. Probabilistic programming environments simplify inference by providing universal inference algorithms; these are usually sample based (MCMC or Gibbs) [@goodman08; @milch05; @pfeffer01], due to their universality and ease of implementation.
Variational inference [@beal2003variational; @jordan99; @winn2006variational] offers a powerful, deterministic approximation to exact Bayesian inference in complex distributions. The goal is to approximate a complex distribution $p$ by a simpler parametric distribution $q_\theta$; inference therefore becomes the task of finding the $q_\theta$ closest to $p$, as measured by KL divergence. If $q_\theta$ is an easy distribution, this optimization can often be done tractably; for example, the mean-field approximation assumes that $q_\theta$ is a product of marginal distributions, which is easy to work with.
Since variational inference techniques offer a compelling alternative to sample based methods, it is of interest to automatically derive them, especially for complex models. Unfortunately, this is intractable for most programs. Even for models which have closed-form coordinate descent equations, the derivations are often complex and cannot be done by a computer. However, in this paper, we show that it *is* tractable to construct a stochastic, gradient-based variational inference algorithm automatically by leveraging compositionality.
Automated Variational Inference {#sec:rl}
===============================
An unconditional probabilistic program $f$ is defined as a parameterless function with an arbitrary mix of deterministic and stochastic elements. Stochastic elements can either belong to a fixed set of known, atomic random procedures called ERPs (for *elementary random procedures*), or be defined as function of other stochastic elements.
The syntax and evaluation of a valid program, as well as the definition of the library of ERPs, define the probabilistic programming language. As the program $f$ runs, it will encounter a sequence of ERPs $x_1,\cdots,x_T$, and sample values for each. The set of sampled values is called the *trace* of the program. Let $x_t$ be the value taken by the $t^{\text{th}}$ ERP. The probability of a trace is given by the probability of each ERP taking the particular value observed in the trace: $$\label{eq:pplik}
p(x) = \prod_{t=1}^T p_t (x_t\:|\:\psi_t(h_t))$$ where $h_t$ is the history $(x_1,\ldots,x_{t-1})$ of the program up to ERP $t$, $p_t$ is the probability distribution of the $t^{\text{th}}$ ERP, with history-dependent parameter $\psi_t(h_t)$. Trace-based probabilistic programs therefore define directed graphical models, but in a more general way than many classes of models, since the language can allow complex programming concepts such as flow control, recursion, external libraries, data structures, etc.
KL Divergence
-------------
The goal of variational inference [@jordan99; @beal2003variational; @winn2006variational] is to approximate the complex distribution $p(x|y)$ with a simpler distribution $p_{\theta}( x )$. This is done by adjusting the parameters $\theta$ of $p_{\theta}( x )$ in order to maximize a reward function $L(\theta)$, typically given by the KL divergence: $$\begin{aligned}
\label{eq:kl}
KL(p_\theta,p(x|y)) & = & \int_x p_{\theta}( x ) \log \left(
\frac{p_\theta(x)}{p(x|y)} \right) \nonumber \\
& = & \int_x p_{\theta}( x ) \log \left(
\frac{p_\theta(x)}{p(y|x)p(x)} \right) + \log p(y)=-L(\theta)+\log p(y)\\
\text{where}\\
L(\theta)& \overset{\Delta}{=}& \int_x p_{\theta}( x ) \log \left(
\frac{p(y|x)p(x)}{p_\theta(x)} \right)\end{aligned}$$
Since the KL divergence is nonnegative, the reward function $L(\theta)$ is a lower bound on the partition function $\log p(y)=\log \int_xp(y|x)p(x)$; the approximation error is therefore minimized by maximizing the lower bound.
Different choices of $p_{\theta}( x )$ result in different kinds of approximations. The popular mean-field approximation decomposes $p_\theta(x)$ into a product of marginals as $p_\theta(x) =
\prod_{t=1}^T p_\theta( x_t | \theta_t )$, where every random choice ignores the history $h_t$ of the generative process.
Stochastic Gradient Optimization
--------------------------------
Minimizing Eq. \[eq:kl\] is typically done by computing derivatives analytically, setting them equal to zero, solving for a coupled set of nonlinear equations, and deriving an iterative coordinate descent algorithm. However, this approach only works for conjugate distributions, and fails for highly structured distributions (such as those represented by, for example, probabilistic programs) that are not analytically tractable.
One generic approach to solving this is (stochastic) gradient descent on $L(\theta)$. We estimate the gradient according to the following computation: $$\begin{aligned}
\label{eq:grad}
-\nabla_\theta\: L(\theta)&= \int_x \nabla_\theta \left(p_{\theta}( x ) \log \left(
\frac {p_\theta(x)} {p(y|x)p(x)}\right) \right)\\
&= \int_x \nabla_\theta p_\theta(x) \left(\log\left(\frac{p_\theta(x)}{p(y|x) p(x)}\right)\right) + \int_x p_\theta(x) \left(\nabla_\theta \log(p_\theta(x))\right)\\
&= \int_x \nabla_\theta p_\theta(x) \left(\log\left(\frac{p_\theta(x)}{p(y|x) p(x)}\right)\right)\label{eq1}\\
&= \int_x p_\theta(x) \nabla_\theta \log(p_\theta(x)) \left(\log\left(\frac{p_\theta(x)}{p(y|x) p(x)}\right)\right)\label{eq2}\\
&= \int_x p_\theta(x) \nabla_\theta \log(p_\theta(x)) \left(\log\left(\frac{p_\theta(x)}{p(y|x) p(x)}\right)+K \right)\label{eq3}\\
& \approx \frac{1}{N} \sum_{x^j} \nabla_{\theta} \log p_{\theta}( x^j
)\left( \log \left( \frac{ p_{\theta}( x^j ) }{ p(y|x^j)p(x^j)} \right)+K \right)\label{eq4}\end{aligned}$$ with $x^j \sim p_{\theta}( x )$, $j=1\ldots N$ and $K$ is an arbitrary constant. To obtain equations (7-9), we repeatedly use the fact that $\nabla \log p_\theta(x)= \frac{\nabla_\theta p_\theta(x)}{p_\theta(x)}$. Furthermore, for equations and , we also use $\int_x p_\theta(x) \nabla \log p_\theta(x)=0$, since $\int_x p_\theta(x) \nabla \log p_\theta(x) =\int_x \nabla_\theta p_\theta(x)= \nabla_\theta \int_x p_\theta(x) =\nabla_\theta 1=0$. The purpose of adding the constant $K$ is that it is possible to approximately estimate a value of $K$ (optimal baseline), such that the variance of the Monte-Carlo estimate of expression is minimized. As we will see, choosing an appropriate value of $K$ will have drastic effects on the quality of the gradient estimate.
Compositional Variational Inference
-----------------------------------
Consider a distribution $p(x)$ induced by an arbitrary, unconditional probabilistic program. Our goal is to estimate marginals of the conditional distribution $p(x|y)$, which we will call the *target program*. We introduce a variational distribution $p_{\theta}( x
)$, which is defined through another probabilistic program, called the *variational program*. This distribution is unconditional, so sampling from it is as easy as running it.
We derive the variational program from the target program. An easy way to do this is to use a *partial mean-field approximation*: the target probabilistic program is run forward, and each time an ERP $x_t$ is encountered, a variational parameter is used in place of whatever parameters would ordinarily be passed to the ERP. That is, instead of sampling $x_t$ from $p_t (x_t\:|\:\psi_t(h_t))$ as in Eq. \[eq:pplik\], we instead sample from $p_t
(x_t\:|\:\theta_t(h_t))$, where $\theta_t(h_t)$ is an auxiliary variational parameter (and the true parameter $\psi_t(h_t)$ is ignored).
Fig. \[fig:ppvp\] illustrates this with pseudocode for a probabilistic program and its variational equivalent: upon encountering the `normal` ERP on line 4, instead of using parameter `mu`, the variational parameter $\theta_3$ is used instead (`normal` is a Gaussian ERP which takes an optional argument for the mean, and `rand(a,b)` is uniform over the set $[a,b]$, with $[0,1]$ as the default argument). Note that a dependency between $X$ and $M$ exists through the control logic, but not the parameterization. Thus, in general, stochastic dependencies due to the parameters of a variable depending on the outcome of another variable disappear, but dependencies due to control logic remain (hence the term *partial mean-field approximation*).
------------------------------------------------------------------------
height2pt 0.04in **Probabilistic program A** 0.04in
------------------------------------------------------------------------
height1pt 0.04in
` M = normal(); ` ` if M>1 ` ` mu = complex_deterministic_func( M ); ` ` X = normal( mu ); ` ` else ` ` X = rand(); ` ` end; `
------------------------------------------------------------------------
height1pt 0.1in
------------------------------------------------------------------------
height2pt 0.04in **Mean-Field variational program A** 0.04in
------------------------------------------------------------------------
height1pt 0.04in
` M = normal( \theta_1 ); ` ` if M>1 ` ` mu = complex_deterministic_func( M ); ` ` X = normal( \theta_3 ); ` ` else ` ` X = rand(\theta_4,\theta_5); ` ` end; `
------------------------------------------------------------------------
height1pt 0.04in -0.1in
This idea can be extended to automatically compute the stochastic gradient of the variational distribution: we run the forward target program normally, and whenever a call to an ERP $x_t$ is made, we:
[$\bullet$]{}[ 0.2in ]{}
Sample $x_t$ according to $p_{\theta_t}(x_t)$ (if this is the first time the ERP is encountered, initialize $\theta_t$ to an arbitrary value, for instance that given by $\psi_t(h_t)$).
Compute the log-likelihood $\log p_{\theta_t}(x_t)$ of $x_t$ according to the mean-field distribution.
Compute the log-likelihood $\log p(x_t|h_t)$ of $x_t$ according to the target program.
Compute the reward $R_t=\log p(x_t|h_t)-\log p_\theta(x_t) $
Compute the local gradient $\psi_t= \nabla_{\theta_t} \log p_{\theta_t}(x_t)$.
When the program terminates, we simply compute $\log p(y|x)$, then compute the gain $R=\sum R_t + \log p(y|x)+K$. The gradient estimate for the $t^{\text{th}}$ ERP is given by $R \psi_t$, and can be averaged over many sample traces $x$ for a more accurate estimate.
Thus, the only requirement on the probabilistic program is to be able to compute the log likelihood of an ERP value, as well as its gradient with respect to its parameters. Let us highlight that being able to compute the gradient of the log-likelihood with respect to natural parameters is the only additional requirement compared to an MCMC sampler.
Note that everything above holds: 1) regardless of conjugacy of distributions in the stochastic program; 2) regardless of the control logic of the stochastic program; and 3) regardless of the actual parametrization of $p(x_t;\theta_t)$. In particular, we again emphasize that we do not need the gradients of deterministic structures (for example, the function `complex_deterministic_func` in Fig. \[fig:ppvp\]).
Extensions
----------
Here, we discuss three extensions of our core ideas.
**Learning inference transfer.** Assume we wish to run variational inference for $N$ distinct datasets $y^1,\ldots,y^N$. Ideally, one should solve a distinct inference problem for each, yielding distinct $\theta^1,\cdots,\theta^N$. Unfortunately, finding $\theta^1,\cdots,\theta^N$ does not help to find $\theta^{N+1}$ for a new dataset $y_{N+1}$. But perhaps our approach can be used to learn ‘approximate samplers’: instead of $\theta$ depending on $y$ implicitly via the optimization algorithm, suppose instead that $\theta_t$ depends on $y$ through some fixed functional form. For instance, we can assume $\theta_t(y)=\sum_j \alpha_{i,j} f_j(y)$, where $f_j$ is a known function, then find parameters $\theta_{t,j}$ such that for most observations $y$, the variational distribution $p_{\theta(y)}(x)$ is a decent approximate sampler to $p(x|y)$. Gradient estimates of $\alpha$ can be derived similarly to Eq. \[eq:kl\] for arbitrary probabilistic programs.
**Structured mean-field approximations.** It is sometimes the case that a vanilla mean-field distribution is a poor approximation of the posterior, in which case more structured approximations should be used [@bouchard2009optimization; @ghahramani1997structured; @bishop2003structured; @geiger2005structured]. Deriving the variational update for structured mean-field is harder than vanilla mean-field; however, from a probabilistic program point of view, a structured mean-field approximation is simply a more complex (but still unconditional) variational program that could be derived via program analysis (or perhaps online via RL state-space estimation), with gradients computed as in the mean-field case.
**Online probabilistic programming** One advantage of stochastic gradients in probabilistic programs is simple parallelizability. This can also be done in an online fashion, in a similar fashion to recent work for stochastic variational inference by Blei et al. [@hoffman2010online; @wangonline; @blei2011stochastic]. Suppose that the set of variables and observations can be separated into a main set $X$, and a large number $N$ of independent sets of latent variables $X_i$ and observations $Y_i$ (where the $(X_i,Y_i)$ are only allowed to depend on $X$). For instance, for LDA, $X$ represents the topic distributions, while the $X_i$ represent the document distribution over topics and $Y_i$ topic $i$. Recall the gradient for the variational parameters of $X$ is given by $K \psi_X$, with $K=R_X+ \sum_i
(R_i+\log P(Y_i | X_i,X)$, where $R_X$ is the sum of rewards for all ERPs in $X$, and $R_i$ is the sum of rewards for all ERPs in $X_i$. $K$ can be rewritten as $X + N E[R_v + \log P(Y_v | X_v,X)$, where $v$ is a random integer in $\{1,\ldots,N\}$. The expectation can be approximately computed in an online fashion, allowing the update of the estimate of $X$ without manipulating the entire data set $Y$.
Experiments: LDA and QMR
========================
\[fig:enac\_results\]
We tested automated variational inference on two common inference benchmarks: the QMR-DT network (a binary bipartite graphical model with noisy-or directed links) and LDA (a popular topic model). We compared three algorithms:
[$\bullet$]{}[ 0.2in ]{}
The first is vanilla stochastic gradient descent on Eq. \[eq:kl\], with the gradients given by Eq. \[eq:grad\].
The Episodic Natural Actor Critic algorithm, a version of the algorithm connecting variational inference to reinforcement learning – details are reserved for a longer version of this paper. An important feature of ENAC is optimizing over the baseline constant $K$.
A second-order gradient descent (SOGD) algorithm which estimates the Fisher information matrix $F_\theta$ in the same way as the ENAC algorithm, and uses it as curvature information.
For each algorithm, we set $M=10$ (i.e., far fewer roll-outs than parameters). All three algorithms were given the same “budget” of samples; they used them in different ways. All three algorithms estimated a gradient $\hat{g}(\theta)$; these were used in a steepest descent optimizer: $\theta = \theta + \alpha \hat{g}(\theta)$ with stepsize $\alpha$. All three algorithms used the same stepsize; in addition, the gradients $g$ were scaled to have unit norm. The experiment thus directly compares the quality of the direction of the gradient estimate.
Fig. \[fig:enac\_results\] shows the results. The ENAC algorithm shows faster convergence and lower variance than steepest descent, while SOGD fares poorly (and even diverges in the case of LDA). Fig. \[fig:enac\_results\] also shows that the gradients from ENAC can be used either with steepest descent or a conjugate gradient optimizer; conjugate gradients converge faster.
Because both SOGD and ENAC estimate $F_\theta$ in the same way, we conclude that the performance advantage of ENAC is *not* due solely to its use of second-order information: the additional step of estimating the baseline improves performance significantly.
Once converged, the estimated variational program allows very fast approximate sampling from the posterior, at a fraction of the cost of a sample obtained using MCMC sampling. Samples from the variational program can also be used as warm starts for MCMC sampling.
Related Work
============
Natural conjugate gradients for variational inference are investigated in [@honkela2008natural], but the analysis is mostly devoted to the case where the variational approximation is Gaussian, and the resulting gradient equation involves an integral which is not necessarily tractable.
The use of variational inference in probabilistic programming is explored in [@harik2010variational]. The authors similarly note that it is easy to sample from the variational program. However, they only use this observation to estimate the free energy of the variational program, but they do not estimate the gradient of that free energy. While they do highlight the need for optimizing the parameters of the variational program, they do not offer a general algorithm for doing so, instead suggesting rejection sampling or importance sampling.
Use of stochastic approximations for variational inference is also used by Carbonetto [@carbonetto2009stochastic]. Their approach is very different from ours: they use Sequential Monte Carlo to refine gradient estimates, and require that the family of variational distributions contains the target distribution. While their approach is fairly general, it cannot be automatically generated for arbitrarily complex probabilistic models.
Finally, stochastic gradient methods are also used in online variational inference algorithms, in particular in the work of Blei et al. in stochastic variational inference (for instance, online LDA [@hoffman2010online], online HDP [@wangonline], and more generally under conjugacy assumptions [@blei2011stochastic]), as a way to refine estimates of latent variable distributions without processing all the observations. However, this approach requires a manual derivation of the variational equation for coordinate descent, which is only possible under conjugacy assumptions which will in general not hold for arbitrary probabilistic programs.
|
[ ]{}
[N. Cribiori$^{1,2}$, G. Dall’Agata$^{1,2}$, F. Farakos$^{1,2}$]{} and M. Porrati$^{3}$
$^1$ [*Dipartimento di Fisica “Galileo Galilei”\
Università di Padova, Via Marzolo 8, 35131 Padova, Italy*]{}
$^2$[*INFN, Sezione di Padova\
Via Marzolo 8, 35131 Padova, Italy*]{}
$^3$[*Center for Cosmology and Particle Physics,\
Department of Physics, New York University,\
4 Washington Place, New York, NY 10003, USA*]{}
[ABSTRACT]{}
We describe minimal supergravity models where supersymmetry is non-linearly realized via constrained superfields. We show that the resulting actions differ from the so called “de Sitter” supergravities because we consider constraints eliminating directly the auxiliary fields of the gravity multiplet.
6 mm
Introduction {#sec:introduction}
============
If supersymmetry is realized in nature, it is likely that its breaking scale is much higher than the one currently probed experimentally. It may also be as high as the string scale. In a scenario of this type, it is then difficult to find a good use to the phenomenological models where supersymmetry has a linear realization on the fields describing elementary particles and their interactions. However, also at energy scales much lower than the supersymmetry breaking scale, supersymmetry could pose visible constraints to interactions, especially if the mass spectrum is split so that are some states that are much lighter than other ones. For this reason, it is interesting to study non-linear realizations of supersymmetry and understand how to construct a general formalism that can be efficiently used to implement them in phenomenological models.
While non-linear realizations have been already studied shortly after the introduction of supersymmetry [@Volkov:1973ix; @others], it is only in the last couple of years that they received a broader attention. This is especially for their possible application to cosmological models (early references on this subject are [@Antoniadis:2014oya; @Ferrara:2014kva; @Kallosh:2014via; @Dall'Agata:2014oka], while [@Ferrara:2015cwa; @Ferrara:2016ajl] provide recent reviews). The recent resurgence of interest in the subject requires a study of non-linear realizations in the context of supergravity theories, that revisits and extends the results obtained in global supersymmetry.
An important step has been the construction of non-linear models of pure supergravity [@Antoniadis:2014oya; @Dudas:2015eha; @Bergshoeff:2015tra; @Hasegawa:2015bza]. These are models where the physical spectrum contains only the graviton, the gravitino and the goldstino. While these models have been dubbed “de Sitter” supergravities, the cosmological constant depends on two parameters, related to the susy-breaking scale $f$ and to the gravitino mass $m_{3/2}$, so that depending on their value it can be positive, negative or vanishing. These models effectively couple the supergravity multiplet, where supersymmetry is linearly realized, to the goldstino, which is the goldstone field of supersymmetry breaking and provides a non-linear realization of the supersymmetry algebra. After integration of the auxiliary fields and in the unitary gauge, where the goldstino is set to zero, the component action for these models is described by $$\label{eq:desittersugra}
e^{-1} {\cal L} = -\frac12 \, R
+ \frac{1}{2} \epsilon^{klmn} (\overline \psi_k \overline \sigma_l {\cal D}_m \psi_n - \psi_k \sigma_l {\cal D}_m \overline\psi_n)
- (m_{3/2} \, \overline \psi_a \overline \sigma^{ab} \overline \psi_b
+ \overline m_{3/2} \, \psi_a \sigma^{ab} \psi_b ) - \Lambda,$$ where $\Lambda= |f|^2 - 3 |m_{3/2}|^2$ is the cosmological constant, which is determined by the only two parameters in the theory: the susy-breaking scale $f$ and the gravitino mass $m_{3/2}$. Throughout this paper we use reduced Planck mass units that set $8\pi G=1$ unless explicitly stated otherwise.
In this note we will construct minimal supergravity models, whose physical spectrum is also given by the graviton, the gravitino and the goldstino, but where supersymmetry is non-linearly realized already on the gravity multiplet itself. We will see that this produces interactions and Lagrangians that differ from those in [@Antoniadis:2014oya; @Dudas:2015eha; @Bergshoeff:2015tra; @Hasegawa:2015bza] and which depend on three independent physical inputs: the susy-breaking scale, the gravitino mass and the cosmological constant. This problem has been already tackled from a different perspective in [@Delacretaz:2016nhw], with the purpose of constructing an effective field theory for supergravity models of inflation. Ref. [@Delacretaz:2016nhw] uses a supersymmetric generalization of the CCWZ construction [@Callan:1969sn]. We will instead perform our analysis in the language of superfields allowing for more general and different constraints. In fact, it is known that non-linear realizations of supersymmetry can be obtained by imposing supersymmetric constraints to superfields, so that some of their component fields get removed from the spectrum. While for a long time these constraints were imposed on a case by case basis, we now have a general procedure [@Dall'Agata:2016yof], which can be used to consistently remove any unconstrained field from a multiplet, both in global and local supersymmetric theories. In the following we are going to apply this procedure to remove the auxiliary fields of the minimal supergravity multiplet, detailing the features of the resulting models.
We will show that the most general models we can construct depend on 3 parameters, related to the scale of supersymmetry breaking $\Lambda_S$, to the gravitino mass $m_{3/2}$ and to the cosmological constant $\Lambda$. Already in the unitary gauge we can see that these models differ from those given in [@Antoniadis:2014oya; @Dudas:2015eha; @Bergshoeff:2015tra; @Hasegawa:2015bza], which are described by (\[eq:desittersugra\]). In fact, after integrating any eventual auxiliary fields and in the unitary gauge, the models we obtain have a cosmological constant given by $$\Lambda = \frac13 |c|^2+ |f|^2 + m_{3/2} \overline c +\overline m_{3/2} c = \Lambda_S - 3 |m_{3/2}|^2\,,$$ where $f$ is the $F$-term of the goldstino multiplet and $c$ is a new parameter which can be introduced when constraining the supergravity auxiliary scalar field.
Minimal constrained supergravity {#sec:minimal_constrained_supergravity}
================================
In the language of superfields, the minimal supergravity multiplet is described by means of two different superfields ${B}_{\alpha \dot \alpha}$ and ${\cal R}$, related by algebraic constraints. Using the conventions of [@Wess:1992cp], to build the Lagrangian we also need the chiral density ${\cal E}$. Its expansion in components is $$\label{calE}
2 {\cal E} = e \left\{1 + i\,\Theta \sigma^a \overline\psi_a - \Theta^2 (m^* + \overline\psi_a \overline{\sigma}^{ab} \overline\psi_b)\right\},$$ where $e$ is the determinant of the vielbein, $\psi_a$ is the gravitino and $m$ is the complex auxiliary scalar field. The curvature superfield ${\cal R}$ is also a chiral superfield $$\label{calR}
\begin{array}{rcl}
{\cal R} &=& \displaystyle - \frac16 \left\{m + \Theta(2\sigma^{ab} \psi_{ab} - i \sigma^a \overline\psi_a \,m + i \psi_a b^a)\right. \\[3mm]
&+& \displaystyle \Theta^2\left(-\frac12 R + i \overline \psi^a\overline{\sigma}^b \psi_{ab} + \frac23 |m|^2 + \frac13 b^2 - i\, {\cal D}_a b^a \right. \\[3mm]
&+& \displaystyle \left. \left. \frac12 \overline{\psi}\overline{\psi}\,m-\frac12 \psi_a \sigma^a \overline\psi_c \,b^c + \frac18\, \epsilon^{abcd}(\overline\psi_a \overline \sigma_b \psi_{cd} + \psi_a \sigma_b \overline \psi_{cd})\right)\right\}.
\end{array}$$ It contains the gravitino field-strength, the Ricci scalar curvature $R$ and the auxiliary vector field $b_a$. The real superfield $B_{\alpha \dot \alpha}$ has the auxiliary vector field $b_a$ as lowest component and it is related to ${\cal R}$ by means of the superspace Bianchi identity $$\label{Bianchi}
{\cal D}^{\alpha} B_{\alpha \dot \alpha} = \overline {\cal D}_{\dot \alpha} \overline{\cal R}.$$ Supersymmetry breaking requires also the existence of a goldstino field, which can be described by means of a chiral superfield $X$, constrained by the nilpotency condition $X^2 = 0$ [@others; @Casalbuoni:1988xh]. The latter solves the scalar field in the lowest component in terms of the goldstino ${\cal G}$ $$\label{Xexp}
X = \frac{{\cal G}^2}{2 F} + \sqrt2\, \Theta {\cal G} + \Theta^2 F,$$ provided ${\cal D}^2 X = -4 F$ is non-zero on the vacuum. The models discussed in [@Antoniadis:2014oya; @Dudas:2015eha; @Bergshoeff:2015tra; @Hasegawa:2015bza] correspond to the coupling of this superfield to the unconstrained supergravity multiplet.
Here we decided to follow a different path and impose supersymmetric constraints on the supergravity superfields in order to remove the auxiliary fields $m$ and $b_a$.
The scalar auxiliary field $m$ can be removed by imposing the constraint $$\label{XR}
X {\cal R} = 0,$$ which fixes the lowest component of ${\cal R}$ in terms of a function of the goldstino, the gravitino, the Riemann curvature and the other auxiliary field $b_a$. This constraint on chiral superfields has been first introduced in global supersymmetry in [@Brignole:1997pe] and then applied to matter superfields in supergravity in [@Dall'Agata:2015zla]. Here we apply it directly to the supergravity curvature superfield and therefore we use it to constrain an auxiliary field. We will see that such constraint, as noted in [@Dall'Agata:2015lek] for other constraints on auxiliary fields, implies that the final form of the potential is going to be different from the one of standard supergravity models. For this reason it is actually interesting to impose a slightly different constraint, namely $$\label{XRc}
X \left({\cal R} + \frac{c}{6}\right) = 0.$$ For a generic chiral superfield this constraint simply adds a constant vacuum expectation value to the scalar field in the ${\cal R}$ multiplet, but for the supergravity curvature superfield this has a non-trivial implication on the effective Lagrangian. As usual, we can consistently solve (\[XRc\]) by inspecting its top component. Given the peculiar structure of the ${\cal R}$ superfield, which contains the auxiliary field $m$ in various places, its solution can be found only after iteratively applying the constraint on $m$. The final result is $$\label{msol}
\begin{array}{rcl}
m &=& \displaystyle N\left(1-\frac{i}{\sqrt2 F} {\cal G}\sigma^a \overline \psi_a\right) + c\left(1-\frac{i}{\sqrt2 F} {\cal G} \sigma^a \overline \psi_a-\frac{1}{2 F^2}({\cal G} \sigma^b \overline\psi_b)^2-\frac{1}{4F^2} \overline\psi\,\overline\psi\,{\cal G}{\cal G}\right)\\[5mm]
&-&\displaystyle \frac{c}{3 F^2} \,{\cal G}{\cal G}\left[\overline N\left(1-\frac{i}{\sqrt2 \overline{F}}\overline{\cal G}\,\overline\sigma^a\psi_a\right)+\overline{c}\left(1-\frac{i}{\sqrt2 \overline{F}} \overline{\cal G}\,\overline\sigma^a\psi_a-\frac{1}{2 \overline F^2}(\overline{\cal G}\,\overline \sigma^b\psi_b)^2-\frac{1}{4 \overline F^2} \overline{\cal G}\overline{\cal G} \,\psi \psi\right)\right.\\[5mm]
&-& \left. \displaystyle\frac{|c|^2}{3 \overline F^2}\, \overline{\cal G}\overline{\cal G}\right],
\end{array}$$ where $$\label{NIN}
\begin{array}{rcl}
N &=& \displaystyle\frac{\sqrt2}{F} {\cal G} \sigma^{ab}\psi_{ab} + \frac{i}{\sqrt2 F} b^a \,{\cal G}\psi_a +\frac{1}{4 F^2} {\cal G}{\cal G} {\cal R} + \frac{i}{2 F^2} {\cal G}{\cal G} {\cal D}_a b^a \\[5mm]
& +&\displaystyle \frac{1}{2 F^2} {\cal G}{\cal G}\left[\frac12 \psi_m \sigma^m \overline\psi_n b^n-\frac13 b^2-i\, \overline\psi^m \overline\sigma^n \psi_{mn}-\frac18\epsilon^{klmn}\left(\overline\psi_k\overline\sigma_l\psi_{mn}+\psi_k\sigma_l\overline\psi_{mn}\right)\right].
\end{array}$$
The minimal supergravity model with non-linear supersymmetry, subject to the constraint (\[XRc\]), is defined by the Kähler potential $$\label{K\"ahler}
K = X \overline{X}$$ and the superpotential $$\label{W}
W = m_{3/2} + f \, X.$$ Using the nilpotency of $X$ and the constraint on ${\cal R}$, we see that the generic superspace Lagrangian [@Wess:1992cp] $$\label{supL}
{\cal L} = \frac{3}{8} \int d^2 \Theta \, 2 {\cal E} \, (\overline {\cal D}^2 - 8 {\cal R} )
e^{- \frac{1}{3} K(X,\overline X)}+ \int d^2 \Theta \, 2 {\cal E} \, W + h.c.,$$ reduces to the sum of three terms: the Einstein–Hilbert action $$\label{L01}
{\cal L}_1 = - 6 \int d^2 \Theta \, {\cal E} \, {\cal R} +h.c.,$$ the Kähler potential[^1] for the nilpotent field $X$ $$\label{L02}
{\cal L}_2 = \int d^2 \Theta \, {\cal E} \, X \left[-\frac14 \left(\overline{\cal D}^2 -8{\cal R}\right)\right]\overline{X} + h.c.$$ and the superpotential $$\label{L03}
{\cal L}_3 = \int d^2 \Theta \, 2 {\cal E} \, W + h.c.\,.$$ The associated component Lagrangian in the unitary gauge, where ${\cal G} = 0$, is $$\label{unitaction}
\begin{array}{rcl}
e^{-1} {\cal L} &=& \displaystyle -\frac12 \, R
+ \frac{1}{2} \epsilon^{klmn} (\overline \psi_k \overline \sigma_l {\cal D}_m \psi_n - \psi_k \sigma_l {\cal D}_m \overline\psi_n) + \frac13 b_a b^a
\\[5mm]
&-& \displaystyle (m_{3/2} \, \overline \psi_a \overline \sigma^{ab} \overline \psi_b
+ \overline m_{3/2} \, \psi_a \sigma^{ab} \psi_b ) - \Lambda,
\end{array}$$ where the cosmological constant $\Lambda$ is $$\label{potential}
\Lambda = \frac13 |c|^2+ |f|^2 + m_{3/2} \overline c +\overline m_{3/2} c = \Lambda_S - 3 |m_{3/2}|^2\,,$$ where we introduced the effective supersymmetry breaking scale $$\label{LambdaS}
\Lambda_S = |f|^2 + \left|\frac{c}{\sqrt3} + \sqrt3\, m_{3/2}\right|^2.$$ We see that we have three independent parameters in the Lagrangian, corresponding to the gravitino mass $m_{3/2}$, the scale of supersymmetry breaking $\Lambda_S$ and the cosmological constant $\Lambda$. Action (\[unitaction\]) depends only on two independent functions of these three parameters, but an action describing supergravity coupled to matter would depend generically on all three parameters. As announced, the cosmological constant does not take the standard form of an ordinary supergravity potential, since the latter is given by the difference of the $F$-terms squared minus (3 times) the squared gravitino mass. The constraint on the supergravity scalar auxiliary field implies a different form, whose numerical value can be positive, negative or zero, depending on the choice of the parameters $f$, $m_{3/2}$ and $c$. When $c=0$ we obtain a pure de Sitter supergravity, with a cosmological constant $\Lambda = |f|^2$. When $c = -3 {W}_0$ on the other hand, we obtain a cosmological constant in the standard form $\Lambda = |f|^2 - 3 |m_{3/2}|^2$ and the supersymmetry breaking scale directly related to $f$, i.e. $\Lambda_S = |f|^2$. This is for instance what one expects from models where supersymmetry is realized linearly on the gravity multiplet and the auxiliary field $m$ gets replaced by its equation of motion $m = -3\, m_{3/2} + \ldots$.
In the construction above, the vector auxiliary field has been left untouched, but we could also impose a further constraint to eliminate it from the spectrum. Since the superfield that contains $b_a$ as its lowest component is a real vector field, we need to use the prescription of [@Dall'Agata:2016yof] and impose the constraint $$\label{Bconst}
X \overline{X} B_{\alpha \dot \alpha} = 0.$$ This can be consistently done because $B_{\alpha \dot \alpha}$ contains $b_a$ nakedly and not through its field-strength. As demonstrated in [@Dall'Agata:2016yof], this constraint has a unique solution for the lowest component, which can be obtained as the $\theta = \bar \theta = 0$ projection of $$B_a = -2 \frac{\overline{\cal D}_{\dot\alpha}\overline X}{\overline{\cal D}^2 \overline X}\overline{\cal D}^{\dot \alpha}B_a - \overline X \frac{\overline{\cal D}^2 B_a}{\overline{\cal D}^2\overline X}-2\frac{{\cal D^\alpha}X}{{\cal D}^2 X\,\overline{\cal D}^2\overline X} {\cal D}_\alpha \overline{\cal D}^2(\overline X B_a) -\frac{X}{{\cal D}^2 X \overline{\cal D}^2 \overline X}{\cal D}^2 \overline{\cal D}^2 (\overline X B_a).$$ This produces an expression for $b_a$ that is a function of the goldstino, the gravitino and the graviton, in a way that it vanishes in the unitary gauge. Clearly, the addition of this constraint further complicates the expression of the other auxiliary field and modifies the couplings of the goldstino in the final Lagrangian.
Non-linear realizations and constrained superfields {#sec:non_linear_realizations_and_constrained_superfields}
===================================================
As we mentioned in the introduction, the authors of [@Delacretaz:2016nhw] also discuss non-linear realizations of minimal supergravity models, though in a different formalism and with a different purpose. We will now comment on the relation between the two approaches and the differences in the results.
The first thing to note is that [@Delacretaz:2016nhw] uses a parametrization for the goldstino field that is different from the one obtained by using the constrained superfield $X$. As it is known, there are actually various different non-linear realizations of the same superalgebra. The one of [@Delacretaz:2016nhw] uses the original Volkov–Akulov formulation [@Volkov:1973ix], where the goldstino $\lambda$ transforms under supersymmetry as $$\delta \lambda_\alpha = - f \, \epsilon_\alpha + \frac{i}{f} \left( \lambda \sigma^m \overline \epsilon - \epsilon \sigma^m \overline \lambda \right) {\cal D}_m \lambda_\alpha + \cdots .$$ The scale of susy-breaking $f$ was set to one in [@Delacretaz:2016nhw]. A different formulation is that of Samuel and Wess [@Samuel:1982uh], where the goldstino $\gamma$ is related to the one of Volkov–Akulov by a non-trivial field redefinition (which can be found in [@Liu:2010sk; @Kuzenko:2010ef] for global supersymmetric theories) and transforms as $$\delta \gamma_\alpha = - f\, \epsilon_\alpha + 2 \frac{i}{f}\, \gamma \sigma^m \overline \epsilon \, {\cal D}_m \gamma_\alpha + \cdots.$$ The one that comes from the constrained superfield representation is then simply a linear field redefinition, where $$\label{gamma}
G_{\alpha} = \sqrt 2 \, \frac{F}{f}\, \gamma_{\alpha}.$$ Once we have this realization, we can see that we can translate the constraints imposed in [@Delacretaz:2016nhw] in terms of constrained superfields using the general recipe given in [@Dall'Agata:2016yof]. Following [@Delacretaz:2016nhw], starting from a given component $\phi$ of a supermultiplet, one can impose a supersymmetric constraint by imposing $$\hat \phi =\left( D_G \Big{[} \text{e}^{\epsilon^\alpha \hat Q_\alpha} \Big{]} \times \phi \right)_{\epsilon=\lambda} = \phi + \delta_\epsilon \phi |_{\epsilon=\lambda}
+ \frac12 \delta_\epsilon \delta_\epsilon \phi |_{\epsilon=\lambda} + \cdots =0,$$ where $D$ is some representation of the group $G$ (which in our case is the supergroup), while $\hat{Q}$ is the infinitesimal generator of the symmetry transformation with parameter $\epsilon(x) = \lambda(x)$. This equation results in a constraint that removes the field $\phi$ from the spectrum. In the language of constrained superfields one can impose the same condition for a generic multiplet $\Phi$, whose lowest component is $\phi$ by setting [@Dall'Agata:2016yof] $$\label{TXC}
X \overline X \Phi = 0.$$ This eliminates the lowest component field of the superfield $\Phi$, namely $\phi$, while also inducing a non-linear realization of supersymmetry for the remaining component fields. It is straightforward to see that the constraint (\[TXC\]) implies $$\label{txc}
({\cal G}{\cal G}) (\overline{\cal G}\overline{\cal G}) \phi = 0,$$ which produces the full constraint once we require that this condition be invariant under supersymmetry. Indeed if the lowest component of a superfield vanishes, the whole superfield vanishes. The two formulations give the same result, because the solution of the constraint $\hat{\phi} = 0$ implies $$(\lambda \lambda)(\overline \lambda \overline \lambda) \phi = 0,$$ which, using the field redefinition from $\lambda$ to ${\cal G}$, is equivalent to (\[txc\]). We therefore conclude that the results of [@Delacretaz:2016nhw] correspond to imposing the constraints (\[XR\]) and (\[Bconst\]) in the constrained superfields language. One can actually be convinced that this is the case by direct computation of the solution of the constraints of [@Delacretaz:2016nhw]. For instance, the constraint $\hat{m} = 0$ gives $$\begin{aligned}
\label{Mgeom}
m &= -\delta_\lambda m +\cdots
&=2\lambda \sigma^{ab}\psi_{ab}+ib^a\lambda \psi_a-2i(\lambda\sigma^a\overline\psi_a)(\lambda \sigma^{ab}\psi_{ab})+\cdots
\end{aligned}$$ which agrees with the first terms in after using the relation between ${\cal G}$ and $\lambda$.
This comparison also shows that imposing the constraint (\[XRc\]) we produce a Lagrangian that differs from the one in [@Delacretaz:2016nhw]. In fact, if we do not introduce the parameter $c$, we see from (\[potential\]) that the cosmological constant can only be positive and proportional to the susy-breaking parameter $f$. Actually, even the coupling of additional matter multiplets would not change this fact, leading to a true “de Sitter” supergravity. The only way to change this fact would be by changing the sign and the overall coefficient to $X \overline X$ in (\[L02\]), which is the term supersymmetrizing the cosmological constant term. If $X$ were a standard chiral superfield this fact would produce a ghost from the scalar component $x$, but here we have a constrained superfield and $x$ is replaced by a bilinear of the goldstino. Still, we expect that introducing this change in sign in front of this term would tell us that any ultraviolet completion of such a model would be pathological.
Discussion {#sec:discussion}
==========
In this letter we considered minimal supergravity models where supersymmetry is non-linearly realized by constraining the auxiliary fields of the supergravity multiplet. The resulting theories depend on three parameters related to the cosmological constant, the mass of the gravitino and the supersymmetry breaking scale. All these three parameters appear in independent combinations in a generic matter-coupled supergravity but the pure broken supergravity constructed in this letter depends of course only on two independent parameters: the gravitino mass $m_{3/2}$ and the cosmological constant $\Lambda$. This is equivalent to say that the actions constructed in this note differ off-shell from those appearing in the literature, either because the latter had supersymmetry realized linearly on the gravity fields, or because the constraints imposed on the auxiliary fields were chosen differently.
An obvious interesting development is the coupling of these models to matter fields, either free or constrained. Already in the case where supersymmetry is linearly realized on the matter fields we see nontrivial effects of the new constraints imposed on the gravity multiplet. Since we impose a constraint on the auxiliary field $m$, the scalar potential cannot be put in the standard supergravity form, where the negative contribution, proportional to the superpotential, comes precisely by the integration of $m$. Also, if one imposes the constraint (\[Bconst\]) on the vector auxiliary field, we expect that the conditions on the scalar manifolds get relaxed. In particular, we expect that the scalar $\sigma$-model will be described by an arbitrary Kähler manifold rather than by a more restrictive Hodge–Kähler manifold. In fact, while the equations of motion of the $b_a$ auxiliary field usually solve it in terms of the composite Kähler connection, the constraint (\[Bconst\]) sets it to zero in the unitary gauge. This should also imply that the fermions are trivial sections of the Kähler bundle. If, in addition, we couple constrained matter fields, then even the Kähler structure of the scalar $\sigma$-model is lost.
Another observation worth making is that the constraints we proposed here can be used to give a natural embedding of the $R^2$ “Starobinsky” model of inflation in supergravity. This can be obtained as a linear combination of the terms ${\cal L}_2$, ${\cal L}_3$, that are defined in eqs. (\[L02\],\[L03\]) and include the goldstino action and the supersymmetry-breaking superpotential (where we set $m_{3/2} =0$ for simplicity), plus a term quadratic in the curvature $${\cal L}_{4} = 54\, \alpha \int d^2 \Theta (2{\cal E})
{\cal R} \left[-\frac14\,\left(\overline{\cal D} - 8 {\cal R}\right)\right] \overline {\cal R} + c.c.\qquad \alpha=\mbox{constant}.$$
Imposing the constraint $X^2 = 0$ and the curvature constraints (\[XRc\]), (\[Bconst\]), we break both conformal invariance and supersymmetry. We also get rid of the auxiliary fields in the gravity action thus producing a bosonic sector which is very simple: $$e^{-1} {\cal L} = - \frac12 |c|^2 \alpha R + \frac{3}{4} \alpha R^2 +
\frac{\alpha}{3} |c|^4 -|f|^2.$$ If we normalize the Einstein–Hilbert term as usual and require the vanishing of the cosmological constant, the parameters are constrained so that for high-scale susy-breaking $\sqrt{|f|} \sim 10^{-3} M_P$ we naturally produce the large coefficient $\alpha=O(10^5)$ needed to make the Starobinsky model consistent with CMB data.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank S. Ferrara, A. Kehagias, S. Rigolin and A. Wulzer for interesting discussions. This work is supported in part by the Padova University Project CPDA119349, by the MIUR grant RBFR10QS5J *“String Theory and Fundamental Interactions”*. MP is supported in part by NSF grants PHY-1316452 and PHY-1620039. We acknowledge hospitality of the GGI institute in Firenze, where this work was performed during the workshop ‘Supergravity: what next?’.
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[^1]: Recall that in supergravity the operator $-1/4 (\overline{\cal D}^2 -8{\cal R})$ transforms an antichiral field into a chiral field.
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---
abstract: 'Recently, ultra diffuse galaxy (UDG) of Dragonfly 44 in the Coma Cluster was observed and observations of the rotational speed suggest that its mass is almost same as the mass of the Milky Way. On the other hand, interestingly, the galaxy emits only 1 % of the light emitted by the Milky Way. Then, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. In this study we try to show that the dark matter that constitutes Dragonfly 44 can form the wormhole or not. Two possible dark matter profiles are used, namely, ultra diffuse galaxy King’s model and generalized Navarro-Frenk-White (NFW) dark matter profile. We have shown that King’s model dark matter profile does not manage to provide wormhole whereas generalized Navarro-Frenk-White (NFW) dark matter profile is managed to find wormholes.'
author:
- Sayeedul Islam
- Ali Övgün
- Farook Rahaman
- Mustafa Halilsoy
title: Formation of Wormholes by Dark Matter in the Galaxy Dragonfly 44
---
Introduction
============
Today, one of the challenging questions in a theoretical physics is the question of the existence of traversable wormholes [@Einstein; @moris; @moris2; @visser1; @ao1; @rahaman; @Richarte:2017iit; @Capozziello:2007ec; @Delgaty:1994vp; @Perry:1991qq; @Cataldo:2017yec; @Cataldo:2016dxq; @Cataldo:2008pm; @Jusufi:2017vta; @Sakalli:2015taa; @Sakalli:2015mka; @ao3; @metric; @outer1; @central; @einasto]. Other mysteries in physics is dark matter (DM) [@Sahni:2004ai; @Feng:2010gw; @Albada:1986roa]. Like black holes there is another miraculous objects of our universe which are wormholes. After the prediction of Einstein-Rosen bridge in 1935 [@Einstein], with a lot of theoretical evidence researchers have proposed the existence of wormholes in space-time which acts like a shortcut path to travel between any two widely separated or infinite region of the universe or between another universes in multi universe model. Structurally it looks like a tunnel (called its throat) with two mouths (most likely spheroidal). Here most interesting thing is exotic matter needed to open its throat (violates null energy conditions [@moris; @moris2; @visser1]. The curiosity about wormholes physics has vigorously been increased since publication of Morris and Thorne’s research article where they proposed the prospect of the existence of traversable wormholes, as a solution of Einstein’s field equations, that does not contain event horizon and traveler can easily move in both the regions in space-time through its straight stretch throat [@moris; @moris2]. By the existence of these hypothetical objects one can realize time machines or shortcut among faraway places of space.
Maximum mass of the universe is made up by the most mysterious substance which do not interact with the electromagnetic force, i.e. can’t absorb, reflect or emit light are dark matter whose nature and composition remain overall question mark [@Sahni:2004ai; @Feng:2010gw]. Researchers hypothesize the existence of dark matter only from the gravitational effect. Observation of the spiral galaxy rotation curves is an enthralling experimental evidence for the existence of dark matter. The dark matter candidate particles are generally classified into cold, warm and hot categories. Interaction of a scalar field whose energy density is dark energy may be caused of the dark matter particle mass. Over several decades it is openly accepted that almost every galaxy contain a large amount of non luminous matter forming massive dark matter halos around the galaxy based on different lines of evidence like flat rotation curves of spiral galaxies [@Albada:1986roa] and strong lensing system [@Keeton:1997by].
The model under consideration predicts that the ultra diffuse galaxies (UDGs) space distribution for 90 globular clusters observed around Dragonfly 44 [@Dragonfly44] has the space density:\
a. King’s model like [@king; @king2]: $$\rho(r)\propto \kappa\left(\frac{r}{r_{0}}^{2}+\lambda\right)^{\eta},$$ where $\eta$ , $\kappa$, $r_0$ and $\lambda$ are parameters . We assume that the averaged relative speed of the galaxies in the Coma cluster is approximately equal to the velocity dispersion in the cluster $v\cong1000$ km/s. This density profile is shown in fig 1.\
b. Generalized Navarro-Frenk-White (NFW) profile [@Navarro:1995iw]: $$\rho(r)\propto\frac{1}{r^\gamma\left[1+\left(\frac{r}{r_0}\right)\right]^{3-\gamma}},$$ where $\gamma $ and $r_0$ are parameters .
![Plot $\rho $ versus $r$ ( $\kappa=1$,$\lambda=1$ and $r_0=1$ ).[]{data-label="Fig. 1"}](rho.eps)
This paper is organized as follows. Sec. II we introduces the traversable wormholes and their basic equations. In Sec. III and IV, we study the possibility of wormholes in the galaxy of Dragonfly 44 by using the UDGs dark matter profile [@king; @king2; @Dragonfly44] and NFWs dark matter profile [@Navarro:1995iw],respectively. Finally, in Sec. V, we present our remarks.
Wormholes formulation
======================
The traversable wormhole space-time is given by following [@moris; @moris2]
$$ds^{2}=-e^{2f(r)}dt^{2}+\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).\label{line}$$
It is noted that the $b(r)$ and $f(r)$ stands for the spatial shape function, and for the redshift function, respectively. The range of the radial coordinate is from +$\infty$ to $b(r_{0})=r_{0}$ where r$_{0},$ is the minimum value of the r (throat of the wormhole).
The Einstein field equations are given by $$G_{\nu}^{\mu}=8\pi T_{\nu}^{\mu}$$ where $T_{\nu}^{\mu}$ and $G_{\nu}^{\mu}$ are the stress-energy tensors and the Einstein tensor, respectively. Then one can calculate the non-zero Einstein tensors: $$G_{t}^{t}=\frac{b^{\prime}}{r^{2}},\label{gtt}$$ $$G_{r}^{r}=\frac{-b}{r^{3}}+2\left(1-\frac{b}{r}\right)\frac{f^{\prime}}{r},$$ $$G_{\theta}^{\theta}=\left(1-\frac{b}{r}\right)\left[f^{\prime\prime}+{f^{\prime}}^{2}+\frac{f^{\prime}}{r}-\left(f^{\prime}+\frac{1}{r}\right)\left\{ \frac{b^{\prime}r-b}{2r(r-b)}\right\} \right],$$ $$G_{\phi}^{\phi}=G_{\theta}^{\theta}$$ where a prime is $\frac{d}{dr}.$
DM is generally defined in the form of general anisotropic energy-momentum tensor $$T_{\nu}^{\mu}=(\rho+p_{r})u^{\mu}u_{\nu}+p_{r}g_{\nu}^{\mu}+(p_{t}-p_{r})\eta^{\mu}\eta_{\nu},$$ where $u^{\mu}u_{\mu}=-\frac{1}{2}\eta^{\mu}\eta_{\mu}=-1$. Note that $p_{t}$ stands for the transverse pressure, $p_{r}$ is the radial pressure and $\rho$ is the energy density. A possible set of $u^{\mu}$ and $\eta^{\mu}$ are given by $u^{\mu}=(e^{2f(r)},0,0,0)$ and $\eta^{\mu}=(0,0,\frac{1}{r},\frac{1}{r\sin\theta})$. Then the stress-energy tensors $T_{\nu}^{\mu}$ are calculated as follows $$T_{t}^{t}=-\rho,\label{Tr}$$ $$T_{r}^{r}=p_{r},$$
$$T_{\theta}^{\theta}=T_{\phi}^{\phi}=p_{t}.\label{Ttheta}$$
Wormholes with the UDG King’s model DM
=======================================
One can find the tangential velocity from the flat rotation curve for the circular stable geodesic motion in the equatorial plane as [@einasto] $$v^{\phi}=\sqrt{rf^{\prime}}\label{v1},$$ which is responsible to fit the flat rotational curve for the DM. Rahaman et. al observe the rotational curve profile in the DM region as follows [@central; @outer1] $$v^{\phi}=\alpha r\exp(-k_{1}r)+\beta\lbrack1-\exp(-k_{2}r)]\label{v2}$$ where $\alpha,$ $\beta,$ $k_{1},$ and $k_{2}$ are constant positive parameters.
Using the Eqns.(\[v1\]) and (\[v2\]), the redshift function is obtained as follows $$\begin{aligned}
f(r) &=&-\frac{\alpha ^{2}r}{2k_{1}e^{(2k_{1}r)}}-\frac{\alpha ^{2}}{%
4k_{1}^{2}e^{(2k_{1}r)}}-\frac{2\alpha \beta }{k_{1}e^{(k_{1}r)}} \notag
\\
&&+\frac{%
2\alpha \beta e^{(-k_{1}r-k_{2}r)}}{k_{1}+k_{2}} +\beta ^{2}\ln (r)+2\beta ^{2}E_{i}(1,k_{2}r) \notag
\\
&&-\beta ^{2}E_{i}(1,2k_{2}r)+D.
\label{fr}\end{aligned}$$where E$_{i}$ and $D$ are the exponential integral and integration constant, respectively. At large scales, it becomes $e^{2f(r)}=B_{0}r^{(4v^{\phi})}$.\
Now we discuss two cases for different values of the parameter $\eta$ by using UDGs king’s dark matter profile for the stability of wormholes in the galaxy Dragonfly 44.
The case I
----------
For the UDGs King’s DM density profile [@king; @king2; @Dragonfly44] $$\rho(r)= \kappa\left[\left(\frac{r}{r_{0}}\right)^{2}+\lambda\right]^{\eta},$$ we assume in this case $\kappa=1$, $\lambda=1$ and $\eta=-3/2$. After one uses the Eq.s (\[v1\]) and (\[v2\]) and the UDG’s dark matter density profile under the Einstein field equations, the shape function is calculated ($8\pi=1$) as $$b(r)=\frac{rr_{0}^{2}}{8(r^{2}+r_{0})}+\frac{r_{o}^{3/2}}{8} \tan^{-1}\left(\frac{r}{\sqrt{r_{0}}}\right)-\frac{r_{0}^{3}r}{4(r^{2}+r_{0})^{2}}+C$$ Note that $C$ is the integration constant and it is chosen as $$C=r_{0}-\frac{r_{0}^{2}}{8(1+r_{0})}+\frac{r_{o}^{3/2}}{8}\tan^{-1}(\sqrt{r_{0}})+\frac{r_{0}^{2}}{4(1+r_{0})^{2}}\label{constant}$$ to satisfy the condition of $b(r_{0})=r_{0}$, then it is checked the flare-out condition ($b^{\prime}<1$), where $r_{0}$ is the radius of throat. $$b^{\prime}=\rho r^{2}=\frac{r_{0}^{3}r^{4}}{(r^{2}+r_{0})^{3}}\label{flr}$$ which is satisfied (see fig.(2)).
![The figure is shown for $b(r)/r$ versus $r$.[]{data-label="Fig. 2"}](br1.eps)
![The figure is shown for $b^\prime(r)$ versus $r$.[]{data-label="Fig. 3"}](db1.eps)
Furthermore from the Eq.(\[Tr\]) the second derivative of $b$ with respect to $r$ is calculated as
$$b^{\prime\prime}=-\frac{2r_{0}^{3}r(2r^{2}-r_{0})}{(r^{2}+r_{0})^{4}}.$$
The radial of pressure is showed as follows by substituting $f(r)$ and $b(r)$ into the solution (Eqns. 5- 12):
$$p_{r}=\frac{-b}{r^{3}}+2\left(1-\frac{b}{r}\right)\frac{f^{\prime}}{r},$$
$$\begin{aligned}
f^{\prime}={\frac{{\alpha}}{{2}r}{ \left( {{\rm e}^{{k_{1}}\,r}} \right) ^{2}}
}-2\,{\alpha}\,{\beta}\,{{\rm e}^{-r \left( {k_{1}}+{k_{2}}
\right) }}+2\,{\frac {{\alpha}\,{\beta}}{{{\rm e}^{{k_{1}}\,r}}}} \notag
&&\\+{
\frac {{{\beta}}^{2}{{\rm e}^{-2\,{k_{2}}\,r}}}{r}} -2\,{\frac {{{
\beta}}^{2}{{\rm e}^{-{k_{2}}\,r}}}{r}}+{\frac {{{\beta}}^{2}}{r}}.\end{aligned}$$
It is showed in the figure (4) that the null energy condition ($\rho+p_{r}<0$) is violated so that one of the essential condition for a wormhole is satisfied. However, the most important criterion for wormhole, namely, $\frac{b(r)}{r} <1$ is violated (see fig.2). So in this case wormhole does not exists.
![Null energy condition for the case I.[]{data-label="Fig. 4"}](A5.eps)
The case II
-----------
In this section we follow the same method using in case I and the shape function is calculated .
![The figure is shown for $b(r)/r$ versus $r$.[]{data-label="Fig. 5"}](br2.eps)
![The figure is shown for $b^\prime(r)$ versus $r$.[]{data-label="Fig. 6"}](db2.eps)
![Null energy condition for the case II.[]{data-label="Fig. 7"}](A3.eps)
It is checked the flare-out condition ($b^{\prime}<1$) is satisfied and plotted in Fig. (6).
It is also showed in the Fig. (7) that the null energy condition ($\rho+p_{r}<0$) is violated as needed to hold a wormhole. However, as in case I, $\frac{b(r)}{r} <1$ is violated(see fig.5), so in this case wormhole does not exists. Thus King’s model dark matter profile that constitutes Dragonfly 44 does not manage to provide wormhole. Now we will perform a study whether generalized NFW dark matter profile that constitutes Dragonfly 44 does manage to provide wormhole or not.
Wormholes with the NFWs DM
===========================
In this section we describe the dark matter halo distribution using the following generalized Navarro-Frenk-White (NFW) profile for the existence of wormholes in the galaxy Dragonfly 44. $$\rho(r)= \frac{1}{r^\gamma\left[1+\left(\frac{r}{r_0}\right)\right]^{3-\gamma}}$$ In this expression, $\gamma$ is the inner slope of the profile ($\gamma=1$ corresponds to the case of a standard NFW profile), and $r_0$ is the scale radius and taking variation constant as unity.\
Here we have discussed three different cases for the several values of the parameter $\gamma$
case I
------
In this case we assume $r_0=10,\rho_0=0.05$ and $\gamma = 3$.\
After using NFW dark matter density profile under the Einstein field equations, the shape function $b(r)$ is calculated $(8\pi=1)$ as $$b(r)=8\pi\rho_0\left[ln(r)+C\right]$$ Note that $C$ is the integration constant and it is choosen as $$C=\frac{r_0}{8\pi\rho_0}-lnr_0\label{constant}$$
![ The figure is shown for $b(r)$ versus $r$.[]{data-label="Fig. 8"}](S1.eps)
![ The figure is shown for $b(r)-r$ versus $r$.[]{data-label="Fig. 9"}](S2.eps)
![ Null energy condition for the case I.[]{data-label="Fig. 10"}](S3.eps)
Here fig.(8) represents the shape function $b(r)$ and we see that the null energy condition ($\rho+p_{r}<0$) is violated ( see fig.10). Also we have checked the most important flare-out condition ($b(r)-r < 0$, after the throat radius i.e. $b^{\prime}<1$) which is satisfied and is plotted in the fig.(9).\
Thus the generalized NFW dark matter profile that constitutes Dragonfly 44 can manage to provide wormhole.
case II
--------
In this case we choose $r_0=10,\rho_0=0.05$ and $\gamma = 4$\
After using NFW dark matter density profile under the Einstein field equations, the shape function $b(r)$ is calculated $(8\pi=1)$ as $$b(r)=8\pi\rho_0\left[\frac{-1}{r}+\frac{ln(r)}{r_0}+C\right]$$ Note that $C$ is the integration constant and it is choosen as $$C=\frac{r_0}{8\pi\rho_0}+\frac{1}{r_0}-\frac{1}{r_0}lnr_0\label{constant1}$$
![ The figure is shown for $b(r)$ versus $r$.[]{data-label="Fig. 11"}](S4.eps)
![ The figure is shown for $b(r)-r$ versus $r$.[]{data-label="Fig. 12"}](S5.eps)
![ Null energy condition for the case II.[]{data-label="Fig. 13"}](S6.eps)
In this section we follow the same method used in case I and the shape function $b(r)$ is plotted in the fig. (11). To form a wormhole, the essential criteria regarding flare-out condition ($b(r)-r < 0$, after the throat radius i.e. $b^{\prime}<1$) and violation of null energy condition ($\rho+p_{r}<0$) are satisfied ( see fig.12 and fig.13 ).\
case III
---------
In this case we assume $r_0=10,\rho_0=0.05$ and $\gamma = 5$\
After using NFW dark matter density profile under the Einstein field equations, the shape function $b(r)$ is calculated $(8\pi=1)$ as $$b(r)=8\pi\rho_0\left[\frac{-2}{rr_0}+\frac{ln(r)}{r_0^{2}}-\frac{1}{2r^{2}}+C\right]$$ Note that $C$ is the integration constant and it is choosen as $$C=\frac{r_0}{8\pi\rho_0}+\frac{2}{r_0^{2}}-\frac{1}{r_0^2}lnr_0+\frac{1}{2r_0^{2}}\label{constant}$$
![ The figure is shown for $b(r)$ versus $r$.[]{data-label="Fig. 14"}](S7.eps)
![ The figure is shown for $b(r)-r$ versus $r$.[]{data-label="Fig. 15"}](S8.eps)
![Null energy condition for the case III.[]{data-label="Fig.16"}](S9.eps)
Here we use the same NFW dark matter density profile like above cases and similarly the shape function $b(r)$ is calculated and plotted in the fig. (14).\
In the fig. (15) and fig. (16), it is clearly shown that the flare-out condition (($b(r)-r < 0$, after the throat radius i.e. $b^{\prime}<1$)) is satisfied and null energy condition ($\rho+p_{r}<0$) is violated to hold a wormhole open.
So we can claim that the generalized NFW dark matter profile that constitutes Dragonfly 44 provides wormhole.
Conclusion
==========
The presence of stable traversable wormholes is a noteworthy issue in theoretical physics. There is doubtlessly that wormholes is the most interesting objects in universe. This work is persuaded primarily by Ref.s [@outer1; @central; @einasto]. In this paper, we use the UDG systems in the Coma Cluster which is known as Dragonfly 44 that astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. Moreover, there is another possibility: in this study we show that this dark matter can form the wormhole, and it affects the observations. All the normal matter might be passed through this wormhole. For this purpose we firstly use the UDG profile and try to construct wormhole solution, then we repeat our calculations for the Navarro-Frenk-White (NFW) profile. We have shown that only Navarro-Frenk-White (NFW) profile provide wormhole solutions. Thus we able to find the solutions of the wormhole in the Dragonfly 44 galaxy so that the dark matter‘s halos around the Dragonfly 44 galaxy is suitable to harbor wormholes.
This work was supported by the Chilean FONDECYT Grant No. 3170035 (AÖ). AÖ is grateful to the CERN theory (CERN-TH) division for hospitality where part of this work was done. FR would like to thank the authorities of the Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing research facilities. FR and SI are also grateful to DST-SERB and DST-INSPIRE, Govt. of India , for financial support respectively.
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---
abstract: 'We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on $\mathbb R$ only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.'
author:
- Serban Teodor Belinschi
title: |
The Lebesgue decomposition of the free additive\
convolution of two probability distributions
---
1truecm
Introduction
============
The notion of freeness (or free independence) has been introduced by Voiculescu in [@Voiculescu1], with the main purpose of better understanding free group factors. As in the classical case, the distribution of a sum of free random variables is uniquely determined by the distributions of the summands, and the resulting distribution of the sum is called the [*free additive convolution*]{} of the distributions of the summands.
Specifically, for the case of probability distributions on $\mathbb R$, let $\mu$ and $\nu$ be Borel probability measures on the real line. We can define the free additive convolution $\mu\boxplus\nu$ of $\mu$ and $\nu$ in the following way. Denote by $\mathbb F[a,b]$ the free group with free generators $a$ and $b$, and consider the group von Neumann algebra $L(
\mathbb F[a,b])$ generated by the left regular representation of $\mathbb F[a,b]$, endowed with the (unique) normal faithful trace $\tau$. Choose two selfadjoint operators $X_\mu$ and $X_\nu$ affiliated with the subalgebras of $L(\mathbb F[a,b])$ generated by the images, via the left regular representation, of $a$ and $b$, respectively, so that their distribution with respect to $\tau$ is $\mu$ and $\nu$, respectively. It has been shown in [@BVIUMJ] by Bercovici and Voiculescu that the distribution of the selfadjoint operator $X_\mu+X_\nu$ with respect to $\tau$ depends only on the distributions $\mu$ and $\nu$ of $X_\mu$ and $X_\nu$, respectively. We denote it by $\mu\boxplus\nu$. For an introduction to the field of free probability we refer to [@VDN].
An analytic method for the computation of free additive convolutions has been devised in [@Voiculescu1] (for compactly supported probabilities) and in [@BVIUMJ] (for the case of probabilities with arbitrary support on $\mathbb R$). We will give below a brief outline of this method.
For any finite positive measure $\sigma$ on $\mathbb R$, define its Cauchy transform $$G_\sigma(z)=\int_\mathbb R \frac{d\sigma(t)}{z-t},\quad z\in\mathbb C
\setminus\mathbb R,$$ and let $F_\sigma(z)=1/G_\sigma(z).$ Since $G_\sigma(\overline{z})=\overline{G_\sigma(z)},$ we shall consider from now on only the restrictions of $F_\sigma$ and $G_\sigma$ to the upper half-plane $\mathbb C^+=\{z\in\mathbb C\colon \Im z>0\}.$
For given $\alpha\geq0,\beta>0,$ let us denote $\Gamma_{\alpha,\beta}=\{z\in\mathbb C^+\colon\Im z>\alpha,|\Re z|<\beta\Im z
\}.$ The following two results appear in [@BVIUMJ].
\[inversion+\] Let $\mu$ be a probability on $\mathbb R$. There exists a nonempty domain $\Omega$ in $\mathbb C^+$ of the form $\Omega=\cup_{\alpha>0}\Gamma_{\alpha,
\beta_\alpha}$ such that $F_\mu$ has a right inverse with respect to composition $F_\mu^{-1}$ defined on $\Omega$. In addition, we have $\Im F_\mu^{-1}(z)\leq\Im z$ and $$\lim_{z\to\infty,z\in
\Gamma_{\alpha,\beta}}\frac{F_\mu^{-1}(z)}{z}=1$$ for every $\alpha,\beta>0.$
Let $\phi_\mu(z)=F_\mu^{-1}(z)-z$, $z\in\Omega.$ The basic property of the function $\phi_\mu$ is described in the following theorem of Voiculescu:
\[R-transform\] Let $\mu,\nu$ be two probability measures supported on the real line. Then $\phi_{\mu\boxplus\nu}(z)=\phi_\mu(z)+\phi_\nu(z)$ for $z$ in the common domain of the three functions.
Thus, the map $\phi$, called the Voiculescu transform, is the free analogue of the logarithm of the Fourier transform from classical probability theory. This function is related to the $R$-transform by the equality $\phi_\mu(z)=R_\mu(1/z).$ (Historically, the $R
$-transform, defined as $R_\mu(z)=G_\mu^{-1}(z)-(1/z)$ was first introduced by Voiculescu in [@Voiculescu1], but the analysis in the context of measures with unbounded support turns out to be simpler when expressed in terms of the Voiculescu transform.)
Another important property for (Cauchy transforms of) free convolutions of probability measures is subordination. It has been shown that $G_{\mu\boxplus\nu}$ is subordinated to $G_\mu$, in the sense that there exists a unique analytic self-map $\omega$ of the upper half-plane $\mathbb C^+$ so that $G_{\mu\boxplus\nu}(z)=G_\mu(\omega(z))$, $z\in\mathbb C^+,$ and $\lim_{y\to+\infty}
\omega(iy)/iy=1.$ This result was proved first in [@V3] under a genericity assumption, then extended to full generality in [@Biane1]. A new proof based on the theory of fixed points of analytic self-maps of the upper half-plane has been given in [@Subord].
Subordination has been until now the most powerful tool for proving regularity results for free convolutions. Pioneering work in this direction has been done by Voiculescu, alone in [@V3] (see, for example, Proposition 4.7), and together with Bercovici in [@BVSemigroup] and [@BercoviciVoiculescuRegQ]. Among the results proved in [@BercoviciVoiculescuRegQ], we mention the description of the atoms of $\mu\boxplus\nu$ (Theorem 7.4): a number $a\in\mathbb R$ is an atom for $\mu\boxplus\nu$ if and only if there exist $b,c\in\mathbb R$ so that $a=b+c$ and $\mu(\{b\})+\nu(\{c\})>1.$ Moreover, $(\mu\boxplus\nu)(\{a\})=\mu(\{b\})+\nu(\{c\})-1$. For the special case when $\mu$ is the semicircular distribution (i.e. $d\mu(t)=\frac{1}{2\pi}\chi_{[-2,2]}(t)\sqrt{4-t^2}dt,$ where $\chi_A$ denotes the characteristic function of the set $A$), Biane [@Biane2] proved several properties of $\mu\boxplus\nu$, from which we mention that the singular continuous part with respect to the Lebesgue measure of $\mu\boxplus\nu$ is always zero, while the density of its absolutely continuous part with respect to the Lebesgue measure is bounded and analytic wherever positive. Similar results have been proved for measures belonging to partially defined semigroups with respect to free additive and multiplicative convolutions (see [@BB] and [@BBercoviciMult]). In [@AIHP] it has been shown that, roughly speaking, free convolutions of two probability measures can be purely singular only if at least one of the two measures is a point mass. Moreover, for compactly supported measures, the support of the singular part, if existing, must be of zero Lebesgue measure.
All these results seem to indicate that free convolutions do not favorize large singular parts. In Theorem \[thm4.1\] of this paper we show that when neither $\mu$ nor $\nu$ is a point mass, the singular continuous part of $\mu\boxplus\nu$ is zero, while the density of the absolutely continuous part of $\mu\boxplus\nu$ with respect to the Lebesgue measure is analytic outside a closed set of zero Lebesgue measure.
The rest of the paper is organized as follows: in Section 2 we give without proof several results from complex analysis that we will use later, in Section 3 we analyze the boundary behaviour of the subordination functions, and in Section 4 we prove the main result of the paper.
[**Acknowledgments.**]{} A weaker result than the one presented in this paper appears in the author’s PhD thesis. I am deeply grateful to Professor Hari Bercovici for the invaluable help as advisor during the doctoral studies, and afterwards, which made this paper possible. My thanks go also to Professor Zhenghan Wang for his help and support, including through an RAship. Many thanks are due also to the Complex Analysis research group from Indiana University, especially Professors Eric Bedford, Norm Levenberg, and Kevin Pilgrim, for numerous useful discussions.
Preliminary results
===================
In the following, unless otherwise specified, the attributes “singular”, “singular continuous”, and “absolutely continuous” will be considered with respect to the Lebesgue measure on $\mathbb R$. Given a finite positive Borel measure $\sigma$ on the real line, we denote by $\sigma^s$ (respectively $\sigma^{sc},$ $\sigma^{ac}$) the singular (respectively singular continuous and absolutely continuous) part of $\sigma$. All measures considered in this paper are assumed to be Borel measures.
The following results characterize the Cauchy transform of $\sigma$. For more details and proofs we refer to [@Achieser].
\[Cauchy\] Let $G\colon\mathbb C^+\longrightarrow\mathbb C^-$, where $\mathbb C^-=-\mathbb
C^+,$ be an analytic function. The following statements are equivalent:
1. There exists a unique positive measure $\sigma$ on $\mathbb R$ such that $G=G_\sigma$;
2. For any $\alpha,\beta>0,$ we have that $$\lim_{z\to\infty,z\in\Gamma_{\alpha,\beta}}zG(z)$$ exists and is finite ($\Gamma_{\alpha,\beta}=\{z\in\mathbb C^+\colon|\Re z|<\alpha\Im z,\Im z>\beta\}$).
3. The limit $\lim_{y\to+\infty}iyG(iy)$ exists and is finite.
Moreover, the limits from 2 and 3 equal $\sigma(\mathbb R).$
Observe also that $$-\frac1\pi\Im G_\sigma(x+iy)=\frac1\pi\int_\mathbb R\frac{y}{(x-t)^2+y^2}d\sigma(t),
\quad x\in\mathbb R,y>0,$$ is the Poisson integral of $\sigma.$
As mentioned also in the introduction, it turns out that in many situations it is much easier to deal with the reciprocal $F_\sigma=1/G_\sigma$ of the Cauchy transform of the measure $\sigma
.$ The following proposition is an obvious consequence of Theorem \[Cauchy\]:
\[Cauchyalt\] Let $F\colon\mathbb C^+\longrightarrow\mathbb C^+$ be an analytic self-map of the upper half-plane. The following statements are equivalent:
1. There exists a positive measure $\sigma$ on $\mathbb R$ such that $F=1/G_\sigma$;
2. For any $\alpha,\beta>0,$ the limit $\lim_{z\to\infty,z\in\Gamma_{\alpha,\beta}}\frac{F(z)}{z}$ exists and belongs to $(0,+\infty)$;
3. The limit $\lim_{y\to+\infty}\frac{F(iy)}{iy}$ exists and belongs to $(0,+\infty)$.
Moreover, both limits form 2. and 3. equal $\sigma(\mathbb R)^{-1}.$
In general, analytic self-maps of the upper half-plane can be represented uniquely by a triple $(a,b,\rho)$, where $a$ is a real number, $b\in[0,+\infty
),$ and $\rho$ is a positive finite measure on $\mathbb R.$ This representation is called the Nevanlinna representation (see [@Achieser]).
\[Nevanlinna\] Let $F\colon\mathbb C^+\longrightarrow\mathbb C^+$ be an analytic function. Then there exists a triple $(a,b,\rho)$, where $a\in\mathbb R$, $b\geq0,$ and $\rho$ is a positive finite measure on $\mathbb R$ such that $$F(z)=a+bz+\int_{\mathbb R}\frac{1+tz}{t-z}d\rho(t),\quad z\in\mathbb C^+.$$ The triple $(a,b,\rho)$ satisfies $a=\Re F(i),$ $b=\lim_{y\to+\infty}\frac{F(iy
)}{iy},$ and $b+\rho(\mathbb R)=\Im F(i)$.
The converse of Theorem \[Nevanlinna\] is obviously true.
\[ImF>Imz\] [An immediate consequence of Proposition \[Cauchyalt\] and Theorem \[Nevanlinna\] is that for any finite measure $\sigma$ on $\mathbb R$, we have $\Im F_\sigma(z)\geq
\sigma(\mathbb R)^{-1}\Im z$ for all $z\in\mathbb C^+,$ with equality for any value of $z$ if and only if $\sigma$ is a point mass. In this case, the measure $\rho$ in the statement of Theorem \[Nevanlinna\] is zero.]{}
As observed above, any finite measure $\sigma$ on the real line is uniquely determined by its Cauchy transform. Moreover, regularity properties of $\sigma$ can be deduced from the behaviour of $G_\sigma$, and hence of $F_\sigma$, near the boundary of its domain. In the following we shall state several classical theorems concerning analytic self-maps of the unit disc $\mathbb D=\{z\in\mathbb C\colon
|z|<1\}$ and their boundary behaviour, i.e. the behaviour near points belonging to the boundary $\mathbb T=\{z\in\mathbb C\colon|z|=1\}$ of $\mathbb D$. Because the upper half-plane is conformally equivalent to the unit disc via the rational transformation $z\mapsto\frac{z-i}{z+i},$ most of these theorems will have obvious formulations for self-maps of the upper half-plane.
We shall consider the set $\mathbb C\cup\{\infty\}$ to be endowed with the usual topology: for any point $z\in\mathbb C$, the family $B_n(z)=\{w\in\mathbb C\colon
|z-w|<1/n\}$, $n\in\mathbb N$, forms a basis of neighbourhoods of $z$, while the family $K_n=\{w\in\mathbb C\colon|w|>n\}\cup\{\infty\}$, $n\in\mathbb N$, forms a basis of neighbourhoods for the point infinity. The notions of limit and continuity will be considered with respect to this topology and, when subsets of $\mathbb C\cup\{\infty\}
$ are involved, we consider on them the topology inherited from $\mathbb C\cup\{\infty
\}$, unless otherwise specified. For a function $f\colon\mathbb C^+\longrightarrow\mathbb
C\cup\{\infty\}$, and a point $x\in\mathbb R$, we say that the nontangential limit of $f$ at $x$ exists if the limit $\lim_{z\to x,z\in\Gamma_\alpha(x)}f(z)$ exists in $\mathbb C\cup\{\infty\}$ for all $\alpha>0$, where $\Gamma_\alpha(x)=\{z\in\mathbb C^+\colon|\Re z-x|<\alpha\Im z\}.$ A similar definition holds for functions defined in the unit disc. We shall denote nontangential limits by $\sphericalangle\lim_{z\to x}f(z),$ or $$\lim_{\stackrel{ z\longrightarrow x}{{
\sphericalangle}}}f(z).$$ The nontangential limit of $f$ at infinity is defined in a similar way: $\sphericalangle\lim_{z\to \infty}f(z)$ is said to exist in $\mathbb C\cup\{\infty\}$ if the limit $\lim_{z\to\infty,z\in\Gamma_\alpha(0)}f(z)$ exists in $\mathbb C\cup\{\infty\}
$ for all $\alpha>0$.
In the following three theorems are described some properties of meromorphic functions in the unit disc related to their nontangential boundary behaviour.
\[Fatou\] Let $f\colon\mathbb D\longrightarrow\mathbb C$ be a bounded analytic function. Then the set of points $x\in\mathbb T$ at which the nontangential limit of $f$ fails to exist is of linear measure zero.
\[Privalov\] Let $f\colon\mathbb D\longrightarrow\mathbb C$ be an analytic function. Assume that there exists a set $A$ of nonzero linear measure in $\mathbb T$ such that the nontangential limit of $f$ exists at each point of $A$, and equals zero. Then $f(z)=0$ for all $z\in\mathbb D$.
\[Lindelof\] Let $f\colon\mathbb D\longrightarrow\mathbb C\cup\{\infty\}$ be a meromorphic function, and let $e^{i\theta}\in\mathbb T$. Assume that the set $(\mathbb C\cup\{\infty\})\setminus f(\mathbb D)$ contains at least three points. If there exists a path $\gamma\colon[0,1)\longrightarrow\mathbb D$ such that $\lim_{t\to1}\gamma(t)=e^{i\theta}$ and $\ell=\lim_{t\to1}f(\gamma
(t))$ exists in $\mathbb C\cup\{\infty\}$, then the nontangential limit of $f$ at $e^{i\theta}$ exists, and equals $\ell$.
Theorem \[Fatou\] is due to Fatou, and Theorem \[Privalov\] to Privalov. Theorem \[Lindelof\] is an extension of a result by Lindelöf. For proofs, we refer to [@CollingwoodL], theorems 2.4, 8.1 and 2.20.
These three theorems have obvious reformulations for meromorphic functions defined on the upper half-plane. For the convenience of the reader we will provide them below.
1. Let $f\colon\mathbb C^+\longrightarrow\mathbb C^+\cup\mathbb R\cup\{\infty\}$ be an analytic function. Then the set of points $x\in\mathbb R$ at which the nontangential limit of $f$ fails to exist in $\mathbb C\cup\{\infty\}$ is of Lebesgue measure zero.
2. Let $f\colon\mathbb C^+\longrightarrow\mathbb C$ be an analytic function. Assume that there exists a set $A\subseteq \mathbb R$ of nonzero Lebesgue measure such that the nontangential limit of $f$ exists at each point of $A$ and equals zero. Then $f(z)=0$ for all $z\in\mathbb C^+$.
3. Let $f\colon\mathbb C^+\longrightarrow\mathbb C\cup\{\infty\}$ be a meromorphic function, and let $x\in\mathbb R\cup\{\infty\}$. Assume that the set $(\mathbb C\cup\{\infty\})\setminus f(\mathbb C^+)$ contains at least three points. If there exists a path $\gamma\colon[0,1)\longrightarrow\mathbb C^+$ such that $\lim_{t\to1}\gamma(t)=x$ and $\ell=\lim_{t\to1}f(\gamma
(t))$ exists in $\mathbb C\cup\{\infty\}$, then the nontangential limit of $f$ at $x$ exists, and equals $\ell$.
The equivalence of statements (2) and (3) above to Theorems \[Privalov\] and \[Lindelof\] follows by simply considering $f\circ T$, where $T$ is the conformal automorphism $T(w)=i\frac{1+w}{1-w},$ and recalling that $T$ preserves angles, carries a set $A\subseteq\mathbb T$ of zero linear measure into a set of zero Lebesgue measure, and vice-versa. The equivalence of Theorem \[Fatou\] with statement (1) follows by considering the conjugation of $f$ with $T$ and, if necessary, a re-scaling. We would also like to mention here that in the context of the above three theorems the point infinity is not in any way a special point; for example, Theorem \[Privalov\] forbids any nonconstant analytic function to have constant - finite as well as infinite - nontangential limit on a non-negligible set. Thus, given a non-constant analytic self-map of $\mathbb C^+$ and a countable set $C\subset
\mathbb C\cup\{\infty\}$, we can always find a set $A\subseteq\mathbb R$ whose complement is negligible so that our map has nontangential limits at all points of $A$ which do not belong to $C$.
Consider a domain (i.e. an open connected set) $D\subseteq\mathbb C\cup\{\infty
\}$ and a function $f\colon D\longrightarrow\mathbb C\cup\{\infty\}$. Assume that $\Gamma\subseteq D$ and $x_0$ is an accumulation point for $\Gamma$. The cluster set $C_\Gamma(f,x_0)$ of the function $f$ at the point $x_0
$ relative to $\Gamma$ is $$\{z\in\mathbb C\cup\{\infty\}\ |\ \exists
\{z_n\}_{n\in\mathbb N}\subset \Gamma\setminus x_0\ {\rm such\ that }
\lim_{n\to\infty}z_n=x_0, \ \lim_{n\to\infty}f(z_n)=z\}.{\rm }$$ If $\Gamma=D$, we shall write $C(f,x_0)$ instead of $C_D(f,x_0).$ The following result is immediate.
\[ClusterConex\] Let $D\subset\mathbb C\cup\{\infty\}$ be a domain and let $f\colon D\longrightarrow\mathbb C\cup\{\infty\}$ be continuous. If $D$ is locally connected at $x\in\overline{D}$, then $C(f, x)$ is connected.
This result appears in [@CollingwoodL], as Theorem 1.1.
It will be useful for our purposes to understand the behaviour of analytic self-maps of $\mathbb C^+$ near open intervals in $\mathbb R$ on which their nontangential limits are real almost everywhere. The following theorem of Seidel can be used to describe the behaviour of such analytic functions near the boundary of their domain of definition. For proof, we refer to [@CollingwoodL], Theorem 5.4.
\[Seidel\] Let $f\colon\mathbb D\longrightarrow\mathbb D$ be an analytic function such that the radial limit $f(e^{i\theta})=\lim_{r\to1}f(re^{i\theta})$ exists and has modulus $1$ for almost every $\theta$ in the interval $(\theta_1,\theta_2)$. If $\theta\in(\theta_1,\theta_2
)$ is such that $f$ does not extend analytically through $e^{i\theta},$ then $C(f,e^{i\theta})=\overline{\mathbb D}.$
This theorem can be applied to self-maps of the upper half-plane, via a conformal mapping, but in that case one must consider meromorphic, instead of analytic, extensions.
A second result refering to the behaviour of $C(f,x)$ for bounded analytic functions $f$ is the following theorem of Carathéodory. (This result appears in [@CollingwoodL], Theorem 5.5.)
\[Caratheodory\] Let $f\colon\mathbb D\longrightarrow\mathbb C$ be a bounded analytic function. Assume that for almost every $\theta\in(\theta_1,\theta_2)$ the radial limit $f(e^{i\theta})$ belongs to a set $W$ in the plane. Then, for every $\theta\in(\theta_1,\theta_2)$ the cluster set $C(f, e^{i\theta})$ is contained in the closed convex hull of $W$.
The following proposition is a consequence of the previous two theorems.
\[SeidelCaratheodory\] Let $f$ be an analytic self-map of $\mathbb C^+$ such that $\lim_{y\to 0}f(x+iy)$ exists and belongs to $\mathbb R$ for almost every $x\in (a,b).$ Suppose that $x_0\in(a,b)$ is such that $f$ cannot be continued meromorphically through $x_0
$. Then for any $c<d$ there is a set $E\subseteq(a,b)$ of nonzero Lebesgue measure such that $\lim_{y\to0}f(x+iy)$ exists for all points $x\in E,$ and the set ${\{\lim_{y\to0}f(x+iy)\colon x\in E\}}$ is dense in the interval $(c,d).$
For proof we refer to [@AIHP], Proposition 1.9.
We will next focus on boundary behaviour of derivatives of analytic self-maps of the unit disk and of the upper half-plane. These results are described in detail by Nevanlinna [@Nev] and Shapiro [@Shapiro]; see also Exercises 6 and 7 in Chapter I of Garnett’s book [@Garnett].
\[JuliaCaratheodory\] Let $f\colon\mathbb D\longrightarrow\mathbb D$ be an analytic function, and let $w\in\mathbb T$. The following statements are equivalent:
1. We have $$\liminf_{z\to w}\frac{|f(z)|-1}{|z|-1}<\infty;$$
2. There exists a number $\zeta\in\overline{\mathbb D}$ such that $$\lim_{\stackrel{ z\longrightarrow w}{{\sphericalangle}}}f(z)=\zeta,$$ and the limit $$\ell=
\frac{w}{\zeta}\lim_{\stackrel{ z\longrightarrow w}{{\sphericalangle}}}\frac{f(z)-\zeta}{z-w}
\label{eq1.1}$$ exists and belong to $(0,+\infty)$.
Moreover, if the equivalent conditions above are satisfied, the limit $\sphericalangle\lim_{z\to w}f'(z)$ exists, and the following equality holds: $$\ell=\frac{w}{\zeta}\lim_{\stackrel{ z\longrightarrow w}{{\sphericalangle}}}
f'(z)=\liminf_{z\to w}\frac{|f(z)|-1}{|z|-1}.$$ If $$\liminf_{z\to w}\frac{|f(z)|-1}{|z|-1}=\infty$$ and $$\lim_{\stackrel{ z\longrightarrow w}{{\sphericalangle}}}f(z)=\zeta,$$ then the limit in equation exists and equal infinity.
The number $\ell$ from the above theorem is called the Julia-Carathéodory derivative of $f$ at $w$. Since it will be useful in the third and fourth section, we discuss below in some detail the formulation of the Julia-Carathéodory Theorem for self-maps of the upper half-plane. In the following lemma we isolate the part of the Julia-Carathéodory Theorem for the upper half-plane that will be used in Sections 3 and 4.
\[JC\] Let $F\colon\mathbb C^+\longrightarrow\mathbb C^+$ be analytic, and let $a\in\mathbb R$. Assume that $$\lim_{\stackrel{ z\longrightarrow a}{{\sphericalangle}}}F(z)=c\in\mathbb R.$$ Then $$\lim_{\stackrel{ z\longrightarrow a}{{\sphericalangle}}}\frac{F(z)-c}{z-a}=
\liminf_{z\to a}\frac{\Im F(z)}{\Im z},$$ where the equality is considered in $\overline{\mathbb C}$. Conversely, if $$\liminf_{z\to a}\frac{\Im F(z)}{\Im z}<\infty,$$ then $\sphericalangle\lim_{z\to a}F(z)$ exists and belongs to $\mathbb R\cup\{\infty\}$. Moreover, if $F$ is not constant, then we have $\liminf_{z\to a}{\Im F(z)}/{\Im z}>0$.
Observe that by replacing $F$ with the function $F(z)-c$ we may assume without loss of generality that $c=0$. Let $T(z)=\frac{z-i}{z+i},$ $z\in\overline{\mathbb C}$. $T$ maps $\mathbb C^+$ conformally onto the unit disc $\mathbb D$, and its composition inverse is $T^{-1}(w)=i\frac{1+w}{1-w}.$ We consider $f\colon\mathbb D\longrightarrow\mathbb D$ defined by $f(w)=T(F(T^{-1}(w))).$ Obviously, $\sphericalangle\lim_{w\to T(a)}f(w)=-1.$ Denote $T(a)=b$. We have $$\begin{aligned}
\lim_{\stackrel{ z\longrightarrow a}{{\sphericalangle}}}\frac{F(z)}{z-a} & = & \lim_{\stackrel{ w
\longrightarrow b}{{\sphericalangle}}}\frac{F(T^{-1}(w))}{T^{-1}(w)-T^{-1}(b)}\\
& = & \lim_{\stackrel{ w\longrightarrow b}{{\sphericalangle}}}\frac{1+f(w)}{1-f(w)}\cdot\frac{(1-w)(1-b)}{2(w-b)}\\
& = & \frac{(1-b)^2}{4}\lim_{\stackrel{ w\longrightarrow b}{{\sphericalangle}}}\frac{f(w)-(-1)}{w-b}\\
& = & \frac{(1-b)^2}{4}\cdot\frac{-1}{b}\liminf_{w\to b}\frac{1-|f(w)|}{1-|w|}\\
& = & \frac{1}{a^2+1}\liminf_{w\to b}\frac{1-|f(w)|}{1-|w|}.\end{aligned}$$ We have used the Julia-Carathéodory Theorem in the next to last equality and the definition of $b$ in the last one. (The situation in which all expressions in the equalities above are infinite is not excluded.)
On the other hand, $$\begin{aligned}
\frac{\Im F(z)}{\Im z}=\frac{\Im F(T^{-1}(w))}{\Im T^{-1}(w)} & = & \frac{\Im\left(i\frac{1+f(w)}{1-f(w)}\right)}{\Im\left(
i\frac{1+w}{1-w}\right)}\\
& = & \frac{\frac{1-|f(w)|^2}{1-2\Re f(w)+|f(w)|^2}}{\frac{1-|w|^2}{1-2\Re w+|w|^2}}\\
& = & \frac{1-|f(w)|}{1-|w|}\cdot\frac{1+|f(w)|}{1+|w|}\cdot
\frac{|1-w|^2}{|1-f(w)|^2}\end{aligned}$$ By taking $\liminf$ in the above equality we obtain $$\begin{aligned}
\label{2}
\liminf_{z\to a}\frac{\Im F(z)}{\Im z} & = & \liminf_{w\to b}\frac{1-|f(w)|}{1-|w|}\cdot\frac{1+|f(w)|}{1+|w|}\cdot
\frac{|1-w|^2}{|1-f(w)|^2}\nonumber\\
& \ge & \liminf_{w\to b}\frac{1-|f(w)|}{1-|w|}\cdot\frac{1+|f(w)|}{1+|w|}\cdot\liminf_{w\to b}
\frac{|1-w|^2}{|1-f(w)|^2}.\end{aligned}$$ Observe that if $|f(w)|$ does not tend to one, the first $\liminf$ in the last row above is infinite. Thus, the first $\liminf$ is realized on a sequence on which $|f|$ tends to 1. The second $\liminf$ is realized when $w$ tends nontangentially to $b$, and equals $
|1-b|^2/4=1/(a^2+1).$ (This follows trivially from the facts that $|f(w)|<1$, $w\in\mathbb
D$ and $\sphericalangle\lim_{w\to b}f(w)=-1$, since this implies directly that the denominator of our expression $\frac{|1-w|^2}{|1-f(w)|^2}$ cannot be greater than $2$, value reached when $w$ tends to $b$ nontangentially.) Thus, $$\begin{aligned}
\liminf_{z\to a}\frac{\Im F(z)}{\Im z} & \ge & \frac{1}{a^2+1}\liminf_{w\to b}\frac{1-|f(w)|}{1-|w|}.\end{aligned}$$ We conclude that $$\lim_{\stackrel{ z\longrightarrow a}{{\sphericalangle}}}\frac{F(z)}{z-a}\le\liminf_{z\to a}\frac{\Im F(z)}{\Im z}.$$ (Again the case in which both sides of the inequality are infinite is included.) But $$\lim_{\stackrel{ z\longrightarrow a}{{\sphericalangle}}}\frac{F(z)}{z-a}
=\lim_{y\downarrow0}\frac{F(a+iy)}{iy}
=\lim_{y\downarrow0}\frac{\Re F(a+iy)}{iy}+\frac{\Im F(a+iy)}{y}
\ge\liminf_{z\to a}\frac{\Im F(z)}{\Im z},$$ as our limit is real or infinite. Thus, $$\lim_{\stackrel{ z\longrightarrow a}{{\sphericalangle}}}\frac{F(z)}{z-a}
=\liminf_{z\to a}\frac{\Im F(z)}{\Im z}.$$
Assume now that $\liminf_{z\to a}\Im F(z)/\Im z=d\in\mathbb R_+.$ Equation (2) above implies that the limit $\liminf_{w\to b}(1-|f(w)|)/(1-|w|)$ is also finite (recall that $b=T(a)\neq1$, so the second $\liminf$ in (\[2\]) is nonzero), so, by the Julia-Carathéodory Theorem $f$ has nontangential limit at $b$, and thus $F$ has nontangential limit at $a$. (Observe that this limit is infinite if and only if $\sphericalangle\lim_{w\to b}f(w)=1$.) Relation (\[2\]) together with the Julia-Carathéodory Theorem guarantees that $d>0$.
Consider now an analytic function $f\colon\mathbb D\longrightarrow\overline{
\mathbb D}.$ A point $w\in\overline{\mathbb D}$ is called a Denjoy-Wolff point for $f$ if one of the following two conditions is satisfied:
1. $|w|<1$ and $f(w)=w$;
2. $|w|=1,$ $\sphericalangle\lim_{z\to w}f(z)=w, $ and $$\lim_{\stackrel{z\longrightarrow w}{\sphericalangle}}\frac{f(z)-w}{z-w}\leq1.$$
The following result is due to Denjoy and Wolff.
\[DenjoyWolff\] Any analytic function $f\colon\mathbb D\longrightarrow\overline{\mathbb D}$ has a Denjoy-Wolff point. If $f$ has more than one such point, then $f(z)=z$ for all $z$ in the unit disc. If $z\in\mathbb D$ is a Denjoy-Wolff point for $f$, then $|f'(z)|\leq1;$ equality occurs only when $f$ is a conformal automorphism of the unit disc.
We refer to Shapiro’s book [@Shapiro], Chapter 5, for a detailed introduction to this subject.
The Denjoy-Wolff point of a function $f$ is characterized also by the fact that it is the uniform limit on compact subsets of the iterates $f^{\circ n}=
\underbrace{f\circ f\circ\cdots\circ f}_{n\ {\rm times}}$ of $f$. We state the following theorem for the sake of completeness (for the original statements, see [@Denjoy] and [@Wolff]):
Let $f\colon\mathbb D\longrightarrow\overline{\mathbb D}$ be an analytic function. If $f$ is not a conformal automorphism of $\mathbb D$, then the functions $f^{\circ n}$ converge uniformly on compact subsets of $\mathbb D$ to the Denjoy-Wolff point of $f$.
The previous two theorems have been used in [@Subord] to give a new proof for the subordination property for free additive and free multiplicative convolutions. We reproduce below the result for the free additive convolution (Theorem 4.1 in [@Subord]):
\[sub\] Given two Borel probability measures $\mu,\nu$ on the real line, there exist unique analytic functions $\omega_1,
\omega_2\colon\mathbb C^+\longrightarrow\mathbb C^+$ such that
1. $\Im \omega_j(z)\ge\Im z$ for $z\in\mathbb C^+$, and $$\lim_{y\uparrow+\infty}\frac{\omega_j(iy)}{iy}=1,\quad j=1,2.$$
2. $F_{\mu\boxplus\nu}(z)=F_\mu(\omega_1(z))=F_\nu(\omega_2(z)),$ and
3. $\omega_1(z)+\omega_2(z)=z+F_{\mu\boxplus\nu}(z),$ for all $z\in\mathbb C^+.$
For $z\in\mathbb C^+,$ the point $\omega_1(z)$ appears as the Denjoy-Wolff point of the function $f_z\colon
\mathbb C^+\longrightarrow\mathbb C^+$ given by $f_z(w)=F_\nu(F_\mu(w)-w+z)-F_\mu(w)+w.$ The function $f_z$ is well-defined for all $z\in\mathbb C^+\cup\mathbb R$ by Remark \[ImF>Imz\]. An immediate consequence of Theorem \[sub\] is the fact that free additive convolution can be defined equivalently by purely complex analytic methods, using equations (2) and (3) from the theorem above. This has been proved independently by different means in [@CG].
We shall use boundary properties of the subordination functions to describe the atomic, singular continuous, and absolutely continuous parts, with respect to the Lebesgue measure on $\mathbb R
$, of the free convolution $\mu\boxplus\nu$ of two probability measures $\mu,
\nu$ on the real line. The following lemma describes the behaviour of the Cauchy transform $G_\mu$ near points belonging to the support of the singular part of the probability measure $\mu.$ For proof, we refer to [@AIHP], Lemma 1.10, [@BercoviciVoiculescuRegQ], Lemma 7.1, and [@SteinWeiss], Theorem 3.16 of Chapter II.
\[sing\] Let $\mu$ be a Borel probability measure on $\mathbb R$.
1. For $\mu^{s}$-almost all $x\in\mathbb R$, the nontangential limit of the Cauchy transform $G_\mu$ of $\mu$ at $x$ is infinite.
2. We have $\mu(\{x\})=\sphericalangle\lim_{z\to x}(z-x)G_\mu(z).$
3. Denote by $f$ the density of $\mu^{ac}$ with respect to the Lebesgue measure on $\RR$. Then for almost all $x\in\mathbb R$, we have $-\pi f(x)=
\sphericalangle\lim_{z\to x}\Im G_\mu(z).$
Finally, we provide a technical lemma, whose proof can be also found in the proof of Theorem 2.3 of [@AIHP]. We shall give here a more conceptual proof.
\[ntg\] Let $f\colon\mathbb C^+\longrightarrow\mathbb C^+$ be a nonconstant analytic function, $x\in
\mathbb R\cup\{\infty\}$, and assume that $C(f,x)\subseteq\mathbb R\cup\{\infty\}$ contains more than one point, and hence, by Lemma \[ClusterConex\], a closed nondegenerate interval or the complement of an open interval. Then for all $c\in C(f,x)$, with the possible exception of at most two points, there exists a sequence $\{z_n^{(c)}
\}_{n\in\mathbb N}\subset\mathbb
C^+$ so that
1. $\displaystyle\lim_{n\to\infty}z_n^{(c)}=x$;
2. $\displaystyle\lim_{n\to\infty}f(z_n^{(c)})=c$, and
3. $\Re f(z_n^{(c)})=c$ for all $n\in\mathbb N$.
Observe first that, by replacing $f(z)$ with $f\left(-\frac{1}{z}\right)$ if necessary, we may assume that $x\in\mathbb R.$ Now pick two arbitrary points $c_1<c_2\in
\mathbb R\cap C(f,x)$, and fix two arbitrary constants $\varepsilon\in(0,|c_1+c_2|/4)$ and $M>\max
\{2+\varepsilon+|c_1|,2+\varepsilon+|c_2|\}.$ Define the compact region
$$X_M=\{z\in\mathbb C^+\colon|\Re z|\leq M,\ 1/M\leq\Im z\leq M\}.$$ Let $\{z_n^j\}_{n\in\mathbb N}\subset\mathbb C^+$, $j\in\{1,2\}$, be two sequences with the following properties:
1. $\lim_{n\to\infty}z_n^j=x$, $j\in\{1,2\}$;
2. $1>|z_n^2-x|>2|z_n^1-x|>4|z_{n+1}^2-x|$ for all $n\in\mathbb N$;
3. $|f(z_n^j)-c_j|<\frac1n$ for all $n\in\mathbb N$, $n>0$ and $j\in\{1,2\}$.
Define a path $\gamma\colon[0,1]\longrightarrow\mathbb C^+\cup\{x\}$ so that $
\gamma(0)=i,$ $\gamma(1)=x$, $\gamma\left(1-\frac{1}{2n}\right)=z_n^2,$ $\gamma\left(1-\frac{1}{2n+1}\right)=z_n^1,$ and $\gamma$ is linear on the intervals $\left[0,\frac12\right],$ $\left[1-\frac{1}{2n},1-\frac{1}{2n+1}\right]$ and $\left[1-\frac{1}{2n+1},1-\frac{1}{2n+2}
\right]$ for all $n\in\mathbb N$, $n>0$. We easily observe that $\gamma$ is a simple curve in $\mathbb C^+\cup\{x\}$ and $\lim_{t\to1}\gamma(t)=x.$ Thus, there exists $
n_M\in\mathbb N$ so that $f\left(\gamma\left(\left[1-\frac{1}{2n_M},1\right)\right)\right)
\cap X_M=\varnothing.$ Indeed, assume towards contradiction that for all $n\in\mathbb
N$ there exists $t_n\in\left(1-\frac{1}{2n},1\right)$ so that $f(\gamma(t_n))\in X_M
\subset\mathbb C^+$. Since $X_M$ is compact, there exists a subsequence $\{t_{n_k}\}_k$ of $\{t_n\}_n$ with the property that $\lim_{k\to\infty}f(\gamma(t_{n_k}))$ exists and belongs to $X_M$. But $t_n\in\left(1-\frac{1}{2n},1\right)$ implies that $\lim_{k\to\infty}
t_{n_k}=1$ and thus $\lim_{k\to\infty}\gamma(t_{n_k})=x$, so that $\lim_{k\to\infty}f(\gamma(t_{n_k}))\in C(f,x)\subseteq\mathbb R\cup\{\infty\}$. Contradiction. Thus indeed the path $f\circ\gamma\colon[0,1)\to\mathbb C^+$ ultimately stays out of $X_M$ as $t\to1$.
For each $n\in\mathbb N$, $n>n_M+\frac1\varepsilon$, define $\Gamma_n$ to be the image of the restriction of $f\circ\gamma$ to $\left[1-\frac{1}{2n},1-\frac{1}{2n+1}\right]$. By the construction of our sequences $\{z_n^j\}_n$ and the path $\gamma$, $\Gamma_n\subset\mathbb C^+\setminus X_M$ unites through $\mathbb C^+
\setminus X_M$ the balls $B(c_2,1/n)$ and $B(c_1,1/n)$ for all $n\in\mathbb N$, $n>n_M+\frac1\varepsilon$. Thus, it must intersect at least one of the following three segments: $$\mathcal S_0=\left\{\frac{c_1+c_2}{2}+is\colon 0<s<1/M\right\},\quad
\mathcal S_1=\{c_1-\varepsilon+is\colon 0<s<1/M\},\quad {\rm or}$$ $$\mathcal S_2=\{c_2+\varepsilon+is\colon 0<s<1/M\},$$ at least once. We conclude that there exists $t_n\in\left(1-\frac{1}{2n},1-\frac{1}{2n+1}
\right)$ so that $f(\gamma(t_n))\in\mathcal S_r$ for some $r=r(n)\in\{0,1,2\}$. Since this holds for all $n\in\mathbb N$, $n>n_M+\frac1\varepsilon$, there is at least one of $\mathcal S_0,\mathcal S_1,\mathcal S_2$, call it $\mathcal S_{r_0}$, which is intersected by infinitely many paths $\Gamma_n$, and thus there exists a subsequence $\{t_{n_k}\}_k$ with $t_{n_k}\in\left(1-\frac{1}{2n_k},1-\frac{1}{2n_k+1}
\right)$ so that $f(\gamma(t_{n_k}))\in\mathcal S_{r_0},$ $k\in\mathbb N$. But $\lim_{k\to\infty}t_{n_k}=1$, so $\lim_{k\to\infty}\gamma(t_{n_k})=x$, and thus $\lim_{k\to\infty}f(\gamma(t_{n_k}))=c$, where $\{c\}=\overline{\mathcal S_{r_0}}\cap\mathbb R\subset
\{c_1-\varepsilon,c_2+\varepsilon,(c_1+c_2)/2\}.$
For the point $c$, $\{c\}=\overline{\mathcal S_{r_0}}\cap\mathbb R$, we have constructed the sequence $z_k^{(c)}=\gamma(t_{n_k})$ satisfying conditions (i), (ii) and (iii) from our lemma. We shall prove next that all points of $C(f,x)$, with at most two exceptions, can be realized in the form of a $c$, with $\{c\}=\overline{\mathcal S_{r_0}}\cap\mathbb R$. Consider two separate cases:
[**Case 1:**]{} $C(f,x)=\mathbb R\cup\{\infty\}.$ If for all $d\in\mathbb R$ we can find a sequence $\{z_n^{(d)}\}_n$ with the properties (i), (ii) and (iii), then we are done. Assume thus that there exists a point $d\in\mathbb R$ for which no sequence with property (iii) exists (observe that properties (i) and (ii) can always be satisfied, by the definition of the cluster set). We claim that for any $c\in\mathbb R\setminus\{d\}$ there is a sequence $\{z_n^{(c)}\}_n$ satisfying properties (i), (ii) and (iii). Indeed, choose such a point $c\neq d$ and pick $c_1<c<c_2<d$ so that $c=(c_1+c_2)/2$. Construct sequences $\{z_n^j\}_n$, $j\in\{1,2\}$, and a path $\gamma$ as above. We claim that there exists an $\eta>0$ so that $f(\gamma([0,1)))\cap\{d+is\colon0<s<\eta\}=\varnothing.$ Indeed, assume towards contradiction that this is not the case. Then for any $n\in\mathbb N$ there exists a point $v_n\in f(\gamma([0,1)))\cap\{d+is\colon0<s<1/n\},$ and so there exists a point $t_n\in[0,1)$ so that $f(\gamma(t_n))=v_n$. Since $\gamma([0,1))\subset
\mathbb C^+$ and $f$ is not constant by hypothesis, the sequence $\{t_n\}_n$ cannot have any accumulation points in $[0,1)$, so $\lim_{n\to\infty}t_n=1$. But then $\lim_{n\to\infty}\gamma(t_n)=x$, $\lim_{n\to\infty}f(\gamma(t_n))=d$ and $\Re f(\gamma(t_n))=d$ for all $n\in\mathbb N$. So the sequence $\{\gamma(t_n)\}_n$ satisfies properties (i), (ii) and (iii) for $d$, a contradiction. This proves the existence of the required $\eta>0$.
Choose now $\varepsilon\in(0,\min\{\eta,|c_1+c_2|/4,1\})$ and $M>\max\{1/\eta
,|c_1|+2+\varepsilon,|c_2|+2+\varepsilon\}$ and define $X_M$ and $\Gamma_n$ as above. It follows immediately then that the paths $\Gamma_n$ must intersect the segment $\mathcal S_0=\left\{\frac{c_1+c_2}{2}+is\colon 0<s<1/M\right\}$ infinitely often, since $c_2$ is separated from $c_1$ by $\mathcal S_0$, the lower edge of $X_M$ and $\{d+is\colon0<s<\eta\},$ and $\Gamma_n$ does not intersect the lower edge of $X_M$ or $\{d+is\colon0<s<\eta\}.$ Thus we have found a sequence $z_n^{(c)}$ as required for our $c=(c_1+c_2)/2$. Since $c\in\mathbb R\setminus\{d\}$ is arbitrary, we have proved our lemma, with the exceptional points $d$ and $\infty$.
[**Case 2:**]{} $\mathbb R\setminus C(f,x)\neq\varnothing$. This case is straightforward: pick $c_1$ and $c_2$ to be, one of them the (or a) finite endpoint of $C(f,x)$, and the other any arbitrary point in $C(f,x)$ except the previously chosen one. It follows trivially that any point $c\in C(f,x)$ in between these two points accepts a sequence $\{z_n^{(c)}\}_n$ as in our lemma. We leave the details of the proof to the reader. Thus, the lemma holds, with the two exceptional points being the endpoints of the cluster set.
Boundary behaviour of the subordination functions
=================================================
In the following we fix two Borel probability measures $\mu$ and $\nu$ on $\mathbb R
$, neither of them a point mass. For technical reasons, we first study the case when $\mu$ and $\nu$ are both convex combinations of two point masses. (We denote by $\delta_a$ the probability which gives mass one to the point $a$.) Let $a\in\mathbb R$ be fixed and define the following two self-maps of the upper half- plane: $h_\mu(w)=F_\mu(w)-w+a,$ $ h_\nu(w)=F_\nu(w)-w+a,$ $w\in\mathbb C^+.$ The following proposition generalizes Lemma 2.1 of [@AIHP].
\[31\] The function $f_a(w)=h_\nu(h_\mu(w))$ is a conformal automorphism of $\mathbb C^+$ if and only if each of $\mu$ and $\nu$ are convex combinations of two distinct point masses. In this case, $\omega_1(z)$ and $\omega_2(z)$, provided by Theorem \[sub\], satisfy quadratic equations having $z$ and real numbers as coefficients.
Assume that $f_a$ is a conformal automorphism of $\mathbb C^+.$ We claim that the analytic functions $h_\mu,h_\nu\colon\mathbb C^+\longrightarrow\mathbb C^+,$ are both conformal automorphisms of the upper half-plane. Observe that by the definition of a conformal automorphism, $h_\mu$ must be injective, and $h_\nu$ surjective. Let $k$ be the inverse with respect to composition of $h_\nu\circ h_\mu,$ so that $$h_\nu(h_\mu(k(z)))=z,\quad z\in\mathbb C^+.$$ Applying $h_\mu\circ k$ to both sides of the above equality gives $h_\mu(k(h_\nu(w)))=w$ for all $w$ in the open subset $(h_\mu\circ k)(
\mathbb C^+)$ of the upper half-plane, so, by analytic continuation, for all $w\in\mathbb C^+.$ This proves surjectivity of $h_\mu$ and injectivity of $h_\nu$ and thus our claim is proved.
Observe that, by Theorem \[Nevanlinna\], $\lim_{y\to+\infty}h_\mu(iy)/iy=\lim_{y\to+\infty}h_\nu(iy)/iy=0.$ Thus, there exist real numbers $b_\mu,c_\mu,d_\mu,b_\nu,c_\nu,d_\nu$ so that $$h_\mu(z)=\frac{b_\mu z+c_\mu}{z+d_\mu},\quad h_\nu(z)=\frac{b_\nu z+c_\nu}{z+d_\nu},\quad z\in\mathbb
C^+,$$ and $${\rm det}\left[\begin{array}{cc}
b_\mu & c_\mu \\
1 & d_\mu
\end{array}\right]>0,\quad {\rm det}\left[\begin{array}{cc}
b_\nu & c_\nu \\
1 & d_\nu
\end{array}\right]>0.$$ By the definition of $h_\mu,$ we obtain $$\begin{aligned}
F_\mu(z)=h_\mu(z)+z-a & = & \frac{z^2+z(d_\mu+b_\mu-a)+c_\mu-d_\mu a}{z+
d_\mu}\\
& = & \left(t\frac{1}{z-u}+(1-t)\frac{1}{z-v}\right)^{-1},\end{aligned}$$ where $$t=\frac{d_\mu-b_\mu+a+\sqrt{(d_\mu+b_\mu-a)^2-4c_\mu+4d_\mu a}}{2\sqrt{(d_\mu
+b_\mu-a)^2-4c_\mu+4d_\mu a}},$$ and $$u=\frac{a-d_\mu-b_\mu+\sqrt{(d_\mu+b_\mu-a)^2-4c_\mu+4d_\mu a}}{2},$$ $$v=\frac{a-d_\mu-b_\mu-\sqrt{(d_\mu+b_\mu-a)^2-4c_\mu+4d_\mu a}}{2}.$$ Thus, $\mu$ is a convex combination of two point masses $\delta_u$ and $\delta_v,$ with weights $t$ and $1-t,$ respectively. The result for $\nu$ follows the same way. Conversely, if $\mu=t\delta_u+(1-t)\delta_v,$ then a direct computation shows that $F_\mu(z)=(z-v)(z-u)(z-tv-(1-t)u)^{-1},$ so that $$h_\mu(z)=\frac{(a-(tu+(1-t)v))z+uv-a(tv+(1-t)u)}{z-(tv+(1-t)u)},\quad z\in\mathbb C^+,$$ and $${\rm det}\left[\begin{array}{cc}
a-(tu+(1-t)v) & uv-a(tv+(1-t)u) \\
1 & -tv-(1-t)u
\end{array}\right]=t(1-t)(u-v)^2>0,$$ for all $0<t<1,u\neq v$. This proves the first statement of the proposition.
Assume now that $\mu=t\delta_u+(1-t)\delta_v,$ $\mu=s\delta_w+(1-s)\delta_x$, with $u\neq v,$ $w\neq x$, and $0<s,t<1.$ The computations above provide real numbers $b_\mu,c_\mu,d_\mu,b_\nu,c_\nu,d_\nu$ which depend polynomially on $u,v,w,x,s,t$ so that $$h_\mu(z)=\frac{b_\mu z+c_\mu}{z+d_\mu},\quad h_\nu(z)=\frac{b_\nu z+c_\nu}{z+d_\nu},\quad z\in\mathbb
C^+.$$ On the other hand, by parts (2) and (3) of Theorem \[sub\] we obtain that $$\begin{aligned}
F_\mu(\omega_1(z)) & = & F_\nu(F_\mu(\omega_1(z))-\omega_1(z)+z)\\
& = & h_\nu(F_\mu(\omega_1(z))-\omega_1(z)+z)+F_\mu(\omega_1(z))-\omega_1(z)+z-a,\end{aligned}$$ so that $$\begin{aligned}
\omega_1(z) & = & h_\nu(F_\mu(\omega_1(z))-\omega_1(z)+z)+z-a\\
& = & h_\nu(h_\mu(\omega_1(z))-a+z)-a+z\\
& = & \frac{b_\nu\left(\frac{b_\mu\omega_1(z)+c_\mu}{\omega_1(z)+d_\mu}+z-a\right)
+c_\nu}{\frac{b_\mu\omega_1(z)+c_\mu}{\omega_1(z)+d_\mu}+z-a+d_\nu}+z-a\\
& = & \frac{b_\nu(b_\mu \omega_1(z)+c_\mu+(z-a)(\omega_1(z)+d_\mu))+c_\nu
(\omega_1(z)+d_\mu)}{b_\mu\omega_1(z)+c_\mu+(z-a+d_\nu)(\omega_1(z)+d_\mu)}
+z-a.\end{aligned}$$ Thus, $\omega_1$ satisfies an equation of degree two with coefficients that are polynomials of $z,s,t,u,v,w,$ and $x$. The same argument shows the required statement for $\omega_2$ This proves the second statement of the proposition.
A consequence of the proposition above is the following
\[2atoms\] With the notations from Theorem \[sub\], if $\mu$ and $\nu$ are both convex combinations of two point masses, then:
1. $\omega_1$ and $\omega_2$ extend continuously to $\mathbb R$ as functions with values in the extended complex plane $\overline{\mathbb C}$;
2. If either $\omega_1(a)\in\mathbb C^+$ or $\omega_2(a)\in\mathbb C^+$ for some $a\in\mathbb R$, then $\omega_1,\omega_2,$ and $F_{\mu
\boxplus\nu}$ extend analytically around $a$;
3. $(\mu\boxplus\nu)^{sc}=0,$ and $\frac{d(\mu\boxplus\nu)^{ac}(t)}{dt}$ is analytic wherever positive and finite.
Parts (1) and (2) of the corollary are obvious. For the part (3) observe that $F_{\mu
\boxplus\nu}$ extends continuously to $\mathbb R$ and the set $F_{\mu\boxplus\nu}^{-1}(\{0\})$ contains only finitely many points, by Theorem \[sub\] (3) and by Proposition \[31\]. Thus, by Lemma \[sing\] (1), $(\mu\boxplus\nu)^{sc}=0$. The statement regarding $\frac{d(\mu\boxplus\nu)^{ac}(t)}{dt}$ follows immediately from Theorem \[sub\] (3), Proposition \[31\], and Lemma \[sing\] (3).
The next theorem describes the boundary behaviour of the subordination functions provided by Theorem \[sub\].
\[Boundary+\] Let $a\in\mathbb R$ be fixed. With the notations from Theorem \[sub\], the following hold:
1. If either $C(\omega_1,a)\cap\mathbb C^+\neq\varnothing,$ or $C(\omega_2,a)\cap\mathbb C^+\neq\varnothing,$ then the functions $\omega_1$, $\omega_2,$ and $F_{\mu\boxplus\nu}$ extends analytically in a neighbourhood of $a$.
2. The functions $\omega_1$ and $\omega_2$ have nontangential limits at $a$.
3. Assume in addition that the sets $I_1=\mathbb R\setminus{\rm supp}
(\mu)$ and $I_2=\mathbb R\setminus{\rm supp}(\nu)$ are nonempty. Then $\lim_{z\to a}\omega_j(z)$, $j=1,2,$ exist in the extended complex plane $\overline{\mathbb C}.$
A slightly less general version of part (3) of this theorem appears implicitly in the proof of Theorem 2.3 in [@AIHP].
Consider a sequence $\{z_n\}_{n\in\mathbb N}\subset\mathbb C^+$ and a number $\ell\in\mathbb C^+$ with the property that $\lim_{n\to\infty}z_n=a$ and $\ell=\lim_{n\to\infty}\omega_1(z_n).$ Define $f\colon(\mathbb C^+\cup\mathbb R)\times\mathbb C^+\longrightarrow\mathbb C^+$ by $f(z,w)=F_\nu(F_\mu(w)-w+z)-F_\mu(w)+w.$ As noted in the comments following Theorem \[sub\], $\omega_1(z)$ is the Denjoy-Wolff point of the function $f_z=f(z,\cdot)$ whenever $z\in\mathbb
C^+.$ Thus, we have $$\ell=\lim_{n\to\infty}\omega_1(z_n)=\lim_{n\to\infty}f(z_n,\omega_1(z_n))=
f(a,\ell).$$ If $f_a=f(a,\cdot)$ is not an automorphism of the upper half-plane, then we can use Theorem \[DenjoyWolff\] to conclude that $|f_a'(\ell)|<1.$ It follows from Remark \[ImF>Imz\] that $f$ can be extended on a bidisc $B(a,\varepsilon)\times B(\ell,\varepsilon)$ for $
\varepsilon>0$ small enough, so by the Implicit Function Theorem there exists $\eta\in(0,\varepsilon)$ and an analytic function $\omega\colon B(a,\eta)\longrightarrow B(\ell,\varepsilon)$ such that $f(z,\omega(z))=
\omega(z)$ for all $z\in B(a,\eta)$. Since $f(z,\omega_1(z))=\omega_1(z)$ for $z\in\mathbb C^+$, we conclude by the uniqueness part of Theorem \[sub\] that $\omega$ extends $\omega_1$ to $\mathbb C^+\cup B
(a,\eta).$ Since $F_{\mu\boxplus\nu}=F_\mu\circ\omega_1$ and $\ell=\omega_1(a)\in
\mathbb C^+,$ $F_{\mu\boxplus\nu}$ also extends analytically to some neighbourhood of $a$. The similar statement for $\omega_2$ follows from part (3) of Theorem \[sub\].
The case when $f_a$ is a conformal automorphism of $\mathbb C^+$ is covered by Corollary \[2atoms\]. This proves (1).
Assume now that the hypothesis of (3) holds and yet $C(\omega_1,a)$ contains more than one point. By part (1), $C(\omega_1,a)\subseteq
\mathbb R\cup\{\infty\},$ and by Lemma \[ClusterConex\], either $C(\omega_1,a)
\setminus\{\infty\}$ is a closed interval in $\mathbb R$ (possibly all of $\mathbb R$), or $\mathbb R\setminus C(\omega_1,a)$ is an open interval in $\mathbb R.$
By Lemma \[ntg\], for any $c$ in $C(\omega_1,a)\setminus\{\infty\},$ with the possible exception of two points, there exists a sequence $\{z_n^{(c)}\}_{n\in\mathbb N}$ converging to $a$ such that $\lim_{n\to\infty}\omega_1(z_n^{(c)})=c,$ and $\Re\omega_1(z_n^{(c)})=c$ for all $n$ (i.e. $\omega_1(z_n^{(c)})$ converges to $c$ nontangentially - in fact approaches $c$ vertically). Thus, if existing, $\lim_{n\to\infty}F_\mu(\omega_1(z_n^{(c)}))=\sphericalangle
\lim_{z\to c}F_\mu(z).$ By Fatou’s theorem (Theorem \[Fatou\]), this limit exists for almost all $c\in C(\omega_1,a).$ Denote it by $F_\mu(c).$ We shall prove that for every $c\in C(\omega_1,a)$ for which $F_\mu(c)$ exists, with at most two exceptions, $
F_\mu(c)\not\in\mathbb C^+.$ Indeed, suppose that $F_\mu(c)\in\mathbb C^+$ for some $c\in C(\omega_1,a)$ for which $\omega_1(z_n^{(c)})$ as above can be constructed. Recall that $\omega_1(z_n^{(c)})$ converges nontangentially to $c$. Then, using parts (2) and (3) of Theorem \[sub\], Remark \[ImF>Imz\], and the fact that $\nu$ is not a point mass, we obtain $$\begin{aligned}
\Im F_\mu(c) & = & \lim_{n\to\infty}\Im F_\mu(\omega_1(z_n^{(c)}))\\
& = & \lim_{n\to\infty}\Im F_\nu(F_\mu(\omega_1(z_n^{(c)}))-\omega_1(z_{n}^{(c)})+
z_{n}^{(c)})\\
& = & \Im F_\nu(F_\mu(c)-c+a)\\
& > & \Im (F_\mu(c)-c+a)\\
& = & \Im F_\mu(c),\end{aligned}$$ which is a contradiction.
Consider now ${\omega_2}(z)=F_\mu(\omega_1(z))-\omega_1(z)+z,$ $z\in\mathbb C^+.$ We shall argue that the set $C(\omega_2,a)\subseteq\mathbb R\cup\{\infty\}$ must be also infinite. Suppose this were not the case. Then for any $c\in{\rm Int}C(\omega_1,a),$ $$\lim_{z\to a}{\omega_2}(z)=\lim_{n\to\infty}F_\mu(\omega_1(z_n^{(c)}))-
\omega_1(z_n^{(c)})+z_n^{(c)}=F_\mu(c)-c+a,$$ so that, by Theorem \[Privalov\], $F_\mu(z)=z-a+\lim_{v\to a}{\omega_2}(v)$ for all $z\in\mathbb C^+$. (We denote by ${\rm Int}A$ the interior of $A\subseteq
\mathbb R$ with respect to the usual topology on $\mathbb R$.) This contradicts the fact that $\mu$ is not a point mass. So $C(\omega_2,a)$ must be an infinite set.
Assume that $c_0\in{\rm Int}C(\omega_1,a)$ is a point where $F_\mu$ does not continue meromorphically. Proposition \[SeidelCaratheodory\] shows that the set $$E=\{c\in C(\omega_1,a)\colon a+F_\mu(c)-c\in I_2\}$$ has nonzero Lebesgue measure. In particular, for all $c\in E$, $$\begin{aligned}
F_\mu(c) & = & \lim_{n\to\infty}F_\mu(\omega_1(z_n^{(c)}))\\
& = & \lim_{n\to\infty}F_\nu(F_\mu(\omega_1(z_n^{(c)}))-\omega_1(z_n^{(c)})+
z_n^{(c)})\\
& = & F_\nu(F_\mu(c)-c+a),\end{aligned}$$ where we used the analiticity of $F_\nu$ on $I_2$ in the last equality. Privalov’s theorem (Theorem \[Privalov\]) implies that $F_\mu(z)=
F_\nu(F_\mu(z)-z+a)$ for all $z\in\mathbb C^+.$ Rewriting this equality gives $$F_\nu(F_\mu(z)-z+a)-(F_\mu(z)-z+a)+a=z,$$ or, equivalently, $$h_\nu(h_\mu(z))=z,\quad z\in\mathbb C^+.$$ Corollary \[2atoms\] and Proposition \[31\] provide now a contradiction.
We conclude that $F_\mu$ extends meromorphically through the whole open interval ${\rm Int}C(\omega_1,a).$ By the same argument, $F_\nu$ must continue meromorphically through all of the (nondegenerate) interval ${\rm Int}C({\omega_2},a)$.
For any sequence $\{z_n^{(c)}\}_{n\in\mathbb N}$ constructed above, we may take a subsequence so that $\lim_{n\to\infty}{\omega_2}(z_n^{(c)})$ exists (we do not claim that ${\omega_2}(
z_n^{(c)})$ converges nontangentially to this limit). Suppose there were a point $d\in C({\omega_2},a)$ and a set $V_{d}
\subset{\rm Int}C(\omega_1,a)$ of nonzero Lebesgue measure such that $\lim_{n\to\infty}{\omega_2}(z_n^{(c)})=d$ for all $c\in V_{d}.$ Taking limit as $n\to\infty$ in the equality $$F_\mu(\omega_1(z_n^{(c)}))+z_n^{(c)}=
\omega_1(z_n^{(c)})+{\omega_2}(z_n^{(c)})$$ gives $$F_\mu(c)+a=c+d,\quad{\rm for\ all}\ c\in V_{d}.$$ Applying again Privalov’s theorem, we obtain that $F_\mu(z)=z-(a-d)$ for all $z\in\mathbb C^+.$ This contradicts the fact that $\mu$ is not a point mass. Thus, there exists a set $E\subseteq{\rm Int}C(\omega_1,a)$ of positive Lebesgue measure such that $\{\tilde{c}=\lim_{n\to\infty}{\omega_2}(z_n^{(c)})
\colon c\in E\}\subseteq{\rm Int}C({\omega_2},a).$ Then, since $F_\nu$ extends meromorphically through ${\rm Int}C({\omega_2},a)$, by Theorem \[sub\] we conclude that $$\begin{aligned}
F_\mu(c) & = &
\lim_{n\to\infty}F_\mu(\omega_1(z_n^{(c)}))\\
& = & \lim_{n\to\infty}F_\nu({\omega_2}(z_n^{(c)}))\\
& = & \lim_{n\to\infty}F_\nu\left({F_\mu(\omega_1
(z_n^{(c)}))}-\omega_1(z_n^{(c)})+z_n^{(c)}\right)\\
& = & F_\nu(a+F_\mu(c)-c)\end{aligned}$$ for all $c\in E$. Privalov’s theorem implies that $F_\mu(z)=F_\nu(a+F_\mu(z)-
z)$ for all $z\in\mathbb C^+.$ As we have proved already, this implies that $h_\mu$ and $h_\nu$ are conformal automorphisms of the upper half-plane. Corollary \[2atoms\] and Proposition \[31\] provide again a contradiction. This proves part (3) of the theorem.
Consider now the case when at least one of the two sets $I_1,I_2$ is empty. Without loss of generality, assume $I_1=\varnothing$. We show that in this case, if any of the two sets $C(\omega_1,a)$, $C(\omega_2,a)$ contains more than one point, we must have $C(\omega_1,a)=C(\omega_2,a)=
\mathbb R\cup\{\infty\}.$ We shall do this in four steps. [**Step 1:**]{} We show that ${\rm Int}C(\omega_1,a)\cap{\rm supp}
(\mu^{ac})=\varnothing.$ Indeed, assume this is not the case. Then (by Lemmas \[sing\] and \[ntg\]) for almost all $c$ in ${\rm Int}C(\omega_1,a)\cap{\rm supp}
(\mu^{ac})$ with respect to the Lebesgue measure, we have $\sphericalangle
\lim_{z\to c}F_\mu(z)\in\mathbb C^+$, and there exists $\{z_n^{(c)}\}_{n\in
\mathbb N}$ so that $\lim_{n\to\infty}z_n^{(c)}=a$, $\Re\omega_1(z_n^{(c)})
=c,$ and $\lim_{n\to\infty}\omega_1(z_n^{(c)})=c.$ Thus, $$\begin{aligned}
\lim_{n\to\infty}\omega_2(z_n^{(c)}) & = & \lim_{n\to\infty}
F_{\mu}(\omega_1(z_n^{(c)}))-\omega_1(z_n^{(c)})+z_n^{(c)}\\
& = & a-c+\lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}
F_\mu(z)\in\mathbb C^+,\end{aligned}$$ contradicting part (1) of the theorem. [**Step 2:**]{} We show that $C(\omega_2,a)=\mathbb R\cup\{\infty\}.$ One inclusion is an immediate consequence of part (1) of this theorem. Thus, it is enough to show that $C(\omega_2,a)$ is dense in $\mathbb R$. By Step 1 and our hypothesis $I_1=\varnothing,$ we have ${\rm Int}C(\omega_1,a)
\subseteq{\rm supp}(\mu^{s})\setminus{\rm supp}(\mu^{ac}).$ Thus, by Proposition \[SeidelCaratheodory\], for any $x\in{\rm Int}C(\omega_1,a)$ and any open interval $I$ containing $x$, the set
$$\{\lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}
F_\mu(z)\colon c\in I, F_\mu\ {\rm has\ nontangential\ limit\ at}\ c\}$$ is dense in $\mathbb R$. So, given $x$ as above, $\varepsilon>0$ and $s\in\mathbb R$, there exists $c\in I\cap(x-\frac{\varepsilon}{2},x+
\frac{\varepsilon}{2})$ such that $|s-x+a-\sphericalangle\lim_{z\to c}F_\mu(z)|
<\varepsilon/2,$ and thus $$\begin{aligned}
|s-\lim_{n\to\infty}\omega_2(z_n^{(c)})| & = & |s-\lim_{n\to\infty}F_{\mu}
(\omega_1(z_n^{(c)}))-\omega_1(z_n^{(c)})+z_n^{(c)}|\\
& = & |s-c+a-\lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}
F_\mu(z)|\\
& \le & |s-x+a-\lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}
F_\mu(z)|+|x-c|\\
& < & \frac{\varepsilon}{2}+\frac{\varepsilon}{2}\\
& = & \varepsilon.\end{aligned}$$ We conclude that $C(\omega_2,a)$ is dense in $\mathbb R$. This proves step 2. [**Step 3:**]{} We show that $\nu=\nu^s.$ This follows from Step 2 and the argument used in Step 1. [**Step 4:**]{} We show that $C(\omega_1,a)=\mathbb R\cup\{\infty\}.$ If $\nu$ is a convex combination of point masses and there exist two consecutive atoms of $\nu$ at $\alpha$ and $\beta$ ($\alpha<\beta$), then obviously $G_\nu$ extends analytically to the interval $(\alpha,\beta)$ and $G_\nu((\alpha,\beta))=\mathbb R$, so that $F_\nu((\alpha,\beta))=(\mathbb R\setminus\{0\})\cup\{\infty\}.$ For any $x\in(\alpha,\beta)$ let $z_n^{(x)}$ converge to $a$ so that $\omega_2(z_n^{(x)})\to x$ as $n$ tends to infinity. Then $$\begin{aligned}
\lim_{n\to\infty}\omega_1(z_n^{(x)}) & = & \lim_{n\to\infty}F_\nu(\omega_2(z_n^{(x)}))
-\omega_2(z_n^{(x)})+z_n^{(x)}\\
& = & F_\nu(x)-x+a,\end{aligned}$$ and thus $$C(\omega_1,a)\supseteq\overline{\{\lim_{n\to\infty}\omega_1(z_n^{(x)})\colon
x\in(\alpha,\beta)\}}=\overline{\{F_\nu(x)-x+a\colon x\in(\alpha,\beta)\}}=\mathbb R
\cup\{\infty\}.$$
If either $\nu$ is not purely atomic, or there exist no consecutive atoms of $\nu$, then there exists at least one point $x_0\in\mathbb R$ so that $F_\nu$ does not extend meromorphically through $x_0$. The argument used in the proof of Step 2, with $I$ an open interval containing $x_0$, assures us that $C(\omega_1,a)=
\mathbb R\cup\{\infty\}.$ This proves Step 4.
We have proved now that if there exists a point $a$ where either $C(\omega_1,a)$ or $C(\omega_2,a)$ is nondegenerate (i.e. contains more than one point), then $\mu=\mu^s$, $\nu=\nu^s$, at least one has total support, and $C(\omega_1,a)=
C(\omega_2,a)=\mathbb R\cup\{\infty\}.$ We assume without loss of generality that $\mu$ has support equal to the whole real line. Choose a point $c\in\mathbb R$ so that $\sphericalangle\lim_{z\to c}F_\mu(z)=0$ and there exists $\{z_n^{(c)}
\}_{n\in\mathbb N}$ converging to $a$ so that $\Re\omega_1(z_n^{(c)})=c$ and $\lim_{n\to\infty}\omega_1(z_n^{(c)})=c.$ By Lemmas \[sing\] and \[JC\] we have $$\begin{aligned}
\frac{1}{\mu(\{c\})}-1 & = & \lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}
\frac{F_\mu(z)}{z-c}-1\\
& = & \liminf_{z\to c}\frac{\Im F_\mu(z)}{\Im z}-1\\
& \le & \liminf_{n\to\infty}\frac{\Im F_\mu(\omega_1(z_n^{(c)}))}{\Im
\omega_1(z_n^{(c)})}+\frac{\Im z_n^{(c)}}{\Im\omega_1(z_n^{(c)})}-1\\
& = & \liminf_{n\to\infty}\frac{\Im \omega_2(z_n^{(c)})}{\Im\omega_1(z_n^{(c)})}\\
& = & \left(\limsup_{n\to\infty}
\frac{\Im \omega_1(z_n^{(c)})}{\Im\omega_2(z_n^{(c)})}\right)^{-1}\\
& = & \left(\limsup_{n\to\infty}\frac{\Im F_\nu(\omega_2(z_n^{(c)}))}{\Im
\omega_2(z_n^{(c)})}+\frac{\Im z_n^{(c)}}{\Im\omega_2(z_n^{(c)})}-1\right)^{-1}\\
& \le & \left(\liminf_{n\to\infty}\frac{\Im F_\nu(\omega_2(z_n^{(c)}))}{\Im
\omega_2(z_n^{(c)})}-1\right)^{-1}\\
& \le & \left(\liminf_{z\to a-c}\frac{\Im F_\nu(z)}{\Im z}-1\right)^{-1}.\end{aligned}$$ By our assumption, $(\mu(\{c\}))^{-1}-1\in(0,\infty]$, so that $$\liminf_{z\to a-c}\frac{\Im F_\nu(z)}{\Im z}<\infty.$$ Lemma \[JC\] implies that $F_\nu$ has a nontangential limit at $a-c$ belonging to $\mathbb R\cup\{\infty\}$. Denote it by $l.$ Moreover, we claim that $\mu(\{c\})>0.$ Indeed, assume this is not the case. Then the above chain of inequalities together with Lemma \[JC\] imply that $$\liminf_{z\to a-c}\frac{\Im F_\nu(z)}{\Im z}=1,$$ so that $h_\nu(z)=F_\nu(z)-z$ is constant, by Lemma \[JC\]. This contradicts the assumption that $\nu$ is not a point mass and proves our claim. We conclude that $\mu$ must be an infinite convex combination of point masses, densely distributed in $\mathbb R$.
Assume towards contradiction that $\omega_1$ has no nontangential limit at $a$. Then there exists an angle $\Gamma\subset\mathbb C^+$ with vertex at $a$ and bisected by the line $a+i\mathbb R_+$ with the property that $C_\Gamma(\omega_1,a)$ is infinite. By Lemma \[ClusterConex\] and the argument above, $C_\Gamma(\omega_1,a)$ must contain a nondegenerate open subinterval $J$ so that for any $c\in J$ there exists a sequence $\{z_n^{(c)}\}_{n\in\mathbb N}\subset\Gamma$ converging to $a$ such that $\Re \omega_1(z_n^{(c)})=c$ and $\Im\omega_1(z_n^{(c)})$ converges to zero as $n\to\infty$. Let $m=\max_{x\in\mathbb R}\nu(\{x\}).$ By hypothesis, $0\le m<1.$ Since the set of atoms of $\mu$ is dense in $\mathbb R$, in particular infinite, there exists $c\in J$ so that $0<\mu(\{c\})<1-m$. Observe that since $1>\mu(\{c\})>0,$ the non-constant function $F_\mu(z)-z$ maps any nontangential path ending at $c$ into a nontangential path. Indeed, since $$\lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}\frac{(F_\mu(z)-z)+c}{z-c}=
\frac{1}{\mu(\{c\})}-1\in(0,+\infty),$$ this follows by simply analyzing the real and the imaginary part of the left-hand term above. Thus, the sequence $\{
\omega_2(z_n^{(c)})\}_{n\in\mathbb N}=\{F_{\mu}(\omega_1(z_n^{(c)}))
-\omega_1(z_n^{(c)})+z_n^{(c)}\}_{n\in\mathbb N}$ must also converge nontangentially to $a-c.$ But we have seen above that $F_\nu$ has nontangential limit at $a-c$. Thus, $$0=\lim_{\stackrel{z\longrightarrow c}{{\sphericalangle}}}F_\mu(z)=
\lim_{n\to\infty}F_\mu(\omega_1(z_n^{(c)}))=\lim_{n\to\infty}F_\nu(
\omega_2(z_n^{(c)}))=\lim_{\stackrel{z\longrightarrow a-c}{{\sphericalangle}}}F_\nu(z),$$ and $$\begin{aligned}
\frac{1}{\mu(\{c\})}-1 & \leq & \left(\liminf_{z\to a-c}
\frac{\Im F_\nu(z)}{\Im z}-1\right)^{-1}\\
& = & \left(\lim_{\stackrel{z\longrightarrow a-c}{{\sphericalangle}}}
\frac{F_\nu(z)}{z-(a-c)}-1\right)^{-1}\\
& = & \left(\frac{1}{\nu(\{a-c\})}-1\right)^{-1}.\end{aligned}$$ Multiplication with $\frac{1}{\nu(\{a-c\})}-1$ in the above inequality gives $$(1-\mu(\{c\}))(1-\nu(\{a-c\}))\leq\mu(\{c\})\nu(\{a-c\}),$$ so $$\mu(\{c\})+\nu(\{a-c\})\ge1.$$ This contradicts the choice $\mu(\{c\})<1-m$ and concludes the proof of part (2) of the theorem.
The main result
===============
We can now prove the main result of this paper. For the sake of completeness, we state in the theorem below the result of Bercovici and Voiculescu describing the atoms of the free additive convolution of two probability distributions.
\[thm4.1\] Let $\mu,\nu$ be two Borel probability measures on $\mathbb R$, neither of them a point mass. Then
1. The point $a\in\mathbb R$ is an atom of the measure $\mu
\boxplus\nu$ if and only if there exist $b,c\in\mathbb R$ such that $a=b+c$ and $\mu(\{b\})+\nu(\{c\})>1.$ Moreover, $(\mu\boxplus\nu)(\{a\})=
\mu(\{b\})+\nu(\{c\})-1.$
2. The absolutely continuous part of $\mu\boxplus\nu$ is always nonzero, and its density is analytic wherever positive and finite. More precisely, there exists an open set $U\subseteq\mathbb R$ so that the density function $f(x)=\frac{d(\mu\boxplus\nu)^{ac}(x)}{dx}$ with respect to the Lebesgue measure in the real line is analytic on $U$ and $(\mu\boxplus\nu)^{ac}(\mathbb R)=\int_{U}f(x)dx$.
3. The singular continuous part of $\mu\boxplus\nu$ is zero.
Part (1) of the theorem is due to Bercovici and Voiculescu (see [@BercoviciVoiculescuRegQ], Theorem 7.4). We shall proceed with the proof of part (2). Suppose that $\mu\boxplus\nu$ is purely singular, and thus for almost all $x\in\mathbb R$ with respect to the Lebesgue measure, we have $$\lim_{y\to0}\Im F_{\mu\boxplus\nu}(x+iy)=\lim_{y\to0}\Im
G_{\mu\boxplus\nu}(x+iy)=0.$$ By part (1), we are assured that $\mu\boxplus\nu$ cannot be purely atomic, so we must have $(\mu\boxplus\nu)^{sc}\neq0,$ and hence, by Lemma \[sing\], $\lim_{y\to0} F_{\mu\boxplus\nu}(x+iy)=0$ for uncountably many $x\in\mathbb R.$ Theorem \[Seidel\] applied to the function $F_{\mu\boxplus\nu}$ yields a point $x_0\in\mathbb R$ such that $C(F_{\mu\boxplus\nu},x_0)=
\overline{\mathbb C^+}.$ Using relations (2) and (3) from Theorem \[sub\] we conclude that at least one of $C(\omega_1,x_0),C(\omega_2,x_0)$, will intersect the upper half-plane. But now we can apply Theorem \[Boundary+\] (1) to obtain a contradiction. Thus, $\mu\boxplus\nu$ cannot be purely singular[^1].
Next we prove that there exists an open subset $U$ of $\mathbb R$ on which the density $f(x)=\frac{d(\mu\boxplus\nu)^{ac}(x)}{dx}$ is analytic. By Theorem \[Fatou\], there exists a subset $E$ of $\mathbb R$ of zero Lebesgue measure such that for all $x\in\mathbb R\setminus E$ the limits $\lim_{y\to0}F_{\mu\boxplus\nu}(x+iy)$, $\lim_{y\to0}\omega_j(x+iy),$ $j\in\{1,2\}$ exist and are finite. Also, by Lemma \[sing\], (3), for almost all $x\in{\rm supp}((\mu\boxplus\nu)^{ac})\setminus E$, with respect to $(\mu\boxplus\nu)^{ac},$ we have $\lim_{y\to0}F_{\mu\boxplus\nu}(x+iy)\in
\mathbb C^+.$ By relation (3) in Theorem \[sub\], at least one of $\lim_{y\to0}\omega_j(x+iy),$ $j\in\{1,2\},$ must also be in $\mathbb C^+$. For definiteness, assume $\omega_1(x)=\lim_{y\to0}\omega_1(x+iy)\in\mathbb C^+.$ Part (1) of Theorem \[Boundary+\] assures us that $\omega_1,\omega_2$ and $F_{\mu\boxplus\nu}$ extends analytically through $x$. We conclude by part (3) of Lemma \[sing\] that the density of $(\mu\boxplus\nu)^{ac}$ must be analytic in $x$. Of course, the set $U$ of points $x$ where $f(x)=\frac{d(\mu\boxplus\nu)^{ac}(x)}{dx}$ is analytic is open in $\mathbb R$, and by Lemma \[sing\] (3) we conclude that $\int_Uf(x)dx=(\mu\boxplus\nu)^{ac}(\mathbb R)$.
To prove (3), assume that $(\mu\boxplus\nu)^{sc}\neq0;$ Lemma \[sing\] provides an uncountable set $H$ of points $c\in\mathbb R$ such that $(\mu\boxplus\nu)(\{c\})=0$ and $\sphericalangle\lim_{z\to c}
F_{\mu\boxplus\nu}(z)=0.$ Theorem \[Boundary+\] (2) assures us that $\omega_1$ and $\omega_2$ have nontangential limits at all points of $\mathbb R$, in particular at each such point $c\in H$. We claim that
1. $\sphericalangle\lim_{z\to c}\omega_j(z)\in\mathbb R$, $j=1,2$. Denote those limits by $v_j$, $j=1,2$;
2. The following equalities hold: $$\lim_{\stackrel{ z\longrightarrow v_1}{{\sphericalangle}}}F_\mu(z)=0
\quad {\rm and}\quad
\lim_{\stackrel{ z\longrightarrow v_2}{{\sphericalangle}}}F_\nu(z)=0.$$
3. $\mu(\{v_1\})+\nu(\{v_2\})=1.$
Parts (2) and (1) of Theorem \[Boundary+\] guarantee the existence of the limits in (i) and the fact that they do not belong to $\mathbb C^+.$ If, say, $\sphericalangle\lim_{z\to c}\omega_1(z)=\infty$, then Theorem \[Lindelof\] assures us that $$\begin{aligned}
0 & = & \lim_{y\downarrow0}F_{\mu\boxplus\nu}(c+iy)\\
& = & \lim_{y\downarrow0}F_\mu
(\omega_1(c+iy))\\
& = & \lim_{s\to\infty,s\in\omega_1(c+i\mathbb R)}F_{\mu}(s)\\
& = & \lim_{\stackrel{ z\longrightarrow \infty}{{\sphericalangle}}}F_\mu(z)\\
& = & \infty,\end{aligned}$$ which is of course a contradiction. Thus (i) holds. To prove (ii) just observe that, by the same Theorem \[Lindelof\] $$\begin{aligned}
0 & = & \lim_{\stackrel{ z\longrightarrow c}{{\sphericalangle}}}F_{\mu\boxplus\nu}(z)\\
& = &\lim_{\stackrel{ z\longrightarrow c}{{\sphericalangle}}}F_\nu(\omega_2(z))\\
& = & \lim_{z\to v_2,z\in\omega_2(c+i\mathbb R)}F_\nu(z)\\
& = & \lim_{\stackrel{ z\longrightarrow v_2}{{\sphericalangle}}}F_\nu(z).\end{aligned}$$ The same argumend provides the proof for $F_\mu$.
We now prove (iii). Recall that by Lemmas \[sing\] and \[JC\], for any $c$ as above we have $$\begin{aligned}
\frac{1}{\mu(\{c\})} & = & \lim_{\stackrel{z\longrightarrow c}{\sphericalangle}}\left[(z-c)
\int_\mathbb R\frac{d\mu(t)}{z-t}\right]^{-1}\\
& = & \lim_{\stackrel{z\longrightarrow c}{\sphericalangle}}\frac{F_\mu(z)}{z-c}\\
& = & \liminf_{z\to c}\frac{\Im F_\mu(z)}{\Im z}.\end{aligned}$$ Thus the following chain of inequalities holds: $$\begin{aligned}
\frac{1}{\nu(\{v_2\})}-1 & = & \liminf_{z\to v_2}\frac{\Im F_\nu(z)}{\Im z}-1\\
& \leq & \liminf_{y\downarrow0}\frac{\Im
F_\nu(\omega_2(c+iy))}{\Im\omega_2(c+iy)}-1\\
& \leq & \limsup_{y\downarrow0}\left(\frac{\Im F_\nu(\omega_2(c+iy))}{\Im
\omega_2(c+iy)}+
\frac{y}{\Im \omega_2(c+iy)}-1\right)\\
& = & \limsup_{y\downarrow0}\frac{\Im \omega_1(c+iy)}{\Im \omega_2(c+iy)}\\
& = & \left(\liminf_{y\downarrow0}\frac{\Im\omega_2(c+iy)}{\Im \omega_1(c+iy)}
\right)^{-1}\\
& = & \left(\liminf_{y\downarrow0}\left(\frac{\Im F_\mu(\omega_1(c+iy))}{\Im
\omega_1(c+iy)}+\frac{y}{\Im \omega_1(c+iy)}\right)-1\right)^{-1}\\
& \leq & \left(\liminf_{y\downarrow0}\frac{\Im F_\mu(\omega_1(c+iy))}{\Im \omega_1(c+
iy)}+
\liminf_{y\downarrow0}\frac{y}{\Im \omega_1(c+iy)}-1\right)^{-1}\\
& \leq & \left(\liminf_{y\downarrow0}\frac{\Im F_\mu(\omega_1(c+iy))}{\Im \omega_1(c
+iy)}-1\right)^{-1}\\
& \leq & \left(\liminf_{z\to v_1}\frac{\Im F_\mu(z)}{\Im z}-1\right)^{-1}\\
& = & \left(\frac{1}{\mu(\{v_1\})}-1\right)^{-1}.\end{aligned}$$ We have assumed that $\mu$ and $\nu$ are not point masses, so the above implies that $1<\frac{1}{\mu(\{v_1\})},\frac{1}{\nu(\{v_2\})}<
\infty.$ Thus, multiplying the above inequality by $\frac{1}{\mu(\{v_1\})}-1$ will give $$(1-\mu(\{v_1\}))(1-\nu(\{v_2\}))\leq\mu(\{v_1\})\nu(
\{v_2\}),$$ or, equivalently, $$\mu(\{v_1\})+\nu(\{v_2\})\geq1.$$ Using relation (3) from Theorem \[sub\] and the fact that $F_{\mu\boxplus\nu}(c)=0$, we obtain $$v_1+v_2=c\quad {\rm for\ all}\ c\in H.$$ Since $c$ has been chosen so that $(\mu\boxplus\nu)(\{c\})=0$, part (1) of the theorem tells us that the inequality above must be an equality. This proves the last point of our claim.
Now, since any probability can have at most countably many atoms, this, together with part ([iii]{}) of our claim contradicts the fact that $H$ is uncountable and concludes the proof.
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MSC: primary, 46L54; secondary, 30D40.
Address:
Department of Pure Mathematics, University of Waterloo, 200 University Street West, Waterloo, ON, N2L 1V9, Canada,\
and\
Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania.\
Tel.: +1-(519)-888-4567 Ext. 32803\
E-mail: [[email protected]]{}\
*Present address:Department of Pure Mathematics, University of Waterloo, 200 University Street West, Waterloo, ON, N2L 1V9, Canada\
*
[^1]: This is a slightly modified version of part of the proof of Corollary 2.5 in [@AIHP]. Of course, the result presented here is much stronger.
|
---
abstract: '**When electron-hole pairs are excited in a semiconductor, it is a priori not clear if they form a fermionic plasma of unbound particles or a bosonic exciton gas. Usually, the exciton phase is associated with low temperatures. In atomically thin transition metal dichalcogenide semiconductors, excitons are particularly important even at room temperature due to strong Coulomb interaction and a large exciton density of states. Using state-of-the-art many-body theory including dynamical screening, we show that the exciton-to-plasma ratio can be efficiently tuned by dielectric substrate screening as well as charge carrier doping. Moreover, we predict a Mott transition from the exciton-dominated regime to a fully ionized electron-hole plasma at excitation densities between $3\times10^{12}$ cm$^{-2}$ and $1\times10^{13}$ cm$^{-2}$ depending on temperature, carrier doping and dielectric environment. We propose the observation of these effects by studying excitonic satellites in photoemission spectroscopy and scanning tunneling microscopy.**'
author:
- 'A. Steinhoff'
- 'M. Florian'
- 'M. Rösner'
- 'G. Schönhoff'
- 'T.O. Wehling'
- 'F. Jahnke'
title: 'Excitons versus electron-hole plasma in monolayer transition metal dichalcogenide semiconductors'
---
Excitons play a prominent role in the optical properties of atomically thin transition metal dichalcogenide (TMDC) semiconductors due to electron-hole Coulomb interaction being strongly enhanced by carrier confinement and reduced dielectric screening. This suggests an interpretation of experimental results as well as theoretical prediction in terms of excitons rather than unbound electrons and holes. [@zhang_absorption_2014; @berghauser_analytical_2014; @sun_observation_2014; @schmidt_ultrafast_2016; @selig_excitonic_2016] On the other hand, it is well-known that at a certain excitation density of electron-hole pairs the Mott transition is observed. [@shah_investigation_1977; @manzke_mott_2012; @chernikov_electrical_2015] Here a phase where excitons and unbound carriers can coexist evolves into a fully ionized electron-hole plasma, which shows the significance of plasma effects to the physics of excited semiconductors. Since excitons are more or less neutral compound particles, many-particle renormalization and screening effects in an exciton gas are very different from those in a plasma of unbound electrons and holes, which we refer to in the following as quasi-free carriers. For this reason it is highly desirable to quantify the relative importance of excitonic and plasma effects over a wide range of electron-hole excitation densities and for various material and externally controllable parameters. It has already been suggested to tune exciton binding energies by electrical doping [@chernikov_electrical_2015] and some effort has been devoted to study the influence of dielectric screening on excitons in TMDC semiconductors. [@lin_dielectric_2014; @ugeda_giant_2014; @latini_excitons_2015; @trolle_model_2017]
In the past, a very powerful scheme has been developed to theoretically describe the so-called ionization equilibrium between the formation of bound particles and their dissociation into quasi-free particles [@kremp_equation_1984; @kremp_ladder_1984; @zimmermann_mass_1985; @kremp_quantum_1993; @kremp_quantum_2005; @semkat_ionization_2009], with applications to atomic plasmas and highly excited semiconductors. In general, a dominance of free-carrier plasma over excitons is expected at very low excitation density due to entropy ionization of excitons [@mock_entropy_1978] and above the so-called Mott density due to pressure ionization of excitons. The fraction of excitons reaches a maximum at elevated densities still below the Mott transition and can assume values of practically $100\%$ in TMDCs, though being sensitive to various externally controllable parameters. The scheme relies on the assumption of a quasi-equilibrium between plasma and excitons being established before electron-hole recombination sets in. Ultrafast equilibration is facilitated by efficient carrier-carrier [@steinhoff_nonequilibrium_2016] and carrier-phonon interaction [@selig_excitonic_2016] as well as exciton formation [@ceballos_exciton_2016; @steinleitner_direct_2017] after optical excitation, see Ref. for a review. Experimental verification of the ionization equilibrium has been achieved in GaAs quantum wells using THz spectroscopy to probe transitions between 1s- and 2p-exciton states. [@kaindl_ultrafast_2003; @koch_semiconductor_2006] A similar technique in the mid-infrared range has been applied recently to monolayer WSe$_2$. [@steinleitner_direct_2017] Alternatively, the fractions of excitons and plasma can be determined from their contributions to photoluminescence (PL) spectra [@chatterjee_excitonic_2004] in combination with additional PL simulations. As we suggest below, other ways to quantify the degree of exciton formation are angular-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). In this paper, we combine calculations of material-realistic band structures and Coulomb matrix elements of the monolayer TMDC materials MX$_2$ (M$=$W,Mo and X$=$S,Se) with the state-of-the-art theory of ionization equilibrium, which we briefly introduce in the following section before discussing the results in detail. The theory is based on GW- and T-matrix self-energies describing the excited carriers and the effect of frequency-dependent screening. We also include excitonic screening that we find to be relevant and that is usually not taken into account. On these grounds, we study the influence of experimental and device-relevant parameters like dielectric screening, temperature and carrier doping on the ionization equilibrium and the Mott density.
Spectral Functions and Exciton Satellites.
==========================================
To examine the equilibrium properties of excited carriers in TMDCs, we follow the approach developed in Refs. and reviewed in Ref. . We use the quantum-statistical expression for the carrier density $n_a$ of the species $a$, which can be electrons or holes, as a function of temperature $T$ and chemical potential $\mu_a$ as a starting point: $$n_a(\mu_a,T)=\frac{i\hbar}{\mathcal{A}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\sum_{{\mathbf{k}}\sigma}f^a(\omega)A_{{\mathbf{k}}\sigma}^a(\omega)
~.
\label{eq:carr_dens}$$ $f^a(\omega)$ denotes the Fermi distribution function depending on $\mu_a$ and $T$, $\mathcal{A}$ is the crystal area and $A_{{\mathbf{k}}\sigma}^a(\omega)=2i\textrm{Im}G^{\textrm{ret},a}_{{\mathbf{k}}\sigma}(\omega)$ is the spectral function of the single-particle state $\ket{{\mathbf{k}}\sigma a}$ related to the retarded single-particle Green’s function $$G^{\textrm{ret},a}_{{\mathbf{k}}\sigma}(\omega)=\frac{1}{\hbar\omega-\varepsilon_{{\mathbf{k}}\sigma}^a-\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(\omega)}
~.
\label{eq:G_ret}$$ The self-energy $\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(\omega)$ accounts for many-particle effects giving rise to renormalizations of the single-particle band structure $\varepsilon_{{\mathbf{k}}\sigma}^a$ as well as contributions of bound states. For a given self-energy, the inversion of Eq. (\[eq:carr\_dens\]) yields the chemical potential $\mu_a(n_a,T)$ for each species and therefore any thermodynamic property of the system in the grand canonical description.
![ **Spectral properties of excited carriers in monolayer WS$_2$.** **(a)** Band structure of freestanding WS$_2$ as obtained from a G$_0$W$_0$ calculation at zero excitation density. **(b)** Bound-state spectrum for WS$_2$ on SiO$_2$ substrate relative to the quasi-particle gap at zero excitation density over the modulus of total exciton momentum $Q$ as obtained from a Bethe-Salpeter equation, see Eq. (\[eq:bse\]) in the Methods section. Excitons involving the $\Gamma$, $K'$ and $\Sigma'$ valleys are included but not marked explicitely. **(c)** Electron spectral function in extended quasi-particle approximation at $T=300$ K and excitation density $n_a=3.2\times10^{12}$ cm$^{-2}$ showing resonances from bound and quasi-free particles in momentum-resolved representation on a logarithmic scale and as normalized local density of states (LDOS). Energies are measured relative to the quasi-particle band gap at zero excitation density. A phenomenological Gaussian broadening of $10$ meV (HWHM) is applied. **(d)** Hole spectral function in extended quasi-particle approximation. **(e)** Electron spectral function as in (c) for spin-down electrons at the $K$-point. The vertical line marks the electron chemical potential. **(f)** Hole spectral function as in (d) for spin-up holes at the $K$-point. []{data-label="fig:spectralf"}](./figure_1_new2.eps){width="1.\columnwidth"}
{width="\textwidth"}
As we describe in detail in the Methods section, by using a T-matrix self-energy in screened ladder approximation and assuming small quasi-particle damping, we obtain a spectral function $A^a(\omega)$ in the so-called extended quasi-particle approximation. It exhibits poles for quasi-free and bound carriers as shown in Fig. \[fig:spectralf\]. From the multiple valleys in the single-particle band structure of electrons and holes (Fig. \[fig:spectralf\](a)), a rich spectrum of bound states emerges (Fig. \[fig:spectralf\](b)), that contains a variety of dark excitons with large total momentum $Q$ besides the bright $K$-valley excitons commonly referred to as A and B. The dark excitons, though playing a minor role in optical experiments, are essential to the description of the ionization equilibrium. Various bound states are reflected in the low-energy satellites of the single-particle spectral function. This shows that excitonic contributions are expected to be observed in experiments that are sensitive to these spectral properties. In ARPES [@damascelli_angle-resolved_2003], momentum-resolved images of the electron spectral function comparable to Fig. \[fig:spectralf\](c) are obtained, weighted with Fermi distribution functions that are defined by the chemical potential $\mu_e$ and temperature $T$. For a fixed quasi-momentum state as shown in Fig. \[fig:spectralf\](e), this is typically referred to as energy distribution curve. On the other hand, STS [@fischer_scanning_2007] probes the local density of states (LDOS) and thus momentum-averaged spectral functions of electrons and holes, which are displayed in Figs. \[fig:spectralf\](c) and (d). We therefore propose to use these experimental techniques to spectrally distinguish between excitons and quasi-free carriers. This opens the possibility to quantify the degree of exciton ionization and verify the results we present in this paper.
Degree of Ionization and Mott Transition.
=========================================
According to Eq. (\[eq:carr\_dens\]), the spectral function can be used to separate the total electron and hole density ($a=e,h$), $$\begin{split}
n_a=n_{\textrm{free}}^a+n_X
\,,
\label{eq:density_final_short}
\end{split}$$ into contributions from quasi-free carriers and from carriers bound as excitons, where the excitons are approximately treated as bosons. Hence the properties of the excited semiconductor at a given temperature and excitation density are defined by the density of electrons $n_e$, the density of holes $n_h$ and the density of excitons $n_X$. A certain degree of ionization of the excited carriers $$\alpha_a=\frac{n_{\textrm{free}}^{a}}{n_a}
\label{eq:ionization}$$ will be established that is determined by the ionization equilibrium between electrons, holes and excitons. While for optical excitation, equal densities of electrons and holes are generated, we distinguish here between electron and hole ionization to also include the effect of carrier doping where electron and hole densities are different.
Using single-particle band structures and bound-state spectra, which are determined as discussed in the Methods section, we solve Eq. (\[eq:density\_final\_short\]) numerically to obtain the degree of ionization $\alpha_a$ in various TMDC materials under different experimental conditions. The results are collected in Fig. \[fig:ionization\] and exhibit the behaviour of ionization degree as a function of excitation density. There are different regimes of ionization to be observed. At high excitation densities between $3\times10^{12}$ cm$^{-2}$ and $1\times10^{13}$ cm$^{-2}$, depending on experimental parameters, efficient screening and many-particle renormalizations lead to a full ionization of excited carriers, which is known as Mott effect. At lower densities around $n_a=1\times10^{12}$ cm$^{-2}$, excitons dominate the physical properties of TMDCs for the parameters studied here due to the large exciton binding energies and density of states that is mostly given by dark excitons. Bright excitons with very small momenta that are optically active make up only a tiny fraction of the total exciton density. As Fig. \[fig:ionization\](b) shows, an efficient tuning knob for the degree of ionization is the dielectric screening by the environment, which can change over a wide range depending on the experimental situation or device realization in which the TMDC monolayer is the active material. The reason is the strong impact of dielectric screening on the exciton binding energies. Typical examples for substrates are Borofloat ($\varepsilon=2$), SiO$_2$ ($\varepsilon=3.9$) and sapphire ($\varepsilon=10$). The dielectric constant of the environment on top of the monolayer is often given by the vacuum value. On the other hand, in devices the TMDC monolayer is fully encapsulated by dielectric material. As an example we consider a full dielectric environment with $\varepsilon=10$, which might be either sapphire or additional layers of TMDC material in a vertical heterostructure whose main influence on the excitons is the dielectric screening. [@andersen_dielectric_2015] We find that the minimal degree of ionization can be tuned from below $0.1\%$ ($99.9\%$ excitons) at weak dielectric screening to about $30\%$ at strong screening, while the Mott density is lowered at the same time by roughly a factor of $3$. The second important parameter that is relevant to applications of TMDC monolayers is the doping with additional carriers which might be either intrinsic or induced by external electric fields in a capacitor structure. Here the fractions of ionized electrons and holes, $\alpha_e$ and $\alpha_h$, are discussed separately as the densities of the species are not equal anymore. We consider hole doping of WS$_2$, but similar results are expected in case of electron doping. Looking at Figs. \[fig:ionization\] (e) and (f), we find that even at weak doping the minority carriers are practically all bound as excitons below the Mott transition. On the other hand, at higher doping levels an increasing fraction of majority carriers exists as quasi-free plasma due to missing partners for exciton formation. As a function of minority-carrier density the Mott transition is lowered by roughly the density of doped excess carriers. From this we conclude that at doping levels above $10^{13}$ cm$^{-2}$ neither dark nor bright excitons will exist in any case. This is supported by the experimental estimate for doping-induced ionization at several $10^{13}$ cm$^{-2}$. [@chernikov_electrical_2015] Another crucial parameter is the temperature, see Fig. \[fig:ionization\](c), which can vary in experiments or devices due to heating of the active material under strong optical or electrical pumping. This effect has been taken into account to explain the observed exciton-to-plasma ratio in monolayer WSe$_2$ in Ref. . At room temperature and even at elevated temperatures up to $700$ K excitons clearly dominate below the Mott transition. At the same time, the Mott density slightly increases with temperature due to weaker renormalizations of the quasi-particle gap. It turns out that strain is no efficient tuning knob as both bright and dark excitons contribute to the ionization equilibrium, although bright excitons are preferred in moderately tensile-strained TMDCs, see the discussion in the Methods section. A comparison of different TMDC materials shows that excitons are slightly more important in molybdenum- than in tungsten-based TMDCs due to the larger binding energies, which leads to larger Mott densities.
{width="\textwidth"}
When approaching the Mott density from the low-density side, many-particle renormalizations become increasingly important, cf. Eq. (\[eq:GW\_energy\]) in the Methods section. Exchange interaction and efficient screening due to free carriers as well as excitons reduce the quasi-particle band gap and the exciton binding energies. More and more excitons are ionized, which leads to an increase of efficient free-carrier screening and thereby to a self-amplification of the ionization effect until all excitons are dissociated into an electron-hole plasma and the degree of ionization becomes $\alpha_a=1$. Note that $\alpha_a$ includes not only bright but also dark excitons with large total momentum for example between $K$ and $\Sigma$ valleys. They may have larger binding energies and be slightly more stable against ionization than bright excitons visible in an optical experiment. Fig. \[fig:spectralf\_density\] shows an illustration of the Mott effect in terms of the spectral functions in extended quasi-particle approximation, which contain both exciton and quasi-free-particle signatures. At low excitation densities, the only spectral contribution stems from quasi-free carriers at the band edge. With increasing density, the quasi-particle peak is shifted to lower energies due to many-particle renormalizations. At the same time, spectral weight is transferred from the quasi-particle to the bound-state peaks as exciton populations increase, see the explicit expression of the spectral function in Eq. (\[eq:spectral\_fct\_extended\_T\_matrix\]). The appearence of several exciton satellites in the hole spectral function is due to different bound states involving electrons either in the $K$- or $\Sigma$-valleys, see Fig. \[fig:spectralf\] (b). The spectral position of a bound-state peak in the spectral function of carrier $a$ is given by the difference of the corresponding exciton energy $E^{ab}$ and the energy of the second carrier $b$ involved in the bound state. The bound resonance might therefore be interpreted as an effective ionization energy of the actual carrier $a$ with respect to its energy in the quasi-particle band structure. This underlines the fact that excitonic signatures are observable in experiments that are sensitive to the single-particle spectral function, such as ARPES and STS. Note that while the amplitudes of bound-state resonances in the spectral functions are relatively small, observables like the carrier density (\[eq:carr\_dens\]) and the photoemission intensity involve weighting with a Fermi function that strongly favors the low-energy resonances over the quasi-free contribution. With increasing excitation density, quasi-particle and excitonic resonances approach until above the Mott density all excitons are ionized and only a quasi-particle peak remains. Fig. \[fig:E\_Gap\] shows the reduction of the quasi-particle gap until the Mott transition appears around $n_a=8\times10^{12}$ cm$^{-2}$.
![**Ionization equilibrium and quasi-particle band-gap renormalization.** Shown is the band-gap renormalization $\Delta E_{\textrm{Gap}}$ for spin-up carriers at the $K$-point in WS$_2$ on SiO$_2$ substrate at $T=300$ K as function of excitation density. The vertical black line marks the Mott transition. The inset shows a schematic of the ionization equilibrium introducing the quasi-particle band gap at zero excitation density $E^0_{\textrm{Gap}}$, which is conventionally set to zero energy. All quantities including the bound-state energies $E_{\nu}$ and the sum of electron and hole chemical potentials are measured relative to $E^0_{\textrm{Gap}}$.[]{data-label="fig:E_Gap"}](./figure_5_ink.eps){width="1.\columnwidth"}
An alternative picture of the interacting electrons and holes, that is consistent with the extended quasi-particle approximation, is the so-called *chemical picture* in which excitons are considered as a new particle species besides electrons and holes. [@kremp_quantum_2005; @semkat_ionization_2009] They are characterized by a chemical potential $$\mu_{X,\nu}=\mu_e+\mu_h-E_{\nu}\,,
\label{eq:gen_MAL}$$ with bound-state energies $E_{\nu}$ that are given by the relative motion of electron and hole, and an ideal Bose distribution function. In the chemical picture, solving Eq. (\[eq:density\_final\_short\]) corresponds to an adaption of the chemical potentials of the different particle species, namely electrons, holes and excitons, as in a chemical reaction. These considerations are consistent with the theory based on spectral functions, that we use to obtain all numerical results presented in this paper. Only for the purpose of illustration, we simplify the theory considering the nondegenerate case ($f^a(E^a_{{\mathbf{k}}\sigma})\ll 1$) and a single band-structure valley for electrons and holes each. Then a Saha equation can be formulated that determines the degree of ionization: $$\frac{n_X}{n_e n_h}\propto e^{\beta I_{\nu}^{\textrm{eff}}} \,.
\label{eq:saha}$$ In analogy to the usual mass action law, $I_{\nu}^{\textrm{eff}}=\Delta E_{\textrm{Gap}}-E_{\nu}$ can be interpreted as an effective ionization potential of excitons that corresponds to the exciton binding energy, see also the inset in Fig. \[fig:E\_Gap\]. It is obvious from Saha’s equation, that a large exciton binding energy and low temperature favor the formation of excitons versus the dissociation into an unbound electron-hole plasma. The ionization potential depends on excitation density as a consequence of the excitation-induced lowering of the band continuum edge given by $\Delta E_{\textrm{Gap}}$ and the shift of the bound-state energy $E_{\nu}$. The bound-state shift on the other hand is a net result of band-gap shrinkage, screening of exciton binding energy and Pauli blocking [@steinhoff_influence_2014] and is much weaker than the band-gap shift due to compensation effects. In the end, the ionization potential is lowered with increasing excitation density until at $I_{\nu}^{\textrm{eff}}=0$ the bound state vanishes and merges with the continuum edge, which is the Mott effect.
A striking observation in Fig. \[fig:ionization\] is the degree of ionization approaching unity at low excitation densities, which is somewhat counter-intuitive but can be understood from a thermodynamical point of view. The potential that is minimized by the many-particle system is the free energy $F=U-TS$. At low densities and fixed temperature, the entropy $S$ gained by a dissociation of an exciton into two separate particles overcompensates the reduction of internal energy $U$ by the exciton binding energy $E_B$. Hence the so-called entropy ionization already discussed by Mock et al. [@mock_entropy_1978] is connected to the huge phase space available for quasi-free carriers in the low-density limit. We may clarify this using the entropy of an ideal gas with $N$ particles in a volume $V$ as given by the Sackur-Tetrode equation: $$\begin{split}
S=N k_B\left[\textrm{ln}\left(\frac{V}{N} c(T)\right) +\frac{5}{2}\right]
\,,
\label{eq:entropy_ideal_gas}
\end{split}$$ where $c(T)$ is a temperature-dependent parameter. Obviously, the dissociation of an exciton gas ($N$ particles) into a free electron-hole plasma ($2N$ particles) yields the entropy $\Delta S=N k_B \textrm{ln}\left(n^{-1}\right)$ with $n=N/V$ up to some additive constant. It follows that the critical density $n_{\textrm{crit}}$ below which the free energy is dominated by entropy essentially scales as $\textrm{exp}\left(-E_B/k_B T\right) $ with temperature. Although the extended quasi-particle approximation and the chemical picture are very descriptive, we have to be aware of their limitations. The approach relies on the assumption of a quasi-equilibrium of both types of carriers, which tends to overestimate the fraction of bound carriers as exciton formation takes a certain time after excitation of electron-hole pairs. Nevertheless, we expect the approach to give a reasonable quantitative description of the ionization equilibrium, particularly of the trends that can be expected under variation of external experimental parameters. A rather fundamental discussion is concerned with the Mott transition as a first-order phase transition between an exciton gas and a fully ionized electron-hole plasma. [@kremp_quantum_2005; @semkat_ionization_2009] The phase transition would be connected to an instability of thermodynamic functions that manifests itself in an ambiguity of $\alpha_a$ in a certain region below the Mott density. Due to excitation-induced broadening of the two-particle states, which is assumed small in our approach, and the shrinkage of the ionization potential towards the Mott transition, quasi-free and bound carriers cannot really be separated in this density regime. We avoid this regime as a more sophisticated theory including full spectral functions and exciton-exciton interaction would be required. Also, screening in a correlated many-particle system near the Mott transition is an intricate problem [@kraeft_quantum_1986]. However, we observe that taking into account excitonic screening is necessary to obtain meaningful results around the minima of $\alpha_a$ in the exciton-dominated regime. Otherwise, coming from the low-density side of the ionization curves in Fig. \[fig:ionization\], there would be no mechanism to efficiently break up the excitonic binding and trigger the transition to an ionized plasma. We discuss the contribution of excitons to screening in the following section.
Another prominent feature of TMDC semiconductors is the formation of trions, which could in principle be included as additional particle species in the spirit of the chemical picture. [@kremp_quantum_1993] In practice, obtaining bound-trion spectra on the same footing as excitons is a very challenging task of its own that is beyond the scope of this paper. Due to the relatively small trion binding energies, we assume that their influence on our results are negligible.
Excitonic Screening.
====================
In the spirit of the extended quasi-particle approximation to the spectral function, there are two types of contributions to excited-carrier screening of the Coulomb interaction, the metal-like free-carrier screening and dipolar screening due to bound excitons.
![**Characterization of excited-carrier screening.** Shown is the plasmon spectral function given by the imaginary part of the inverse dielectric function including the 2-d-Coulomb singularity $\varepsilon^{-1,\textrm{ret}}_{{\mathbf{q}}}(\omega)/|{\mathbf{q}}|$ for momenta along the contour $\Gamma$-$K$ in WS$_2$ on SiO$_2$ substrate at $T=300$ K. A quasi-particle broadening of $10$ meV is used. The carrier densities are **(a)** $n_{\textrm{free}}=2.7\times10^{10}$ cm$^{-2}$, $n_X=3.1\times10^{12}$ cm$^{-2}$ and **(b)** $n_{\textrm{free}}=1.8\times10^{13}$ cm$^{-2}$, $n_X=0$ corresponding to points on the ionization equilibrium curve in Fig. \[fig:ionization\] in the exciton-dominated and the plasma-dominated regime, respectively.[]{data-label="fig:plasmon"}](./figure_2.eps){width="\columnwidth"}
The screening can be characterized by the plasmon spectral function, see Eq. (\[eq:W\_spec\]), that contains excitations in the interacting electron-hole plasma as poles in the $q$-$\omega$-plane, see Fig. \[fig:plasmon\]. In the exciton-dominated regime shown in Fig. \[fig:plasmon\](a), besides the usual 2-d free-carrier contribution at small energies and small momenta a broad resonance above $150$ meV appears. It stems from transitions between 1s- and 2s-like exciton states, see Fig. \[fig:spectralf\](b), and also from comparable transitions between exciton states with large momenta. There are contributions at smaller energies as well that can not be as easily distinguished from free-carrier screening. At large densities beyond the Mott transition the plasmon spectral function shows a pronounced peak structure with a square-root-like behaviour at small momenta which has been discussed for TMDCs in Ref. and which is typical for a two-dimensional electron gas. [@haug_quantum_1993] Although excitons are expected to be much less polarizable than a free electron-hole plasma and hence contribute less to screening, at elevated excitation densities with a large fraction of carriers bound as excitons, their contribution can be essential. From the many-particle renormalization caused by free-carrier and excitonic screening, we deduce that in monolayer TMDC semiconductors excitonic screening is less efficient by two to three orders of magnitude for comparable excitation densities. Nevertheless, in a regime where more than $99\%$ of carriers are bound as excitons, excitonic screening still yields a significant contribution. As the plasmon spectral function is directly observable by electron energy loss spectroscopy (EELS) [@kogar_exciton_2016], we suggest to use this technique to explore exciton signatures in the dielectric function eperimentally.
Conclusion.
===========
The exciton ionization equilibrium in monolayer TMDC semiconductors has been studied for various material as well as experimentally and device-relevant external parameters on the basis of an ab initio description of the electronic band structure and Coulomb interaction. We observe entropy ionization of excitons at low excitation densities and a Mott transition to a fully ionized plasma at high densities between $3\times10^{12}$ cm$^{-2}$ and $1\times10^{13}$ cm$^{-2}$ depending on experimental parameters. Below the Mott transition, excitons become dominant in all cases with maximal fractions of excitons between $70\%$ and more than $99.9\%$. The most efficient tuning knobs are dielectric screening of the Coulomb interaction via the choice of dielectric environment and carrier doping that can induce complete ionization above a level of $10^{13}$ cm$^{-2}$. We suggest that fingerprints of excitonic contributions can be observed in ARPES and STS experiments, which are sensitive to the single-particle spectral functions. Moreover, we find that excitonic screening, although two to three orders of magnitude less efficient than free-carrier screening at comparable excitation densities, plays an important role in the description of ionization equilibrium. Exciton signatures in the dielectric function suggest EELS as another way to study the ionization equilibrium in excited semiconductors.
Methods.
========
Theory of Ionization Equilibrium.
---------------------------------
We start from the general expression for the carrier density (\[eq:carr\_dens\]) and the spectral function $$A_{{\mathbf{k}}\sigma}^a(\omega)=2i\textrm{Im}\,\frac{1}{\hbar\omega-\varepsilon_{{\mathbf{k}}\sigma}^a-\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(\omega)}
~.
\label{eq:G_spec}$$ In the limit of small quasi-particle damping $\textrm{Im}\,\Sigma^{\textrm{ret},a}\ll\textrm{Re}\,\Sigma^{\textrm{ret},a}$, the spectral function can be expanded in linear order of $\textrm{Im}\,\Sigma^{\textrm{ret},a}$ yielding the carrier density in so-called extended quasi-particle approximation $$\begin{split}
&n_a(\mu_a,T)\\ & =\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}\sigma}f^a(E^a_{{\mathbf{k}}\sigma}) \\ & -\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}\sigma}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}
\frac{2}{\hbar}\textrm{Im}\,\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(\omega)\left[f^a(E^a_{{\mathbf{k}}\sigma})-f^a(\omega)\right]\\
&\frac{d}{d\omega}\frac{\mathcal{P}}{\omega-E^a_{{\mathbf{k}}\sigma}/\hbar}\\
&=n_a^{\textrm{QP}}+n_a^{\textrm{corr}}
~,
\label{eq:n_ext_qp_approx}
\end{split}$$ where the quasi-particle energy $E^a_{{\mathbf{k}}\sigma}$ is given by $E^a_{{\mathbf{k}}\sigma}=\varepsilon_{{\mathbf{k}}\sigma}^a+\textrm{Re}\,\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(E^a_{{\mathbf{k}}\sigma})$ and $\mathcal{P}$ denotes the Cauchy principal value. [@kremp_equation_1984; @semkat_ionization_2009] The total density is divided into contributions from quasi-free particles and correlated particles, the latter being either in bound or scattering many-particle states.
The spectral function in extended quasi-particle approximation corresponding to this separation into free and correlated carriers is given by $$\begin{split}
A_{{\mathbf{k}}\sigma}^a(\omega) = &-2\pi i\delta(\hbar\omega-E^a_{{\mathbf{k}}\sigma})(1-Z_{{\mathbf{k}}\sigma}^a) \\
&- 2\pi i \Gamma_{{\mathbf{k}}\sigma}^a(\omega)
~,
\label{eq:spectral_fct_extended}
\end{split}$$ with $ \Gamma_{{\mathbf{k}}\sigma}^a(\omega) = \textrm{Im}\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(\omega)\frac{1}{\pi}\frac{d}{d\hbar\omega}\frac{\mathcal{P}}{\hbar\omega-E^a_{{\mathbf{k}}\sigma}} $ and the renormalization factor $Z_{{\mathbf{k}}\sigma}^a = \int d\hbar\omega \Gamma_{{\mathbf{k}}\sigma}^a(\omega)$. The first term describes quasi-free particles at renormalized energies. Their spectral weight is reduced according to the renormalization factor to account for correlated carriers, which are spectrally described by the second term. To evaluate the expressions (\[eq:n\_ext\_qp\_approx\]) and (\[eq:spectral\_fct\_extended\]), we have to choose an approximation for the self-energy $\Sigma^{\textrm{ret},a}(\omega)$. The real and imaginary parts of $\Sigma$ determine the quasi-particle energies and the correlated part of the carrier density, respectively. An appropriate choice is the screened ladder approximation [@stolz_correlated_1979; @kremp_equation_1984; @kremp_quantum_1993] $\Sigma(\omega)=\Sigma^{\textrm{H}}+\Sigma^{\textrm{GW}}(\omega)+\Sigma^{\textrm{T}}(\omega)$ that takes into account screening of Coulomb interaction due to excited carriers as well as the formation of bound two-particle states and consists of Hartree, GW and T-matrix contributions. We assume that renormalizations due to the Hartree self-energy are small compared to exchange and correlation effects. In the T-matrix contribution, we neglect exchange terms and assume static screening so that the T-matrix depends only on one instead of three frequency arguments. Thus we obtain for the imaginary part of the self-energy using the generalized Kadanoff-Baym ansatz [@kremp_quantum_2005]: $$\begin{split}
&\textrm{Im}\,\Sigma_{{\mathbf{k}}\sigma}^{\textrm{ret},a}(\omega) \\
&=\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}b}V^{ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}\textrm{Im}\,\varepsilon^{-1,\textrm{ret}}_{{\mathbf{k}}-{\mathbf{k}}'}(\omega-E^b_{{\mathbf{k}}'\sigma}/\hbar)\\
&\left[f^b(E^b_{{\mathbf{k}}'\sigma})-n^B(E^b_{{\mathbf{k}}'\sigma}-\hbar\omega)\right]\\
&+\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}b\sigma'}\textrm{Im}\,T''^{,\textrm{ret},ab}_{{\mathbf{k}}{\mathbf{k}}'\sigma\sigma'}(\omega+E^b_{{\mathbf{k}}'\sigma'}/\hbar)\\
&\left[f^b(E^b_{{\mathbf{k}}'\sigma'})+n^B_{ab}(\hbar\omega+E^b_{{\mathbf{k}}'\sigma'})\right]
~.
\label{eq:im_self_en_screened_ladder}
\end{split}$$ Here we applied thermal equilibrium relations for the screened Coulomb interaction [@kremp_quantum_2005]: $$\begin{split}
& V^{\textrm{S},<,ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(\omega)=n^B(\omega)V^{ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}2i\textrm{Im}\,\varepsilon^{-1,\textrm{ret}}_{{\mathbf{k}}-{\mathbf{k}}'}(\omega), \\
& V^{\textrm{S},>,ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(\omega)=(1+n^B(\omega))V^{ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}2i\textrm{Im}\,\varepsilon^{-1,\textrm{ret}}_{{\mathbf{k}}-{\mathbf{k}}'}(\omega)\,.
\label{eq:thermal_plasmon_prop}
\end{split}$$ $\varepsilon^{-1,\textrm{ret}}_{{\mathbf{q}}}(\omega)$ is the longitudinal dielectric function describing screening due to excited carriers and $n^B(\omega)$ is the Bose distribution function of the elementary plasma excitations called plasmons. $V^{ab}$ denotes Coulomb matrix elements between species $a$ and $b$ which contain dielectric screening due to carriers in the ground state and due to the environment but no screening due to excited carriers. $T''$ denotes the T-matrix with the two lowest-order terms subtracted from the ladder expansion and is discussed in the following subsection.
T-matrix and Bound Carriers.
----------------------------
The T-matrix in statically screened ladder approximation describing bound and scattering two-particle states between carrier species $a$ and $b$ obeys a Lippmann-Schwinger equation (LSE) $$T^{\textrm{ret},ab}(\omega)=V^{\textrm{S},\textrm{ret},ab}+i\hbar V^{\textrm{S},\textrm{ret},ab}\,\mathcal{G}^{\textrm{ret},ab}(\omega)\, T^{\textrm{ret},ab}(\omega)
\,,
\label{eq:LSE}$$ where $\mathcal{G}^{\textrm{ret},ab}(\omega)$ is the free two-particle Green’s function in the particle-particle channel. The corresponding interacting two-particle Green’s function $G^{\textrm{ret},ab}_2(\omega)$ fulfills a Bethe-Salpeter equation, that has been discussed in detail in [@kraeft_quantum_1986; @bornath_two-particle_1999] and is equivalent to the LSE. We will exploit this fact later when solving the LSE and evaluating the T-matrix self-energy. In its homogeneous form, the BSE in static ladder approximation is given by $$\begin{split}
&0=(E_{{\mathbf{k}}\sigma}^a+E_{{\mathbf{k}}+{\mathbf{Q}}\sigma'}^b)G^{\textrm{ret},ab}_{2,{\mathbf{k}},{\mathbf{k}}+{\mathbf{Q}}\sigma\sigma'} \\
-&(1-f_{{\mathbf{k}}\sigma}^a-f_{{\mathbf{k}}+{\mathbf{Q}}\sigma'}^b)\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}'}V^{\textrm{S},ab}_{{\mathbf{k}}+{\mathbf{Q}},{\mathbf{k}}',{\mathbf{k}},{\mathbf{k}}'+{\mathbf{Q}}}G^{\textrm{ret},ab}_{2,{\mathbf{k}}',{\mathbf{k}}'+{\mathbf{Q}}\sigma\sigma'}
\,.
\label{eq:bse}
\end{split}$$ Diagonalization yields bound states $\ket{\nu\sigma\sigma'{\mathbf{Q}}}$ and eigenenergies $E^{\sigma\sigma'}_{\nu{\mathbf{Q}}}$. We drop the indices $a$ and $b$ here, assuming that only two-particle states between different carrier species are involved. Due to the translational invariance of the crystal, the bound states can be classified by the total exciton momentum ${\mathbf{Q}}$ as discussed in [@qiu_nonanalyticity_2015]. Here we neglect the effect of electron-hole exchange interaction that leads to a fine-structure splitting of excitons and trions [@qiu_nonanalyticity_2015; @jones_excitonic_2016; @plechinger_trion_2016] in the meV range, which is small compared to the exciton binding energies of several hundred meV. As a consequence, electron and hole spins, which are already good quantum numbers in monolayer TMDC materials due to crystal symmetry, also classify the bound states. For each total momentum and spin combination a series of excitons exists, which is labeled here by $\nu$, analogue to the angular momentum states of Hydrogen-like Wannier excitons. Due to the two-dimensional nature of monolayer TMDCs and the related strong momentum dependence of dielectric screening, nontrivial exciton series deviating from a Hydrogen-like spectrum are observed. [@qiu_optical_2013; @chernikov_exciton_2014; @berghauser_analytical_2014] The eigenenergies decompose into a part from the relative motion of electron and hole and a kinetic part depending on the total momentum: $E^{\sigma\sigma'}_{\nu{\mathbf{Q}}}=E^{\sigma\sigma'}_{\textrm{rel},\nu}+E^{\sigma\sigma'}_{\textrm{kin},{\mathbf{Q}}}$. We can use Bloch basis functions to find a representation of the bound states corresponding to exciton wave functions $$\begin{split}
& \psi^{\sigma\sigma'}_{\nu{\mathbf{Q}}}({\mathbf{k}})= \big< {\mathbf{k}}{\mathbf{k}}'\sigma\sigma'ab \big| \nu\sigma\sigma'{\mathbf{Q}}\big> \delta_{{\mathbf{k}}',{\mathbf{k}}+{\mathbf{Q}}}\,,
\label{eq:wave_functions}
\end{split}$$ where ${\mathbf{k}}$ conventionally denotes the hole momentum, while the electron momentum is fixed via the total momentum.
An explicit expression for the T-matrix can be obtained by writing the LSE (\[eq:LSE\]) in the basis of two-particle eigenstates $\ket{\nu\sigma\sigma'{\mathbf{Q}}}$ as shown in detail in Ref. . Since the BSE represents a generalized eigenvalue problem, the eigenstates form a biorthogonal basis. The procedure yields a spectral representation of the T-matrix in operator form that is referred to as *bilinear expansion*: $$\begin{split}
&T^{\textrm{ret},ab}(\omega)= \\
&\sum_{\nu\sigma\sigma'{\mathbf{Q}}}N_{ab}^{-1}(E_{\textrm{kin}}-\hbar\omega)\frac{\ket{\nu\sigma\sigma'{\mathbf{Q}}}\bra{\nu\widetilde{\sigma\sigma'}{\mathbf{Q}}}}{\hbar\omega - E^{\sigma\sigma'}_{\nu{\mathbf{Q}}}}(E_{\textrm{kin}}-E^{\sigma\sigma'}_{\nu{\mathbf{Q}}})
\label{eq:bilinear}
\end{split}$$ with the Pauli blocking factor $N_{ab}=1-f^a-f^b$, the operator of kinetic energy $E_{\textrm{kin}}$ and the eigenstate of the adjoint BSE $\bra{\nu\widetilde{\sigma\sigma'}{\mathbf{Q}}}$. The bilinear expansion is used in the following to evaluate the imaginary part of the self-energy (\[eq:im\_self\_en\_screened\_ladder\]) and thereby the contribution of correlated carriers.
Separation of Bound and Quasi-Free Carriers.
--------------------------------------------
Inserting Eq. (\[eq:im\_self\_en\_screened\_ladder\]) into Eq. (\[eq:n\_ext\_qp\_approx\]) and noting that neither the GW self-energy nor the two lowest T-matrix terms contribute to the carrier density [@stolz_correlated_1979], we obtain [@kremp_quantum_1993; @semkat_ionization_2009] $$\begin{split}
n_a(\mu_a,T)&=\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}\sigma}f^a(E^a_{{\mathbf{k}}\sigma})\\
&+\frac{1}{\mathcal{A}^2}\sum_{{\mathbf{k}}{\mathbf{k}}' b\sigma'}\int_{-\infty}^{E_{\textrm{Gap}}}\frac{d\omega}{\pi}n^B_{ab}(\omega)\\
&\textrm{Im}
\,T^{\textrm{ret},ab}_{{\mathbf{k}}{\mathbf{k}}'\sigma\sigma'}(\omega)\frac{d}{d\omega}i\hbar\mathcal{G}^{\textrm{ret},ab}_{{\mathbf{k}}{\mathbf{k}}'\sigma\sigma'}(\omega)\\
&+n_{\textrm{scatt}}
\,.
\label{eq:density_full}
\end{split}$$ $n^B_{ab}(\omega)=\big[\textrm{exp}(\beta(\hbar\omega-\mu_a-\mu_b))-1\big]^{-1}$ is the Bose distribution function depending on the chemical potentials of both carrier species. Eq. (\[eq:density\_full\]) contains contributions of both bound two-particle states (below the single-particle gap $E_{\textrm{Gap}}$) and scattering two-particle states, the latter being explicitely given in Refs. . The renormalization factor $Z_{{\mathbf{k}}\sigma}^a$ of the quasi-particle resonance in the spectral function (\[eq:spectral\_fct\_extended\]) enters the contribution of correlated carriers as Pauli-blocking factor and as correction to the two-particle scattering spectrum. To simplify the following discussion, we neglect the contribution $n_{\textrm{scatt}}$ of scattering states beyond the quasi-free carriers and consider only the bound-state contribution given by the real-frequency poles of the T-matrix [@kremp_ladder_1984]: $$\begin{split}
&\frac{1}{\mathcal{A}^2}\sum_{{\mathbf{k}}{\mathbf{k}}' b\sigma'}\textrm{Im}
\,T^{\textrm{ret},ab}_{{\mathbf{k}}{\mathbf{k}}'\sigma\sigma'}(\omega)\frac{d}{d\omega}i\hbar\mathcal{G}^{\textrm{ret},ab}_{{\mathbf{k}}{\mathbf{k}}'\sigma\sigma'}(\omega)\Big|_{\textrm{bound}} \\
&=\pi\frac{1}{\mathcal{A}}\sum_{\nu\sigma'{\mathbf{Q}}}\delta(\hbar\omega - E^{\sigma\sigma'}_{\nu{\mathbf{Q}}})\,.
\label{eq:dens_bound}
\end{split}$$ Using Eq. (\[eq:dens\_bound\]), we arrive at the final expression for the carrier density: $$\begin{split}
n_a(\mu_a,T)&=\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}\sigma}f^a(E^a_{{\mathbf{k}}\sigma})\\
&+\frac{1}{\mathcal{A}}\sum_{b\neq a}\sum_{\sigma\sigma'}\sum_{\nu{\mathbf{Q}}}n^B_{ab}(E^{\sigma\sigma'}_{\nu{\mathbf{Q}}})\\
&=n_{\textrm{free}}^{\textrm{GW},a}+n_X
\,.
\label{eq:density_final}
\end{split}$$ The total carrier density separates into contributions from quasi-free carriers and from carriers bound as excitons according to the two poles in the spectral function $A^a(\omega)$. For a specific material, the ionization equilibrium has to be computed numerically. The electron and hole chemical potentials are determined by adapting the Fermi functions $f^a(E^a_{{\mathbf{k}}\sigma})$ of electrons and holes to a given density of quasi-free carriers at the quasi-particle energies $E^a_{{\mathbf{k}}\sigma}$. As the chemical potentials also enter the bound-carrier density via the Bose function $n^B_{ab}$, Eq. (\[eq:density\_final\]) represents an implicit equation for the fraction of quasi-free carriers $\alpha_a=n_{\textrm{free}}^{a}/n_a$, that has to be solved self-consistently with the quasi-particle energies in GW approximation, see Eq. (\[eq:GW\_energy\]), and the bound-state energies $E^{\sigma\sigma'}_{\nu{\mathbf{Q}}}$. To simplify the procedure, we exploit the fact that shifts of excitonic resonances are naturally much smaller than band-gap shifts, which is due to compensation effects between gap shrinkage and binding-energy reduction. [@bornath_two-particle_1999; @steinhoff_influence_2014] Hence we assume that the exciton spectrum depends only weakly on the excitation density so that we can limit ourselves to the BSE (\[eq:bse\]) in the limit of zero excitation density.
Consistent with the imaginary part of the self-energy (\[eq:im\_self\_en\_screened\_ladder\]), the quasi-particle energies $E^a_{{\mathbf{k}}\sigma}$ contain GW- and T-matrix contributions: $$\begin{split}
E^a_{{\mathbf{k}}\sigma}&=\varepsilon_{{\mathbf{k}}\sigma}^a+\textrm{Re}\,\Sigma_{{\mathbf{k}}\sigma}^{\textrm{GW},\textrm{ret},a}(E^a_{{\mathbf{k}}\sigma})+\textrm{Re}\,\Sigma_{{\mathbf{k}}\sigma}^{\textrm{T},\textrm{ret},a}(E^a_{{\mathbf{k}}\sigma})
\,.
\label{eq:GW_energy_full}
\end{split}$$ The GW self-energy is separated into the Fock term and the so-called Montroll-Ward term containing all contributions beyond bare exchange interaction. The T-matrix contribution is explicitely given in Ref. and leads to a blue shift of single-particle energies that is in the nondegenerate case ($f^a(E^a_{{\mathbf{k}}\sigma})\ll 1$) caused by the bound-carrier population. At the same time, the Fock self-energy contains exchange interaction with both quasi-free and bound carriers via the extended spectral functions that leads to a lowering of single-particle energies. This can be seen by using the T-matrix self-energy in Eq. (\[eq:im\_self\_en\_screened\_ladder\]) to obtain an excitonic contribution to the spectral function (\[eq:spectral\_fct\_extended\]) given by $$\begin{split}
&\Gamma_{{\mathbf{k}}\sigma}^a(\omega)= \frac{1}{\mathcal{A}}\sum_{b\neq a} \sum_{\nu\sigma'{\mathbf{Q}}}\sum_{{\mathbf{k}}'}\\
&(|\psi^{\sigma\sigma'}_{\nu{\mathbf{Q}}}({\mathbf{k}}) |^2\delta_{a,h}\delta_{{\mathbf{k}}',{\mathbf{k}}+{\mathbf{Q}}} + |\psi^{\sigma\sigma'}_{\nu{\mathbf{Q}}}({\mathbf{k}}-{\mathbf{Q}}) |^2\delta_{a,e}\delta_{{\mathbf{k}}',{\mathbf{k}}-{\mathbf{Q}}}) \\
&\delta(\hbar\omega+E^b_{{\mathbf{k}}'\sigma'}-E^{\sigma\sigma'}_{\nu{\mathbf{Q}}})[n^B_{ab}(E^{\sigma\sigma'}_{\nu{\mathbf{Q}}})+f^{b}(E^b_{{\mathbf{k}}'\sigma'})]
~.
\label{eq:spectral_fct_extended_T_matrix}
\end{split}$$ It yields a sharp resonance for each bound state weighted by its Bose population function and the exciton wave functions at the corresponding position in k-space. Note that the spectral positions of the resonances are not given by the bound-state energies $E^{\sigma\sigma'}_{\nu{\mathbf{Q}}}$, which are two-particle quantities, but by an effective binding energy of the carrier in state $\ket{{\mathbf{k}}\sigma a}$, as $\Gamma$ represents a single-particle spectral function. The Fock self-energy [@kremp_quantum_2005] can then be expressed in terms of the spectral function using the Kubo-Martin-Schwinger relation for the propagators $G^{<}(\omega)$ in thermal equilibrium: $$\begin{split}
\Sigma_{{\mathbf{k}}\sigma}^{\textrm{F},a}=&i\hbar\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}'}V^{aa}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}G^{<,a}_{{\mathbf{k}}'\sigma}\\
=-&i\hbar\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}'}V^{aa}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}\int\frac{d\omega}{2\pi}f^a(\omega)A_{{\mathbf{k}}'\sigma}^a(\omega) \\
=-&\frac{1}{\mathcal{A}}\sum_{{\mathbf{k}}'}V^{aa}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}\Big(f^a(E^a_{{\mathbf{k}}'\sigma})+f^{a,\textrm{bound}}_{{\mathbf{k}}'\sigma}\Big)\,~.
\label{eq:HF_self_energy}
\end{split}$$ The first contribution to the Fock self-energy scales, besides the Coulomb matrix elements, with the free-carrier density while the second contribution scales with the density of bound carriers. It turns out that similar to exchange interaction with free carriers, bound-carrier exchange leads to k-dependent renormalizations according to the exciton wave functions and populations that are contained in the population factor $f^{a,\textrm{bound}}_{{\mathbf{k}}'\sigma}$. As a conclusion, the real part of the self-energy (\[eq:GW\_energy\_full\]) contains quasi-particle renormalizations due to exciton populations via the T-matrix in two different places that act in opposite directions. We assume that these renormalizations cancel to a large degree and focus on the free-carrier contributions in accordance with Refs. . Then we obtain for the quasi-particle energies: $$\begin{split}
E^a_{{\mathbf{k}}\sigma}&\approx\varepsilon_{{\mathbf{k}}\sigma}^a+\textrm{Re}\,\Sigma_{{\mathbf{k}}\sigma}^{\textrm{GW},\textrm{ret},a}(E^a_{{\mathbf{k}}\sigma})\Big|_{\textrm{free}}\\
&=\varepsilon_{{\mathbf{k}}\sigma}^a+\Sigma_{{\mathbf{k}}\sigma}^{\textrm{F},a}\Big|_{\textrm{free}}+i\hbar\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\\
\sum_{{\mathbf{q}}}&\frac{(1-f^a(E^a_{{\mathbf{q}},\sigma}))V^{\textrm{S},>,aa}_{{\mathbf{k}}{\mathbf{q}}{\mathbf{k}}{\mathbf{q}}}(\omega')+
f^a(E^a_{{\mathbf{q}},\sigma})V^{\textrm{S},<,aa}_{{\mathbf{k}}{\mathbf{q}}{\mathbf{k}}{\mathbf{q}}}(\omega')}{E^a_{{\mathbf{k}}\sigma}-E^a_{{\mathbf{q}},\sigma}-\hbar\omega'}
\,.
\label{eq:GW_energy}
\end{split}$$ In a similar manner as for the Fock self-energy, the full spectral functions could be used to evaluate the Montroll-Ward self-energy in Eq. (\[eq:GW\_energy\]). Due to the spectral structure of the self-energy, however, renormalizations of the single-particle band structure caused by bound carriers involve a denominator of the order of the exciton binding energy, which is very off-resonant. Therefore the Montroll-Ward term is evaluated using spectral functions for quasi-free carriers.
Screening due to Excited Carriers.
----------------------------------
In the spirit of the extended quasi-particle approximation, screening of the Coulomb interaction due to both free carriers and bound excitons is taken into account. The free-carrier screening is treated in RPA with a Lindhard dielectric function [@steinhoff_influence_2014], while the excitonic polarizibilities are calculated as described in [@ropke_influence_1979; @ropke_greens_1981]: $$\begin{split}
&\varepsilon^{\textrm{ret}}_{{\mathbf{q}}}(\omega)=\varepsilon^{\textrm{ret},\textrm{RPA}}_{{\mathbf{q}}}(\omega)\\
&-V_{{\mathbf{q}}}\frac{1}{\mathcal{A}}\sum_{\nu\nu'\sigma\sigma'{\mathbf{Q}}}\left|M^{\sigma\sigma'}_{\nu\nu'{\mathbf{Q}}}({\mathbf{q}})\right|^2\frac{n^B_{ab}(E^{\sigma\sigma'}_{\nu{\mathbf{Q}}})-n^B_{ab}(E^{\sigma\sigma'}_{\nu'{\mathbf{Q}}-{\mathbf{q}}})}
{E^{\sigma\sigma'}_{\nu{\mathbf{Q}}}-E^{\sigma\sigma'}_{\nu'{\mathbf{Q}}-{\mathbf{q}}}-\hbar\omega-i\gamma}
\label{eq:screening}
\end{split}$$ with matrix elements $$\begin{split}
&M^{\sigma\sigma'}_{\nu\nu'{\mathbf{Q}}}({\mathbf{q}})=\\
&\frac{1}{\mathcal{A}}\sum_{{\mathbf{p}}}\psi^{\sigma\sigma'}_{\nu{\mathbf{Q}}}({\mathbf{p}})\Big[(\psi^{\sigma\sigma'}_{\nu'{\mathbf{Q}}-{\mathbf{q}}}({\mathbf{p}}+{\mathbf{q}}))^*-(\psi^{\sigma\sigma'}_{\nu'{\mathbf{Q}}-{\mathbf{q}}}({\mathbf{p}}))^*\Big]
\label{eq:screening_ME}
\end{split}$$ and exciton wave functions $\psi^{\sigma\sigma'}_{\nu{\mathbf{Q}}}({\mathbf{p}})$ as defined above. The momentum and frequency dependence of screening is characterized by the plasmon spectral function $$\begin{split}
\hat{V}^{\textrm{S},ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(\omega)&=V^{\textrm{S},>,ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(\omega)-V^{\textrm{S},<,ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(\omega) \\
&=V^{ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}2i\textrm{Im}\,\varepsilon^{-1,\textrm{ret}}_{{\mathbf{k}}-{\mathbf{k}}'}(\omega)\,.
\label{eq:W_spec}
\end{split}$$
Ab initio Single- and Two-Particle Properties.
----------------------------------------------
{width="\textwidth"}
For a material-realistic description of the strong Coulomb interaction in atomically thin systems, we use band structures $\varepsilon^{a}$ and Coulomb matrix elements $V^{ab}$ from material-realistic G$_0$W$_0$ calculations for freestanding slabs of various monolayer TMDCs as a basis, see the Supporting Information and Ref. . The matrix elements already take into account dielectric screening due to charge carriers in the ground state of the system. Furthermore, we account for additional screening provided by a possible dielectric environment, like a substrate, as described in the Supporting Information. The main effect of the additional screening is a reduction of exciton binding energies, which are key quantities for the exciton-plasma balance. The corresponding reduction of quasi-particle band gaps on the other hand plays no role here, as all bound-state energies and excited-carrier chemical potentials are measured relative to the gap. By convention, we choose the gap between the spin-up conduction and valence bands at $K$ as reference which belong to the A-exciton resonance. The calculation of ionization equilibrium requires knowledge of two-particle energies, which are obtained by solving the Bethe-Salpeter equation (\[eq:bse\]). Fig. \[fig:spectralf\] (b) shows the full exciton spectrum for monolayer WS$_2$ on a SiO$_2$ substrate, for which we assume a dielectric constant $\varepsilon=3.9$. Besides the $K$-excitons at $Q=0$ the spectra exhibit a rich structure of various inter-valley excitons with large total momenta. Even for direct-band-gap materials, inter-valley excitons between $K$- and $\Sigma$-valleys at $Q\approx6$ nm$^{-1}$ may be lower than intra-valley $K$-excitons due to a larger exciton binding energy. As can be seen in Fig. \[fig:bound\_states\] (a) the binding energies of the lowest $Q=0$-excitons (“1s”) are comparable for all considered materials, the main difference being the splitting between excitons involving different conduction bands, which is larger for tungsten-based materials. For MoS$_2$ bound states belonging to the B exciton are visible at $-175$ meV corresponding to the valence-band splitting of about $140$ meV. Higher exciton states (“2s”) appear as well. Fig. \[fig:bound\_states\] (b) shows that dielectric screening from the environment has a strong impact on the exciton spectrum as it can reduce exciton binding energies significantly [@latini_excitons_2015]. This has a major influence on the ionization equilibrium, as we discuss in the main text. Another tunable parameter in experiments is the application of strain to the TMDC monolayer, which leads to changes in the band structure [@conley_bandgap_2013]. This is reflected by MoS$_2$ changing from an indirect to a direct semiconductor in the exciton picture under tensile strain, see Fig. \[fig:bound\_states\] (c).
Numerical Details.
------------------
We calculate the ionization equilibrium from the fraction of quasi-free carriers as root of the implicit equation (\[eq:density\_final\]). The two highest valence and two lowest conduction bands are considered to cover all excitons that are relevant in a quasi-equilibrium situation. Likewise, we limit the Brillouin zone to circles with radius $2.7$ nm$^{-1}$ around the $K$,$K'$,$\Sigma$,$\Sigma'$ and $\Gamma$ points using a Monkhorst-Pack mesh with $30$ mesh points along $\Gamma$-$M$, which yields reasonable convergence of all results. The frequency integrals involved in the Montroll-Ward self-energy (\[eq:GW\_energy\]) are extended from $-600$ to $600$ meV exploiting the relation $V^{\textrm{S},<,ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(-\omega)=V^{\textrm{S},>,ab}_{{\mathbf{k}}{\mathbf{k}}'{\mathbf{k}}{\mathbf{k}}'}(\omega)$. For simplicity, we use a dielectric function (\[eq:screening\]) which is isotropic in momentum by evaluating its dependence on $|{\mathbf{q}}|$ along the contour $\Gamma$-$K$ and using Coulomb matrix elements $V_{|{\mathbf{q}}|}$ that are averaged over Wannier orbitals. Both the Lindhard and the excitonic dielectric function (\[eq:screening\]) are evaluated using ground-state energies and extrapolated to the limit of vanishing phenomenological quasi-particle broadening $\gamma$. For a sum over bound states, only those states that are below the local (in ${\mathbf{k}}$-space) renormalized band structure measured at the maximum of the corresponding exciton wave functions are taken into account.
**Acknowledgement** We acknowledge financial support from the Deutsche Forschungsgemeinschaft (JA 14-1 and RTG 2247 “Quantum Mechanical Materials Modelling”) and the European Graphene Flagship as well as resources for computational time at the HLRN (Hannover/Berlin). We thank Michael Lorke, Christopher Gies and Dirk Semkat for fruitful discussions.
**Supporting Information available**, containing a description of the ab initio procedures to obtain band structures and screened Coulomb matrix elements.
Supporting Information.
=======================
Ab-initio based parametrization of the TMDCs \[app:modeling\] {#ab-initio-based-parametrization-of-the-tmdcs-appmodeling .unnumbered}
-------------------------------------------------------------
To get an *analytic description* of the Coulomb interaction in semiconducting TMDC monolayers, we apply the approach as presented in [@steinhoff_influence_2014; @PhysRevB.88.085433; @schonhoff_interplay_2016] for MoS$_2$ also to MoSe$_2$, WS$_2$, and WSe$_2$. We start with ab initio calculations for density-density like bare $U_{\alpha\beta} (q)$ and screened $V_{\alpha\beta}(q)$ Coulomb interaction matrix elements in the Wannier basis (with $\alpha, \beta \in [d_{z^2}, d_{xy}, d_{x^2-y^2}]$ ) for the freestanding, undoped TMDC slabs using the FLEUR and SPEX codes [@FLEUR; @friedrich_efficient_2009; @friedrich_efficient_2010] on discrete $18 \times 18 \times 1$ q grids. To interpolate the resulting $3 \times 3$ matrices analytically we make use of the (sorted) eigenbasis of the bare Coulomb interaction $\mathbf{U}$ by diagonalizing it $$\begin{aligned}
\mathbf{U}_\mathrm{diag}(q) =
\left(
\begin{array}{ccc}
U_1(q) & 0 & 0\\
0 & U_2 & 0 \\
0 & 0 & U_3
\end{array}
\right),
\label{eq:BareCoulombInteractionMatrixEigenbasis}\end{aligned}$$ where the diagonal matrix elements are given by $$U_i = \braket{ e_i | \mathbf{U} | e_i }$$ using the eigenvectors of $\mathbf{U}$ in their long-wavelength limits $$e_1 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},
e_2 = \frac{1}{\sqrt{6}} \begin{pmatrix} +2 \\ -1 \\ -1\end{pmatrix},
e_3 = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ +1 \\ -1 \end{pmatrix}.
\label{eq:vEV}$$ While the leading eigenvalue $U_1(q)$ is a function of $q$ the other two eigenvalues can be readily approximated as constants (see Fig. \[fig:CompSpexFit\]). For the analytic description of the former we use $$U_1(q) = \frac{3 e^2}{2 \varepsilon_0 A} \frac{1}{q(1 + \gamma q + \delta q^2)},
\label{eq:v1}$$ where $A=\frac{\sqrt{3}}{2}a^2$ is the area of the hexagonal unit cell, $a$ is the lattice constant, and $\varepsilon_0$ is the vacuum permittivity.
![Examplary data for the fit and ab initio calculation of the Coulomb interaction in MoSe$_2$. **(a)** Bare Coulomb matrix elements in its eigenbasis. Red dots, blue squares and green triangles correspond to the leading, second and third eigenvalue of $U(q)$ as obtained from ab initio calculations. Dashed lines show the corresponding fits using Eq. (\[eq:v1\]) and Tab. \[tab:Fits\]. **(b)** Eigenvectors of the bare Coulomb matrix (from left to right corresponding to the leading, second and third eigenvalue). The corresponding vector elements of the $d_{z^2}$ (red), $d_{xy}$ (green) and $d_{x^2-y^2}$ (blue) orbitals are shown. Dashed lines indicate constant analytic values for $U(q\rightarrow 0)$, see Eq. (\[eq:vEV\]). **(c)** Matrix elements of the diagonal dielectric function. Markers indicate ab initio results and dashed lines show the fits using Eq. \[eq:BackgroundScreening\] and Tab. \[tab:Fits\].[]{data-label="fig:CompSpexFit"}](U_eigenvalues.eps "fig:"){width="50.00000%"} ![Examplary data for the fit and ab initio calculation of the Coulomb interaction in MoSe$_2$. **(a)** Bare Coulomb matrix elements in its eigenbasis. Red dots, blue squares and green triangles correspond to the leading, second and third eigenvalue of $U(q)$ as obtained from ab initio calculations. Dashed lines show the corresponding fits using Eq. (\[eq:v1\]) and Tab. \[tab:Fits\]. **(b)** Eigenvectors of the bare Coulomb matrix (from left to right corresponding to the leading, second and third eigenvalue). The corresponding vector elements of the $d_{z^2}$ (red), $d_{xy}$ (green) and $d_{x^2-y^2}$ (blue) orbitals are shown. Dashed lines indicate constant analytic values for $U(q\rightarrow 0)$, see Eq. (\[eq:vEV\]). **(c)** Matrix elements of the diagonal dielectric function. Markers indicate ab initio results and dashed lines show the fits using Eq. \[eq:BackgroundScreening\] and Tab. \[tab:Fits\].[]{data-label="fig:CompSpexFit"}](U_eigenvectors.eps "fig:"){width="50.00000%"} ![Examplary data for the fit and ab initio calculation of the Coulomb interaction in MoSe$_2$. **(a)** Bare Coulomb matrix elements in its eigenbasis. Red dots, blue squares and green triangles correspond to the leading, second and third eigenvalue of $U(q)$ as obtained from ab initio calculations. Dashed lines show the corresponding fits using Eq. (\[eq:v1\]) and Tab. \[tab:Fits\]. **(b)** Eigenvectors of the bare Coulomb matrix (from left to right corresponding to the leading, second and third eigenvalue). The corresponding vector elements of the $d_{z^2}$ (red), $d_{xy}$ (green) and $d_{x^2-y^2}$ (blue) orbitals are shown. Dashed lines indicate constant analytic values for $U(q\rightarrow 0)$, see Eq. (\[eq:vEV\]). **(c)** Matrix elements of the diagonal dielectric function. Markers indicate ab initio results and dashed lines show the fits using Eq. \[eq:BackgroundScreening\] and Tab. \[tab:Fits\].[]{data-label="fig:CompSpexFit"}](Eps_eigenvalues.eps "fig:"){width="50.00000%"}
The matrix elements of the screened interaction $\mathbf{V}(q)$ in the eigenbasis of the bare interaction $\mathbf{U}(q)$ are then obtained via $$V_i(q) = \varepsilon_i^{-1}(q) \ U_i(q)
\label{eq:CoulombInteractionPartiallyScreened}$$ where $\varepsilon_i(q)$ accounts for both the material-specific internal polarizability and the screening by the environment. Its diagonal representation is given by $$\mathbf{\varepsilon}_\mathrm{diag}(q) =
\left(
\begin{array}{ccc}
\varepsilon_1(q) & 0 & 0\\
0 & \varepsilon_2 & 0 \\
0 & 0 & \varepsilon_3
\end{array}
\right).$$ Once again, the leading eigenvalue $\varepsilon_1(q)$ is a function of $q$ while the other elements are described sufficiently well as constants. Here, the former can be expressed by $$\varepsilon_1(q) =
\varepsilon_\infty (q)
\frac{ 1 - \beta_1 \beta_2 e^{-2q\,h}}
{ 1 + (\beta_1 + \beta_2) e^{-q\,h} + \beta_1 \beta_2 e^{-2q\,h}}
\label{eq:BackgroundScreening}$$ which describes the macroscopic dielectric function of a two-dimensional semiconductor. The parameters $\beta_i$ include the screening effects of substrates (see Ref. [@rosner_wannier_2015]) via $$\beta_i = \frac{\varepsilon_\infty (q) - \varepsilon_{\mathrm{sub},i}}{\varepsilon_\infty (q) + \varepsilon_{\mathrm{sub},i}}$$ where the dielectric constants of the substrate ($i=1$) and the superstrate ($i=2$) are introduced. In order to describe the original ab initio data as close as possible we fit $\varepsilon_\infty (q)$ using $$\varepsilon_\infty (q)= \frac{a + q^2}{\frac{a \sin(q c)}{q b c} + q^2 } + e
\label{eq:EpsInf}$$ and $\varepsilon_{\mathrm{sub},1}=\varepsilon_{\mathrm{sub},2}=1$ as the ab intio calculations were performed for freestanding layers.
As soon as all fitting parameters are obtained (see Tab. \[tab:Fits\]) the screening of a dielectric environment can be included by choosing $\varepsilon_{\mathrm{sub},1}$ or $\varepsilon_{\mathrm{sub},2}$ correspondingly. Thus we have a closed analytic description of the screened Coulomb interaction $\mathbf{V}(q)$ in the eigenbasis of the bare interaction $\mathbf{U}(q)$ at arbitrary momenta $q$ in the first Brillouin zone. In order to transform it to the original orbital basis we make use of the eigensystem given in Eq. (\[eq:vEV\]). In fact, this model is appropriate for every two-dimensional semiconductor.
Besides the analytical description of the screened Coulomb matrix elements we make use of a tight-binding Hamiltonian to describe the electronic band structure (as obtained from $G_0 W_0$ calculations) of the TMDC slab. To this end, we utilize the same Wannier basis as described before (see Ref. [@groenewald_valley_2016]) and derive a minimal three-band model describing the highest valence and two lowest conduction bands using the Wannier90 package [@mostofi_updated_2014]. Thereby we solely disentangle our target bands from the rest without performing a maximal localization in order to preserve the original transition metal d-orbital characters. The latter is crucial for the subsequent addition of first and second order Rashba spin-orbit coupling following Ref. [@PhysRevB.88.085433], which takes into account the large spin-orbit splitting in the conduction- and the valence-band K valleys.
Throughout the whole model-building process we assume that the substrate mainly affects the internal Coulomb interaction and neglect its influence on the band structure. We have used relaxed lattice constants as given in Tab. \[tab:Fits\]. The values of the bare and screened Coulomb interaction were extrapolated from vacuum heights of $16\,$Å to $32\,$Å.
MoS$_2$ MoSe$_2$ WS$_2$ WSe$_2$
------------------------- --------- ---------- -------- ---------
lattice constant $a$(Å) 3.180 3.320 3.191 3.325
bare Interaction $U$
$\gamma$ (Å) 1.932 2.232 2.130 2.297
$\delta$ (Å$^2$) 0.395 -0.356 0.720 0.174
A (Å$^2$) 8.758 9.546 8.818 9.574
$U_2$ (eV) 0.810 0.837 0.712 0.715
$U_3$ (eV) 0.367 0.376 0.354 0.360
screening $\varepsilon$
$a$ (1/Å$^2$) 2.383 2.856 3.947 2.430
$b$ 17.836 11.635 29.931 20.764
$c$ (Å) 5.107 1.979 5.440 5.761
$h$ (Å) 2.740 4.298 1.578 2.489
$e$ 5.739 6.303 4.497 5.305
$\varepsilon_2$ 3.077 3.148 2.979 3.028
$\varepsilon_3$ 2.509 2.510 2.494 2.481
: Parameters describing the Coulomb interaction and the static screening in the semiconducting TMDCs $MX_2$.[]{data-label="tab:Fits"}
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abstract: 'We consider examples of $D=4$ string theory vacua which, although globally non-geometric, admit a local description in terms of $D=10$ supergravity backgrounds. We analyze such backgrounds and find that the supersymmetry spinors vary non-trivially along the internal manifold, reproducing the interpolating supergravity solutions found by Frey and Graña. Finally, we propose a simple, local expression for non-geometric fluxes in terms of the internal spinors of the compactification.'
author:
- Fernando Marchesano
- Waldemar Schulgin
title: 'Non-geometric fluxes as supergravity backgrounds'
---
Undoubtedly, fluxes have played a major role in recent attempts to connect string theory to observable physics. Why is this the case may be understood from two simple facts. First, string compactifications with background fluxes enlarge non-trivially the set of $D=4$ string vacua based on conventional Calabi-Yau compactifications, and so the set of effective field theories that one can obtain. Second, flux compactifications are rich sources of $D=4$ effective superpotentials, allowing to address two important issues of string phenomenology, moduli stabilization and supersymmetry breaking, in a controlled manner [@reviews].
A celebrated class of flux vacua is given by type IIB string theory compactified on a warped Calabi-Yau with orientifold planes [@drs99; @gkp01]. Indeed, such constructions admit the inclusion of internal background 3-form fluxes, usually denoted by $H_3$ and $F_3$, which back-react on the Calabi-Yau geometry and produce an effective superpotential that lifts certain compactification moduli [@gvw99]. In many senses, one can treat the resulting $D=4$ effective theory as that obtained from the well-understood fluxless Calabi-Yau compactification, to which we add a flux-induced superpotential and a non-trivial warping. This provides a simple global picture of this class of compactifications, which even allows to treat them as an ensemble as opposed to consider each vacuum individually [@ddk07]. Finally, a nice feature of this set of $D=4$ vacua is that in general they can be understood in terms of $D=10$ supergravity. The geometric intuition that one obtains from this fact has triggered several phenomenologically interesting scenarios, in particular some related to early universe cosmology [@cosmo].
While warped CY’s have received a lot of attention, they are not the only class of flux compactifications in the literature, even if we restrict to type IIB $\cn = 1$ $D=4$ Poincaré invariant vacua. Another well-known family is based on compact non-Kähler manifolds, also containing O-planes, and dual to the solutions found in the context of heterotic strings [@fluxhet]. Here the set of fluxes not only consists of field strengths like $F_3$, but also of torsion factors that deviate the metric from the Calabi-Yau one, and so they are usually referred to as geometric fluxes [@glmw02; @kstt02].
While geometric fluxes are less intuitive than the usual ones, an even more exotic kind of fluxes are those named non-geometric fluxes, which arise in the context of non-geometric compactifications [@dh02hmw02fww04; @Hull04; @stw05; @dh05; @acfi06]. Such compactifications are possible because, as string theory has a larger duality group than plain supergravity, it can be compactified in spaces where (classical) gravity would not even make sense. Perhaps the best understood class of non-geometric vacua are those based on T-folds [@Hull04], where locally there is a standard spacetime description of the compactification but, as we relate two overlapping patches, we must use transition functions that mix the metric with the antisymmetric B-field and interchange KK with winding modes, just as T-dualities do.
Clearly, non-geometric compactifications are a novel and exciting area where to develop string phenomenology, as they are genuine string compactifications that cannot be described as $D=10$ supergravity plus localized sources. Moreover, as non-geometric fluxes are combined with the previous fluxes, the effective superpotential that one obtains generalizes that of a Calabi-Yau with fluxes [@stw05], in such a way that compactifications with all moduli stabilized are easily obtained [@allmoduli].
On the other hand, our current understanding of non-geometric compactifications is still too weak in order to perform a detailed analysis of their phenomenological possibilities. Part of the problem is that, unlike for other fluxes, there is no simple, intuitive definition of what non-geometric fluxes mean. In practice, they appear as the structure constants of $D=4$ effective gauged supergravities, or as internal monodromies between the metric and B-field [@kstt02; @stw05; @dh05]. In fact, very few examples of non-geometric compactifications have been built beyond Scherk-Schwarz compactifications and close relatives and, because of the issues above, it is hard to figure out how the space of non-geometric vacua looks like.
The purpose of this note is to shed further light into our understanding of non-geometric fluxes by describing them in terms of supergravity backgrounds. More precisely, we focus on some simple examples of T-folds, where a local $D=10$ description is possible, and study in detail the supergravity solutions associated to them. This will not only provide us with a simple definition of non-geometric flux, but also unveil many local properties of this kind of backgrounds. Notice that the latter point is essential in order to build realistic scenarios from non-geometric compactifications. Indeed, most of the cosmological models constructed from warped CY’s with fluxes are based on D-brane inflation, where only a local description of the supergravity background (like, e.g., the KS throat [@ks00]) is needed. Hence, based on the results below, similar scenarios could presumably be constructed for non-geometric vacua.
In order to perform our analysis, let us first construct a non-geometric vacuum. As pointed out in [@kstt02] this can be done by performing appropriate T-dualities on simple geometric flux compactifications. In particular, let us consider type IIB string theory compactified on the warped toroidal background [^1] \[fullmetric1\] ds\^2 & = & Z\^[-1/2]{} ds\^2\_[M\_4]{} + Z\^[1/2]{} ds\^2\_[[\^6]{}]{}\
\[metric1\] ds\^2\_[\^6]{} & = & \_[i=1]{}\^3 ds\^2\_[(\^2)\_i]{}, ds\^2\_[(\^2)\_i]{} = R\_i\^2 |dx\^i + \^i dx\^[i+3]{}|\^2\
\[H31\] H\_3 & = & N (dx\^1 dx\^5 - dx\^4 dx\^2) dx\^6\
\[F31\] F\_3 & = & N (dx\^4 dx\^2 - dx\^1 dx\^5) dx\^3\
\[F51\] \_5 & = & (1 + \*\_[10]{}) d[vol]{}\_[M\_4]{} dh\
\[dilaton1\] & = & C\_0 + ie\^[-\_0]{} = const. which is closely related to the vacua considered in [@kstt02], and a simple example of a warped Calabi-Yau with $H_3$ and $F_3$ fluxes of [@gkp01]. Here $ds^2_{M_4}$ stands for the $D=4$ Minkowski metric, $\tau$ for the type IIB axio-dilaton, the warp factor $Z$ only depends on the internal coordinates $x^i \sim x^i +1$, $N$ is an even integer [@fp02], and we must impose $dh = e^{-\phi_0} dZ^{-1}$ [@gkp01]. In order to find a solution to the Bianchi identity of $\tilde{F}_5$ and Einstein’s equations we need to add negative tension objects to the compactification [@gkp01], which in our case will be 64 O3-planes expanding $M_4$. Finally, by adding $32 -2N^2$ D3-branes and no further localized objects all consistency conditions will be satisfied.
The presence of the background fluxes $H_3$ and $F_3$ generates a superpotential [@gvw99] W = \_[\^6]{} (F\_3 - H\_3) \_3 = N (\^1 - \^2) (\^3 - ) \[super\] and so we also need to impose $\tau^1 = \tau^2$ and $\tau^3 = \tau$. This will guarantee that the complexified 3-form flux $G_3 = F_3 -\tau H_3$ is an imaginary-self-dual (ISD), primitive (2,1)-form, and so we will have a supersymmetric vacuum [@gp00]. In fact, due to the simplicity of our example, the present compactification yields a $D=4$ $\cn = 2$ effective theory. The supersymmetry generators are of the form \[spinorL\] \_L & = & (u \_L + u\^\* \^\*\_L )\
\[spinorR\] \_R & = & (u \_R + u\^\* \^\*\_R ) where $u$ stands for the external and $\chi$ for the internal spacetime spinor, the latter chosen of negative chirality. While the above ansatz is valid for general supersymmetric type IIB compactifications, here the presence of the flux $G_3$ and the O3-planes imposes the following relation \_L = \_[O3]{} \_R = -i \^[(6)]{} \_R \[pO3\] between left ($\eps_L$) and right ($\eps_R$) spinors. Here $\G^{(6)}$ stands for the internal chirality operator, and so we have that \_L = i \_R = \_L + i \_R = u \_L \[typeB\] which is the usual B-type ansatz for warped Calabi-Yau compactifications with fluxes [@bb96]. In the present background, the internal spinors are of the form $\chi_{R,i} = Z^{-1/8} \eta_i$, where \_1 = (—),\_2 = (++-) \[spinors1\] in the usual spinor notation.
As pointed out in [@kstt02], by performing two consecutive T-dualities along two directions of a $H_3$ component one obtains a non-geometric compactification. In particular, following the notation of [@stw05], after two T-dualities along $\{x^5, x^1\}$ the flux $H_3 = dB$ partially becomes a non-geometric Q-flux via H\_[x\^1x\^5x\^6]{} = N Q\_[x\^6]{}\^[x\^5x\^1]{} = N \[Qflux\] whereas the other component of $H_3$ remains untouched.
In order to obtain the new background one can apply Buscher’s T-duality rules [@Buscher87], but first one needs have the directions $\{x^5, x^1\}$ as isometries of the background. This is achieved by choosing an appropriate gauge for the B-field, like $B = N x^6 (dx^1 \wedge dx^5 - dx^4 \wedge dx^2)$, and by smoothing out the localized sources (i.e., the D3-branes and O3-planes) along $\{x^5, x^1\}$. This means in particular that the warp factor $Z$ will be given by a function $\D \equiv \D(x^2,x^3,x^4,x^6)$, which by Einstein’s equations must satisfy - e\^[-\_0]{} \^2\_[\^4]{} = + \_[i]{} q\_i \_[\^4]{}(\_[i]{}) \[warpeq\] where $\T^4$ stands for the four-torus expanded by $\{x^2,x^4,x^3,x^6\}$ and the delta functions represent the sources localized in the same $\T^4$, with a charge $q = 1$ in the case of a D3-brane and $q = -2$ for a (partially delocalized) O3-plane. Notice that on $\T^4$ the O3-planes are at the fixed locations $x^i \in \mathbb{Z}/2$ ($i = 2, 3, 4, 6$), and in the following we will focus our attention to the neighborhood containing the one at the origin. Finally, in order to simplify the discussion below, we will choose the compactification moduli to be $R_1 = R_2 = 1$ and $\tau^1 = \tau^2 = i$, although more general choices consistent with the supersymmetry condition $\tau^1 = \tau^2$ may also be chosen.
Taken into account these prescriptions and directly applying Buscher’s rules along $\{x^5, x^1\}$ one again obtains type IIB string theory, but this time on the background \[fullmetric2\] ds\^2 & = & \^[-1/2]{} ds\^2\_[M\_4]{} + \^[1/2]{} ds\^2\_[[\_6]{}]{}\
\[metric2\] ds\^2\_[\_6]{} & = & (dx\^2)\^2 + (dx\^4)\^2 + dz\^3 d|[z]{}\^3\
& & + e\^[2(- \_0)]{} ((dx\^1)\^2 + (dx\^5)\^2)\
\[B22\] B & = & N x\^6 (dx\^2 dx\^4 + e\^[2(-\_0)]{} dx\^5 dx\^1)\
\[F12\] F\_1 & = & Ndz\^3\
\[F32\] \_3 & = & F\_1 B + e\^[2-\_0]{} \*\_[\_6]{} d e\^[-2]{} dx\^5 dx\^1\
\[F52\] \_5 & = & (1 + \*\_[10]{}) d[vol]{}\_[M\_4]{} d (N x\^6/) e\^[-\_0]{}\
\[dilaton2\] e\^[- \_0]{} & = & (+ (Nx\^6)\^2)\^[-1/2]{} where for simplicity we have defined $dz^3 = dx^3 + \tau dx^6$, with $\tau$ given by (\[dilaton1\]). When compactifying this theory, one should perform the identifications $x^3 \sim x^3 + 1$ and $x^6 \sim x^6 +1$. However, for reasons that will become clear below, we will initially not impose such identifications. On the other hand, we will still impose $x^i \sim x^i + 1$ for $i = 1,2,4,5$. The background fluxes $\tilde{F}_3 = F_3 - C_0 H_3$, $F_1 = dC_0$ stand for the generalized field strength of type IIB supergravity. Notice that neither the axion $C_0$ nor the dilaton $e^{-\phi}$ are constant in this new background. Nevertheless, as usual one can still define an average string coupling by setting $g_s = e^{\phi_0}$, with $e^{\phi_0}$ given by (\[dilaton1\]), that can be taken to arbitrary small values.
Just as the above background has been T-dualized we must also transform the localized sources, namely the orientifold planes and the open string sector of the theory. By the usual T-duality rules we would now expect to have O5-planes and D5-branes wrapped on the two-torus expanded by $\{x^5, x^1\}$. This intuition is confirmed by looking at the Bianchi identities of the new background. While $dF_1 = dH_3 = 0$ are clearly satisfied, for $\tilde{F}_3$ we have that [^2] d\_3 - H\_3 F\_1 & = & g\_s\^[-1]{} \^2\_[\^4]{} e\^[-2 (-\_0)]{} d[vol]{}\_[\_4]{}\
\[BIF32\] & = & - \_[i]{} q\_i \_[\^4]{}(x\_[i]{}) d[vol]{}\_[\_4]{} where we have used (\[F32\]) and (\[warpeq\]), and we are defining $d{\rm vol}_{\T^4} = R_3^2 \pim \tau\, dx^2 \wedge dx^4 \wedge dx^3 \wedge dx^6$. Hence we have a set of localized sources to which the RR field strength $F_3$ couples. As we expected, these are D5-branes and O5-planes, expanding $M_4 \times \T^2_{(x^5, x^1)}$ and being pointlike in $\{x^2, x^4, x^3, x^6\}$. In particular, there will be an O5-plane at the origin $x^2 = x^3 = x^4 = x^6 = 0$ and, if we treat such coordinates as non-compact, one at each location given by $x^i \in \mathbb{Z}/2$ ($i = 2, 3, 4, 6$), as we would also expect from T-duality. The remaining Bianchi identity can be easily computed if we realize that the internal component of $\tilde{F}_5$ satisfies \_5 - B \_3 + B\^2 F\_1 = - e\^[-2(-\_0)]{} N \*\_[\_6]{} d z\^3 \[relF52\] and so the r.h.s. of (\[relF52\]) is closed. This implies that d\_5 - H\_3 \_3 = B ( d\_3 -H\_3 F\_1 ) \[BIF52\] and so, by (\[BIF32\]), we again have sum of delta functions. The presence of the $B$-field is easy to understand from the fact that the l.h.s. of (\[BIF52\]) measures the D3-brane charge of localized sources, and this same charge is induced on D5-branes in the presence of a non-trivial B-field [@Douglas95]. Hence the r.h.s. of (\[BIF52\]) takes into account that we are in the presence of D5-branes magnetized by a non-trivial B-field. What is perhaps more surprising is the fact that the O5-planes which are not located at $x^6 = 0$ also seem to be ‘magnetized’. In that case we should be dealing with exotic orientifold planes, analogous to the ones analyzed in [@adm02]. Finally, in the same way that we have solved the Bianchi identities, one can check that the equations of motion for the metric, axio-dilaton and background fluxes are satisfied. As a result, even if one is skeptical about applying Buscher’s on the initial type IIB background, it is clear that eqs.(\[fullmetric2\])-(\[dilaton2\]) describe a genuine type IIB supergravity vacuum.
Notice that in this new background the warp factor $Z'$ and the 4-form potential $C_4' = h' d{\rm vol}_{M_4}$ are given by Z’ = ,h’ = Nx\^6 e\^[-\_0]{} \^[-1]{} \[newforms\] and so the usual relation $d(h - e^{-\phi}Z^{-1})=0$ of type IIB compactifications with ISD $G_3$ fluxes, D3-branes and O3-planes is no longer satisfied. Such relation is only satisfied in the limit $x^6 \raw \infty$, where the $B$-field is so strong and the two-torus $\{x^5, x^1\}$ so small that a D5-brane wrapped on it looks like a D3-brane. On the other hand, on the limit $x^6 \raw - \infty$ we recover the relation $d(h + e^{-\phi}Z^{-1})=0$, satisfied by type IIB backgrounds with IASD $G_3$ fluxes, anti-D3-branes and anti-O3-planes. Finally, at $x^6 = 0$ we have that $h=0$ and $Z = e^{-2(\phi- \phi_0)}$, which is typical of type IIB flux compactifications containing O5-planes and geometric fluxes.
It thus seems that, as we move along $x^6$, our type IIB background interpolates between different classes of type IIB compactifications. We can parameterize such interpolation by defining an angle $\a$ as = \^[1/2]{} e\^[-\_0]{}, = Nx\^6 e\^[-\_0]{} \[defangle\] and analyzing how does the background depend on it. A non-trivial dependence will appear on the forms $J$ and $\Om$ describing the metric, which can be taken to be & = & e\^[2(-\_0)]{} dx\^5dx\^1 + dx\^2 dx\^4 + i dz\^3 d|[z]{}\^3\
& = & i e\^[-\_0]{} R\_3 (dx\^1 + idx\^5) (dx\^4 + i dx\^2) dz\^3 which naturally define a complex and a symplectic structure. In terms of these definitions, one can check that the ISD, (2,1)-form in this background is \_3 = \_3 - i e\^[-]{} (e\^[i]{} ) \[inter3formflux\] which, at specific values of $\a$ takes a more familiar form \[limitinf2\] 0 & & \_3 = \_3 - i e\^[-]{} H\_3 = G\_3\
\[limit02\] = /2 & & \_3 = \_3 - i e\^[-2]{} d(e\^ J)\
\[limit-inf2\] & & \_3 = \_3 + i e\^[-]{} H\_3 = |[G]{}\_3 Finally, a D5-brane wrapped on $\{x^5, x^1\}$ satisfies the conditions \_X |\_[D5]{} = (e\^[-i]{} (B+iJ))|\_[D5]{} = 0 \[susyD5\] which are the usual BPS conditions for a D5-brane in a Calabi-Yau compactification [@mmms99], notice however that in the Calabi-Yau case $\a$ is a constant parameter, while here it varies along the compactification manifold.
But the easiest way to see that this supergravity background interpolates between different classes of standard type IIB compactifications is by analyzing the supersymmetry spinors. These spinors can also be obtained by T-duality, by applying the rules of [@Hassan99]. One then obtains that the new spinors no longer satisfy the relation (\[typeB\]) but instead \_L’ = ([cos]{} \_[O3]{} + [sin]{} \_[O5]{} ) \_R’ \[typeBC\] where $\G_{O5}$ is the chirality operator on the coordinates $\{x^2, x^4, x^3, x^6\}$ transverse to $M_4 \times \T^2_{(x^5, x^1)}$, as well as the orientifold projection that an O5-plane wrapping $M_4 \times \{x^5, x^1\}$ would implement in standard compactifications.
Now, since our initial background contained two independent spinors so will the new one. In order to see how the $D=10$ spinors (\[spinorL\]) and (\[spinorR\]) now look like, let us consider a particular linear combination of them. Namely, we choose \[killingnew\] \_[R,]{} = \^[-1/8]{} \_, \_ = (\_1 \_2)\
then, taking into account that \[actionO3\] \_[O3]{} \_ = i \_& & \_[O3]{} \_\^\* = - i \_\^\*\
\[actionO5\] \_[O5]{} \_ = \_& & \_[O5]{} \_\^\* = \_\^\* we obtain the following $D=10$ spinors \[spinorRnew\] \_[R,]{} & = & (u \_[R,]{} + u\^\* \^\*\_[R,]{} )\
\[spinorLnew\] \_[L,]{} & = & ( i e\^[i]{} u \_[R,]{} -i e\^[i]{} u\^\* \^\*\_[R,]{} ) precisely matching the spinor ansatz considered in [@fg03], where type IIB supergravity solutions interpolating between standard compactification ansätze were shown to be possible.
In particular, the interpolating solutions analyzed in [@fg03] connected backgrounds where D3-branes are BPS objects to those where D5-branes are BPS. We see that, if we extend our supergravity solution along $x^6 \in (-\infty,\infty)$ the same situation will happen, with the difference that now anti-D3-branes will also be included. So, just as expected, our supergravity background is such that for $x^6 \raw -\infty$ anti-D3-branes are BPS, for $x^6 = 0$ D5-branes are BPS and for $x^6 \raw \infty$ D3-branes are BPS. At intermediate points of $x^6$ the supersymmetry preserved by the background is that of a magnetized D5-brane.
This description fits exactly with the D5-brane BPS conditions found in (\[susyD5\]), which can be derived as follows. First, one follows the computations of [@fg03] in order to see that (\[spinorLnew\]) and (\[spinorRnew\]) are indeed supersymmetry generators of the new background. Second, one chooses one of the two spinors, say $\eta_+$, and constructs the spinor bilinears J\_[mn]{} = - i \_+\^\_[mn]{} \_+,\_[mnp]{} = - \_+\^T \_[mnp]{} \_+ \[bilinears\] familiar from $SU(3)$-structure compactifications [@reviews], and obtains the same forms $J$ and $\Om$ as above. Finally, by repeating the $\kappa$-symmetry computations of [@mmms99] one arrives at (\[susyD5\]). Had we chosen the internal spinor $\eta_-$, we would have obtained a different two-form $J$, but nevertheless the same final condition for a D5-brane wrapped on $\{x^5, x^1\}$. Alternatively, the same conclusion can be reached by using the results in [@ms05].
To sum up, we have analyzed the type IIB supergravity backgrounds associated to a simple set of non-geometric backgrounds, and found that they correspond to solutions that interpolate between standard type IIB compactifications. This interpolation is particularly manifest in the internal part of the space-time supersymmetry generators, which fit into the ansatz used in [@fg03]. Although our background is particularly simple and yields $D=4$ extended supersymmetry, we do not expect this to be the general case. In fact, by analyzing other simple non-geometric backgrounds we find that the ansatz (\[spinorRnew\]) and (\[spinorLnew\]) is not always realized, but rather more general interpolating ansätze like that used in [@Dall'agata04] and generalizations. What does remain valid is the relation (\[typeBC\]) between left and right-handed spinors, in the sense that we have a rotation between two usual orientifold projections $\G_{Op}$ and $\G_{Oq}$ with a rotation parameter $\a$ varying along the internal manifold. This should not only be true for toroidal-like non-geometric vacua, but also for more general ones that can be obtained from fiberwise T-duality on a geometric flux background. It would be interesting to verify this by generalizing the computations performed in [@fmt03]. Remarkably, the rotation ansatz (\[typeBC\]) is compatible with the presence of orientifold planes, as our example explicitly shows. Indeed, by construction we know that there is an O5-plane located at $x^6 = 0$. At this point $\a = \pi/2$ and so (\[typeBC\]) reduces to the usual O5-plane projection (in flat space), but notice that this is no longer true as soon as $x^6 \neq 0$.
When describing this background as a non-geometric compactification, one considers $\{ x^2, x^4, x^3, x^6\}$ to expand a four-torus $\T^4$, over which the $\T^2$ expanded by $\{x^5, x^1\}$ is fibered. As we perform the identification $x^6 \sim x^6 + 1$, $\T^2_{(x^5, x^1)}$ will suffer a non-geometric monodromy (namely a T-duality transformation) that signals the presence of the non-geometric Q-flux (\[Qflux\]). Notice that this is a global definition, and that there is no obvious local definition of a Q-flux, like the one for a 3-form flux $H_3$ via $H_3 = dB$. However, our background example above suggests how such definition could be. Consider the quantity q\_[a]{}\^[b c]{} = \_[a]{} ( [ \_R\^ \^[bc]{} \_[Op]{} \_L e\^[-\_0]{} \_L\^\_L]{}) \[localdef\] defined in a local, geometric neighborhood containing an $Op$-plane, where $\g^{bc}$ is the antisymmetrized product of two $\g$-matrices. The usual Q-flux is then obtained by integrating (\[localdef\]) over a one-cycle $\g_a$: Q\_[a]{}\^[bc]{} = \_[\_a]{} q\_a\^[bc]{} \[globaldef\] (see [@defs] for alternative, possibly related definitions). It is easy to see that, by plugging (\[defangle\]) and (\[typeBC\]) in the above expressions and setting $p = 5$ one recovers the Q-flux component (\[Qflux\]) present in our background.
It would be interesting to compute (\[localdef\]) and (\[globaldef\]) for more involved non-geometric compactifications, as well as to generalize the notion of Q-charge, along the lines of [@gh05; @glw06; @dft07]. In any case, it should be clear what the intuition behind the above definitions is. The local quantity (\[localdef\]) measures how the relative rotation generated by $\g^{bc}$ between left and right spinors varies along the internal manifold. Hence it will always vanish for the standard, geometric ansätze [@fluxhet; @bb96]. On the other hand, if (\[globaldef\]) does not vanish for some closed path $\g_a$, then there is a spin monodromy that acts differently for the internal spinors $\chi_L$ and $\chi_R$. This could not possibly be for a geometric compactification, since in that case both $\chi_L$ and $\chi_R$ are the same kind of objects, i.e., sections of the same spin bundle, and should transform in the same way when going around $\g_a$. Thus, (\[localdef\]) and (\[globaldef\]) provide a suitable way to measure global non-geometrical aspects of string compactifications. The hope is that, with these definitions at hand, one can better understand what the space of non-geometric vacua is.
**Acknowledgements**
We wish to thank A. Font, M. Graña, L. E. Ibáñez and P. Koerber for useful discussions. The work of F.M. is supported by the European Network “Constituents, Fundamental Forces and Symmetries of the Universe", under the contract MRTN-CT-2004-005104. W.S. would like to thank the University of Munich for hospitality.
[99]{}
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[^1]: We are using the convention $4\pi^2\a' = 1$.
[^2]: Following the Hodge star conventions in [@gkp01], for a $p$-form $\a$ on a manifold $\cm$ we have that $\a \wedge *_{\cm} \a = - \a\cdot\a\, d{\rm vol}_\cm$.
|
---
abstract: |
An origin and necessity of so called conformal (or, Penrose-Chernikov-Tagirov) coupling of scalar field to metric of $n$-dimensional Riemannian space-time is discussed in brief. The corresponding general-relativistic field equation implies a one-particle (quantum mechanical) Schrödinger Hamiltonian which depends on $n$, contrary to the Hamiltonian constructed by quantization of geodesic motion, which is the same for any value of $n$. In general, the Hamiltonians can coincide only for $n =
4$, the dimensionality of the ordinarily observed Universe. In view of the fundamental role of a scalar field in various cosmological models, this fact may be of interest for models of brane worlds where $n > 4 $.\
PACS numbers: 04.20.Cv, 04.50., 04.62.+v
address: |
N N Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980, Russia,\
e–mail: [email protected]
author:
- 'E. A. Tagirov,'
title: A mystery of conformal coupling
---
=10000
In papers [@cht; @tg1], it was found that, in the framework of the canonical quantum field theory (QFT), wave equation determining quantum motion of a neutral scalar particle in the general $n$-dimensional Riemannian space-time $V_n$ with the metric form $$ds^2 = g_{\alpha\beta} (x) dx^\alpha dx^\beta,
\quad \alpha, \beta, ... = 0, 1, ..., n-1,$$ should [*necessarily*]{} be of the form $$\Box_g\varphi + \xi\, R(x)\, \varphi +
\left(\frac{mc}{\hbar}\right)^2 \varphi = 0. \label{r}$$ Here $\varphi \equiv \varphi(x)$ is the real scalar field, $\Box_g $ is the ordinary generalization of the d’Alembert operator to $V_n$, i.e. $$\Box_g {\stackrel{\rm def}{=}}\frac1{\sqrt{-g}} \frac\partial{\partial x^\alpha}
\left(\sqrt{-g} g^{\alpha\beta} \frac\partial{\partial x^\beta}\right),
\quad g {\stackrel{\rm def}{=}}{\rm det}\|g_{\alpha\beta}\|,$$ $R(x)$ is the scalar curvature of $V_n$, and $$\xi = \xi_c (n) {\stackrel{\rm def}{=}}\frac{n-2}{4(n-1)}\ .$$ This equation is referred as the equation of with *conformal coupling* (CC) to external gravitation or as the Penrose–Chernikov–Tagirov (PCT) equation. If $m = 0$, the equation is *covariant* with respect to the *conformal mapping* of manifold $V_n$ to $V'_n$ simultaneously with $\varphi$ : $$V_n \rightarrow V'_n : g_{\alpha\beta}(x)\ \rightarrow \ g'_{\alpha\beta}(x)
= \Omega^2 (x) g_{\alpha\beta}(x),\\
\varphi(x)\ \rightarrow \ \varphi'(x)=
\Omega^\frac{2-n}{2} \varphi(x) \label{cm}$$ and is *invariant* with respect to *conformal transformations of coordinates* $x^\alpha$ if $V_n$ admits such transformations which is not the generic case, see more details on the conformal symmetries, e.g., in [@tt]. Thus, the CC equation (1) with $\xi = \xi_c $ and $ m = 0$ has the same conformal symmetry as the wave equations for a photon and neutrino and the equation of isotropic geodesic lines (the classical equation of motion of massless particles) have. This symmetry feature of CC equation for $ n =4$, i.e., for $\xi_c = 1/6 $, had been noted by R. Penrose [@pen] but only as a mathematical possibility, with no discussion and physical consequence.
However, eq.(1) violates explicitly the well-known Einstein principle of local equivalence according to which it should take the form of the ordinary Klein-Gordon equation at the origin of the normal Riemannian (locally quasi-Lorentzian) space-time coordinates fixed at a given point $\{x^\alpha\}_0\, \in \,V_n$, see below. The latter requirement is satisfied generally only if $\xi = 0$, that is for the minimal coupling (MC) of $\phi$ to gravitation. In itself, the conformal symmetry mentioned above was not a sufficient argument in 1960’s to break, in favor of CC , the fundamental principle sanctified by the name of the originator of General Relativity. The main arguments of [@cht; @tg1] were based not on the conformal symmetry in the particular case of $m = 0$; they resulted from the physical idea that motion of a high-energy quantum particle is asymptotically close to that of the classical one and, thus, to the geodesic lines in the case of $V_n$. In view of the radical change which is caused by these arguments and their importance for the final conclusion of the present paper, let us consider in brief the logic of which lead us (N. Chernikov and the present author) to $\xi = \xi_c$.
We considered canonical quantization of the field in the $n$-dimensional de Sitter space-time $dS_n$ of radius $r$ and, at first, had naturally started with the MC version of eq.(1), i.e with $\xi = 0$, in accord with the principle of equivalence. We looked for a Fock space the cyclic state of which is invariant with respect to the de Sitter group $SO (1, n)$ (the symmetry of $dS_n$) and the subspace of the *one-quasi-particle* states of which realizes an irreducible representation of this group. As a result, we found an one-parameter family of such Fock spaces. They are unitarily non-equivalent for different values of the parameter. The question was: is there a physical reason to distinguishes one of them? Our reasoning was that such representation space should include states corresponding to the quasi-classical motion of *a particle*. The phases of such states should asymptotically satisfy the Hamilton-Jacobi equation for large eigenvalues of the Casimir operator of the representation of $SO_{1,n}$. However, it had turned out that no such representation space exists. Instead, we observed that such space would exist and be unique if the mass term $(mc/\hbar)^2 \phi $ in the initial MC version of eq. (1) were shifted by $(n(n-2)/(4 r^2)) \phi $. We had just done it and realized that this term is just the value of the second term of eq.(1) for $dS_n$ if $\xi = \xi_c (n) $.
Further, it is easy to see that $R(x)$ is a unique scalar of the dimension ${\rm [length]}^{-2}$ constructed of the metric tensor and its derivatives in the generic $V_n$, which takes on the value $ n(n-1)/r^2$ in $dS_n$ [@tg1]. It should be noted also that, in the Friedmann–Robertson–Walker models, the approach of [@cht] singles out a unique family of unitary equivalent Fock spaces [@bt].
As soon as eq.(1) is accepted with $\xi \neq 0$ (this case, more general than that of $\xi = \xi_c$, is appropriate to refer as the non-minimal coupling (NMC) version of eq.(1) ), a very important physical consequence follows. Namely, variation of the corresponding action integral by $g^{\alpha\beta}(x) $ gives a new energy-momentum tensor $T_{\alpha\beta}(x;\, \xi)$ which differs from the traditional one $T_{\alpha\beta}(x; 0)$ for the minimal coupling: $$T_{\alpha\beta}(x; \xi) = T_{\alpha\beta}(x; 0) \\
- \xi (R_{\alpha\beta } - \frac12 R g_{\alpha\beta } + \nabla_\alpha
\partial_\beta - g_{\alpha\beta }\Box_g ) \phi^2, \label{t}$$ where $\nabla_\alpha$ is the covariant derivative and $T_{\alpha\beta}(x;0)$ is the energy-momentum tensor. If $\xi = \xi_c(n) $ , one has $$T^\alpha_\alpha (x; \xi_c ) = \left(\frac{mc\phi}{\hbar}\right)^2$$ *on solutions of* eq.(1). Thus, the trace of $T_{\alpha\beta}(x; \xi)$ vanishes for $m = 0$ contrary to the one of $T_{\alpha\beta}(x; 0)$. The vanishing trace is necessary for definition of the conserved quantities of massless fields, which correspond to the conformal symmetry in $V_n$’s having such a symmetry, see details in [@cht]. Here, the most remarkable fact is that the difference between $T_{\alpha\beta}(x; \xi)$ and $T_{\alpha\beta}(x;
0)$ retains even in the flat space-time $E_n$, where $R(x)\equiv 0$ and it is of no importance which value the constant $\xi$ has. In particular, this result eliminated a certain perplexity existed until then and consisted in that the conformal invariance of the ordinary d’Alembert equation $\Box\phi = 0 $ does not lead to the standard expressions for the corresponding conserved quantities in terms of the canonical energy-momentum tensor contrary to the Maxwell, Dirac-Weyl and isotropic geodesic world-lines equations. It was known (private communication by V.I. Ogievetsky) that this paradoxical situation can be resolved by some extra terms to the canonical tensor but but an origin of them was not known.
Shortly later, it was also shown [@ccj] that “the new improved energy-momentum tensor” has finite matrix elements in the ordinary *Poincare-invariant* quantum theory of the non-linear scalar field in $E_4$ with self-interaction $ \lambda \phi^4$ (in the Lagrangean).
As concerns the principle of equivalence, a more deep look at eq.(1) taking into account its quantum nature shows its actual accord with the principle of equivalence formulated in terms of the Feynman propagator [@grb].
By now, eq.(1) came to the ordinary and wide use for the scalar field in $V_n$. The most models of cosmic inflation include a fundamental inflaton scalar field and there are serious arguments that it should necessarily obey the NMC equation, very probably with the conformal covariant self-interaction term [@far]. Now, even prospectives of detection of the value of $\xi$ from astrophysical observational data are under discussion, see, e.g., [@prkp] and references therein.
In view of the importance of the question, it should be emphasized that $\xi
= \xi_c$ had arose in the process of determination of the one-particle subspace of the particle-interpretable Fock space, that is in the process of extraction of quantum mechanics (QM) of a particle from the QFT in $dS_n$. This is a particular case of that which may be called the field–theoretical (FT) approach to construction of QM in $V_n$; development of the approach to the general $V_n$ is given in [@tg2; @tg3].
However, there is another approach to the same problem: quantization of the corresponding classical mechanics (CM). For the free scalar particle, the latter is the Hamilton theory of the geodesic lines in $V_n$. Which value of $\xi$ does this approach suggest? There are different formalisms of quantization of the finite-dimensional Hamiltonian dynamics’. Those of them which reproduce, under appropriate conditions, the mathematical structure of the standard non-relativistic QM require *logically* $\xi = 1/6$ for *any* value of dimensionality $n$ of $V_n$ ! That is the values of $\xi $ to which lead the two approaches coincide only if $n = 4$, the space-time dimensionality of the Universe which we observe. This conclusion is so surprising that some explanations are necessary on the way which leads to it.
In fact, one should confront the two approaches only on the level of the standard non-relativistic quantum mechanics’ which follow from them. To this end, a particular frame of reference should be introduced, that is a representation of $V_n$ as a foliation by space-like hypersurfaces $\Sigma_t
(x) = const $ (spatial sections of $V_n$) enumerated by values of an evolution parameter $t$, i.e. the time coordinate. Since a value of the constant $\xi$ only is in interest, the task can be simplified essentially by restriction to the case of the globally static $V_n$ and the normal geodesic frame of reference in which the metric has the form $$ds^2 = c^2 dt^2 - \omega_{ij} (u) du^i du^j, \quad
u\in \Sigma_t,$$ where $i, j, ... = 1,..., n-1$. It is convenient that scalar curvatures $R(u)$ for the metric tensors $g_{\alpha\beta}(u)$ and $\omega_{ij}(u) $ coincide in the globally static case.
If the mentioned assumptions are made, the systems under consideration (the scalar field and the particle moving along a geodesic line) have conserved energy. Therefore the vacuum and quantum particle states can be determined, which are stable. In the both approaches, the quantum one-particle state space can be represented as the space of functions $\psi (t,\, u )$ which are square integrable (as functions of $u$) over $\Sigma_t$ with its invariant measure and therefore are the probability amplitudes to detect the particle at the point $u$ of $\Sigma_t$. They are solutions of the following Schrödinger equation [@tg2; @tg3]: $$i\hbar \frac{\partial}{\partial t} \psi
= mc^2 \sqrt{1 + \frac{2\hat H_0}{mc^2}}\ \psi, \label{ham}$$ $\hat H_0$ being the non-relativistic Hamiltonian $$\hat H_0 = - \frac{\hbar^2}{2m}(\triangle_\Sigma - V^{(q)}(u)),$$ where $\triangle_\Sigma$ a is the Laplace–Beltrami operator on $\Sigma_t$ and $V^{(q)} (u) $ is the so called *quantum potential*.
In the FT–approach, where eq.(\[ham\]) arises as a result of restriction to the positive energy solutions of eq.(1), see [@tg2], one has $V^{(q)} (u) = - \xi R(u)$ with $\xi = \xi_c$ to satisfy the requirements of [@cht; @tg1].
In the CM-approach, the situation is more complicate, since the function $V^{(q)} (\xi)$ depends not only on a formalism of quantization, but also on choice of coordinates $u^i$, that is $V^{(q)} (u)$ is not a scalar. This circumstance looks very strange, but in the CM–approach coordinates $u^i$ together with the canonically conjugate momenta $p_i$ are the basic *observables* contrary the FT-approach where the corresponding observables are quadratic functionals of the field [@tg2]. According to analysis by C. Rovelli [@rov], information on quantum system unavoidably includes some information on the classical devices which are used to observe the system and one may think that the quantum potential includes information on the system of coordinates used *to observe* a position of the particle. Therefore, it is not so suprising that quantum dynamics depends on which system of the basic observables $u^i,\ p_i$ is taken to describe it.
Then, to compare quantum potentials arising in different formalisms of quantization in the CM-approach a concrete system of coordinates $u^i$ should be chosen. Such system is suggested by B. DeWitt’s construction [@dw] of the WKB-propagator in $V_n$, which is equivalent, in fact, to introduction of *Riemannian* coordinates $y^i $ in a neighborhood of a point of observation $y^i_0 = y^i(u_0)$, see details in [@tg3]. They define a position of the point $u$ through the geodesic distance $s(u,\, u_0 )$ and the unit tangent vector $(du^i/ds)_0 $ along the geodesic line at $u_0$ : $$y^i (u) {\stackrel{\rm def}{=}}s(u,\,u_0)\left(\frac{du^i}{ds}\right)_0 \label{y}$$ In this notation, the quantum potential $(\hbar^2/2m) V^{(q)} (y)$ in the Schrödinger equation (\[ham\]) corresponding to DeWitt’s propagator has the form $$V^{(q)}(y) = \frac16 R(y) + O(y^i - y_0^i). \label{vr}$$ Thus, in this form, the non-minimal term in the non-relativistic Hamiltonian appeared as early as 1957. However, DeWitt considered it as an unfavorable phenomenon and preferred to avoid it changing the Lagrangean in the action integral. This curious story shows once more the radical nature of transition to conformal coupling in [@cht].
Further, it was shown in paper [@tg3] that one can use ambiguities in the Beresin–Shubin [@bsh] and Feynman quantizations of the geodesic motion so that the quantum potential will be of the form (\[vr\]) when the Riemannian coordinates are taken as observables of space position. This condition removes the ambiguities and fixes these formalisms in their application to the geodesic motion as well as establishes their concordance with the WKB formalism. However, the potential (\[vr\]) does not depend on $n$, the dimensionality of $V_n$. This is consistent with the field-theoretically deduced $V^{(q)} (u) = -\xi_c (n) R(u)$ only if $n=4 $. This fact is just that was meant above as logical inconsistency of the FT- and CM-approaches to formulation of quantum mechanics in $V_n$. Attempts to explain it by topological difference between $dS_n$ and $V_n$ (in the simple versions of the CM approach [@tg3], the latter is supposed to be topologically trivial) or by that the two approaches should not be compatible because only one of them is correct will meet the question: then, why they are compatible for $n = 4$?
A pragmatically inclined physicist might be satisfied with that the matters are OK, at least, in the four-dimensional Universe observed ordinarily and consider the problem with $n \neq 4 $ as having only an academic interest. However, recent time very interesting models of “the world on a brane” are in intensive discussion, according to which the ordinary matter “lives” (evolves in time) on a three dimensional space while gravitation and, probably, some other exotic fields act in an embracing space of a higher dimensionality, see a review of these models, e.g., in [@rub]. Is a scalar field, which plays a fundamental role in various cosmological models, an “ordinary” matter or not, this is a question of the concrete model.
N.A. Chernikov and E.A. Tagirov, [*Annales de l’Institute Henry Poincarè*]{} [**A9**]{} (1968), 109. E.A. Tagirov, [*Ann.Phys. (N.Y.)*]{} [**76**]{} (1973), 561. R. Penrose, in: [*Relativity, Groups and Topology*]{}, B.S. DeWitt ed., Gordon and Breach, London, 1964, p.255 E.A. Tagirov and I.T. Todorov, [*Acta Phys. Austriaca*]{}, [**51**]{} (1979), 135. K.A. Bronnikov and E.A. Tagirov, Preprint JINR - P2 - 4151, Dubna, 1968 (in Russian); \[English translation with comments: [*Gravitation & Cosmology*]{}, [**10**]{} (2004), 1; arXiv:[*gr - qc/0412128*]{}\] . C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. (N.Y.), [**59**]{} (1970) 42. A.A. Grib and E.A. Poberii, [*Helv.Phys.Acta*]{} [**68**]{} (1995), 380. T. Prokopec and E. Puchwein, arXiv:[*astro - ph/0403335*]{}. V.Faraoni, [*Int.J.Theor.Phys.*]{} [**40**]{} (2002), 2259. E.A. Tagirov, [*Grav.Class.Quan.*]{} [**16**]{} (1999), 2165; arXiv: [*gr - qc/9812084*]{}. E.A. Tagirov, [*Int.J.Theor.Phys.*]{} [**42**]{} (2003), 465; arXiv: [*gr - qc/0212076*]{}. C. Rovelli, [*Int.J.Theor.Phys.*]{} [**35**]{} (1996), 1637. F.A. Berezin and M.A. Shubin, [*Schrödinger Equation*]{} (in Russian), Moscow: Moscow University Press, 1985. B.S. DeWitt, [*Rev.Mod.Phys.*]{} [**29**]{} (1957), 377. V.A. Rubakov, [*Phys.Uspekh.*]{} [**44**]{} (2001), 871; arXiv: [*hep - th/0104152*]{}.
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---
abstract: 'We describe a new 512-CPU Beowulf cluster with Teraflop performance dedicated to problems in computational astrophysics. The cluster incorporates a cubic network topology based on inexpensive commodity 24-port gigabit switches and point to point connections through the second gigabit port on each Linux server. This configuration has network performance competitive with more expensive cluster configurations and is scaleable to much larger systems using other network topologies. Networking represents only about 9% of our total system cost of USD\$561K. The standard Top 500 HPL Linpack benchmark rating is 1.202 Teraflops on 512 CPUs so computing costs by this measure are \$0.47/Megaflop. We also describe 4 different astrophysical applications using complex parallel algorithms for studying large-scale structure formation, galaxy dynamics, magnetohydrodynamic flows onto blackholes and planet formation currently running on the cluster and achieving high parallel performance. The MHD code achieved a sustained speed of 2.2 teraflops in single precision or 44% of the theoretical peak.'
address: ' Department of Astronomy and Astrophysics and Canadian Institute for Theoretical Astrophysics, [email protected], [email protected], [email protected], [email protected], and [email protected]'
author:
- 'John Dubinski, Robin Humble, Ue-Li Pen, Chris Loken and Peter Martin'
title: ' High Performance Commodity Networking in a 512-CPU Teraflop Beowulf Cluster for Computational Astrophysics '
---
Introduction
============
The Olympic motto “Citius, altius, fortius” (“Swifter, higher, stronger”) succinctly describes the direction of 21st century parallel supercomputing - swifter processors, higher resolution and stronger fault-tolerant parallel algorithms. In computational astrophysics, the need for all of these qualities is perhaps greater than most supercomputing applications. While the bulk of the physics of the formation of the planets, stars, galaxies and the large-scale structure in the universe are now largely understood with the initial conditions well posed in some cases, the challenge of computing the formation of objects and structures in the universe is difficult. The standard methods of computational fluid dynamics (including magnetohydrodynamics) and gravitational N-body simulation are stressed by the large dynamic range in density, pressure, and temperature that exist in nature. Ever-finer computational meshes and greater numbers particles in N-body simulations are needed to capture the physics of the formation of things in the universe correctly.
In recent years, the relative low cost of commodity servers running the Linux operating system and more importantly switches and network interface cards has led to the growth in performance and competitiveness of Beowulf clusters versus vector machines and symmetric multi-processors. The recent list of the Top 500 supercomputers [@top500] last year reveals that 55 out of the top 100 are clusters. Most high performance Beowulfs now use low-latency, proprietary networking with costs several times that of current commodity gigabit networking. Large-scale 200+ port gigabit switches are a slightly cheaper option but still eat significantly into the cost of a cluster. Networking costs alone can dominate cluster costs.
The spirit of building a Beowulf cluster is to get the highest performance on a limited budget so the incorporation of expensive networking solutions defeats this fundamental motivation. We describe here a cheaper networking configuration based on off-the-shelf 24-port commodity gigabit switches in a 256-node/512-CPU cluster. We achieve a total cross-sectional bandwidth of 128 Gbit between 256 nodes. The innovation to achieving this performance is a networking topology involving both a stack of commodity 24-port gigabit SMC switches and the configuration of each Linux server as a network router through a second gigabit network port. Our economical networking strategy competes directly with both large-scale 200+ port switches and proprietary low-latency networking at a fraction of the cost. The high throughput provided by Linux routing capability makes this performance possible. We ran the HPL LINPACK standard benchmark and achieved 1.202 Tflops placing it in the top 50 computers according to the Top 500 list of November 2002.
We also describe the performance and results of 4 parallel applications for astrophysical problems that are currently running on the cluster spanning problems in large-scale structure formation, galactic dynamics, magnetohydrodynamic flows on blackholes and planet formation.
Cluster Hardware Specifications
===============================
In this section, we briefly describe the design and hardware specifications of the cluster. The hardware was assembled and installed by Mynix Technologies, Montreal, PQ at the University of Toronto [@mynix]. The installation of the Linux operation system was done using the OSCAR [@oscar] cluster package by our group.
Compute Nodes
-------------
The CITA Beowulf Cluster dubbed McKenzie is comprised of 268 dual Xeon 2.4 Ghz rack-mounted format Linux servers (536 processors in total) distributed over 7 racks. All the nodes are 1U format with the exception of two 2U format head nodes. The hardware specifications for each server are given in Table 1. From the total of 268 nodes, 256 nodes are dedicated to parallel, message-passing supercomputing, 8 nodes for code development and smaller scale applications, 2 spare nodes also running to act as hot replacements in case of hardware failures on the compute nodes and 2 head nodes. The main head node “bob" contains the home disk space as well as the root space images and Linux kernels for the compute slave nodes. All slave nodes boot through the network from bob using PXElinux allowing easy kernel upgrades and maintenance for the cluster nodes. A secondary master node “doug" mirrors bob’s root and home directories on a daily basis and acts a backup in case of a bob crash. This allows quick recovery and continued operation if bob fails.
Switches and Cables
-------------------
The cluster is networked using 19 SMC Tiger 24-port configurable gigabit switches. Seventeen switches are used for the main 256-node cluster network while the development cluster is connected to its own switch and the 19th switch is reserved as a spare and connected to the running spare nodes. We defer the discussion of the network topology to the section below.
Rack Configuration
------------------
The nodes and switches are mounted on 7 44U high racks. The main 256 compute nodes are mounted on the first 3 and last 3 racks. The central rack is reserved for the head nodes, development nodes and spares as well as the stack of switches. The switches are mounted on the back of the racks to simplify the cabling.
--------------- -------------------------------------
Nodes: 256 main compute nodes +
8 development nodes +
2 head nodes + 2 spares = 268 total
Chassis: Chenbro RM11802 1U format
Motherboard: Intel Westville SE7500WV2
CPUS: Dual Xeon 2.4GHz processors
E7500 chipset/512KB L2 cache,
400 MHz system bus
536 CPUs in total
Memory: 1 Gbyte DDR-200 RAM
268 Gbytes total
Disk Drives: 2X80 Gbyte Seagate IDE drives
Networking: Dual Intel Pro/1000XT gigabit
Switches: 19 SMC Tiger SMC8624T 24-port
managed gigabit switches 1U format
Storage: 3TB Arena RAID array
Node cost: USD\$513K
Network cost: USD\$48
Total cost: USD\$561K
--------------- -------------------------------------
: Mckenzie Part List and Cost
Cost
----
The cost for the compute nodes plus storage and incidentals was USD\$513K. The cost for the gigabit switches plus cabling was USD\$48K representing only 9% of the total cluster cost of \$561K.
Networking
==========
When considering options for networking the cluster, we decided against proprietary low-latency networking and large-scale 200+ port switches after realizing that they would consume a significant fraction of our modest budget. The prices of 24-port gigabit switches this past year had dropped significantly so we considered networking schemes that could take advantage of this inexpensive hardware.
Fat-tree networks that link switches together through a hierarchy are a straightforward configuration that provide connectivity but quite low cross-sectional bandwith. Each node comes with a second gigabit port, however, and could be exploited in some way using point to point connections with other machines. With this in mind we came up with the following scheme. Switches (or pairs of trunked switches) can be thought of as separate tightly coupled networks of up to a few dozen nodes. These nodes can be assigned to vertices in some more complex network topology. How can one connect these vertices to form a high-bandwidth global network linking all the compute nodes? After loading a switch with compute nodes there are few ports left to link to other vertices. However, each compute node provides a single connection through its second port that can in principle be connected to a compute node on an adjacent network vertex. If each linux node has routing enabled, network traffic can be relayed through the second port to other nodes. This was the strategy we ultimately employed in setting up a high-bandwidth global network.
The Fat-Tree Maintenance Network
--------------------------------
Before setting up our high performance global network, we first set up a simple fat-tree that we treat as a robust but lower-performance maintenance network. This network was used to install the cluster originally but also acts as a bootstrap for configuring a high-performance network. In our final configuration, we trunk pairs of switches together making network vertices containing 48 ports with a total of 8 vertices using 16 switches. We connect 32 compute nodes to each vertex. For 6 vertices, we run a 4-port trunk to a 17th master switch filling it to capacity. The fat-tree networking is completed by connecting the last two vertices with 4-port trunks to 2 vertices trunked to the master switch. The head nodes bob and doug, plus the development and spare nodes are all plugged into this network as well allowing direct communication between all available nodes.
This fat-tree network configuration is robust and provides connectivity to all the 256 nodes and is adequate for maintenance, installation and embarassingly parallel applications. However, the cross-sectional bandwidth is not at all ideal for heavy duty MPI applications. We now describe how we use this fat-tree network to bootstrap to our high-bandwidth network which runs predominantly through point to point connections between the second network ports of all machines.
The Cross-Diagonally Connected Cube Network
-------------------------------------------
The network vertices described above can be thought of as independent networks that we can connect to each other according to some topology. The use of a master switch to build a fat tree is one such topology but it has quite low cross-sectional bandwidth. How can we use the second port on each compute node to connect these vertices to increase the network bandwidth and minimize network hops to reduce latency? There are many possibilities but we finally settled on a cubic network topology (Figure \[cdcc\]).
In our chosen topology, the 8 network vertices are arranged on the corners of a cube. Communication between vertices can occur along the edges of the cube but we also connect opposite corners through diagonals that cross through the cube centroid. We call this topology a Cross-Diagonal Connected Cube (CDCC). Each vertex then has 4 outgoing lines of communication to its adjacent corners and opposite along the diagonal.
The CDCC topology requires hard-wired routing tables between nodes which we constructed using a variety of scripts. We have 32 gigabit lines on each vertex provided by the second network port on each compute node and 4 vertex connections. We therefore assign 8-gigabit lines to each vertex connection. These lines connect 8 compute nodes on one vertex to another 8 on a neighbouring vertex through direct port to port connections with a straight-thru cable. In this way, there is an 8-gigabit pipeline connecting each vertex which must be shared between 32 compute nodes. Also, the 4-port trunks connecting pairs of switches are 4- gigabit pipelines relaying traffic for 16 compute nodes per switch.
In this way, the CDCC topology with 256 nodes approximately represents a fully-switched network running at 250 Mbit, although it is slightly better since nodes on the same switch are fully-switched at 1 Gbit. The full-duplex cross-sectional bandwidth is 128 Gbit for 256 nodes. When constructing the routing tables, the maximum number of network hops to get from one node to another is 4 so latency is greater than a fully-switched system but we find in practice for our applications and standard benchmarks this is not a large hindrance. The performance of this configuration is about half the 500 Mbit performance that is expected for large gigabit switches that typically connect 16 ports to an 8- gigabit backplane internally. Latency is also about 4 times greater. However, the total cost for the 17 required 24-port switches plus cabling is much less than the cost of a single 256-port capable switch. We show below that our cluster provides competitive benchmark speeds for similar clusters with better networking and performs well for our applications.
Cabling
-------
The task of cabling the cluster also proved to be formidable but we came up with some ways to minimize confusion and tangling. Connections from the first ethernet port to the switches is straightforward if the switches are centralized to one location at the centre of the rack array in a stack. Network cables are run over and under the cluster in small bundles and colour coded by rack number and then plugged into their appropriate ports in the central stack of switches. Although tedious, this stage of the cabling took a small team of people two days to complete. We point out that the same cabling effort is required even for more expensive multi-port switches so there is no significant overhead in manual labour.
In the second stage, it was necessary to wire the CDCC network using the point to point connections through the second network port on each node. Wiring these connections could potentially lead to a tangled mess of cables if the nodes are ordered sequentially according to their vertex numbers. A simple solution to this wiring problem is to re-order the nodes on the racks in logically connected pairs so that they are also physically connected to one another. In this way, only a short 1 foot length of cable is required for each contiguous pair and wiring can be done quickly. The logic of the connectivity of the CDCC network was used to create the correct ordered list of nodes on the rack for this arrangement as well as a wiring diagram for the connecting cables to the right switches in the fat-tree diagram. The only extra level of complexity for doing things this way in comparison to 200+ port monster switches is the node ordering and the need to make sure each cable is connected to the correct port in the switch stack. In practice, we have re-wired the network a couple of times to test the performance of different topologies and it generally only takes a few hours to do.
Routing
-------
Another essential (and complex) part of getting the CDCC network to perform was the configuration of the routing tables for each Linux server. Network packets relayed through each server on this server have to know where to go according to our defined network topology.
A useful part of our design is that we have a back-up maintenance network (the fat-tree) that we can always use to gain access to each node in case of routing bugs or broken links in the CDCC network. This allowed us to experiment extensively with the scripts for generating the routing tables of the CDCC and led to an optimized system. Our strategy is to generate purely static routing tables on each server with a simple script at startup. Each of these scripts contains routing commands that establish about 50 static routes to individual nodes and sub-networks (vertices) in the CDCC network. Each node has its own unique routing table. We find this system works very well in practice and can be taken up and down quickly if necessary.
Our arrangement is also very fault-tolerant in case of node failures or broken links. If one or more nodes fails in the system, it will create holes in the CDCC network but as a contingency we can route around the broken nodes simply by using the fat-tree network. We have written a simple network repair script that pings all neighbours on the CDCC - if a node fails to answer we attempt to establish a direct route to it through the fat-tree - if that fails the node is assumed to be dead. In this way, we can continue to run jobs on the cluster even when main compute nodes go down. Either the hot spares, or development nodes can fill in while we replace the failed nodes. Since our networking is similarly fragmented into many switches, the hardware failure of one switch will only take out those nodes connected to it so most of the cluster can still run while the broken switch is being replaced.
Figure \[cdcc\] shows some final details of our routing scheme that are worth pointing out. It turned out that the network performance of the trunks was not as good as we expected. We therefore modified our routing scheme to avoid using the trunks as much as possible. Network traffic from one node to another on different vertices used cross links that avoid the trunks. The only network traffic going through the trunks is between the nodes on bonded pairs of switches.
HPL Benchmark
-------------
As an initial test of performance, we ran the High-performance Linpack Benchmark (HPL), a portable implementation for distributed-memory computers to measure the speed of our cluster and compare with others on the Top 500 list [@top500]. This code uses the Message Passing Interface (MPI) libraries. The actual benchmark involves the inversion of the largest matrix that can be stored on your cluster. In our case, the matrix size was $160000 \times 160000$ elements. We compiled the code using the Intel C compiler icc version 7.0 and linked to the Intel MKL math libraries and Kazushige Goto’s optimized BLAS libraries [@goto]. We used the LAM version 6.6b1 as the MPI library. We achieved a sustained speed of 1.202 Teraflops about 48% the theoretical peak speed of 2.46 Teraflops for 256 Xeon 2.4 GHz chips. This ratio of sustained to theoretical peak speed is comparable to many of the machines quoted on the November 2002 Top 500 list including similar sized clusters running with proprietary networking such as Myrinet. Our ranking on the list would currently be number 34. Of course, it remains to be seen how we will rank on the next list released in a few months since these trends tend to evolve very quickly!
Astrophysical Applications
==========================
The Beowulf cluster Mckenzie was constructed for problems in computational astrophysics. We describe here several applications that are being used to investigate areas on many astronomical scales from planets to the entire universe. We also highlight the performance of the various codes we have started to use.
Cosmology
---------
Cosmological N-body simulation is the main numerical tool used to study the formation of structure in the universe. Dubinski et al. [@dub03] have developed a new parallel algorithm that combines the standard techniques of particle-mesh methods (PM) [@hockney81] with the Barnes-Hut (BH) oct-tree algorithm [@barnes86]. The PM method is used to measure the long-range gravitational forces from particles within a periodic cube representing a typical volume in the universe. These forces are refined on the sub-mesh scale using the BH algorithm by building a grid of oct-trees within the simulation domain. The new code is called GOTPM for Grid-(of)-Oct-Trees-Particle-Mesh. The code uses slab domain decomposition to distribute the particles among the CPUs. During the PM phase, Poisson’s equation for the gravitational potential is solved using Fourier methods. We use the MPI implementation of FFTW [@frigo98] to do the FFT’s during this phase which assumes slabs of equal width. A grid of BH oct-trees is then constructed within each slab and used to refine the forces at short range. Force accuracies are typically 0.5% using this method. To ensure load balance, the slab widths are allowed to vary during the tree phase with the width determined by the amount of computational work in the previous timestep. A fair amount of communication is required to move particle data at each step since they must move back and forth from an equal width slab decomposition for the PM phase and variable width slabs for tree phase. Despite this high demand for bandwidth, communication overhead is about 25% even with our inexpensive networking configuration.
Figure \[fig-cosmo\] shows one snapshot from a cosmological simulation using a standard model ($\Omega = 0.3$ and $\Lambda=0.7$) in a 200 $h^{-1}$ Mpc (650 million light years) box. This simulation used a $1536^3$ mesh to solve Poisson’s equation using Fourier methods in single precision. A total of $N=768^3$ (453M) particles are used and simulated for 5000 timesteps. An animation of a fly-through of this dark matter simulation as it structure develops is available at our website [@animations]. In a 256 processor run, the wallclock time per timestep grows from 80 seconds at the beginning of the simulation to about 200 seconds once clustering develops with CPU time being dominated by the use of trees. The spatial resolution of these simulations is set by the gravitational softening radius used to shut down the $1/r^2$ Newtonian force at small separations. For this simulation, we set the softening radius to 1 kpc (3000 light years) so we achieve a spatial resolution about 65 times greater than what could be achieved using a PM method by itself.
Galaxy Dynamics
---------------
We have also applied a pure parallelized N-body treecode [@dub96] to study galaxy interactions. The main mode by which spiral galaxies are transformed into ellipticals is through merging. When galaxies interact, the ordered energy of rotation and bulk motion of the two spiral galaxies is redistributed in the form of randomly oriented orbits. This process only requires a few dynamical times and the final structure of the merger remnant resembles and elliptical galaxy.
To study this process at high resolution, we built two models of spiral galaxies composed of a disk, a central bulge and surrounding dark halo using Kuijken & Dubinski’s method [@kd95]. To make the calculation more interesting, we used initial conditions chosen to represent a future configuration of the Milky Way and the Andromeda Galaxy which will likely merge in about 3 billion years. Each model is composed of 128M particles representing stars and 25.6 million particles representing the surrounding dark matter halos for a total of 307.2M particles. The two galaxies are on a collision course and we computed the trajectories for all particles for 5300 timesteps representing a physical timescale of 2.3 billion years. A sequence of 9 images showing the evolution of the system over this timescale is shown in Figure \[fig-gal\]. An animation showing the galaxy collision is available here [@animations].
The code used for these calculations is PARTREE [@dub96], a parallelized treecode based on Salmon’s [@salmon90] original algorithm and ideas of locally essential trees. The code has recently been modified to incorporate asynchronous message-passing in the construction of the locally essential trees to minimize communication overhead which becomes dominant for large numbers of CPUs. With current modifications, communication overhead only used 10% of the wallclock time of 180 seconds per step in the 307M particle simulation. The code should scale well to thousands of processors in its current form assuming the problem size grows as well. A several thousand CPU machine with 2GB/CPU is now large enough to follow the trajectory of about 10 billion particles, the same number of stars in some galaxies and only a factor of 40 away from the number of stars in the Milky Way. Galactic dynamics simulations will reach their ultimate resolution in terms of the number particles representing stars in less than a decade.
Blackhole Accretion
-------------------
We have undertaken MHD simulations with 1400$^3$ zones arrayed in a uniform Cartesian grid, the largest MHD simulations to date (Figure \[fig:mag\]). At this resolution, each full dimensional sweep corresponding to two timesteps took 40 seconds. A series of optimizations allows the code to exploit the hardware’s vector units, hyperthreading OpenMP parallelism, and uses the Message Passing Interface (MPI) to communicate between nodes. The code [@pen03a] is based on a 2nd order accurate in space and time high resolution Total-Variation-Diminishing (TVD) algorithm; it explicitly conserves the sum of kinetic, thermal and magnetic energy; hence magnetic dissipation (at the grid scale) heats gas directly. No explicit resistivity or viscosity is added, and reconnection and shocks occur through the solution of the flux conservation laws and the TVD constraints. Magnetic flux is conserved to machine precision by storing fluxes perpendicular to each cell face.
Each time step requires about 20 seconds, corresponding to 2.2 Tflop sustained in single precision. The operation count was obtained by adding the number of operations in the source code by hand. This is about 44% of theoretical peak speed achievable with the using the SSE2 vector units. The main parallelization overhead is the need for 16 buffer zones between computational domains. This increases the floating point count by 65%. Fatter nodes with more processors and more memory would decrease the surface to volume ratio. The domain decomposition was matched to the physical three dimensional connectivity of the network. Most traffic does not require any routing, and up both gigabit ports per node can be effectively exploited. All message passing latency is hidden by staggering the computation phase: the processes do not need to wait for data, and communication is asynchronous.
Dusty disks and the Formation of planets
----------------------------------------
The earliest stages of the planet formation process, in which micron sized particles of dust somehow come together to form meter and kilometer sized objects and eventually planets, is very poorly understood. Humble, Maddison and Murray [@rjh:03] have developed a new parallel two-phase (dust and gas) smoothed particle hydrodynamics [@jjm:97] and self-gravity code, so that for the first time we can model the non-linear evolution of two coupled fluids in a 3D protoplanetary dusty disk.
A typical calculation uses 125k gas and 125k dust particles and resolves the motion of the entire disk from $2$ to $200$AU. We investigate how dust evolves in a so-called minimum mass solar nebula in which the total mass of the dust is $1$% of the gas phase, which in turn is $1$% the mass of the protostar. Figure \[fig-contour-1\] shows the distribution of $1$cm grains after $10^4$ years.
Our simulations clearly capture the expected settling and density enhancement of the dust in the midplane of the disk where grain growth is most likely to occur, and where gravitational instabilities may emerge. The simulations also reveal the new result that the pressure supported differential rotation between gas and dust is maintained only in a relatively narrow region in radius. Rapid evolution of the dust occurs mostly in this narrow band, the location of which is a strong function of the dust grain size. This results in large variations of structure and dynamics between dusty disks of different characteristic grain radii.
The code is based on the hashed oct tree algorithms of Salmon and Warren [@sw], is written in C and MPI and uses HDF5 [@hdf5] for i/o. The code has evolved to be a flexible parallel tree framework, being tunable for minimum bandwidth or maximum speed, and incorporating asynchronous MPI latency hiding techniques using multiple simultaneous tree traversals from each processor. The hash table allows O($1$) random access to all data in the tree and acts as a serialisation point for remote data retrieval, whilst more traditional pointers are used for rapid traversal of all locally cached tree data.
Most calculations performed to date utilise around $32$ processors with $90$% parallel efficiency. This number of processors is currently adequate as this is a new field and there are large regions of parameter space to explore. A relatively inexpensive machine like the McKenzie cluster is vital so that many runs with different configurations and disk physics can be explored simultaneously in a reasonable time. We are working on further scalability and speed enhancements of the code.
Conclusions
===========
We have built Mckenzie, 512-CPU Beowulf cluster that incorporates commodity gigabit networking in a new topology that permits 128 Gbit cross-sectional bandwidth. Networking costs represent 9% the total cost of the cluster of \$561K. The machine has been benchmarked with the Top 500 list HPL code and achieved a sustained speed of $1.202$ Teraflops or 48% of the theoretical peak speed making it competitive with clusters with more expensive networking solutions. By this measure, our computing costs are just \$0.47/Megaflop. We note that the idea of using a Linux node as a router through a secondary network or even tertiary network is a general concept that can be applied to different network topologies with larger numbers of nodes. The main limitation of this method is not so much the bandwidth but the extra latency inherent to commodity switches and perhaps multiple hops in the chosen topology. Nevertheless, complex parallel algorithms with high bandwidth requirements can still run on systems like this based on our experience.
We have also described 4 astrophysical parallel applications running on the cluster that span the scale of the universe from solar system scale to the size of the universe. We have been able to run some of the largest simulations to date in cosmology, galaxy dynamics and MHD accretion flows using MPI parallel codes. N-body simulations of hundreds of millions of particles in cosmology and galaxy dynamics can be done routinely. The MHD code achieved a sustained speed of 2.2 teraflops in single precision or 44% of the theoretical peak speed in $1400^3$ zone calculation. Our low-cost networking also justifies using the machine in a parameter survey mode on smaller sub-units of the cluster while such strategies might be considered a waste of expensive resources on a large-scale SMP. The ability to run on 512 processors has allowed us to tune and optimize our codes to hide latency for large processor numbers and paves the way for porting codes to clusters of thousands of processors.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Foundation for Innovation (CFI) for funding this project.
Barnes, J. & Hut, P. 1986, Nature, 324, 446 Dubinski, J. 1996, New Astronomy, 1, 133-147 Dubinski, J., Kim, J., Park, C., Humble, R.J. 2003, astro-ph/0304467 Kazushige Goto’s Optimized BLAS Library, http://www.cs.utexas.edu/users/flame/goto/ Frigo, M. and Johnson S.G. 1998, FFTW: An adaptive software architecture for the FFT. In ICASSP ’98, 3, 1381, 1998. http://www.fftw.org. http://hdf.ncsa.uiuc.edu/ Hockney, R. W., & Eastwood, J. W. 1981, Computer Simulation Using Particles (New York: McGraw-Hill) Kuijken, K., Dubinski, J. 1995, MNRAS, 277, 1341 Humble, R. J., Maddison, S. T., Murray, J. R., 2003, in preparation Monaghan, J. J., 1997, JCompPhys, 1997, 138, 801 http://www.mynix.com The OSCAR cluster installation package, http://oscar.sourceforge.net Pen, U.-L., Matzner, C.D., Wong, S. 2003, astro-ph/0304227 Pen, U.-L., Arras, P., Wong, S. 2003, in prep) Salmon, J.K., Warren, M.S., 1994, JCompPhys, 111, 136 Salmon, J.K. 1990, PhD Thesis, Caltech http://www.top500.org Link to animations, http://www.cita.utoronto.ca/$\sim$dubinski/sc2003
|
---
author:
- 'V. Silva Aguirre,'
- 'J. Ballot,'
- 'A.M. Serenelli,'
- 'A. Weiss'
bibliography:
- 'references.bib'
date: 'Received —; accepted —'
subtitle: 'Impact of semiconvection in low-mass stars'
title: Constraining mixing processes in stellar cores using asteroseismology
---
[The overall evolution of low-mass stars is heavily influenced by the processes occurring in the stellar interior. In particular, mixing processes in convectively unstable zones and overshooting regions affect the resulting observables and main sequence lifetime.]{} [We aim to study the effects of different convective boundary definitions and mixing prescriptions in convective cores of low-mass stars, and to discriminate the existence, size, and evolutionary stage of the central mixed zone by means of asteroseismology.]{} [We implemented the Ledoux criterion for convection in our stellar evolution code, together with a time-dependent diffusive approach for mixing of elements when semiconvective zones are present. We compared models with masses ranging from 1 M$_{\sun}$ to 2 M$_{\sun}$ computed with two different criteria for convective boundary definition and including different mixing prescriptions within and beyond the formal limits of the convective regions. Using calculations of adiabatic oscillations frequencies for a large set of models, we developed an asteroseismic diagnosis using only $l=0$ and $l=1$ modes based on the ratios of small to large separations $r_{01}$ and $r_{10}$ defined by Roxburgh & Vorontsov (2003). We analyzed the sensitivity of this seismic tool to the central conditions of the star during the main sequence evolution.]{} [The seismic variables $r_{01}$ and $r_{10}$ are almost linear in the expected observable frequency range, and we show that their slope depends simultaneously on the central hydrogen content, the extent of the convective core, and the amplitude of the sound-speed discontinuity at the core boundary. By considering about 25 modes and an accuracy in the frequency determinations as expected from the CoRoT and *Kepler* missions, the technique we propose allows us to detect the presence of a convective core and to discriminate the different sizes of the homogeneously mixed central region without the need of a strong a priori for the stellar mass.]{}
Introduction {#s:intro}
============
Understanding the physical processes dominating stellar interiors and correctly characterizing their impact on the evolution of stars is one of the main goals of stellar astrophysics. For a long time it has been acknowledged that different mixing processes in stars such as convection, microscopic diffusion and rotational mixing strongly influence the overall evolution of stars [e.g. @jpz92; @amgm00; @at94; @st98 and references therein], with subsequent effects in observable quantities such as luminosity, effective temperature and surface abundances. This has a direct impact on the determinations of derived parameters relying still to a large extent on evolutionary models, as in the case of stellar masses and ages.
The treatment of convectively unstable zones and overshooting regions in evolutionary calculations severely affects the interior structure of the stellar models. This depends on how the boundaries of the convective regions are defined and convective zones themselves treated from the point of view of mixing of elements. The resulting luminosity, temperature, and main sequence lifetime of the star are largely affected by these processes. In the case of massive star evolution, one of the mixing processes that has been extensively studied is semiconvection. During the hydrogen-burning phase the convective core retreats leaving behind a non-uniform chemical profile determined by the composition of the core at the moment each layer is detached from it. Depending on the criterion considered for defining convective boundaries, these layers could maintain their chemical composition or undergo some mixing process, known as semiconvective mixing. Since the pioneering work of @sh58, several authors have investigated the occurrence of semiconvective mixing and its effects on stellar evolution [e.g. @rs70; @sc75; @cc78; @nl85]. Although it was initially thought to occur only in massive stars, semiconvection was also predicted to take place in low-mass stars [e.g. @rm72; @fc73; @hs75; @gn77] but with a larger impact during the helium-burning phase.
During the main sequence evolution, stars with masses larger than $\sim$1.1 M$_{\sun}$ develop a convective core whose extent is determined by their temperature stratification. As the timescale for mixing of elements in this convective region is much shorter than the nuclear timescale, the core is believed to be homogeneously mixed and a discontinuity in density appears at the edge of the fully mixed core. This discontinuity is produced either by convective core expansion due to the increasing importance of the *CNO* cycle over the *pp* chain [@rm72; @hs75], or by the retreating convective core leaving behind a non-uniform chemical profile [@fc73]. Both cases produce higher opacities outside the convective core and allow a semiconvective region to develop. If the resulting density barrier is sustained throughout the main sequence evolution the homogeneously mixed central region will be either restricted in its growth or not allowed to develop at all, drastically changing the behavior of the evolutionary tracks especially close to the main sequence turn-off. This has important implications for the use of the Color-Magnitude Diagram (CMD) morphology and the existence of a hook-like feature at the turn-off to estimate properties of stars at the end of the main sequence and the age of a given stellar population [see for instance @am74a; @am74b; @mm91].
While it is not possible to obtain direct information about stellar interiors, helioseismology has provided the best example of indirect observations from the interior of a star by piercing the outer layers of the Sun using the oscillations observed at its surface. These data have served to extract the adiabatic sound speed and density profiles of the Sun, determine the location of the solar helium second ionization zone and the base of the convective envelope, as well as to constrain the solar surface abundances [see @jcd02; @ba08 and references therein]. With this principle in mind, efforts are being addressed to extend the techniques applied to the Sun to other stars by means of asteroseismology, in the light of the data currently being obtained by the CoRoT [@ab06] and *Kepler* [@wb09a] missions.
We address in this paper the issue of the determination of convective boundaries and the treatment of zones which present a gradient in the molecular weight by the inclusion of the Ledoux criterion for convective instability, and a diffusive approach for semiconvective mixing in our stellar evolution code. We explore the possibility of determining the presence of a convective core and discriminating different sizes of homogeneously mixed central regions and evolutionary stages by means of asteroseismology. We do this by exploring the effects that different types of convective mixing processes have on asteroseismic frequency combinations, in agreement with the expected quality of data being obtained by the aforementioned space missions.
The paper is organized as follows. In Sect. \[s:convec\] we describe the differences in the definition of the convective boundaries for each stability criterion, and the prescription for mixing in convective, semiconvective and overshooting regions. Sect. \[s:semiconv\] focuses in the impact of these boundary definitions and mixing processes during the main sequence evolution. In Sect. \[s:ast\] we discuss the method of isolating the influence of the stellar core by means of different asteroseismic tools. We assess the possibility of disentangling between the different mixing prescriptions using the expected observable range of frequencies. A discussion on the open issues and detection possibilities is given in Sect. \[s:disc\]. We conclude and state final remarks in Sect. \[s:conc\].
Stellar Modeling {#s:convec}
================
We used the Garching Stellar Evolution Code [GARSTEC, @ws08] for our calculations. The interested reader can find in that reference a more detailed description of the numerics and input physics included in the code. For our model calculations we used the 2005 version of the OPAL equation of state [@fr96; @rn02] complemented with the MHD equation of state for low temperatures [@hm88], low temperature opacities from @jf05 and OPAL opacities for high temperatures [@ir96], the @gs98 solar mixture, and the NACRE compilation for thermonuclear reaction rates [@ca99]. Convective zones are treated with the mixing-length theory (MLT), no matter which criterion is used to define their boundaries. Within these zones, the chemical composition is modified either instantaneously or by a diffusive process using the convective velocity estimated from the MLT as described, for instance, in @kw90.
For the frequency computations of the specific models analyzed, we have used the Aarhus adiabatic oscillation package [ADIPLS, @jcd08].
Boundaries of convective and semiconvective zones {#ss:cases}
-------------------------------------------------
The convective zones within a stellar model are defined using a stability criterion. The convectively unstable regions are usually determined by the Schwarzschild criterion [@sh58] by comparing the radiative and adiabatic temperature gradients, denoted $\nabla_{\mathrm{rad}}$ and $\nabla_{\mathrm{ad}}$ respectively. A layer is convective when $$\nabla_{\mathrm{ad}} < \nabla_{\mathrm{rad}}.$$ However, gradients of molecular weight $\mu$ can have a stabilizing – or destabilizing – effect on the convection process. The Ledoux criterion [@pl47] takes into account this effect. Within this prescription, a layer is convective when $$\nabla_{\mathrm{L}} < \nabla_{\mathrm{rad}},$$ with $\nabla_{\mathrm{L}}$, the Ledoux temperature gradient, defined as $$\label{eqn:led_ori}
\nabla_{\mathrm{L}}=\nabla_{\mathrm{ad}}+\frac{\varphi}{\delta}\nabla_{\mu},$$ where $$\label{eqn:partials}
\varphi = \left(\frac{\partial\ln\rho}{\partial\ln\mu}\right)_{\mathrm{P}, T}, \delta = -\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{\mathrm{P, \mu}}, \nabla_{\mu}= \left(\frac{d\ln\mu}{d\ln P}\right).$$ For an equation of state appropriate to a mixture of an ideal gas and black body radiation, Eq. \[eqn:led\_ori\] reduces to: $$\label{eqn:led}
\nabla_{\mathrm{L}}=\nabla_{\mathrm{ad}}+\frac{\beta}{4-3\beta}\nabla_{\mu},$$ where $\beta$ is the ratio of gas pressure to total pressure [@kw90].
The inclusion of changes in the molecular weight in the definition of the relevant temperature gradients leads to the appearance of zones whose energy transport process will depend on the convective criterion considered. There are regions which would be considered convective if the Schwarzschild criterion were used, but are in turn radiative if the Ledoux criterion were applied. These zones are called semiconvective zones, and are defined as the regions where $$\label{eqn:sc}
\nabla_{\mathrm{ad}} < \nabla_{\mathrm{rad}} < \nabla_{\mathrm{L}}\,.$$
![Internal structure of a 1.5 M$_{\sun}$ stellar model in the main sequence. The hydrogen abundance profile X (black solid line), the adiabatic gradient $\nabla_{\mathrm{ad}}$ (red dotted line), the radiative gradient $\nabla_{\mathrm{rad}}$ (green dash-dotted line), and the Ledoux gradient $\nabla_{\mathrm{L}}$ (blue dashed line) are plotted as a function of the mass fraction. From left to right, tones of gray fill regions of different energy transport processes: convective zone, semiconvective zone, and radiative zone.[]{data-label="Grad"}](gradientes2.eps){width="\linewidth"}
As an example, a model of a 1.5 M$_{\sun}$ star displaying a semiconvective zone during its main sequence evolution is depicted in Fig. \[Grad\]. The hydrogen profile in the stellar interior is constant inside the homogeneously mixed convective core, and presents a sharp variation at the position where the edge of the core is located. This feature produces both a step in the radiative gradient $\nabla_{\mathrm{rad}}$ due to the change in the opacities, and the departure of the Ledoux gradient $\nabla_{\mathrm{L}}$ from the adiabatic gradient $\nabla_{\mathrm{ad}}$ as a result of the change in the molecular weight. Therefore, a significant semiconvective layer appears at the boundary of the convective core. We notice that, in this example, the semiconvective zone is 60% as massive as the convective region itself. If the Schwarzschild criterion were used instead, the semiconvective layer would be part of the convective core.
Prescription for semiconvective layers {#ss:mixing}
--------------------------------------
Throughout the years, several approaches have been proposed to deal with semiconvective regions when they appear in stellar models [@ds79; @nl83; @hs92; @gt96]. According to the used prescription, the semiconvective zone is considered to be mixed more or less efficiently [see the comparison made by @wm95]. The amount of mixing in semiconvective layers is crucial: an efficient mixing process can reduce the molecular weight gradient sufficiently, and then trigger convective instability. For the present work, we have implemented in GARSTEC the prescription proposed by @nl83 [@nl85], based on the description of vibrational instability made by @sk66. With this method, the true temperature gradient $\nabla$ in a semiconvective region is calculated from the relation $$\label{eqn:real}
\frac{L_{\mathrm{sc}}}{L_{\mathrm{rad}}}=\alpha_{\mathrm{sc}} \frac{\nabla-\nabla_{\mathrm{ad}}}{2\nabla\left(\nabla_{\mathrm{L}}-\nabla\right)}\left[(\nabla-\nabla_{\mathrm{ad}})-\frac{\beta\left(8-3\beta\right)}{32-24\beta-\beta^{2}}\nabla_{\mathrm{\mu}}\right]\,,$$ where $L_{\mathrm{sc}}$ and $L_{\mathrm{rad}}$ are the semiconvective and radiative luminosities, respectively.
The mixing is treated as a time-dependent diffusive process, with the diffusion coefficient calculated as $$\label{eqn:dif_cons}
D_{\mathrm{sc}}=\alpha_{\mathrm{sc}} \frac{\kappa_\mathrm{r}}{6\ c_{\mathrm{p}}\ \rho} \frac{\nabla-\nabla_{\mathrm{ad}}}{\nabla_{\mathrm{L}}-\nabla}\,,$$ where $\kappa_{\mathrm{r}}$ is the radiative conductivity, $c_{\mathrm{p}}$ the specific heat at constant pressure, and $\rho$ the density. Both the diffusion coefficient and the temperature gradient in this prescription depend on an efficiency parameter of semiconvection ($\alpha_{\mathrm{sc}}$). The meaning of this parameter is further discussed in Sect. \[ss:mainseq\]. We stress the point that this diffusive and time-dependent approach significantly differs from the ones used in previous efforts to treat semiconvection in low-mass stars. In previous studies, the mixing in the semiconvective layer is either performed by adjusting the composition until convective neutrality according to the Schwarzschild criterion is again reached [@cm82; @am08], or approximating the results obtained by @hs92 with a two-step function [@pd05].
Overshooting {#ss:ov}
------------
In some of our models we have considered mixing of chemical elements beyond the formal convective boundaries. We assume that this process induces mixing but does not modify the thermal structure of the layers [*overshooting*, if we use the terminology given by @jpz91]. It is implemented in our code as a diffusive process consisting of an exponential decline of the convective velocities within the radiative zone [@bf96]. The diffusion constant is given by $$\label{eqn:ove}
D_{\mathrm{ov}}\left(z\right) = D_0 \ \exp \ \left(\frac{-2z}{\xi H_p}\right)\,,$$ where $\xi$ corresponds to an efficiency parameter, $H_p$ is the pressure scale height, $z$ is the distance from the convective border, and the diffusion constant $D_0$ is derived from MLT-convective velocities [@kw90]. We have used an overshooting efficiency of $\xi=0.016$, which is in the range expected for main sequence stars [@fh97] and corresponds to the value obtained by calibrating the parameter $\xi$ with open clusters. The size of the overshooting region is further limited in the case of small convective cores. This is done in GARSTEC using a geometrical cutoff factor, allowing the overshooting region to extend only to a fraction of the convective zone [see @ws08; @zm10 for details].
Impact on the Stellar Models {#s:semiconv}
============================
Several models were computed using the different mixing prescriptions and definition of the convective boundaries. Within a convective zone the mixing is performed diffusively using the MLT while in a semiconvective region the mixing is carried out as explained in Sect. \[ss:mixing\]. Calculations were made for both the Schwarzschild and the Ledoux criterion for the definition of the convective zones, with and without including extra mixing due to overshooting and semiconvection. All the models considered here are computed for solar metallicity. We explore the effects of these processes for masses ranging from 1 to 2 M$_{\sun}$ starting at the pre-Zero Age main sequence and evolved until hydrogen exhaustion in the core.
Convective boundary definition criteria {#ss:criteria}
---------------------------------------
Applying either the Schwarzschild or the Ledoux criterion influences the appearance and the size of convective cores as a function of stellar mass. To study these effects, we computed models with no convective overshoot and no mixing throughout the semiconvective regions when these were present ($\alpha_{\mathrm{sc}}=0$).
In Fig. \[Ccore1\] we present the evolution during the main sequence of convective and semiconvective regions in the interiors of three representative models:
- a model where the inclusion of the Ledoux criterion inhibits the growth of the convective core in the late phase of the main sequence evolution (1.2 M$_{\sun}$ model, Fig. \[Ccore1\] *top panel*);
- a model with a convective core increasing in size during the hydrogen burning phase whose growth is constrained if the Ledoux criterion is applied (1.5 M$_{\sun}$ model, Fig. \[Ccore1\] *middle panel*);
- a model where the convective core recedes during the main sequence leaving behind a chemical discontinuity (2.0 M$_{\sun}$ model, Fig. \[Ccore1\] *bottom panel*).
![Evolution of the convective core size in mass coordinate as a function of central hydrogen content during the main sequence phase for the two considered convective boundary definition criteria: Ledoux (solid blue lines) and Schwarzschild (solid red lines). Blue dotted lines depict semiconvective zones when the Ledoux criterion is applied (by definition there are no semiconvective zones when the Schwarzschild criterion is applied). The panels represent different masses: 1.2 M$_{\sun}$ (*top*), 1.5 M$_{\sun}$ (*middle*) and 2.0 M$_{\sun}$ (*bottom*).[]{data-label="Ccore1"}](cores.eps){width="\linewidth"}
There are clear differences in the convective core behavior for the two considered criteria. In the case of growing convective cores (Fig. \[Ccore1\] top and middle panels) the molecular weight discontinuity at the edge of the convective zone produces a steep increase in the opacity, which translates into a sharp discontinuity in the radiative gradient (as shown in Fig. \[Grad\]).
If the Schwarzschild criterion is applied, the profile of the convective core presents wrinkles throughout the evolution, signature of a convective process not properly taken into account [@yl08]. The exact position of the convective core edge will depend largely in the numerics of the evolutionary code and the way the mesh points are placed where the discontinuity in the molecular weight is located [@am08]. Slight variations in the radiative gradient are induced by, for instance, the interpolation in the opacity tables, the allowed time-step and the accuracy of the calculations, resulting in a small region right outside the core edge becoming convective and supplying fresh hydrogen to the outer layer of the core. The core expands and produces the crumple profiles (backwards loops) observed in the top and middle panels of Fig. \[Ccore1\], in a phenomena similar to the so-called *breathing pulses* in horizontal branch stars [see for example @mc07 for a detailed discussion and further references on the topic].
If the Ledoux criterion is applied instead, the convective core profile remains flat and continuous throughout the main sequence evolution, as the molecular weight discontinuity at its edge inhibits its growth. The semiconvective zone appearing outside the core is not mixed through and thus becomes larger during the hydrogen burning phase. At the end of the main sequence evolution, the convective core recedes leaving a radiative layer between it and the semiconvective zone, causing a change in the temperature gradients and making the semiconvective region to ultimately disappear.
In a model with a receding core the convective central region reaches its largest size during the initial phase of hydrogen-burning, the Schwarzschild criterion allowing it to grow to a slightly larger extent than the Ledoux criterion (Fig. \[Ccore1\], bottom panel). The chemical discontinuity left behind by the retreating core leaves an imprint in the molecular weight which permits a semiconvective zone to develop, but the radiative gradient decreases following the shrinking of the core transforming the zone into a radiative one. Since the effect is almost negligible for this case, we will focus for the rest of the paper on the models where the inclusion of the Ledoux criterion plays an important role (masses below $\sim$1.7 M$_{\sun}$ with growing convective cores).
Main sequence evolution {#ss:mainseq}
-----------------------
In the previous section we have shown the impact that different mixing prescriptions have on the size of the homogeneously mixed central region during the main sequence evolution. The size (and existence) of the convective core will determine the appearance (or lack) of a hook-like feature in the CMD at the end of the hydrogen-burning phase. This feature at the turn-off is used in the determination of the age of stellar populations by fitting isochrones reproducing the shape of the hook at a certain metallicity, thus setting constraints on the stellar mass at the end of the main sequence for a given set of input physics [as an example, see @dv07; @zm10 for the specific case of M67]. As discussed in Sect. \[ss:criteria\], using the Ledoux criterion can inhibit the growth of the convective core for the 1.2 M$_{\sun}$ model and therefore the appearance of the hook feature in the evolutionary track. This is the case when no mixing is performed within the developing semiconvective region ($\alpha_{\mathrm{sc}}=0$) and the molecular weight barrier caused by hydrogen burning does not allow the core to grow.
The amount of semiconvective mixing is controlled by the efficiency parameter $\alpha_{\mathrm{sc}}$ and has a value restricted to $\alpha_{\mathrm{sc}} < 1$, but has been used usually in the range $0.001 < \alpha_{\mathrm{sc}} < 0.1$ [@nl85; @nl91; @wm95]. Higher values of $\alpha_{\mathrm{sc}}$ mean faster mixing velocities, achieving instantaneous mixing when $\alpha_{\mathrm{sc}}\rightarrow\infty$. In Fig. \[hrdms\] we have plotted evolutionary tracks for 1.2 and 1.5 M$_{\sun}$ cases calculated with different mixing prescriptions and semiconvective efficiency parameters. By increasing the value of $\alpha_{\mathrm{sc}}$ the convective core develops to a larger extent and the evolutionary tracks resemble the one calculated with the Schwarzschild criterion. For the 1.2 M$_{\sun}$ case the track computed with $\alpha_{\mathrm{sc}} = 0.001$ does not increase the size of the convective core with respect to the track calculated with no mixing in the semiconvective zone, thus their tracks overlap in the Hertzprung-Russel Diagram (HRD). One interesting aspect is that a position in the HRD can be shared by stars with the same mass but different evolutionary stages, internal structures and stellar ages.
![Hertzprung-Russell Diagram for 1.2 M$_{\sun}$ (*left panel*) and 1.5 M$_{\sun}$ (*right panel*) models. Computations done with the Ledoux criterion and no mixing in the semiconvective regions are plotted with dashed blue lines, Schwarzschild criterion with solid red lines, and dotted black lines correspond to tracks with Schwarzschild criterion and overshooting. Two values for the semiconvective efficiency $\alpha_\mathrm{sc}$ are also plotted in thin green dash-dotted line ($\alpha_{\mathrm{sc}} = 0.001$; not shown for the 1.2 M$_{\sun}$ as it overlaps with the dashed track) and thick light blue dash-dotted line ($\alpha_{\mathrm{sc}} = 0.01$). The tracks span the evolution from the Zero-Age main sequence until hydrogen exhaustion in the core.[]{data-label="hrdms"}](hrd.eps){width="\linewidth"}
Changing the prescription for convection has an impact on the stellar age at which the end of the main sequence is reached. In the diffusive prescription we use for semiconvective zones, $\alpha_{\mathrm{sc}}$ modifies the value of the diffusion coefficient and also the true temperature gradient in the semiconvective zone. The growth of the convective core is not enhanced but restricted due to the molecular weight gradient and no extra fuel supply is added to the core, which translates into shorter main sequence lifetimes. In Fig. \[age\] we present the hydrogen contents at the center of the models as a function of age, where the differences are clearly visible for the considered mixing prescriptions and can amount to 30% of the main sequence lifetimes.
![Central hydrogen content of the selected models during their main sequence evolution: 1.2 M$_{\sun}$ (blue) and 1.5 M$_{\sun}$ (red). *Solid lines*: Schwarzschild criterion without overshooting. *Dashed lines*: Ledoux criterion with no semiconvective mixing ($\alpha_{\mathrm{sc}} = 0$). *Dash-dotted lines*: Ledoux criterion and a semiconvective efficiency of $\alpha_{\mathrm{sc}} = 0.01$. *Dotted lines*: models with convective overshooting.[]{data-label="age"}](ages.eps){width="\linewidth"}
Effects of mixing in semiconvective layers {#ss:alpha}
------------------------------------------
In Fig. \[chem\_prof\] we present hydrogen profiles near the center for 1.5 M$_{\sun}$ models calculated with the Ledoux criterion and different semiconvective mixing efficiency values, with the Schwarzschild criterion and with overshooting. The sizes of the homogeneously mixed zones are different as well as the shape of the chemical profile at the edge of the convective core. It is worth noticing that for the 1.5 M$_{\sun}$ case considered in this paper, a semiconvective coefficient of $\alpha_{\mathrm{sc}}\sim 0.01$ is already efficient enough to very closely reproduce the results obtained when the Schwarzschild criterion is applied (see the evolutionary tracks in Fig. \[hrdms\]). The same is true for $\alpha_{\mathrm{sc}}\sim 0.1$ in the 1.2 M$_{\sun}$ model. This suggests that the value of the semiconvective efficiency required to reproduce the results obtained with the Schwarzschild criterion decreases with mass. In relative terms, the size of the semiconvective zone compared to the total convective core size is smaller at higher masses, leading to a shorter mixing timescale.
Convective overshooting has for long been thought of as a natural process capable of smoothing out molecular weight gradients close to the convective core [e.g. @am74a; @am74b; @nl91; @an10]. We computed models using the Ledoux criterion and the overshooting prescription described in Sect. \[ss:ov\] and realized that this is also the case for low-mass stars using our calibrated value for the overshooting efficiency. One should keep in mind, though, that the calibration itself was based on certain assumptions about convection [e.g @ap04; @dv06].
![Hydrogen profiles for similar central hydrogen contents of the 1.5 M$_{\sun}$ models and different mixing prescriptions: Ledoux criterion and no mixing in the semiconvective zones (dashed blue line), $\alpha_{\mathrm{sc}} = 0.001$ (thin dash-dotted green line), $\alpha_{\mathrm{sc}} = 0.01$ (thick dash-dotted light blue line), Schwarzschild criterion (solid red line) and one model including convective overshooting (dotted black line).[]{data-label="chem_prof"}](chem_prof2.eps){width="\linewidth"}
Asteroseismic diagnoses {#s:ast}
=======================
We have highlighted in Sect. \[s:semiconv\] the importance of the different mixing prescriptions considered in this paper on structure and evolution of low-mass stars. In practical terms, these convective processes translate into different convective core sizes, ages, and evolutionary stages for compatible positions in the HRD. The quest of detecting the variations in stellar interiors produced by differences in the mixing and energy transport processes can be fulfilled by a technique capable of piercing the outer layers of stars and which is sensitive to density discontinuities. In this respect, asteroseismology is thought to be the most powerful tool to study the inner properties of stars through the observation of stellar oscillations [e.g @mcu07; @ca08; @jcd10a and references therein].
Many efforts have been conducted to discover the physics of stellar interiors by understanding the influence that different physical phenomena have on the observed frequency spectra. In particular, it is known that a discontinuity in the chemical profile of a star produces a sharp variation in the adiabatic sound speed, which in turn introduces an oscillatory component in the frequencies [@sv88; @dg90; @jp93]. This fact has already been applied to determine the position of the base of the convective envelope in the Sun [@jcd91], to constrain the amount of overshoot below it [@mm94; @rv94c; @jcd95], and to constrain the properties of the solar helium second ionization zone [@jcdph91; @ab94; @mmmt05]. The same technique has been proposed to determine the position of the base of the convective envelope and estimate the helium abundance for stars other than the Sun [@mm00; @mmmt98].
Following this principle, convective cores in low-mass stars have been the subject of several studies by means of asteroseismology aiming to determine the evolutionary state and size of the mixed central region in stars [e.g @ap94; @na95; @ma01; @am06; @cm07]. Most of these investigations have focused their attention on constraining the size and chemical composition of the convective core, while only a few of them have acknowledged the impact of its boundary definition and extra mixing processes such as semiconvection and overshooting in the oscillation frequencies [@pd05; @mg07; @am08; @yl09]. In this section we aim at finding a seismic tool that isolates the stellar core and which is sensitive to its size and central hydrogen content, allowing to disentangle between the different mixing prescriptions.
Seismic variables suited to isolate the core {#ss:ratios_def}
--------------------------------------------
In order to study the inner structure of a star using asteroseismology, first we need to find an appropriate seismic variable which allows us to probe the desired region of the star and extract the required information. Different combinations of low-degree p-modes have been suggested as suitable probes of the physical characteristics of a star [e.g. @jcd84], the most commonly used being the so-called large and small frequency separations defined as: $$\begin{aligned}
\label{eqn:diff}
\Delta_{l}(n) & = & \nu_{n,l}-\nu_{n-1,l}\\
d_{l,l+2}(n) & = & \nu_{n,l}-\nu_{n-1,l+2}\,,\end{aligned}$$ where $\nu_{n,l}$ is the mode frequency of angular degree $l$ and radial order $n$. These combinations are affected by the outer layers of the star where turbulence in the near-surface is almost never taken into account into star modeling [e.g. @jb04].
In our case, we need to isolate from surface contamination the signal arising from the interior of the star in order to properly quantify the effects in the frequency spectra of the presence of a convective core. @rv03a proposed to use the smooth 5 points small frequency separations and the ratio of small to large separations, and showed that these quantities are mainly determined by the inner structure of the star. They are constructed as: $$\label{eqn:d01}
d_{01}(n)=\frac{1}{8}(\nu_{n-1,0}-4\nu_{n-1,1}+6\nu_{n,0}-4\nu_{n,1}+\nu_{n+1,0})$$ $$\label{eqn:d10}
d_{10}(n)=-\frac{1}{8}(\nu_{n-1,1}-4\nu_{n,0}+6\nu_{n,1}-4\nu_{n+1,0}+\nu_{n+1,1})$$ $$\begin{aligned}
\label{eqn:rat}
r_{01}(n)=\frac{d_{01}(n)}{\Delta_{1}(n)},& & r_{10} = \frac{d_{10}(n)}{\Delta_{0}(n+1)}\,.\end{aligned}$$ The small frequency separations have already been used to identify the location of the convective envelope and the helium second ionization zone in the Sun by applying them to observational data [@ir09], while it has been shown that the ratios fairly cancel out the influence in the frequencies of the outer layers [@ir04; @ir05; @of05]. As mentioned before, sharp variations on the adiabatic sound speed produce an oscillatory signal in the p-mode spectra whose period is related to the position of such a discontinuity. In particular, the period of this oscillation relates to the travel time of the wave through the acoustic cavity, and therefore to the radial coordinate where the discontinuity is located [@mm94; @rv94a; @rv01]. This travel time is represented by the acoustic radius, given by: $$\begin{aligned}
\label{eqn:time}
t & = & \int_{0}^{r}\frac{\mathrm{d}r}{c_s}\,\end{aligned}$$ where $c_s$ is the adiabatic sound speed. There is an alternative representation of the acoustic radius called the acoustic depth ($\tau$), which measures the travel time from the surface towards the interior. Therefore, if the total acoustic radius is given by $\tau_c = t(R)$, then clearly $\tau = \tau_c - t$. If the location of the density discontinuity is given by, say, $r_1$ in radial coordinates, and that same position is represented in acoustic radius and acoustic depth by $t_1$ and $\tau_1$ respectively, the periods of the oscillation in frequency space induced by this sharp variation are 1/(2$t_1$) and 1/(2$\tau_1$), according to the considered variable. That being the case, if either the ratios or the small separations are affected only by the position of the convective core, we could extract this information from the frequency data.
Sensitivity of $r_{10}$ and $r_{01}$ to the core {#ss:rat_examp}
------------------------------------------------
If the differences and ratios defined in Eqs. \[eqn:d01\], \[eqn:d10\] and \[eqn:rat\] are indeed sensitive to the deep stellar interior, they should be strongly influenced by the presence of a convective core or a steep gradient in the adiabatic sound speed near the center [@rv07]. To test this, we produced two sets of evolutionary tracks for a 1.1 M$_{\sun}$ star, one using the normal Schwarzschild criterion and the other one including overshooting without the geometrical restriction described in Sect. \[ss:ov\]. This way, we produced a 1.1 M$_{\sun}$ model with a growing convective core during the hydrogen burning phase. In order to compare the effects of the presence of the core, we plotted the evolution during the main sequence of these models in Fig. \[core\_comp\]. The left panels present the ratios as a function of frequency, while the center panels show the adiabatic sound speed near the center for the selected models, whose position in the observational plane is depicted by a filled circle in the right panels of the figure.
The frequency ratios present a common component in the low frequency domain (see left panels), which therefore should be determined by the global seismic properties of the star: the selected models have the same large frequency separation. At higher frequencies, the ratios deviate from each other revealing the structural differences between the models. This part of the frequency domain presents two contributions: one given by the presence of a convective core and another one dominated by the evolutionary state of the model (equivalent to the central hydrogen content). The oscillatory component observed in the ratio of the model calculated with overshooting is related to the presence of a convective core.
A straight line can be drawn between the point where the ratios start deviating from each other and the frequency value where the oscillatory component of the model with the convective core shows its first minimum. We will refer to this part of the frequency domain as the *linear range* from here on. Independently of the presence of a convective core or not, we observe an increase of the slope – strictly speaking of its absolute value – with evolution, suggesting that it traces the central hydrogen content of the star (and hence stellar age). Thus, the sole presence of a negative slope does not ensure the existence of a convective core [see also @ib10].
However, in the last row of panels in Fig. \[core\_comp\] the models plotted have very different central hydrogen abundances ($X_{c}\sim0.14$ versus $X_{c}\sim0.4$ for the model without and with convective core, respectively). In the model without convective core, the increase of the slope is mainly due to the growth of the density gradient in the center, built during hydrogen burning. Nevertheless, the slope is sensibly higher for the model with a convective core, even if this star is less evolved. This difference is a clue to the presence or absence of a convective core and will be discussed in Sect. \[ss:Pop\_plots\].
The global behavior of the small frequency separations is the same as the one observed in the frequency ratios. Thus, we will focus the analysis in the ratios keeping in mind that the same conclusions are applicable to the case of the separations.
Influence of mixing prescriptions on $r_{10}$ and $r_{01}$ {#ss:rat_ms}
----------------------------------------------------------
When considering only models which do have a convective core, we can study the sensitivity of the ratios to the position and size of the convective core, and to the size of the density discontinuity. We explore the behavior of 1.3 M$_{\sun}$ models in the three most extreme cases in terms of the size of the mixed region: Ledoux criterion with no extra mixing in the semiconvective region, the normal Schwarzschild criterion, and one evolutionary sequence including overshooting. An error of 0.2 $\mu$Hz in the frequency determinations was assumed, as expected from the CoRoT and *Kepler* missions [@ab06; @jcd07].
In Fig. \[rat\_evol\] we present a similar diagram as in Fig. \[core\_comp\] showing the changes of this seismic diagnose as a function of the central hydrogen content, mapping therefore the main sequence evolution. When considering one mixing prescription (color) we notice that the absolute value of the slope of the frequency ratios increases with evolution. The considered models have the same central hydrogen content, so when observing this behavior of the ratios nothing can be said about the exact evolutionary stage of the star. However, the size of the discontinuity in the adiabatic sound speed is indeed different, as well as the size of the cores, the largest being the one with the most negative slope.
One is tempted to interpret this as a direct relation between the size of the discontinuity and the absolute slope of the ratios, but two details must be taken into account: for the case of the Ledoux criterion, the change in the slope during the main sequence evolution is much smaller than for the other cases, and the acoustic radius of the fully mixed region remains almost constant (while for the other prescriptions it increases). It is clear that the value of the absolute slope has contributions coming from the position of the convective core and the size of the sound speed discontinuity.
Bearing this in mind, instead of using the central hydrogen content as a proxy for the evolution of the ratios we considered models with the same large frequency separation, as it is the parameter first and most easily obtained from seismic observations. Figure \[rat\_large\] shows this case for the 1.3 M$_{\sun}$ models with large separations equal to $\Delta\nu=$110, 100, 95 and 88 $\mu$Hz. Due to the differences in the mixing prescriptions, the selected models do not have the same central hydrogen content but almost the same age.
The behavior at the low frequency range ($\nu\lesssim$ 1200 $\mu$Hz) is the same for the three mixing prescriptions for a given large frequency separation value. As already mentioned in Sect. \[ss:rat\_examp\], this is an indication of this part of the frequency spectrum being dominated by the global seismic properties of the star, while the frequency values above $\sim$ 1200 $\mu$Hz seem to be sensitive to the differences in the interior (the *linear range*).
Within this frequency region where the information of the core is contained, we can compare in Fig. \[rat\_large\] a case opposite to that in Fig. \[rat\_evol\]: the last row presents models with the same size of discontinuity of the adiabatic sound speed ($\sim 0.27\times 10^7$ cm/s). The hydrogen content of the Ledoux (blue) model is only $\sim$ 2$\%$ while for the the Schwarzschild criterion and the model with overshooting it is $\sim$ 28$\%$ and $\sim$ 32$\%$, respectively. The model calculated with the Ledoux criterion again has the smallest absolute slope value and size of the convective core.
We stress the fact that the absolute value of the slope cannot be attributed only to the size of the adiabatic sound speed discontinuity. In fact, we can distinguish three effects directly linked to the global behavior of the ratios:
- a decrease in the central hydrogen content produces an increase in the slope in the *linear range*, which is visible no matter if a convective core is present or not;
- when a convective core is present, an increase of the size of the discontinuity in the adiabatic sound speed at the core boundary increases the amplitude of the oscillatory signature, making the slope of the ratios steeper;
- the extent of the convective core also affects the slope: a larger convective core reduces the period of the oscillatory component in the ratios, increasing in turn the absolute value of the slope.
It is not straightforward to determine how much influence on the slope each of these effects have. A differential comparison must be done in order to assess where the major contributions are coming from. Nevertheless, now that the key processes affecting the frequency ratios have been identified, we can use this information to disentangle between models with different central mixed zones sizes and evolutionary stages.
Breaking the degeneracy: splitting mixing prescriptions and finding convective cores {#ss:Pop_plots}
------------------------------------------------------------------------------------
[c c c c c c c c c c c]{}\
$\Delta\nu (\mu$Hz) & & T$_\mathrm{{eff}}$ (K) & log(g) & Age (Myr) & X$_\mathrm{c}$ & Jump (10$^7$ $\mathrm{cm\,s^{-1}}$) & Arad$_\mathrm{{cc}}$ (s) & Slope ($\mathrm{\mu Hz^{-1}}$)& Mean\
\
96 & 1.10 M$_{\sun}$ & Sch & 6048 & 4.22 & 6130.4 & 1e-5 & —– & — & $-23\times 10^{-4}$ & 0.049\
& & Ove & 5955 & 4.22 & 6531.1 & 0.216 & 0.610 & 150 & $-71\times 10^{-4}$ & 0.007\
& 1.25 M$_{\sun}$ & Led & 6379 & 4.22 & 2636.2 & 0.016 & 0.225 & 62 & $-8\times 10^{-4}$ & 0.046\
& & Sch & 6332 & 4.22 & 2636.2 & 0.245 & 0.434 & 115 & $-21\times 10^{-4}$ & 0.041\
& & Ove & 6343 & 4.22 & 2630.4 & 0.287 & 0.430 & 125 & $-26\times 10^{-4}$ & 0.034\
& 1.30 M$_{\sun}$ & Led & 6463 & 4.23 & 1812.8 & 0.159 & 0.370 & 75 & $-6\times 10^{-4}$ & 0.043\
& & Sch & 6496 & 4.23 & 1812.8 & 0.347 & 0.338 & 117 & $-15\times 10^{-4}$ & 0.038\
& & Ove & 6499 & 4.23 & 1810.5 & 0.376 & 0.329 & 128 & $-19\times 10^{-4}$ & 0.033\
& 1.35 M$_{\sun}$ & Led & 6628 & 4.24 & 1146.8 & 0.346 & 0.255 & 90 & $-4\times 10^{-4}$ & 0.041\
& & Sch & 6651 & 4.24 & 1146.8 & 0.444 & 0.213 & 122 & $-10\times 10^{-4}$ & 0.035\
& & Ove & 6655 & 4.24 & 1144.6 & 0.465 & 0.228 & 131 & $-12\times 10^{-4}$ & 0.034\
In the examples above we have shown that the frequency ratios are sensitive to the central hydrogen content, the existence and extent of a convective core, and the amplitude of the discontinuity in the adiabatic sound speed. Thus the importance of being able to disentangle between the different mixing prescriptions. An attempt to do so was presented by @pd05, where they characterized the size of the mixed zone in stellar models using the small frequency separation. The authors studied differences between two types of semiconvective mixing and two values of convective overshooting by averaging over a range of radial orders the small frequency separation and using its slope. We will apply a similar approach to the frequency ratios using the radial orders n=15–27, a typical range which is expected to be detected by the CoRoT and *Kepler* missions (see horizontal lines in the left panels of Figs. \[core\_comp\], \[rat\_evol\] and \[rat\_large\]).
In Fig. \[popMass\] we present a slope versus mean diagram in the observable range of the frequency ratios, for the complete main sequence evolution of different masses and mixing prescriptions. For each mass value and mixing prescription we have also selected one model with a large frequency separation value of 96 $\mu$Hz, their properties presented in Table \[table:pop\_mass\]. The position of the zero-age main sequence model for a given mass value is the same independent of the applied mixing prescription, as the interiors are until that point the same. Once the models start evolving, the different behaviors in their frequency ratios make them clearly distinguishable. The tracks reach a minimum value in slope near the exhaustion of the hydrogen in the center (when the convective core starts receding if present).
For the 1.1 M$_{\sun}$ tracks the two cases present very different values of slopes throughout the evolution, allowing us to disentangle between a star with a convective core and one that has a radiative interior. This is particularly interesting as the stellar parameters are almost the same for both stars, so their positions in the CMD overlap within the usual observational uncertainties (see Table \[table:pop\_mass\]). The potential of this technique has only recently started to be exploited [@sd10; @pdm10].
For more massive models, we show how this type of diagram can help us identifying different convective core sizes, evolutionary stages, and therefore mixing prescriptions. Keeping in mind the results presented in Fig. \[rat\_large\], for a given mass value the diagram plotted in Fig. \[popMass\] can disentangle between different convective core sizes, as tracks of a given mass and composition neatly separate according to the mixing prescription applied. With an accurate measurement of the large frequency separation, as expected from the space missions, we can identify the size of the convective core and, as the mass and composition are known, the evolutionary stage.
Nevertheless, mass values determined from a combination of asteroseismic measurements and modelling are usually precise within 5-10% when metallicity is known [@tm10; @ng10]. In Table \[table:pop\_mass\] we present the parameters of models for each mixing prescription with a mass difference of up to $\sim$8%. These are neatly distinguishable in Fig. \[popMass\] for the considered error bars, showing that our method is able to break the degeneracy between mass value and mixing process. We stress again the fact that this analysis relies on known seismic and stellar parameters, in particular frequencies and metallicity. The effective temperature differences for the mass range considered in Table \[table:pop\_mass\] within a mixing prescription are $\sim$150 K, a typical observational error bar. Stars with a larger mass difference for this composition will have an effective temperature not compatible with the observations at the same large frequency separation, which allows us to exclude them from the analysis.
As a closing remark, we notice that in Fig. \[popMass\] the track representing the evolution according to the Schwarzschild criterion shows abrupt jumps in its slope and mean values. This effect is a consequence of the convective process not properly taken into account at the edge of the core, and the abrupt variations correspond to the loops in the convective core profiles (see Sect. \[ss:criteria\]). However, the same general behavior as in the other mixing prescription is present, so a mean value can be considered for comparisons.
![Slope versus mean value diagram of the frequency ratios in the assumed observable range throughout the main sequence evolution. *Top panel*: 1.1 M$_{\sun}$ models without overshooting (solid red line) and with overshooting (dashed green line). *Bottom panel*: different mass values for the three considered mixing prescriptions: Ledoux (blue), Schwarzschild (red) and Overshooting (green). A representative 1-$\mathrm{\sigma}$ error bar is plotted on the top left. Models marked with a filled circle have the same large frequency separation value, and error bars correspond to 3-$\mathrm{\sigma}$ uncertainties in the frequencies. See text for details.[]{data-label="popMass"}](pop_plotmass2.eps){width="\linewidth"}
Discussion {#s:disc}
==========
While investigating different convective zone boundary definitions, we have seen that using the Schwarzschild criterion for models with growing convective cores leads to inconsistencies in the exact location of these regions. The layers at the top of the core may suddenly be mixed into the central region increasing the central hydrogen content and developing loops in the convective core boundary evolution (*breathing pulses* phenomenon). The question that arises then is how to treat these zones. In the study made by @cm82, the authors took into account these layers by mixing them until convective neutrality according to the Schwarzschild criterion was recovered. The convective core grows beyond its previously established boundary and a new supply of fresh hydrogen is added to the core, also increasing the timespan a star spends transforming hydrogen into helium.
Another option for dealing with this *breathing pulses* is using convective overshoot as implemented in GARSTEC, producing a smooth convective core profile during the main sequence evolution. This prescription does not modify the thermal structure of the overshooting layers. When an adiabatic stratification of the convective zone is considered instead, small fluctuations at the edge of the core still appear but of smaller size than the ones presented in Fig. \[Ccore1\] (M. Salaris, private communication). These fluctuations are not likely to affect the computation of adiabatic frequencies.
Overshooting is an interesting topic by itself, and a detailed analysis of it goes beyond the scope of this paper. However, we can mention two important aspects we think deserve attention. One is the temperature gradient in the overshooting region. We have assumed in our study that the temperature stratification in the radiative zone is not affected by the convective motions beyond the formal boundary, while convective penetration assumes an adiabatic stratification in this layer. Currently there is no consensus on the issue, and it is expected that numerical simulations and asteroseismology will cast some light on the topic [@mg07]. The other important aspect is the size of the overshooting region. An exponential decay of the convective velocities is an alternative to an adiabatic extension to a fraction of a scale height of the convective region [see @pv07 for details]. However, numerical simulations are not yet able to reproduce the conditions of stellar interiors [@bd09].
Regarding semiconvection, our calibrated value for the the overshooting efficiency washes out the molecular weight discontinuity that produces semiconvective regions. Nevertheless, it has been suggested that the presence of a molecular weight barrier at the edge of the core might reduce the amount of overshooting from it [@vc99]. It might be possible for a semiconvective region to exist even with the presence of overshooting.
We have shown that the frequencies in the *linear range* are sensitive to the central regions of the star. However, it is not straightforward to determine if the modes are reaching the convective core or not. The radial ($l=0$) modes propagate down to the center of the star, but the situation is different for the $l=1$ modes. Different estimations of the turning points can be made if the perturbation in the gravitational potential is neglected [the cowling approximation, @tc41] or taken into account [@jj28]. In fact, it has been suggested that for stars with convective cores the dipolar modes reach the very center of the star [@mt05; @mt06]. This is an interesting topic where further development is surely needed.
From the results presented in Sect. \[s:ast\] we can say that it is possible to estimate the size of the homogeneously mixed zone using seismic data. In particular, we can discriminate between stars with and without a convective core. There are a number of assumptions behind this prediction. Besides the expected frequency data, we assume that there are determinations of stellar parameters from either spectroscopy or photometry, with an uncertainty in effective temperature of $\sim$150 K. Given these observables we are only limited by the value of the stellar mass, which might be estimated within 5-10% precision for a given mixing prescription using isochrones and asteroseismic scaling relations [@ds09a; @sb10; @ng10].
The frequency ratios seem to sufficiently isolate the signal arising from the stellar core allowing us to study the central conditions of a star. Another asteroseismic tool aimed to study small convective cores was proposed by @cm07, where a combination between small and large separations was constructed and analyzed using modes of angular degree $l=0,1,2,3$. In their work, @cm07 found that the sharp discontinuity formed at the edge of the convective core has a direct impact on the slope of their proposed quantity when plotted against frequency, showing that the absolute value of the slope increases as a function of the size of the chemical discontinuity (and therefore with decreasing amount of central hydrogen content). Since observing $l=3$ modes is very challenging due to severe amplitude reduction, @cb10 extended this study using the small frequency separations defined in Eqs. \[eqn:d01\] and \[eqn:d10\] and analyzing the changes in the slope, reaching the same conclusions as @cm07.
The behavior with evolution is the same for the small frequency separations, the frequency ratios, and the asteroseismic tool proposed by @cm07. Thus, our findings are applicable to all of them. We have shown in Sect. \[ss:rat\_ms\] that, although there is a relation between the size of the discontinuity in the adiabatic sound speed and the absolute slope of the ratios, the total extent of the convective core and the central hydrogen content heavily influence the frequency ratios. Three models with the same large frequency separation and similar size of chemical discontinuity at the edge of the core have very different absolute slopes due to their different convective core sizes and central hydrogen contents. On the other hand, models which barely change the size of the convective core during the hydrogen burning phase show the mildest change in the absolute slope of the ratios.
From the analysis performed in this paper, we have a clear theoretical understanding of the different contributions in the *linear range* of the frequency domain and how to disentangle between the effects of mass and type of mixing for realistic error expectations in the observables. We have based our study of the frequency ratios assuming that the oscillations are pure p-modes, but this assumption breaks down when modes with mixed p-mode and g-mode character appear [see @ca10 and references therein]. These mixed modes carry information from the deep interior and are very sensitive to the core conditions, making them ideal probes for age calibrations in sub-giant stars and mixing processes occurring during the main sequence [e.g @dp91; @cs05; @dm10; @tm10]. For instance, they could be detected in main sequence F-type stars with shallow convective envelopes, where this avoided crossings could more easily reach the visible frequency range. However, different analysis tools from the ones presented in this paper must be developed to fully exploit their potential.
Summary and conclusions {#s:conc}
=======================
We have presented results of our study of the influence of different convective boundary definitions and prescriptions of chemical mixing on main sequence evolution of low-mass stars. This has important consequences for the overall evolution of low-mass stars. We have focused our study on the effects on stellar ages and CMD morphology of the existence and size of convective cores.
The age of a star at the end of the main sequence phase depends on the amount of fuel available for core burning. In the prescriptions we have considered, this is governed by the size of the homogeneously mixed zone in the center. The implementation of the Ledoux criterion for convection and no mixing beyond the formal boundaries of the central convective region produces the smallest size of convective core and consequently the shortest main sequence age for a given stellar mass and composition. The time-dependent diffusive mixing in the semiconvective region increases the amount of hydrogen in the center, reaching the same convective core size as the Schwarzschild criterion for high enough mixing efficiencies. Convective overshoot can increase even further the extension of the mixed zone and the associated age value at the end of the main sequence.
The appearance of a hook-like feature at the end of the central hydrogen burning phase can be shifted to higher mass values by the inclusion of the Ledoux criterion, as we have shown for a 1.2 M$_{\sun}$ model. This has relevant consequences when isochrones are used to reproduce the shape of the CMD of a stellar population given a chemical composition and a set of input physics [@dv07; @zm10]. Previous methods of analysis do not allow to ensure the existence or absence of a semiconvective region in stellar interiors. Further development of numerical simulations and asteroseismic tools, such as the one presented in this paper, can be the key to solve this interesting issue [@gb07; @am08]. In the same line, constraining the existence and size of the central mixed region is another step towards a comprehensive understanding of mixing processes.
We have focused our attention upon using frequency combinations sensitive to the convective core size and the evolutionary stage of the star to discriminate between the different mixing prescriptions. For this purpose, we have chosen the frequency ratios and analyzed their behavior throughout the main sequence evolution for different masses and mixing prescriptions. The ratios present a low-frequency domain which is governed by the global seismic properties of the star (the large frequency separation). However, for higher frequencies the behavior of the ratios changes together with the inner conditions of the star. These are all high order low-degree modes and correspond to what we have termed the *linear range*.
Using this *linear range*, our findings predict that we can discriminate between stars with and without convective cores, constrain the size of the homogeneously mixed central region and disentangle the effects from stellar mass. All of these provided we use an adequate frequency combination to isolate the signal coming from the core. The method relies on our knowledge of the stellar composition, where large uncertainties can still be present. In order to fully assess this and other potentially important effects, such as helium abundance and microscopic diffusion, a hare and hounds exercise with extension to other metallicities and applications to real data is currently underway to further test the method (Silva Aguirre et al., in preparation). Nevertheless, the extensive and highly accurate sets of observations obtained by the CoRoT and *Kepler* missions [e.g @em08; @ds10; @wc10; @jcd10b] open an exciting possibility for stellar physics: to apply differential analysis to stars with a common property, such as metallicity, and pin down the differences arising from the inner layers of their structure.
The authors would like to the anonymous referee for useful suggestions that helped improving the manuscript. They would also like to thank S. Charpinet and M. Salaris for providing evolutionary models and pulsation frequencies for comparisons.
|
**Detection of Genuine Multipartite Entanglement in Multipartite Systems**
Jing Yun Zhao,$^1$ Hui Zhao,$^1$ Naihuan Jing,$^{2,3}$ and Shao-Ming Fei$^{4,5}$
*$^1$ College of Applied Sciences, Beijing University of Technology, Beijing 100124, China*
$^2$ Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
$^3$ Department of Mathematics, Shanghai University, Shanghai 200444, China
$^4$ School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
$^5$ Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
We investigate genuine multipartite entanglement in general multipartite systems. Based on the norms of the correlation tensors of a multipartite state under various partitions, we present an analytical sufficient criterion for detecting the genuine four-partite entanglement. The results are generalized to arbitrary multipartite systems.
[**Keywords:**]{}
genuine multipartite entanglement, correlation tensor
03.65.Ud, 03.67.Mn
Introduction
============
Quantum entanglement is one of the most fascinating features in quantum physics, with numerous applications in quantum information processing, secure communication and channel protocols \[1,2,3\]. In particular, the genuine multipartite entanglement appears to have more significant advantages than the bipartite ones in these quantum tasks \[4\].
The notion of genuine multipartite entanglement (GME) was introduced in \[5\]. Let $H_i^d$, $i=1,2,...,n$, denote $d$-dimensional Hilbert spaces. An $n$-partite state $\rho \in H_1^d \otimes ... \otimes H_n^d$ can be expressed as $
\rho = \sum {{p_\alpha }} \left| {{\psi _\alpha }} \right\rangle \left\langle {{\psi_\alpha }} \right|,
$ where $0<p_\alpha\leq 1$, $\sum {{p_\alpha }} = 1$, $\left| {{\psi _\alpha }} \right\rangle \in H_1^d \otimes ...\otimes H_n^d$ are normalized pure states. $\rho$ is said to be fully separable if it can be written as $\rho=\sum_i q_i\ \rho^1_i\otimes\rho^2_i\otimes\cdots\otimes\rho^n_i$, where $q_i$ is a probability distribution and $\rho^j_i$ are density matrices with respect to the subsystem $H_j$. On the other hand, $\rho$ is called genuine $n$-partite entangled if $\left| {{\psi _\alpha }} \right\rangle$ are not separable under any bipartite partitions.
The genuine multipartite entangled states exist in physical systems like the ground state of the XY model \[6\]. However, it is extremely difficult to identify the GME for general mixed multipartite states. The GME concurrence and its lower bound were studied in \[7-9\]. Some sufficient or necessary conditions of GME were presented in \[10-12\]. As for detection of GME, the common criterion is the entanglement witnesses \[13-16\]. Using correlation tensors, the authors in \[17\] have provided a general framework to detect different classes of GME for quantum systems of arbitrary dimensions. In \[18\] the genuine multipartite entanglement has been investigated in terms of the norms of the correlation tensors and multipartite concurrence. The relations between the norms of the correlation tensors and the detection of GME in tripartite quantum systems have been established in \[19\].
We need to use some simple mathematical concepts in this paper, let’s briefly review them here. The elements of a vector space are called vectors. As we known, tensor product is a way of putting vector spaces together to form larger vector spaces. Suppose $W$ and $V$ are Hilbert spaces of dimension $m$ and $n$ respectively. Then $W\otimes V$ is an $mn$ dimensional vector space. The elements of $W\otimes V$ are liner combinations of ‘tensor products’ $u\otimes v$ of elements $u$ of $W$ and $v$ of $V$. The outer product of $u$ and $v$ is equivalent to a matrix multiplication $uv^t$, provided that $u$ is represented as a $m\times 1$ column vector and $v$ as a $n\times 1$ column vector (which makes $v^t$ a row vector).
In this paper, we analyze the relationship between the norms of the correlation tensors and various bipartitions of multipartite quantum systems, and present sufficient conditions of GME for four partite and multipartite quantum systems.
We generalize some inequalities of the norms of the correlation tensors for four-partite states and give a criterion to detect GME of four-partite quantum systems in Section 2. In Section 3, we generalize these concepts and conclusions to multipartite quantum systems. Comments and conclusions are given in Section 4.
Detection of GME for Four-partite Quantum States
================================================
We first consider the GME for four-partite qudit states $\rho \in H_1^d \otimes ... \otimes H_4^d$. Let $\lambda_i$, $i=1,\cdots,d^2-1$, denote the mutually orthogonal generators of the special unitary Lie algebra $\mathfrak{su}(d)$ under a fixed bilinear form \[20\], and $I$ the $d\times d$ identity matrix. Then $\rho$ can be expanded in terms of $\lambda_i$s, $$\begin{aligned}
\rho&=&\frac{1}{d^4}I\otimes I\otimes I \otimes I+\frac{1}{2d^3}\sum^{4}_{f=1}\sum^{d^2-1}_{i_1=1} t_{i_1}^{(f)} \lambda_{i_1}^{(f)}\otimes I\otimes I \otimes I+\cdots\nonumber\\&&+\frac{1}{16}\sum^{d^2-1}_{i_1,i_2,i_3,i_4=1} t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)} \lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)} \otimes\lambda_{i_3}^{(3)} \otimes\lambda_{i_4}^{(4)},
\label{C}\end{aligned}$$ where $\lambda_{i_1}^{(f)}$($(f)$ represents the position of $\lambda_{i_1}$ in the tensor product) stand for the operators with $\lambda_{i_1}$ on $H_{f}$ and $I$ on the rest spaces, $t_{i_1}^{(f)}=tr(\rho\,\lambda_{i_1}^{(f)}\otimes I\otimes I \otimes I)$, $\cdots, t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}=tr(\rho\,\lambda_{i_1}^{(1)}\otimes \lambda_{i_2}^{(2)}\otimes \lambda_{i_3}^{(3)} \otimes \lambda_{i_4}^{(4)} )$.
Let $T^{(f)},\cdots,T^{(1,2,3,4)}$ denote vectors with entries $t_{i_1}^{(f)},\cdots, t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}$$(i_1,i_2,i_3, i_4$ $=1,\cdots, d^2-1;f=1,2,3,4)$, respectively. From $T^{(f)},\cdots,T^{(1,2,3,4)}$ we further define the following matrices under different partitions.
We denote $T_{f|ghl}$ the ${(d^2-1)\times(d^2-1)^3}$ matrices with entries $t_{i_f,(d^2-1)^2(i_g-1)+(d^2-1)(i_h-1)+i_l}=t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}$, $T_{fg|hl}$ the ${(d^2-1)^2\times(d^2-1)^2}$ matrices with entries $t_{(d^2-1)(i_f-1)+i_g,(d^2-1)(i_h-1)+i_l}=t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}$, $T_{fgh|l}$ the ${(d^2-1)^3\times(d^2-1)}$ matrices with entries $t_{(d^2-1)^2(i_f-1)+(d^2-1)(i_g-1)+i_h,i_l}=t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}$, where $f\neq g\neq h\neq l=1,2,3,4; i_f,i_g,i_h,i_l=1,\cdots,d^2-1$. If the state is fully separable, we denote $T_{1|2|3|4}$ the ${(d^2-1)\times(d^2-1)^3}$ matrices with entries $t_{i_1,(d^2-1)^2(i_2-1)+(d^2-1)(i_3-1)+i_4}=t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}$.
Let $T^{(\underline{f},g)}$ and $T^{(f,\underline{g})}$ be ${(d^2-1)\times(d^2-1)}$ matrices with entries $t_{i_1,i_2}=t_{i_1,i_2}^{(f,g)}$ and $t_{i_2,i_1}=t_{i_1,i_2}^{(f,g)}$, respectively. We denote $T^{(\underline{f},g,h)}$, $T^{(f,\underline{g},h)}$ and $T^{(f,g,\underline{h})}$ the ${(d^2-1)\times(d^2-1)^2}$ matrices with entries given by $t_{i_1,(d^2-1)(i_2-1)+i_3}={t_{i_1,i_2,i_3}^{(f,g,h)}}$, $t_{i_2,(d^2-1)(i_1-1)+i_3}={t_{i_1,i_2,i_3}^{(f,g,h)}}$ and $t_{i_3,(d^2-1)(i_1-1)+i_2}={t_{i_1,i_2,i_3}^{(f,g,h)}}$, respectively. We denote $T^{(\underline{f,g},h)}$, $T^{(\underline{f},g,\underline{h})}$ and $T^{(f,\underline{g,h})}$ the ${(d^2-1)^2\times(d^2-1)}$ matrices with entries given by $t_{(d^2-1)(i_1-1)+i_2,i_3}=t_{i_1,i_2,i_3}^{(f,g,h)}$, $t_{(d^2-1)(i_1-1)+i_3,i_2}=t_{i_1,i_2,i_3}^{(f,g,h)}$ and $t_{(d^2-1)(i_2-1)+i_3,i_1}=t_{i_1,i_2,i_3}^{(f,g,h)}$, respectively.
The Frobenius norm is matrix norm of an $m\times n$ matrix $M$ defined as the square root of the sum of the absolute squares of its elements, $\parallel M\parallel$=$\sqrt{\sum_{i,j}|M_{ij}|^2}$. It is also equal to the square root of the matrix trace of $MM^\dag$, where $M^\dag$ is the conjugate transpose, i.e., $\parallel M\parallel=\sqrt{tr(MM^\dag)}$. Since trace is invariant under unitary equivalence, this shows $\parallel M\parallel=\sqrt{\sum_i\sigma_i^2}$. The sum of the $k$ largest singular values of $M$ is a matrix norm, the Ky Fan $k$-norm of $M$, i.e., $\parallel M\parallel_k=\sum_i^k\sigma_i$, where $\sigma_i$, $i=1,\cdots,min(m,n)$, are the singular values of the matrix $M$ arranged in descending order.
For any pure state $\rho\in H^d_1\otimes H^d_2 \otimes H^d_3$, $\rho=\frac{1}{d^3}I\otimes I\otimes I +\frac{1}{2d^2}(\sum^{d^2-1}_{i_1} t_{i_1}^{(1)} \lambda_{i_1}^{(1)}\otimes I\otimes I+\sum^{d^2-1}_{i_2} t_{i_2}^{(2)} I\otimes \lambda_{i_2}^{(2)}\otimes I+\sum^{d^2-1}_{i_3} t_{i_3}^{(3)} I \otimes I\otimes \lambda_{i_3}^{(3)})+\frac{1}{4d}(\sum^{d^2-1}_{i_1,i_2} t_{i_1,i_2}^{(1,2)} \lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)} \otimes I+\sum^{d^2-1}_{i_2,i_3} t_{i_2,i_3}^{(2,3)} I\otimes\lambda_{i_2}^{(2)} \otimes \lambda_{i_3}^{(3)}+\sum^{d^2-1}_{i_1,i_3} t_{i_1,i_3}^{(1,3)} \lambda_{i_1}^{(1)}\otimes I \otimes \lambda_{i_3}^{(3)})+\frac{1}{8}\sum^{d^2-1}_{i_1,i_2,i_3} t_{i_1,i_2,i_3}^{(1,2,3)} \lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)} \otimes \lambda_{i_3}^{(3)}$, we have $tr(\rho^2)=\frac{1}{d^3}+\frac{1}{2d^2}[\sum(t_{i_1}^{(1)})^2+\sum(t_{i_2}^{(2)})^2+
\sum(t_{i_3}^{(3)})^2]+\frac{1}{4d}[\sum(t_{i_1,i_2}^{(1,2)})^2+\sum(t_{i_1,i_3}^{(1,3)})^2+
\sum(t_{i_2,i_3}^{(2,3)})^2]+\frac{1}{8}\sum(t_{i_1,i_2,i_3}^{(1,2,3)})^2=1$. Therefore $$\begin{aligned}
\nonumber
\sum(t_{i_1,i_2,i_3}^{(1,2,3)})^2&&=\frac{8(d^3-1)}{d^3}-\Big\{\frac{4}{d^2}\left[\sum(t_{i_1}^{(1)})^2+\sum(t_{i_2}^{(2)})^2+
\sum(t_{i_3}^{(3)})^2\right]+\nonumber\\&&~~\frac{2}{d}\left[\sum(t_{i_1,i_2}^{(1,2)})^2+ \sum(t_{i_1,i_3}^{(1,3)})^2+
\sum(t_{i_2,i_3}^{(2,3)})^2\right]\Big\}\nonumber\\&&\leq\frac{8(d^3-1)}{d^3}.\nonumber\end{aligned}$$ Thus, $\parallel T^{(1,2,3)}\parallel=\sqrt{\sum(t_{i_1,i_2,i_3}^{(1,2,3)})^2}\leq\frac{2}{d}\sqrt{\frac{2(d^3-1)}{d}}$. Concerning the relations between the correlation tensors and the separability under various partitions, we have the following results:
Let $\rho\in H^d_1\otimes H^d_2 \otimes H^d_3 \otimes H^d_4$ be a pure state. If $\rho$ is fully separable, then for any $k=1,\cdots,d^2-1$, $$\begin{aligned}
\parallel T_{1|2|3|4}\parallel_k= {\frac{4(d-1)^2}{d^2}}.\end{aligned}$$
Since $\rho$ is fully separable, $\rho=\rho_{1}\otimes \rho_{2}\otimes \rho_{3}\otimes \rho_{4}$, where $\rho_{1} ,\rho_{2},\rho_{3},\rho_{4}$ are the reduced density matrices of $\rho$. By the calculation, we obtain $t_{i_1,i_2,i_3,i_4}^{(1,2,3,4)}=t_{i_1}^{(1)}t_{i_2}^{(2)}t_{i_3}^{(3)}t_{i_4}^{(4)}$. According to the inequality for 1-body correlation tensors, $\parallel T^{(f)}\parallel\leq\sqrt{\frac{2(d-1)}{d}}$ \[17\], $f=1, 2, 3, 4$, with the equality holding iff the state is pure, we have $$\begin{aligned}
\parallel T_{1|2|3|4}\parallel_k&&=\parallel T^{(1)}(T^{(2)}\otimes T^{(3)}\otimes T^{(4)})^t\parallel_k=\parallel T^{(1)}\parallel\cdot\parallel( T^{(2)}\otimes T^{(3)}\otimes T^{(4)})^t\parallel_k\nonumber\\&&=\parallel T^{(1)}\parallel\cdot\parallel ( T^{(2)}\otimes T^{(3)}\otimes T^{(4)})^t\parallel=\parallel T^{(1)}\parallel\cdot\parallel T^{(2)}\otimes T^{(3)}\otimes T^{(4)}\parallel\nonumber\\&&=\parallel T^{(1)}\parallel\cdot\parallel T^{(2)}\parallel\cdot\parallel T^{(3)}\parallel\cdot\parallel T^{(4)}\parallel=\frac{4(d-1)^2}{d^2},\end{aligned}$$ which proves the Theorem.
Let $f$, $g$, $h$ and $l$ be any subsystem in a four-partite quantum system. $f\neq g\neq h\neq l\in \{1,2,3,4\}$ means that any two subsystems are not repeatedly selected.
Let $\rho\in H^d_1\otimes H^d_2 \otimes H^d_3 \otimes H^d_4$ be a pure state such that $\rho$ is separable under at least one bipartition. Then for any $k=1,\cdots,d^2-1$, and $f\neq g\neq h\neq l\in \{1,2,3,4\}$, we have\
(i) if $\rho$ is separable under bipartition $f|ghl$, then $$\begin{aligned}
\parallel T_{f|ghl}\parallel_k\leq {\frac{4(d-1)\sqrt{d^2+d+1}}{d^2}};\end{aligned}$$ (ii) if $\rho$ is entangled under bipartition $f|ghl$, then $$\begin{aligned}
\parallel T_{f|ghl}\parallel_k\leq {\frac{4\sqrt{k}(d^2-1)}{d^2}}.\end{aligned}$$
\(i) If $\rho$ is separable under bipartition $f|ghl$, $\rho=\rho_{f}\otimes\rho_{ghl}$, it follows from $\parallel T^{(f,g,h)}\parallel\leq\frac{2}{d}\sqrt{\frac{2(d^3-1)}{d}}$ that $$\begin{aligned}
\parallel T_{f|ghl}\parallel_k&&=\parallel T^{(f)}(T^{(g,h,l)})^t\parallel_k=\parallel T^{(f)}\parallel\cdot\parallel(T^{(g,h,l)})^t\parallel_k\nonumber\\&&=\parallel T^{(f)}\parallel\cdot\parallel (T ^{(g,h,l)})^t\parallel=\parallel T^{(f)}\parallel\cdot\parallel T^{(g,h,l)}\parallel\nonumber\\&&\leq\frac{4(d-1)\sqrt{d^2+d+1}}{d^2}.\end{aligned}$$
\(ii) $\rho$ is entangled under bipartition $f|ghl$, without loss of generality, say, under the bipartition $1|234$. If $\rho$ is separable under some bipartition of one subsystem vs the rest three subsystems, we have $$\begin{aligned}
\parallel T_{f|ghl}\parallel_k\leq\frac{4(d-1)\sqrt{d^2+d+1}}{d^2}.\end{aligned}$$ If $\rho$ is separable under some bipartition of two subsystems vs the rest two subsystems, from the inequality of 2-body correlation tensors $\parallel T^{(f,g)}\parallel\leq\sqrt{\frac{4(d^2-1)}{d^2}}$ \[17\], we have $$\begin{aligned}
\parallel T_{f|ghl}\parallel_k&&=\parallel T^{(\underline{f},{g})}\otimes(T^{({h},{l})})^t\parallel_k=\parallel T^{(\underline{f},{g})}\parallel_k\cdot\parallel (T^{({h},{l})})^t\parallel_k\nonumber\\&&\leq\sqrt{k}\parallel T^{(\underline{f},{g})}\parallel\cdot\parallel T^{({h},{l})}\parallel\leq{\frac{4\sqrt{k}(d^2-1)}{d^2}},\end{aligned}$$ where we have used the inequality $\parallel M\parallel_k\leq k\parallel M\parallel$ for any matrix M. If $\rho$ is separable under some bipartition of three subsystems vs the rest one subsystem, we have $$\begin{aligned}
\parallel T_{f|ghl}\parallel_k&&=\parallel T^{(\underline{f},g,h)}\otimes(T^{(l)})^t\parallel_k=\parallel T^{(\underline{f},g,h)}\parallel_k\cdot\parallel (T^{(l)})^t\parallel_k\nonumber\\&&\leq\sqrt{k}\parallel T^{(\underline{f},g,h)}\parallel\cdot\parallel T^{(l)}\parallel\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}.\end{aligned}$$ Hence, if $\rho$ is entangled under bipartition $1|234$, we have $\parallel T_{f|ghl}\parallel_k\leq max\{\frac{4(d-1)\sqrt{d^2+d+1}}{d^2}, \frac{4\sqrt{k}(d^2-1)}{d^2},$\
$\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}\}$ $={\frac{4\sqrt{k}(d^2-1)}{d^2}}$. Similar discussion applies to other bipartitions $2|134,$ $3|124$ and $4|123$. It indicates that these norms have the same upper bound. Hence, $\parallel T_{f|ghl}\parallel_k\leq {\frac{4\sqrt{k}(d^2-1)}{d^2}}$, if $\rho$ is entangled under bipartition $f|ghl$.
We may analyze the bipartition $fgh|l$ by using similar methods above and obtain the following Lemma.
Let $\rho\in H^d_1\otimes H^d_2 \otimes H^d_3 \otimes H^d_4$ be a pure state such that $\rho$ is separable under at least one bipartition. Then for any $k=1,\cdots,d^2-1$, and $f\neq g\neq h\neq l\in \{1,2,3,4\}$, we have\
(i) if $\rho$ is separable under bipartition $fgh|l$, then $$\begin{aligned}
\parallel T_{fgh|l}\parallel_k\leq {\frac{4(d-1)\sqrt{d^2+d+1}}{d^2}};\end{aligned}$$ (ii) if $\rho$ is entangled under bipartition $fgh|l$, then $$\begin{aligned}
\parallel T_{fgh|l}\parallel_k\leq{\frac{4\sqrt{k}(d^2-1)}{d^2}}.\end{aligned}$$
Now we consider the relations between the correlation tensors and the separability under the bipartition $fg|hl$.
Let $\rho\in H^d_1\otimes H^d_2 \otimes H^d_3 \otimes H^d_4$ be a pure state such that $\rho$ is separable under at least one bipartition. Then for any $k=1,\cdots,d^2-1$, and $f\neq g\neq h\neq l\in \{1,2,3,4\}$, we have\
(i) if $\rho$ is separable under bipartition $fg|hl$, then $$\begin{aligned}
\parallel T_{fg|hl}\parallel_k\leq {\frac{4(d^2-1)}{d^2}};\end{aligned}$$ (ii) if $\rho$ is entangled under bipartition $fg|hl$, then $$\begin{aligned}
\parallel T_{fg|hl}\parallel_k\leq {\frac{4k(d^2-1)}{d^2}}.\end{aligned}$$
\(i) If $\rho$ is separable under bipartition $fg|hl$, $\rho=\rho_{fg}\otimes\rho_{hl}$, then $$\begin{aligned}
\parallel T_{fg|hl}\parallel_k&&=\parallel T^{(f,g)}(T^{(h,l)})^t\parallel_k=\parallel T^{(f,g)}\parallel\cdot\parallel(T^{(h,l)})^t\parallel_k=\parallel T^{(f,g)}\parallel\cdot\parallel (T^{(h,l)})^t\parallel\nonumber\\&&=\parallel T^{(f,g)}\parallel\cdot\parallel T^{(h,l)}\parallel\leq\frac{4(d^2-1)}{d^2},\end{aligned}$$ by using the inequality for 2-body correlation tensors.
\(ii) $\rho$ is entangled under bipartition $fg|hl$, say, $12|34$. If $\rho$ is separable under some bipartition of one subsystem vs the rest three subsystems, we have $$\begin{aligned}
\parallel T_{fg|hl}\parallel_k&&=\parallel T^{(f)}\otimes T^{(\underline{{g}},h,l)}\parallel_k=\parallel T^{(f)}\parallel\cdot\parallel T^{(\underline{g},h,l)}\parallel_k\nonumber\\&&\leq\sqrt{k}\parallel T^{(f)}\parallel\cdot\parallel T^{(\underline{g},h,l)}\parallel\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}.\end{aligned}$$ If $\rho$ is separable under some bipartition of two subsystems vs the rest two subsystems, we have $$\begin{aligned}
\parallel T_{fg|hl}\parallel_k\leq{\frac{4(d^2-1)}{d^2}}.\end{aligned}$$ If $\rho$ is separable under some bipartition of three subsystems vs the rest one subsystem, we have $$\begin{aligned}
\parallel T_{fg|hl}\parallel_k&&=\parallel T^{(\underline{f,g},h)}\otimes(T^{(l)})^t\parallel_k=\parallel T^{(\underline{f,g},h)}\parallel_k\cdot\parallel (T^{(l)})^t\parallel_k\nonumber\\&&\leq\sqrt{k}\parallel T^{(\underline{f,g},h)}\parallel\cdot\parallel T^{(l)}\parallel\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}.\end{aligned}$$ Hence, if $\rho$ is entangled under bipartition $12|34$, we have $\parallel T_{fg|hl}\parallel_k\leq$max$ \{\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2},\\ \frac{4(d^2-1)}{d^2}\}=\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2},$ $k\geq2$. If $k=1$, $\parallel T_{fg|hl}\parallel_1\leq\frac{4(d^2-1)}{d^2}$.
Similarly, if $\rho$ is entangled under bipartition $13|24$, $14|23$ $23|14$, $24|13$ and $34|12$, we have the upper bound of the norm as follows. Let $i\ vs\ j$ denote that $\rho$ is separable under some bipartition of $i$ subsystem vs the rest $j$ subsystems.
[|c|c|c|c|c|c|]{} & $1\ vs\ 3$ & $2\ vs\ 2$ & $3\ vs\ 1$\
$13|24$ &
----------------------------------------------------------------
$\parallel T_{13|24}\parallel_k~~~~~~~~~~~$
$=\parallel T^{(f)}\otimes T^{(g,\underline{h},l)}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~~$
----------------------------------------------------------------
&
------------------------------------------------------------------------------
$\parallel T_{13|24}\parallel_k~~~~~~~~~~$
$=\parallel T^{(\underline{{f}},g)}\otimes T^{(\underline{h},l)}\parallel_k$
$\leq\frac{4k(d^2-1)}{d^2}~~~~~~~~~~~~~~~$
------------------------------------------------------------------------------
&
--------------------------------------------------------------------------------
$\parallel T_{13|24}\parallel_k~~~~~~~~~~~~~~$
$=\parallel T^{(\underline{f},g,\underline{h})}\otimes (T^{(l)})^t\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~~~~~$
--------------------------------------------------------------------------------
\
$14|23$ &
----------------------------------------------------------------
$\parallel T_{14|23}\parallel_k~~~~~~~~~$
$=\parallel T^{(f)}\otimes T^{(g,h,\underline{l})}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~$
----------------------------------------------------------------
&
----------------------------------------------------------------------------
$\parallel T_{14|23}\parallel_k~~~~~~~~~$
$=\parallel T^{(\underline{f},g)}\otimes T^{(h,\underline{l})}\parallel_k$
$\leq\frac{4k(d^2-1)}{d^2}~~~~~~~~~~~~~~~$
----------------------------------------------------------------------------
&
----------------------------------------------------------------
$\parallel T_{14|23}\parallel_k~~~~~~~~~~~~$
$=\parallel T^{(\underline{f},g,h)}\otimes T^{(l)}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~$
----------------------------------------------------------------
\
$23|14$ &
------------------------------------------------------------------
$\parallel T_{23|14}\parallel_k~~~~~~~~~~~$
$=\parallel T^{(f)^t}\otimes T^{(\underline{g,h},l)}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~$
------------------------------------------------------------------
&
----------------------------------------------------------------------------
$\parallel T_{23|14}\parallel_k~~~~~~~~~~$
$=\parallel T^{(f,\underline{g})}\otimes T^{(\underline{h},l)}\parallel_k$
$\leq\frac{4k(d^2-1)}{d^2}~~~~~~~~~~~~~~~$
----------------------------------------------------------------------------
&
--------------------------------------------------------------------
$\parallel T_{23|14}\parallel_k~~~~~~~~~~~~~~$
$=\parallel T^{(f,\underline{g,h})}\otimes (T^{(l)})^t\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~~~~$
--------------------------------------------------------------------
\
$24|13$ &
------------------------------------------------------------------------------
$\parallel T_{24|13}\parallel_k~~~~~~~~~~$
$=\parallel T^{(f)^t}\otimes T^{(\underline{g},h,\underline{l})}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~$
------------------------------------------------------------------------------
&
----------------------------------------------------------------------------
$\parallel T_{24|13}\parallel_k~~~~~~~~~~~$
$=\parallel T^{(f,\underline{g})}\otimes T^{(h,\underline{l})}\parallel_k$
$\leq\frac{4k(d^2-1)}{d^2}~~~~~~~~~~~~~~~$
----------------------------------------------------------------------------
&
----------------------------------------------------------------
$\parallel T_{24|13}\parallel_k~~~~~~~~~~~$
$=\parallel T^{(f,\underline{g},h)}\otimes T^{(l)}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~$
----------------------------------------------------------------
\
$34|12$ &
--------------------------------------------------------------------
$\parallel T_{34|12}\parallel_k~~~~~~~~~~~~~~$
$=\parallel (T^{(f)})^t\otimes T^{(g,\underline{h,l})}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~~~~$
--------------------------------------------------------------------
&
------------------------------------------------------
$\parallel T_{34|12}\parallel_k~~~~~~~~~~~$
$=\parallel (T^{f,g})^t\otimes T^{(h,l)}\parallel_k$
$\leq\frac{4(d^2-1)}{d^2}~~~~~~~~~~~~~~~~~~$
------------------------------------------------------
&
----------------------------------------------------------------
$\parallel T_{34|12}\parallel_k~~~~~~~~~~~$
$=\parallel T^{(f,g,\underline{h})}\otimes T^{(l)}\parallel_k$
$\leq\frac{4(d-1)\sqrt{k(d^2+d+1)}}{d^2}~~~~~$
----------------------------------------------------------------
\
\
Altogether we have $\parallel T_{fg|hl}\parallel_k\leq {\frac{4k(d^2-1)}{d^2}}$ if $\rho$ is entangled under bipartition $fg|hl$.
Next we present a sufficient condition to detect GME for four-partite systems. By the Lemma 2 we have that $\parallel T_{f|ghl}\parallel_k\leq {\frac{4(d-1)\sqrt{d^2+d+1}}{d^2}}$ if $\rho$ is separable, and $\parallel T_{f|ghl}\parallel_k\leq{\frac{4\sqrt{k}(d^2-1)}{d^2}}$ if $\rho$ is entangled. However, $\parallel T_{fg|hl}\parallel_k\leq{\frac{4k(d^2-1)}{d^2}}$ is a rather weak condition. We define the average matricization norm, $M_k=\frac{1}{4}(\parallel T_{1|234}\parallel_k+\parallel T_{2|134}\parallel_k+\parallel T_{3|124}\parallel_k+\parallel T_{4|123}\parallel_k)$.
If $\rho$ is a four-qudit state, and $$\begin{aligned}
\label{thm1}
M_k(\rho)>\frac{(d-1)[\sqrt{d^2+d+1}+3(d+1)\sqrt{k}]}{d^2}\end{aligned}$$ for any $k\in\{1,2,3,\cdots,d^2-1\}$, then $\rho$ is genuine multipartite entangled. \[1\]
[Remark 1:]{} Compared with the Theorem 3 in \[17\] for four-qubit states, our result detects GME for any general four-qudit states.
Detection of GME for Multipartite Quantum States
=================================================
In this section, we study the GME for multipartite qudit states. Any n-partite density matrix $\rho\in H_1^d\otimes H_2^d\otimes \cdots\otimes H_n^d$ can be expressed as $$\begin{aligned}
\rho&&=\frac{1}{d^n}I\otimes \cdots\otimes I+\frac{1}{2d^{n-1}}\sum_{j_1=1}^n\sum^{d^2-1}_{i_1=1} t_{i_1}^{(j_1)} \lambda_{i_1}^{(j_1)}\otimes I\otimes\cdots\otimes I+\cdots\nonumber\\&&~~
+\frac{1}{2^n}\sum_{i_1,\cdots,i_n=1}^{d^2-1} t_{i_1,\cdots,i_n}^{(1,\cdots,n)}\lambda_{i_1}^{(1)}\otimes \lambda_{i_2}^{(2)}\otimes \cdots\otimes \lambda_{i_n}^{(n)},\end{aligned}$$ where $(j_1)$ represents the position of $\lambda_{i_1}$ in the tensor product, $t_{i_1}^{(j_1)}=tr(\rho\lambda_{i_1}^{(j_1)}\otimes I\otimes\cdots\otimes I), \cdots,t_{i_1,\cdots,i_n}^{(1,\cdots,n)}=tr(\rho\lambda_{i_1}^{(1)}\otimes \lambda_{i_2}^{(2)}\otimes \cdots\otimes \lambda_{i_n}^{(n)})$, and $T^{(j_1)},$ $\cdots,T^{(1,\cdots, n)}$ are the vectors (tensors) with elements $t_{i_1}^{(j_1)},\cdots,t_{i_1,\cdots,i_n}^{(1,\cdots,n)}$, respectively.
For a pure state $\rho$, one has $$\begin{aligned}
tr(\rho^2)&&=\frac{1}{d^n}+\frac{1}{2d^{n-1}}\sum_{j_1}^n\sum_{i_1}^{d^2-1}(t_{i_1}^{(j_1)})^2+
\cdots+{\frac{1}{2^n}\sum_{i_1,\cdots,i_n}^{d^2-1}(t_{i_1,\cdots,i_n}^{(1,\cdots,n)})^2=1.}\end{aligned}$$ Hence $$\begin{aligned}
\sum_{i_1,\cdots,i_n}^{d^2-1}(t_{i_1,\cdots,i_n}^{(1,\cdots,n)})^2&&={2^n-\frac{2^n}{d^n}-\cdots-
\frac{2^n}{2d^{n-1}}\sum_{j_1}^n\sum_{i_1}^{d^2-1}(t_{i_1}^{(j_1)})^2\leq \frac{2^n(d^n-1)}{d^n},}\end{aligned}$$ which implies that $$\label{l3}
\parallel T^{(1,2,\cdots,n)}\parallel=\sqrt{\sum_{i_1,\cdots,i_n}^{d^2-1}(t_{i_1,\cdots,i_n}^{(1,\cdots,n)})^2}
\leq\sqrt{\frac{2^n(d^n-1)}{d^n}}.$$ We now consider multipartite systems and their $T$ matrices.
Let $\rho\in H^d_1\otimes\cdots \otimes H^d_n $ be a pure state. If $\rho$ is fully separable, then for any $k=1,\cdots,d^2-1$, $$\begin{aligned}
\parallel T_{1|\cdots|n}\parallel_k=\sqrt{\frac{2^n(d-1)^n}{d^n}}.\end{aligned}$$
According to the Proposition 1 of Ref. \[21\], i.e., if $\rho$ is fully separable then $t_{i_1,\cdots,i_n}^{(1,\cdots,n)}=t_{i_1}^{(1)}\cdots t_{i_n}^{(n)}$, using the bound $\parallel T^{(j_1)}\parallel\leq\sqrt{\frac{2(d-1)}{d}}$, $j_1=1,\cdots,n$, we have $$\begin{aligned}
\parallel T_{1|\cdots|n}\parallel_k&&=\parallel T^{(1)}(T^{(2)}\otimes\cdots\otimes T^{(n)})^t \parallel_k=\parallel T^{(1)}\parallel\cdot\parallel(T^{(2)}\otimes\cdots\otimes T^{(n)})^t\parallel_k\nonumber\\&&=\parallel T^{(1)}\parallel\cdot\parallel T^{(2)}\otimes\cdots\otimes T^{(n)}\parallel_k=\parallel T^{(1)}\parallel\cdot\parallel T^{(2)}\otimes\cdots\otimes T^{(n)}\parallel\nonumber\\&&=\parallel T^{(1)}\parallel\cdot\parallel T^{(2)}\parallel\cdots \parallel T^{(n)}\parallel=\sqrt{\frac{2^n(d-1)^n}{d^n}}.\end{aligned}$$ Hence, if $\rho$ is fully separable, then $\parallel T_{1|\cdots|n}\parallel_k=\sqrt{\frac{2^n(d-1)^n}{d^n}}.$
Let $A_1$ be subsets of the set $\{H_1, H_2,\cdots, H_n\}$ and $A_2$ the complement of $A_1$, $n_{A_1}$ and $n_{A_2}$ be the number of spaces contained in $A_1$ and $A_2$, respectively. For the bipartition $A_1|A_2=j_1\cdots j_{n_{A_1}}|j_{n_{A_1+1}}\cdots j_n$, $j_1\neq j_2\neq \cdots\neq j_n\in \{1,2,\cdots,n\}$ $($this means that any two subsystems are not repeatedly selected$)$, let $T_{A_1|A_2}$ be a matrix with entries $t_{a,b}=t_{i_1,\cdots,i_n}^{(1,\cdots,n)}$, where $a=(d^2-1)^{n_{A_1}-1}(i_{j_1}-1)+\cdots+i_{j_{n_{A_1}}}$, $b=(d^2-1)^{n_{A_2}-1}(i_{j_{n_{A_1}}+1}-1)+\cdots+i_{j_n}$, $i_{j_1}, i_{j_2}, \ldots, i_{j_n}=1, 2,\ldots, d^2-1$.
Let $\rho\in H^d_1\otimes\cdots \otimes H^d_n $ be a pure state. If $\rho$ is separable under bipartition $A_1|A_2$, then for any $k=1,\cdots,d^2-1$, $$\begin{aligned}
\parallel T_{A_1|A_2}\parallel_k\leq \sqrt{\frac{2^n(d^{n_{A_1}}-1)(d^{n_{A_2}}-1)}{d^n}}.\end{aligned}$$
If $\rho$ is separable under bipartition $A_1|A_2$, then $\rho_{A_1}\otimes \rho_{A_2}$. Using the inequality (22), we get $$\begin{aligned}
\parallel T_{A_1|A_2}\parallel_k&&=\parallel T^{(A_1)}(T^{(A_2)})^t \parallel_k=\parallel T^{(A_1)}\parallel\cdot\parallel (T^{(A_2)})^t\parallel_k\nonumber\\&&=\parallel T^{(A_1)}\parallel\cdot\parallel (T^{(A_2)})^t\parallel=\parallel T^{(A_1)}\parallel\cdot\parallel (T^{(A_2)})\parallel\nonumber\\&&\leq\sqrt{\frac{2^n(d^{n_{A_1}}-1)(d^{n_{A_2}}-1)}{d^n}}.\end{aligned}$$
Let $\rho\in H^d_1\otimes\cdots \otimes H^d_n $ be a pure state such that $\rho$ is separable under at least one bipartition. For any $k=1,\cdots,d^2-1$ and $j_1\neq j_2\neq \cdots\neq j_n\in \{1,2,\cdots,n\}$, we have
$(i)$ if $\rho$ is entangled under a certain bipartition $j_1|j_2\cdots j_n$, then
$\parallel T_{j_1|j_2\cdots j_n}\parallel_k\leq \sqrt{\frac{2^nk(d^{[\frac{n}{2}]}-1)(d^{n-[\frac{n}{2}]}-1)}{d^n}}$ $([]$ denotes integer function$)$, when $n$ is odd;
$\parallel T_{j_1|j_2\cdots j_n}\parallel_k\leq\sqrt{\frac{2^nk(d^{\frac{n}{2}}-1)^2}{d^n}}$, when $n$ is even;
$(ii)$ if $\rho$ is entangled under a certain bipartition $j_1\cdots j_{n-1} |j_n$, then
$\parallel T_{j_1\cdots j_{n-1} |j_n}\parallel_k\leq \sqrt{\frac{2^nk(d^{[\frac{n}{2}]}-1)(d^{n-[\frac{n}{2}]}-1)}{d^n}}$, when $n$ is odd;
$\parallel T_{j_1\cdots j_{n-1} |j_n}\parallel_k\leq\sqrt{\frac{2^nk(d^{\frac{n}{2}}-1)^2}{d^n}}$, when $n$ is even.
$(i)$ If $\rho$ is entangled under bipartition $j_1|j_2\cdots j_n$, then there is at least one bipartition ${j'_1}\cdots {j'_p}|{j'_{p+1}}\cdots {j'_n}$ $(p=1,2\cdots,n-1)$ such that $\rho$ is separable. Let ${j'_1}\cdots {j'_p}|{j'_{p+1}}\cdots {j'_n}=A_1|A_2$, then $n_{A_1}=p$.
$\textcircled{1}$ $j_1=1$. If $p=1$ and $j'_1\neq1$, we have $$\begin{aligned}
\label{u1}
\parallel T_{j_1|j_2\cdots j_n}\parallel_k&=&\parallel T^{({j'_1})}(T^{({j'_2},\cdots,{j'_n})})^t\parallel_k=\parallel T^{({j'_1})}\parallel\cdot\parallel(T^{({j'_2},\cdots,{j'_n})})^t\parallel_k\nonumber \\ &=&\parallel T^{({j'_1})}\parallel\cdot\parallel T^{({j'_2},\cdots,{j'_n})}\parallel\leq\sqrt{\frac{2^n(d-1)(d^{n-1}-1)}{d^n}}.\end{aligned}$$
If $p=2,3,\cdots,n-1$, we get $$\begin{aligned}
\label{u2}
\parallel T_{j_1|j_2\cdots j_n}\parallel_k&=&\parallel T^{(\underline{j'_1},\cdots,j'_p)}\otimes(T^{(j'_{p+1},\cdots,j'_n)})^t\parallel_k=\parallel T^{(\underline{j'_1},\cdots,j'_p)}\parallel_k\cdot\parallel(T^{(j'_{p+1},\cdots,j'_n)})^t\parallel_k\nonumber \\ &\leq&\sqrt{k}\parallel T^{(\underline{j'_1},\cdots,j'_p)}\parallel\cdot\parallel T^{(j'_{p+1},\cdots,j'_n)}\parallel\leq\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}}.\end{aligned}$$
$\textcircled{2}$ $j_1=2,\cdots,n-1$. For any $p$ we have $$\begin{aligned}
\label{u3}
\parallel T_{j_1|j_2\cdots j_n}\parallel_k&=&\parallel T^{(j'_1,\cdots,\underline{j'_{j_1}},\cdots,j'_p)}\otimes(T^{(j'_{p+1},\cdots,j'_n)})^t\parallel_k=\parallel T^{(j'_1,\cdots,\underline{j'_{j_1}},\cdots,j'_p)}\parallel_k\cdot\parallel(T^{(j'_{p+1},\cdots,j'_n)})^t\parallel_k\nonumber \\ &\leq&\sqrt{k}\parallel T^{(j'_1,\cdots,\underline{j'_{j_1}},\cdots,j'_p)}\parallel\cdot\parallel T^{(j'_{p+1},\cdots,j'_n)}\parallel\leq\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}}.\end{aligned}$$ $\textcircled{3}$ $j_1=n$. If $p=1,\cdots,n-2$, we have $$\begin{aligned}
\label{u5}
\parallel T_{j_1|j_2\cdots j_n}\parallel_k&=&\parallel (T^{(j'_1,\cdots,j'_p)})^t\otimes T^{(j'_{p+1},\cdots,\underline{j'_n})}\parallel_k=\parallel (T^{(j'_1,\cdots,j'_p)})^t\parallel_k\cdot\parallel T^{(j'_{p+1},\cdots,\underline{j'_n})}\parallel_k\nonumber \\ &\leq&\sqrt{k}\parallel T^{(j'_1,\cdots,j'_p)}\parallel\cdot\parallel T^{(j'_{p+1},\cdots,\underline{j'_n})}\parallel\leq\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}}.\end{aligned}$$
If $p=n-1$, we get $$\begin{aligned}
\label{u6}
\parallel T_{j_1|j_2\cdots j_n}\parallel_k&=&\parallel (T^{(j'_1,\cdots,j'_{n-1})})^t\otimes T^{(j'_n)}\parallel_k=\parallel (T^{(j'_1,\cdots,j'_{n-1})})^t\parallel\cdot\parallel T^{(j'_n)}\parallel_k\nonumber \\
&=&\parallel T^{(j'_1,\cdots,j'_{n-1})}\parallel\cdot\parallel T^{(j'_n)}\parallel\leq\sqrt{\frac{2^n(d-1)(d^{n-1}-1)}{d^n}}.\end{aligned}$$
Now consider max$\{\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}},\sqrt{\frac{2^n(d-1)(d^{n-1}-1)}{d^n}}\}$ $p=1,\cdots,n-1$. Let $y=(d^h-1)(d^{n-h}-1)$ $(h>0)$ be a continuous function. Then the maximal value is $y_{max}=(d^{\frac{n}{2}}-1)^2$. If $n$ is odd, $\parallel T_{A_1|A_2}\parallel_k\leq
\sqrt{\frac{2^nk(d^{[\frac{n}{2}]}-1)(d^{n-[\frac{n}{2}]}-1)}{d^n}}$. If $n$ is even, $\parallel T_{A_1|A_2}\parallel_k\leq\sqrt{\frac{2^nk(d^{\frac{n}{2}}-1)^2}{d^n}}$.
$(ii)$ If $\rho$ is entangled under bipartition $j_1\cdots j_{n-1}|j_n$, then there is at least one bipartition ${j'_1}\cdots {j'_p}|{j'_{p+1}}\cdots {j'_n}$ $p=1,2\cdots,n-1$, such that $\rho$ is separable. Similarly, let ${j'_1}\cdots {j'_p}|{j'_{p+1}}\cdots {j'_n}=A_1|A_2$, then $n_{A_1}=p$. The proof can be done in three cases.
$\textcircled{1}$ $j_n=1$. If $p=1$, we have $$\begin{aligned}
\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k&=&\parallel (T^{(j'_1)})^t\otimes T^{(j'_2,\cdots,j'_n)}\parallel_k=\parallel (T^{(j'_1)})^t\parallel_k\cdot\parallel T^{(j'_2,\cdots,j'_n)}\parallel_k\nonumber \\ &=&\parallel T^{(j'_1)}\parallel\cdot\parallel T^{(j'_2,\cdots,j'_n)}\parallel\leq\sqrt{\frac{2^n(d-1)(d^{n-1}-1)}{d^n}}.\end{aligned}$$
If $p=2,\cdots,n-1$, we get $$\begin{aligned}
\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k&=&\parallel T^{(j'_1,\underline{j'_2,\cdots,j'_p})}\otimes T^{(j'_{p+1},\cdots,j'_n)}\parallel_k=\parallel T^{(j'_1,\underline{j'_2,\cdots,j'_p})}\parallel_k\cdot\parallel T^{(j'_{p+1},\cdots,j'_n)}\parallel_k\nonumber \\ &\leq&\sqrt{k}\parallel T^{(j'_1,\underline{j'_2,\cdots,j'_p})}\parallel\cdot\parallel T^{(j'_{p+1},\cdots,j'_n)}\parallel\leq\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}}.\end{aligned}$$
$\textcircled{2}$ $j_n=2,\cdots,n-1$. For any $p$ we have $$\begin{aligned}
\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k&=&\parallel T^{(\underline{{j'_1},\cdots},{j'_{j_n}},\underline{\cdots,{j'_p}})}\otimes T^{(j'_{p+1},\cdots,{j'_n})}\parallel_k=\parallel T^{(\underline{{j'_1},\cdots},{j'_{j_n}},\underline{\cdots,{j'_p}})}\parallel_k\cdot\parallel T^{(j'_{p+1},\cdots,{j'_n})}\parallel_k\nonumber \\ &\leq&\sqrt{k}\parallel T^{(\underline{{j'_1},\cdots},{j'_{j_n}},\underline{\cdots,{j'_p}})}\parallel\cdot\parallel T^{(j'_{p+1},\cdots,{j'_n})}\parallel\leq\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}}.\end{aligned}$$ $\textcircled{3}$ $j_n=n$. If $p=1,\cdots,n-2$, we get $$\begin{aligned}
\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k&=&\parallel T^{({j'_1},\cdots,{j'_p})}\otimes T^{(\underline{{j'_{p+1}},\cdots,{j'_{n-1}}},{j'_n})}\parallel_k=\parallel T^{({j'_1},\cdots,{j'_p})}\parallel_k\cdot\parallel T^{(\underline{{j'_{p+1}},\cdots,j'_{n-1}},{j'_n})}\parallel_k\nonumber \\ &\leq&\sqrt{k}\parallel T^{({j'_1},\cdots,{j'_p})}\parallel\cdot\parallel T^{(\underline{{j'_{p+1}},\cdots,j'_{n-1}},{j'_n})}\parallel\leq\sqrt{\frac{2^nk(d^{p}-1)(d^{n-p}-1)}{d^n}}.\end{aligned}$$
If $p=n-1$ and $j'_n\neq n$, we have $$\begin{aligned}
\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k&=&\parallel T^{(j'_1,\cdots,j'_{n-1})} (T^{(j'_n)})^t\parallel_k=\parallel T^{(j'_1,\cdots,j'_{n-1})}\parallel\cdot\parallel(T^{(j'_n)})^t\parallel_k\nonumber \\
&=&\parallel T^{(j'_1,\cdots,j'_{n-1})}\parallel\cdot\parallel T^{(j'_n)}\parallel\leq\sqrt{\frac{2^n(d-1)(d^{n-1}-1)}{d^n}}.\end{aligned}$$ If $n$ is odd, $\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k\leq
\sqrt{\frac{2^nk(d^{[\frac{n}{2}]}-1)(d^{n-[\frac{n}{2}]}-1)}{d^n}}$. If $n$ is even, $\parallel T_{j_1\cdots j_{n-1}|j_n}\parallel_k\leq\sqrt{\frac{2^nk(d^{\frac{n}{2}}-1)^2}{d^n}}$.
Conclusion
==========
We have studied genuine multipartite entanglement in four-partite and multipartite qudit quantum systems, and derived the relationship between the norms of the correlation tensors and the specific matrix $T$. Based on these relations we have presented a criterion to detect GME in four-partite quantum systems. These results are generalized to multipartite systems. Our main results concern with special inequalities that bound the various norms of the correlation tensors, upon which our criterion is presented to detect GME in multipartite systems. These results can help distinguishing genuine multipartite entangled states. Genuine multipartite entanglement plays significant roles in many quantum information processing. Our approach and results may highlight further researches on the theory of genuine multipartite entanglement.
$\textbf{Acknowledgments}$ This work is supported by the National Natural Science Foundation of China under grant Nos. 11101017, 11531004, 11726016 and 11675113, and Simons Foundation under grant No. 523868, the NSF of Beijing under Grant No. KZ201810028042.
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abstract: 'This paper is a next step in the project of systematic description of colored knot and link invariants started in [@MMfing; @Rama2]. In this paper, we managed to explicitly find the Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at [@knotebook]. Using the obtained inclusive Racah matrices, we also calculated the Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.'
author:
- |
[**C. Bai$^a$**]{}, [**J. Jiang$^b$**]{}, [**J. Liang$^a$**]{}, [**A. Mironov$^{c,d,e}$**]{},\
[**A. Morozov$^{d,e}$**]{}, [**An. Morozov$^{d,e}$**]{}, [**A. Sleptsov$^{d,e,f}$**]{}
date:
---
FIAN/TD-31/17\
IITP/TH-02/18\
ITEP/TH-02/18\
$^a$\
$^b$\
$^c$\
$^d$\
$^e$\
$^f$
Introduction
============
Knot and link invariants are now widely studied objects. In part, this is due to their many connections to other areas of physics. One of the most important connections was first realized by E.Witten [@Wit] for a particular set of knot polynomials, Jones polynomials [@Jones], which, as he claimed, were equal to the Wilson-loop averages of the Chern-Simons theory with the gauge group $SU(2)$ [@CS]. This was generalized to much wider classes of polynomials and consequently gauge groups, that is, to the HOMFLY-PT polynomials [@HOMFLY] and the gauge group $SU(N)$ or the Kauffman polynomials [@Kauf] and the gauge group $SO(N)$.
What makes knot/link invariants interesting from the point of view of quantum field theories is that these are exact answers for the averages, i.e. they are calculated in their exact form without using any kind of perturbation theory. This makes them a rare example of such observables in quantum field theories.
Also widely known are connections between knot polynomials and transformations of conformal blocks in 2D CFT [@Wit; @inds]. Due to this relation, the exclusive Racah, which we discuss later in this paper can be used to describe modular transformations of the corresponding conformal blocks. Quite important are connections with topological strings [@OV] which in particular implies certain integrality conjectures for knot and link invariants (see [@LMOV]), and the results of the present paper could help in further checking this conjecture for links and consequently check relations with topological strings. Another important connection is possible topological quantum computing [@QC]. It can be based on anyons, which are quasi-particles interacting with the Chern-Simons action. The observables (programs) for this quantum computer should be interpreted as some knot invariants. Thus calculations made in the present paper can provide a further insight in how this quantum computer can work.
The present paper is a continuation of the long program devoted to studies of knot invariants [@MMfing; @Rama2]. In our previous papers, we managed to calculate inclusive and exclusive Racah matrices needed for calculations of knot polynomials in representations up to size four. In the present paper, we further generalize these results to include not only knots but links with distinct representations on distinct components. First, we calculate the inclusive Racah matrices and then, using these matrices, we also calculate the exclusive Racah matrices with the help of the trick earlier suggested in the case of knots [@MMMSint]. This gives a check of this method, which is proved to be effective and allows us to reproduce the exclusive Racah matrices earlier obtained by another method [@RacahRama].
The present paper is organized as follows. Section \[s.rac\] is devoted to the description of Racah matrices with s.\[incrac\] dealing with inclusive Racah matrices and s.\[exrac\] with exclusive Racah matrices. Section \[incrac\] also deals with the definition of $\mathcal{R}$-matrices. Section \[s.raccalc\] describes the methods to calculate the Racah matrices with s.\[s.exin\] providing details on how to find the exclusive matrices from the inclusive ones, while s.\[hwc\] recalls the highest weight calculus method for inclusive Racah matrices. Section \[s.raclist\] provides the list of all Racah matrices for representations up to size 3. Section \[s.signs\] describes how the signs of $\mathcal{R}$-matrices eigenvalues are defined. Section \[s.eig\] recalls the eigenvalue hypothesis for the Racah matrices which immediately provides answers for a number of Racah matrices. As an example of application of the obtained Racah matrices, we consider three essentially different links that admit 3-stand braid representation: the Whitehead link, the three-component Borromean rings, and link $L7a3$, which has two components, one of them being not unknot. Section \[s.HOMFLY\] introduces the links invariants that we evaluate, with the whole list of answers contained in Appendix \[a.pol\]. Section \[s.exex\] provides an example of evaluating the exclusive Racah matrix from the inclusive ones using the approach of s.\[s.exin\].
Racah matrices\[s.rac\]
=======================
The Racah matrices are crucial elements for evaluating invariants of knots and links. There are two distinct sets of Racah matrices which appear in different approaches to studying knot polynomials: and Racah matrices.
Inclusive Racah matrices\[incrac\]
----------------------------------
One of the approaches is originally due to N. Reshetikhin and V. Turaev [@RT], hence the name Reshetikhin-Turaev (RT) approach. According to this approach [@MMMI; @RTmod], in order to construct a knot/link invariant for the knot/link presented by a closed braid, one has to associate each crossing in this braid with a particular $\mathcal{R}$-matrix. If the studied object is a knot colored with a representation $R$, each strand in the braid carries the same representation $R$, all the $\mathcal{R}$-matrices act on the same tensor product $R\otimes R$, and the eigenvalues of all these $\mathcal{R}$-matrices are the same. $\mathcal{R}$-matrices acting on different pairs of strands are connected by rotation matrices. These rotation matrices appear to be Racah matrices [@MMMI]. Let us discuss in more details the three-strand case, which was mostly studied in our recent papers [@MMMI; @MMMS21; @MMMS31; @IMMMec; @Univ; @BJLMMMS] and which we study in the present paper.
Within our approach, the object discussed is in fact not just the HOMFLY-PT polynomial but its character expansion. If one studies a three-strand knot in representation $R$, then the character expansion of its HOMFLY-PT polynomial includes all the representations $Q$ from the decomposition of the triple tensor cube of the representation $R$: $Q\vdash R^{\otimes 3}$. The HOMFLY-PT polynomial is then given by $$H^K_R=\sum\limits_{Q\vdash R^{\otimes 3}} S^*_Q C_Q,$$ where $S^*_Q$ are characters of representations $Q$ and $C_Q$ are coefficients constructed from the $\mathcal{R}$-matrices and Racah matrices. Thus, for each $Q$ for the three-strand knots, there exist two $\mathcal{R}$-matrices (the one corresponding to the crossing between upper pair of strands and the one corresponding to the lower pair) and they are related by rotation with a Racah matrix. One of the $\mathcal{R}$-matrices can be chosen diagonal, for the sake of definiteness, we choose the upper one as such, and the eigenvalues of the $\mathcal{R}$-matrix are described, up to a factor, by the eigenvalue of the second Casimir operator[^1] $\varkappa_Y$: $$\begin{array}{l}
\mathcal{R}_{Q;1}=diag(\epsilon_Y q^{\varkappa_Y}),\ Y\vdash R^{\otimes 2},\ Q\vdash Y\otimes R;
\\
\mathcal{R}_{Q;2}=U_Q \mathcal{R}_{Q;1} U^{\dagger}_Q;\ \ \ \ \ \ \ \ C_Q=Tr(\prod\limits_i \mathcal{R}_{Q;i}).
\end{array}$$ Also needed for the definition of $\mathcal{R}$-matrices are $\epsilon_Y=\pm 1$, which describe the signs of these eigenvalues. At least for knots, the signs depend on if the representation $Y$ comes from symmetric ($+1$) or antisymmetric ($-1$) square of the initial representation $Q$, see [@MMMS21] for details. We discuss the issue of signs in more details in s.\[s.signs\].
We call the set of Racah matrices for representations $Q\vdash R^{\otimes 3}$ Racah matrices. In traditional 6-j symbol notations, they can be written as $$\label{UQ}
U_Q=U_{XY}\left[ \begin{array}{cc} R&R \\ R & Q \end{array} \right],\ X,Y\vdash R^{\otimes 2},\ Q\vdash X,Y\otimes R.$$
If one discusses links rather than knots, the situation becomes slightly more complicated. Instead of one representation $R$ on all three strands, one could have three different representations $R_1$, $R_2$ and $R_3$. Consequently, there are two more diagonal $\mathcal{R}$-matrices for each $Q\vdash R_1\otimes R_2\otimes R_3$, i.e. there are three diagonal matrices in total: $\mathcal{R}_{12}$, $\mathcal{R}_{13}$ and $\mathcal{R}_{23}$. There are three independent Racah matrices as well: $U_{123}$, $U_{132}$ and $U_{213}$. Matrices corresponding to three remaining orderings coincide with the transposed matrices of the first three.
![Trefoil knot $3_1$ presented as a braid (on the left) and as a two-bridge knot(on the right).[]{data-label="knots"}](knot_2.jpg "fig:"){width="6"} ![Trefoil knot $3_1$ presented as a braid (on the left) and as a two-bridge knot(on the right).[]{data-label="knots"}](knot_1.jpg "fig:"){width="6"}
Exclusive Racah matrices\[exrac\]
---------------------------------
Another approach to evaluating knot invariants is to study the two-bridge knots or their generalization, the arborescent knots [@Cau; @arbor; @Rama1]. Such knots have also representations different from the braid representation: the closure of the braid is made differently. From the point of view of representation theory, this leads to a different representation structure. E.g. if one studies the two-bridge knots, among all representations from the product of these four, only the trivial representation contributes, i.e. the four strands carry representations $R$, $R$, $\bar{R}$ and $\bar{R}$. This means that, in the decomposition of the product of three representations out of four $R$, $R$, $\bar{R}$ and $\bar{R}$ discussed in s.\[incrac\], also only the single one contributes: the conjugate to the remaining fourth representation. In its turn, this implies that, in this case, there is no set of the inclusive Racah matrices $U_Q$ (\[UQ\]), and only two matrices for each representation $R$ survive in the two-bridge case depending on the choice of the three representations out of four $R$, $R$, $\bar{R}$ and $\bar{R}$. Indeed, there are two $\mathcal{R}$-matrices for two bridge knots: one, which we call $T_R$ is associated with the crossing of two strands with either representations $R$ on the both strands or representations $\bar{R}$; another one, which we call $\bar{T}_R$ is associated with the crossing between different representations $R$ and $\bar{R}$. Consequently, in this case, there are, indeed, two Racah matrices, which we call . These matrices relate $\mathcal{R}$-matrices acting on two strands in the middle and the ones, on two strands on the side (in the case of representations described by rectangular Young diagrams, both matrices acting on either upper pair of strands or on lower one coincide, while for other cases they typically differ). We denote $S$ the exclusive Racah matrices that intertwine the product $(R\otimes R)\otimes \bar{R}$ and the product $R\otimes (R\otimes \bar{R})$, and, $\bar{S}$ those intertwining the product $(R\otimes \bar{R})\otimes R$ and the product $R\otimes (\bar{R}\otimes R)$. From this definition, it immediately follows that $\bar{S}^{\dagger}=\bar{S}$. In the traditional 6-j symbol notation, these exclusive Racah matrices are written as $$\begin{array}{l}
S_R=U_{XY}\left[ \begin{array}{cc} R&R \\ \bar{R} & R \end{array} \right],\
\begin{array}{l}
X\vdash R\otimes\bar{R},\ R\vdash R\otimes X, \\
Y\vdash R\otimes R,\ R\vdash Y\otimes \bar{R}.
\end{array}
\\
\bar{S}_R=U_{XY}\left[ \begin{array}{cc} R&\bar{R} \\ R & R \end{array} \right],\
\begin{array}{l}
X\vdash R\otimes\bar{R},\ R\vdash R\otimes X, \\
Y\vdash \bar{R}\otimes R,\ R\vdash Y\otimes R.
\end{array}
\end{array}$$
For links, the picture is again more complicated. In this case, due to the construction, there can appear only two-component links colored by $R_1$ and $R_2$. Four strands now carry in any order representations $R_1$, $R_2$, $\bar{R}_1$ and $\bar{R}_2$. This provides four $\mathcal{R}$-matrices: $$T_{12}=T_{\bar{1}\bar{2}},\ T_{1\bar{1}},\ T_{1\bar{2}}=T_{\bar{1}2},\ T_{2\bar{2}}.
$$ There are also three independent Racah matrices: $$S_{1\bar{1}2}= S^{\dagger}_{\bar{1}2\bar{2}},\ S_{1\bar{1}\bar{2}}= S^{\dagger}_{\bar{1}\bar{2}2},\ S_{12\bar{1}}= S^{\dagger}_{2\bar{1}\bar{2}}.$$ All other Racah matrices either coincide or are transposed of one of these three.
![Knot presented as an arborescent diagram.[]{data-label="knotarb"}](Arbor.jpg){width="6"}
Calculating Racah matrices\[s.raccalc\]
=======================================
The main approach to calculating Racah matrices uses representation theory of quantum groups and the highest weight vector calculations. This approach is described in detail in section \[hwc\].
Unfortunately, it is much harder to calculate exclusive Racah matrices using the same approach, since among other reasons the highest weight vectors of conjugate representations essentially depend on the group $SU(N)$ in contrast with non-conjugate representations. However, there is a “trick” [@MMMSint] which allows one to find the exclusive Racah matrices from the inclusive ones.
Exclusive Racah through inclusive Racah\[s.exin\]
-------------------------------------------------
This trick suggested in [@MMMSint] for knots is based on studying a particular series of knots which are both three-strand and arborescent. One such example is given by the three-strand knots $(m,-1|\pm n,-1)$ in the notation of [@MMMI], which, at the same time, are the Pretzel knots $Pr(m,n,\pm \bar{2})$ [@MMSpret].
One can use the evolution method [@evo; @MMSpret] to write down the answer for the whole series of knots. For the three-strand representation, one gets $$H_R^{(m,-1|\pm n, -1)}=\sum\limits_{Y,Z\vdash R^{\otimes 2}} h_{YZ}\cdot \lambda_Y^m\lambda_Z^n.$$ On the other side for the same series described as Pretzel knots [@MMSpret; @Rama1], one gets: $$H_R^{Pr(m,n,\pm \bar{2})}=d^2_R \sum\limits_{\bar{X}\vdash R\otimes\bar{R}}
\cfrac{(ST^mS^{\dagger})_{\emptyset \bar{X}}(ST^nS^{\dagger})_{\emptyset \bar{X}}(\bar{S}\bar{T}^{\pm 2}\bar{S})_{\emptyset \bar{X}}}{\bar{S}_{\emptyset \bar{X}}}=
\sum\limits_{{\bar{X}\vdash R\otimes\bar{R}}\atop{Y,Z\vdash R\otimes R}}\sqrt{d_Y d_Z} K_{\bar{X}}S_{\bar{X}Y}S_{\bar{X}Z}\cdot\lambda_Y^m\lambda_Z^n,$$ where $\lambda_Y$ and $\lambda_Z$ are the eigenvalues of $\mathcal{R}$-matrices in the three strand case, they form the diagonal matrix $T$ from the point of view of the Pretzel knots (they coincide, because they are associated with the crossing with the same representations).
The equality between these two formulae can be rewritten as $$\sum\limits_{\bar{X}} \left(K_{\bar{X}} (\bar{S}\bar{T}^{\pm 2}\bar{S})_{\emptyset\bar{X}}\right)S_{\bar{X}Y}S_{\bar{X}Z}=\cfrac{h_{YZ}}{\sqrt{d_Y d_Z}}.$$ Thus, the matrix $S$ can be found as the matrix that diagonalizes the matrix at the r.h.s.
In the case of links, the situation is as always more difficult. One can use the same series to study links as well. To this end, $m$ and $n$ should be even (we will just put $2m$ and $2n$ instead). Then, the three-strand braid $(2m,-1|2n,\pm 1)$ is a three component link. If we color it with representations $R_1$, $R_2$ and $R_3$, the resulting polynomial is $$H_{123}^{(2m,-1|2n,\pm 1)}=\sum\limits_{Q\vdash R_1\otimes R_2\otimes R_3}S^*_Q \text{Trace}\left(
\mathcal{R}^{2m}_{12} U_{123} \mathcal{R}^{-1}_{23} U_{132}^{\dagger} \mathcal{R}^{2n}_{13} U_{132} \mathcal{R}^{-1}_{23} U_{123}^{\dagger}
\right)_Q=\sum\limits_{{Y\vdash R_1\otimes R_2}\atop{Z\vdash R_1\otimes R_3}} h_{YZ}\lambda_Y^{2m}\lambda_Z^{2n}.$$ At the same time, evaluating the same links from the Pretzel representation gives the following expression $$\begin{array}{r}
H_{123}^{Pr(2m,2n,\pm \bar{2})}=d^2_{R_1} \sum\limits_{\bar{X}\vdash R_1\otimes\bar{R_1}}
\cfrac{(S_{1\bar{1}\bar{2}}T_{12}^{2m}S_{1\bar{1}\bar{2}}^{\dagger})_{\emptyset \bar{X}}(S_{1\bar{1}\bar{3}}T_{13}^{2n}S_{1\bar{1}\bar{3}}^{\dagger})_{\emptyset \bar{X}} (S_{2\bar{2}3}T_{2\bar{3}}^{\pm 2}S_{2\bar{2}3}^{\dagger})_{\emptyset \bar{X}}}{\bar{S}_{\emptyset \bar{X}}}=
\\ \\
=\sum\limits_{{{\bar{X}\vdash R_1\otimes\bar{R_1}}\atop{Y\vdash R_1\otimes R_2}}\atop{Z\vdash R_1\otimes R_3}}\sqrt{d_Y d_Z} K_{\bar{X}}S_{1\bar{1}\bar{2}|\bar{X}Y}S_{1\bar{1}\bar{3}|\bar{X}Z}\cdot\lambda_Y^{2m}\lambda_Z^{2n}.
\end{array}$$ Here we assume that, in the case of symmetric representations, $R_1$ is the smallest representation and dimensions are defined by this smallest representations. This is due to the fact that the set of representations $\bar{X}$ is defined by the smallest of the representations between $R_1$, $R_2$ and $R_3$. If representations are not symmetric then the set of $\bar{X}$ and dimensions are defined by the intersection of sets of representations $R_1\otimes \bar{R}_1$, $R_2\otimes \bar{R}_2$ and $R_3\otimes \bar{R}_3$. Comparing these relations, one gets the equation $$K_{\bar{X}}S_{1\bar{1}\bar{2}|\bar{X}Y}\lambda_Y^{2m}S_{1\bar{1}\bar{3}|\bar{X}Z}\lambda_Z^{2n}=\cfrac{h_{YZ}}{\sqrt{d_Y d_Z}}.$$ Repeating this procedure for other placements of colors, one can get three equations for two matrices each. Solving them will provide the three exclusive Racah matrices $S_{1\bar{1}\bar{2}}$, $S_{1\bar{1}\bar{3}}$ and $S_{2\bar{2}\bar{3}}$. Similar equations can be constructed for other matrices $S$.
We provide a non-trivial essentially link-related example of calculating the exclusive Racah matrices in s.\[s.exex\].
Symmetries of Racah matrices
----------------------------
The Racah matrices possess some symmetry properties, which allow one to greatly reduce the number of Racah coefficients that require direct computation. The main property is general for all the constructions in quantum groups: the substitution $q\rightarrow -q^{-1}$ corresponds to the transposition of all the Young diagrams associated with the representations. This means that if one has calculated the Racah matrix $U_{XY}\left[ \begin{array}{cc} R&R \\ R & Q \end{array} \right]$, it also has provided the answer for $U_{X^TY^T}\left[ \begin{array}{cc} R^T&R^T \\ R^T & Q^T \end{array} \right]$.
Another important symmetry property of the Racah matrix is that, for the inclusive Racah matrices, one can reverse the order of multiplication of all representations. This property was already mentioned earlier, basically it means that the Racah matrix appearing in the product $R_1\otimes R_2\otimes R_3$ is inverse to the one appearing in the product $R_3\otimes R_2\otimes R_1$.
The highest weight calculus \[hwc\]
-----------------------------------
Here we repeat the method to calculate the Racah matrices used in our previous papers [@MMMI; @MMMS21; @MMMS31; @IMMMec; @Univ; @BJLMMMS].
The Racah matrix $U$ is a transformation matrix from one orthonormal basis (I) to another one (II), which are defined as follows: $$U_{XY}\left[ \begin{array}{cc} R_1&R_2\\R_3&R_4 \end{array} \right]: \left( \underbrace{R_1 \otimes R_2}_X \right) \otimes R_3 \xrightarrow{ \ I \ } Q \ \ \rightarrow \ \ R_1 \otimes \left( \underbrace{R_2 \otimes R_3}_Y \right) \xrightarrow{ \ II \ } Q.$$ For example, in the case $R_1=[3],R_2=[2,1],R_3=[1,1,1]$ and $Q$ is arbitrary, most of the Racah matrices are equal to the identity matrix except for few ones. To find $Q$ giving non-trivial contributions, one can use the Littlewood-Richardson rule: $$\begin{array}{ccl}\label{[3][21][111]dec}
\chi_{R_1} \cdot \chi_{R_2} &=& \sum_Q C_{R_1,R_2}^{Q} \cdot \chi_Q \\
\chi_{[3]} \cdot \chi_{[2,1]} \cdot \chi_{[1,1,1]} &=& \chi_{[6,2,1]}+\chi_{[6,1,1,1]}+\chi_{[5,3,1]}+3\,\chi_{[5,2,1,1]}+2\,\chi_{[5,1,1,1,1]}+\chi_{[4,3,2]}+2\,\chi_{[4,3,1,1]} \\ &+& 2\,\chi_{[4,2,2,1]}+ 3\,\chi_{[4,2,1,1,1]}+\chi_{[4,1,1,1,1,1]} + \chi_{[3,3,2,1]}+\chi_{[3,3,1,1,1]}+\chi_{[3,2,2,1,1]} \\ &+&\chi_{[3,2,1,1,1,1]},
\end{array}$$ where $\chi_R$ is the character of the irreducible representation, which is the Schur function in the case of $SU(N)$, while $R$’s in this case are labelled by the Young diagrams. From now on, we identify the representations with the Young diagrams.
The coefficients $C_{R_1,R_2}^{Q}$ count how many times the irreducible representation $Q$ appears in the decomposition, therefore they determine the size of the corresponding Racah matrix. Decomposition (\[\[3\]\[21\]\[111\]dec\]) shows us that there are two matrices of size $3\times3$, three matrices of size $2\times2$ and ten trivial “matrices” of size $1\times1$.
We calculate the Racah matrix using its definition, i.e. as a transformation matrix from the orthonormal basis (I) to the orthonormal basis (II). To this end, we construct the highest weight vectors in the basis (I) for each representation $Q$ from (\[\[3\]\[21\]\[111\]dec\]) and same in the basis (II). To proceed, we need to describe manifestly the action of lowering and raising operators $T_k^{\pm}$ on representations of $U_q(sl_N)$.
To this end, we use the Schur-Weyl duality and, first, realize the representation of $U_q(sl_N)$ in the space of tensors. With each representation labeled by the Young diagram $Y = \{ Y_1\geq Y_2\geq\ldots\geq Y_l>0 \}$, we associate the following tensor with all possible permutations of indices: $$\label{elY}
V_{i_1,\dots,i_{Y_1},j_1,\dots,j_{Y_2},k_1,\dots,k_{Y_3},\ldots}, \text{where}\
i_1=\dots=i_{Y_1}=0, \ j_1=\dots=j_{Y_2}=1, \ k_1=\dots=k_{Y_3}=2, \ldots .$$ In other words, the number of zeros is equal to $Y_1$, the number of units is equal to $Y_2$, the number of deuces is equal to $Y_3$ and so on. Thus, every vector of the representation $Y$ can be written as a linear combination of elements (\[elY\]).
Second, let us define the action of lowering and raising operators $T_k^{\pm}$. It is clear that for 1-tensors they act as follows: $$\begin{array}{l}
T_k^{+}: \left\{ \begin{array}{lcl}V_{k-1} & \longrightarrow & V_k, \\ V_i &\longrightarrow & 0 (i\neq k-1), \end{array}\right.\\
T_k^{-}: \left\{ \begin{array}{lcl}V_k & \longrightarrow & V_{k-1}, \\ V_i & \longrightarrow & 0 (i\neq k). \end{array}\right.
\end{array}$$
To extend this action to higher rank tensors, one needs a uniquely defined comultiplication $\Delta$ on $U_q(sl_N)$: $$\begin{array}{lcl}
\Delta(E_i) &=& 1\otimes E_i + E_i\otimes q^{H_i}, \\
\Delta(F_i) &=& F_i\otimes 1 + q^{-H_i}\otimes F_i, \\
\Delta(q^{H_i}) &=& q^{H_i}\otimes q^{H_i},
\end{array}$$ where $E_i, F_i, H_i(1\leq i \leq N-1)$ are generators of $U_q(sl_N)$.
Since $Y$ is a representation of $U_q(sl_N)$, it means there is a given algebra homomorphism between them. If we denote $T_k^{+}$ and $T_k^-$ to be the images of $E_k$ and $F_k$ respectively, one gets for 2-tensors: $$\begin{array}{ll}
T_k^{+}: V_{i,j} \longrightarrow & \delta_{k-1,j}V_{i,j+1} + \delta_{k-1,i} q^{H_k}(V_j) V_{i+1,j}, \\
T_k^{-}: V_{i,j} \longrightarrow & \delta_{k,j}q^{-H_k}(V_i) V_{i,j-1} + \delta_{k,i} V_{i-1,j}.
\end{array}$$ Here $\delta_{i,j}$ is the usual Kronecker symbol, and the notation $q^{H_k}(V_i)$ is defined as follows: $$q^{H_k}: \left\{ \begin{array}{lcl}V_{k-1} & \longrightarrow & q, \\ V_k &\longrightarrow & q^{-1}, \\ V_i &\longrightarrow & 1 \ (i\neq k-1 \ or \ k). \end{array}\right.$$ Actually it coincides with the action of $H_k$ in a representation space of $U_q(sl_N)$: $$H_k: \left\{ \begin{array}{lcl}v_{k-1} & \longrightarrow & v_{k-1}, \\ v_k &\longrightarrow & -v_k, \\ v_i &\longrightarrow & 0 \ (i\neq k-1 \ or \ k). \end{array}\right.$$
Since $\Delta$ is co-associative, it is easy to extend $T_k^{\pm}$ actions to any rank tensors.
Example 1. $[1]\otimes[1]\otimes[2]$ \[ex112\]
----------------------------------------------
The decomposition in this case takes the form $$[1]\otimes[1]\otimes[2] = [4] \oplus 2[3,1] \oplus [2,2] \oplus [2,1,1].$$ The Racah matrices for $[1]\otimes[1]\otimes[2]\to \{[4], [2,2], [2,1,1]\}$ are just equal to 1. The only non-trivial matrix is for $[1]\otimes[1]\otimes[2]\to [3,1]$, and it relates the bases $$([1]\otimes [1])\otimes [2]=([2]\oplus[1,1])\otimes[2]$$ and $$[1]\otimes ([1]\otimes [2])=[1]\otimes([3]\oplus[2,1]).$$ The corresponding highest weight vectors are $$\label{u1}
\begin{array}{l}
u_1 = \dfrac{1}{\sqrt{q^6+q^4+q^2+1}}\left(q^3v_{0 0 1 0}+q^2v_{0 0 0 1}-qv_{1 0 0 0}-v_{0 1 0 0}\right), \\
u_2 = \dfrac{1}{\sqrt{q^2+1}}\left(qv_{0 1 0 0}-v_{1 0 0 0}\right), \\
u^{'}_1 = \dfrac{1}{\sqrt{(q^6+q^4+q^2+1)(q^4+q^2+1)}}\left(q^5v_{0 1 0 0}+q^4v_{0 0 1 0}-q^4v_{1 0 0 0}+q^3v_{0 0 0 1}-q^2v_{1 0 0 0}-v_{1 0 0 0}\right), \\
u^{'}_2 = \dfrac{1}{\sqrt{(q^4+q^2+1)(q^2+1)}}\left(q^3v_{0 0 1 0}+q^2v_{0 0 0 1}-q^2v_{0 1 0 0}-v_{0 1 0 0}\right).
\end{array}$$
In order to demonstrate how to find the highest weight vectors, let us obtain $u_1$ as an example. From the definition, it is obvious that $v_{00}$ is just the highest weight vector in representation $[2]$. The action of $T_1^{+}$ on $v_{00}$ gives $v_{01}+qv_{10}$. Since $u_1$ is the highest weight vector in $[3,1]\vdash[2]\otimes[2]$, it should be a linear combination of two vectors $w_1$ and $w_2$, obtained by the action of $T_1^{+}$ on the first two indices of $v_{0000}$ and on the second two indices of $v_{0000}$ correspondingly: $u_1=\alpha w_1+\beta w_2$, $w_1=v_{0100}+qv_{1000}$, $w_2=v_{0001}+qv_{0010}$. To determine $\alpha$ and $\beta$, one requires that all lowering operators $T_k^-$ cancel on arbitrary vector from this space: $$T_k^-\left( \alpha \left( v_{0100}+qv_{1000} \right)+\beta \left( v_{0001}+qv_{0010}\right)\right) = 0 \Rightarrow \alpha=-c,\beta=q^2\cdot c,$$ where $c$ is an arbitrary constant.
By definition, the Racah matrix is a transformation matrix from one orthonormal basis to another one, hence, all the highest weight vectors have unit norms. This determines $c$ uniquely up to a sign: $$c = \pm\dfrac{1}{\sqrt{q^6+q^4+q^2+1}}.
\label{c31}$$ In the answer above, we choose the sign to be ’+’. This gives us the highest weight vector $u_1$ in (\[u1\]).
It is immediate now to obtain the Racah matrix: $$U\left[ \begin{array}{cc} [1]&[1] \\ \text{[2]} & [3,1] \end{array} \right] = \left( \begin{array}{cc} \dfrac{1}{\sqrt{[3]}}&\dfrac{\sqrt{[4]}}{\sqrt{[2][3]}} \\ \dfrac{\sqrt{[4]}}{\sqrt{[2][3]}} & -\dfrac{1}{\sqrt{[3]}} \end{array} \right).
\label{u112ans}$$
Example 2. $[1]\otimes[2]\otimes[1]$
------------------------------------
The decomposition in this case takes the form $$[1]\otimes[2]\otimes[1] = [4] \oplus 2[3,1] \oplus [2,2] \oplus [2,1,1]$$
The Racah matrices for $[1]\otimes[2]\otimes[1]\to\{[4], [2,2], [2,1,1]\}$ are just equal to 1. The only non-trivial matrix is for $[1]\otimes[2]\otimes[1]\to [3,1]$ and it relates the bases $$([1]\otimes [2])\otimes [1]=([3]\oplus[2,1])\otimes[1]$$ and $$[1]\otimes ([2]\otimes [1])=[1]\otimes([3]\oplus[2,1]).$$ The corresponding highest weight vectors are $$\begin{array}{l}
u_1 =\dfrac{1}{\sqrt{(1+q^2)(1+q^4)(1+q^2+q^4)}}\left(q(1+q^2+q^4)v_{0 0 0 1}-v_{0 0 1 0}-qv_{0 1 0 0}-q^2v_{1 0 0 0}\right),
\\
u_2 =\dfrac{1}{\sqrt{(1+q^2)(1+q^2+q^4)}}\left(q^2v_{0 0 1 0}+q^3v_{0 1 0 0}-(q^2+1)v_{1 0 0 0}\right),
\\
u^{'}_1 =\dfrac{1}{\sqrt{(1+q^2)(1+q^4)(1+q^2+q^4)}}\left(q^3v_{0 0 0 1}+q^4v_{0 0 1 0}+q^5v_{0 1 0 0}-(1+q^2+q^4)v_{1 0 0 0}\right),
\\
u^{'}_2 =\dfrac{1}{\sqrt{(1+q^2)(1+q^2+q^4)}}\left(-q(q^2+1)v_{0 0 0 1}+v_{0 0 1 0}+qv_{0 1 0 0}\right).
\end{array}$$
It is now immediate to calculate the Racah matrix: $$U\left[ \begin{array}{cc} [1]&[2] \\ \text{[1]} & [3,1] \end{array} \right] = \left( \begin{array}{cc} \dfrac{1}{[3]}&\dfrac{\sqrt{[4][2]}}{[3]} \\ \dfrac{\sqrt{[4][2]}}{[3]} & -\dfrac{1}{[3]} \end{array} \right).
$$
Example 3. $[1]\otimes[2]\otimes[2,1]$
--------------------------------------
The decomposition in this case takes the form $$[1]\otimes[2]\otimes[2,1] = [5,1]\oplus 2[4,2] \oplus 2[4,1,1] \oplus [3,3] \oplus 3[3,2,1] \oplus [3,1,1,1] \oplus [2,2,2] \oplus [2,2,1,1].$$ The Racah matrices for $[1]\otimes[2]\otimes[2,1]\to\{[5,1], [3,3], [3,1,1,1], [2,2,2], [2,2,1,1]\}$ are just equal to 1. The only non-trivial matrices are for $[1]\otimes[2]\otimes[2,1]\to\{[4,2], [4,1,1], [3,2,1]\}$ and they relate the bases $$([1]\otimes [2])\otimes [2,1]=([3]\oplus[2,1])\otimes[2,1]$$ and $$[1]\otimes ([2]\otimes [2,1])=[1]\otimes([4,1]\oplus[3,2]\oplus[3,1,1]\oplus[2,2,1]).$$ The corresponding highest weight vectors of representations $[4,2]$ are $$\begin{array}{ll}
u_1 =&\dfrac{1}{(q^2+1)\sqrt{(1+q^4)(1+q^2+q^4)}}(q(q^4+q^2+1)(qv_{000011}-v_{000101})-q^3v_{100010}+q^2v_{100100}\\
&+qv_{010100}-q^2v_{010010}+v_{001100}-qv_{001010}),\\
u_2 =&\dfrac{1}{(q^2+1)\sqrt{(1+q^2+q^4)}}(q^4v_{010010}-q^3v_{010100}+q^3v_{010100}\\
&-q^2v_{001100}-q(q^2+1)v_{100010}+(q^2+1)v_{100100}),\\
u^{'}_1 =&\dfrac{1}{(q^2+1)\sqrt{(1+q^4)(1+q^2+q^4)}}(-(1+q^2+q^4)(qv_{100010}-v_{100100})+q^4v_{000011}+q^5v_{001010}\\
&+q^6v_{010010}-q^3v_{000101}-q^4v_{001100}-q^5v_{010100}),\\
u^{'}_2 =&\dfrac{1}{(q^2+1)\sqrt{(1+q^2+q^4)}}\left(qv_{001010}+q^2v_{010010}-v_{001100}-qv_{010100}-q(1+q^2)(qv_{000011}-v_{000101})\right).
\end{array}$$ It is now immediate to find out the Racah matrix for $[4,2]$: $$U\left[ \begin{array}{cc} [1]&[2] \\ \text{[2,1]} & [4,2] \end{array} \right] = \left( \begin{array}{cc} \dfrac{1}{\sqrt{[3]}}&-\dfrac{\sqrt{[4][2]}}{[3]} \\ \dfrac{\sqrt{[4][2]}}{[3]} & \dfrac{1}{\sqrt{[3]}} \end{array} \right).
$$ The highest weight vectors in representations $[3,2,1]$ and $[4,1,1]$ are too cumbersome to be written down here. They can be found in [@knotebook]. The Racah matrix for $[3,2,1]$ is $$U\left[ \begin{array}{cc} [1]&[2] \\ \text{[2,1]} & [3,2,1] \end{array} \right] = \left( \begin{array}{ccc} -\dfrac{1}{\sqrt{[2][4]}}&\dfrac{1}{\sqrt{[2]}}&\dfrac{\sqrt{[5]}}{\sqrt{[2][4]}} \\ \dfrac{\sqrt{[5]}}{\sqrt{[4]!}} & -\dfrac{\sqrt{[5]}}{[2]\sqrt{[3]}} & \dfrac{\sqrt{[3]}}{\sqrt{[2][4]}} \\ \dfrac{\sqrt{[4]}}{\sqrt{[2][3]}} & \dfrac{1}{\sqrt{[3]}} &0 \end{array} \right),
$$ and for $[4,1,1]$ is $$U\left[ \begin{array}{cc} [1]&[2] \\ \text{[2,1]} & [4,1,1] \end{array} \right] = \left( \begin{array}{cc} \dfrac{1}{\sqrt{[5]}}&-\dfrac{\sqrt{[4][2]}}{\sqrt{[3][5]}} \\ \dfrac{\sqrt{[4][2]}}{\sqrt{[3][5]}} & \dfrac{1}{\sqrt{[5]}} \end{array} \right).
$$
List of Racah matrices\[s.raclist\]
===================================
We have calculated all the inclusive Racah matrices for the representations described by Young diagrams with 3 boxes. In fact, the case of $U_{XY}\left[ \begin{array}{cc} [2,1]&[2,1] \\ \text{[2,1]} & Q \end{array} \right]$ has been calculated earlier [@MMMS21] (see also [@GJ] for the exclusive Racah matrices in the $[2,1]$ case), hence, here we calculate the following Racah matrices: [$$\label{Rac}
\begin{array}{l}
U_{XY}\left[ \begin{array}{cc} [3]&[2,1] \\ \text{[3]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [3]&[1,1,1] \\ \text{[3]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [2,1]&[3] \\ \text{[3]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [2,1]&[2,1] \\ \text{[3]} & Q \end{array} \right], \\
U_{XY}\left[ \begin{array}{cc} [2,1]&[1,1,1] \\ \text{[3]} & Q \end{array} \right] \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[3] \\ \text{[3]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[2,1] \\ \text{[3]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[1,1,1] \\ \text{[3]} & Q \end{array} \right], \\
U_{XY}\left[ \begin{array}{cc} [3]&[3] \\ \text{[2,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [3]&[2,1] \\ \text{[2,1]} & Q \end{array} \right] \
U_{XY}\left[ \begin{array}{cc} [3]&[1,1,1] \\ \text{[2,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [2,1]&[3] \\ \text{[2,1]} & Q \end{array} \right], \\
U_{XY}\left[ \begin{array}{cc} [2,1]&[1,1,1] \\ \text{[2,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[3] \\ \text{[2,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[2,1] \\ \text{[2,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[1,1,1] \\ \text{[2,1]} & Q \end{array} \right], \\
U_{XY}\left[ \begin{array}{cc} [3]&[3] \\ \text{[1,1,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [3]&[2,1] \\ \text{[1,1,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [3]&[1,1,1] \\ \text{[1,1,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [2,1]&[3] \\ \text{[1,1,1]} & Q \end{array} \right], \\
U_{XY}\left[ \begin{array}{cc} [2,1]&[2,1] \\ \text{[1,1,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [2,1]&[1,1,1] \\ \text{[1,1,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[3] \\ \text{[1,1,1]} & Q \end{array} \right], \
U_{XY}\left[ \begin{array}{cc} [1,1,1]&[2,1] \\ \text{[1,1,1]} & Q \end{array} \right].
\end{array}$$ ]{} Decompositions of the products of the representations involved in these Racah matrices are $$\begin{array}{lcl}
[3]\otimes [3] &=& [6]\oplus [5, 1]\oplus [4, 2]\oplus [3, 3], \\ {}
[3]\otimes [2,1] &=& [5,1]\oplus [4,2] \oplus [4,1,1] \oplus [3,2,1], \\ {}
[3]\otimes [1,1,1] &=& [4,1,1]\oplus [3,1,1,1], \\ {}
[2,1]\otimes [2,1] &=& [4,2] \oplus [4,1,1] \oplus [3,3] \oplus 2[3,2,1] \oplus [3,1,1,1] \oplus [2,2,2] \oplus [2,2,1,1], \\ {}
[2,1]\otimes [1,1,1] &=& [3,2,1]\oplus [3,1,1,1] \oplus [2,2,1,1]\oplus [2,1,1,1,1], \\ {}
[1,1,1]\otimes [1,1,1] &=& [2,2,2]\oplus [2,2,1,1] \oplus [2,1,1,1]\oplus [1,1,1,1,1,1].
\end{array}$$ The maximal size of the Racah mixing matrices will be $5\times 5$: for $U_{XY}\left[ \begin{array}{cc} [2,1]&[2,1] \\ \text{[3]} & [5,2,1,1] \end{array} \right]$ and $U_{XY}\left[ \begin{array}{cc} [2,1]&[2,1] \\ \text{[3]} & [5,3,1] \end{array} \right]$. All representations come without multiplicities, except for the $[2,1]$ case.
As an example, consider the product $$\begin{array}{lcl}
[3]\otimes [3]\otimes[2,1] &=&
[8, 1]+2[7, 2]+2[7, 1, 1]+2[6, 3]+4[6, 2, 1]+[6, 1, 1, 1]+2[5, 4]+4[5, 3, 1]+ \\
&+&2[5, 2, 2]+2[5, 2, 1, 1]+2[4, 4, 1]+2[4, 3, 2]+2[4, 3, 1, 1]+[4, 2, 2, 1]+[3, 3, 2, 1].
\end{array}$$ Hence, the inclusive Racah matrices $U_{XY}\left[ \begin{array}{cc} [3]&[3] \\ \text{[2,1]} & Q \end{array} \right]$ form the collection
[$$\begin{array}{|c|p{14cm}|c|}
\hline
&&\text{number of}\\
\text{matrix size} & \hspace{6.2cm} Q & \\
&&\text{matrices}\\ \hline && \\
1 & [8,1],[6,1,1,1],[4,2,2,1],[3,3,2,1] & 4 \\
&&\\ \hline && \\
2 & [7,2],[7,1,1],[6,3],[5,4],[5,2,2],[5,2,1,1],[4,4,1],[4,3,2],[4,3,1,1] & 9 \\
&&\\ \hline && \\
3 & & 0 \\
&&\\ \hline && \\
4 & [5,3,1],[6,2,1] & 2 \\
&&\\ \hline && \\
5 & & 0 \\
&&\\ \hline
\end{array}$$ ]{}
All the Racah matrices (\[Rac\]) were calculated with the help of the highest weight method. The results are available online at [@knotebook]. Here we list only two Racah matrices as an illustration: $$U_{XY}\left[ \begin{array}{cc} [3]&[3] \\ \text{[2,1]} & [5,4] \end{array} \right]=
\left(\begin{array}{rr}
\sqrt{\cfrac{[3]}{[2][4]}} & \sqrt{\cfrac{[5]}{[2][4]}} \\
& \\
\sqrt{\cfrac{[5]}{[2][4]}} & -\sqrt{\cfrac{[3]}{[2][4]}}
\end{array}\right),$$ $$U_{XY}\left[ \begin{array}{cc} [3]&[2] \\ \text{[2]} & [5,2] \end{array} \right]=
\left(\begin{array}{rrr}
\cfrac{[2]}{[4]\sqrt{[5]}} & \cfrac{\sqrt{[2][3][6]}}{[4]\sqrt{[5]}} & \sqrt{\cfrac{[6]}{[3][4]}} \\
&& \\
\cfrac{[2]^2}{[3]}\sqrt{\cfrac{[6]}{[4][5]}} & \cfrac{[8]}{[4]}\sqrt{\cfrac{[2]}{[3][4][5]}} & -\cfrac{1}{\sqrt{[3]}} \\
&& \\
\cfrac{\sqrt{[2][5][6]}}{[3][4]} &-\cfrac{[2]}{[4]}\sqrt{\cfrac{[5]}{[3]}}& \sqrt{\cfrac{[2]}{[3][4]}}
\end{array}\right).$$
Signs of the $\mathcal{R}$-matrix eigenvalues\[s.signs\]
========================================================
Now we will discuss the issue of signs of the $\mathcal{R}$-matrix eigenvalues. While their absolute values were already discussed in s.\[incrac\], the sign issue is more complicated. Let us first discuss the knot case.
The $\mathcal{R}$-matrices for knots act on the tensor square of the representation associated with the strands: $$\mathcal{R}:\ \ T\otimes T \rightarrow T\otimes T.$$ The sign of eigenvalue depends on whether the representation under consideration, $Q\vdash T\otimes T$ in the classical case $q=1$ is symmetric or antisymmetric under permutation of two representations $T$. It is same to say that the representation $Q$ belongs to the symmetric or antisymmetric squares of representation $T$, see also [@MMMS21; @Rama2].
For links, the representations on which the $\mathcal{R}$-matrix acts may differ: $$\mathcal{R}:\ \ T_1\otimes T_2 \rightarrow T_2\otimes T_1.$$ Let us look at the structure of representations. As can be seen from the examples in the previous section, e.g. (\[c31\]) the highest weight vectors and consequently the whole representation vectors are defined up to a sign. For the knot case, it makes no difference since the change of signs of representation vectors changes for the both spaces which the $\mathcal{R} $ acts on. However, for the link case, the two spaces are different, and the signs of the $\mathcal{R}$-matrix in a unique way. They in fact depend on the choice of the signs of the representations.
This leads to another interesting property. Since the Racah matrices come from the products of three representations which are again defined up to a sign, it means that, by changing the signs of the basis vectors on one or another side of the Racah-matrix one can change the sign of a column or a row in the Racah matrix. For knots, this would not change the diagonal (or, in fact, any) $\mathcal{R}$-matrix, since its eigenvalues are uniquely defined. However, for links, the signs of the eigenvalues also depend on the signs of representation vectors. This means that one should define the signs in both the $\mathcal{R}$-matrices and the Racah matrices at once.
Eigenvalue conjecture and Racah matrices\[s.eig\]
=================================================
In [@IMMMec], an eigenvalue conjecture for the Racah matrices was suggested. This conjecture states that the inclusive Racah matrices in the three-strand knots depend only on normalized eigenvalues of the corresponding $\mathcal{R}$-matrix. The idea behind this conjecture, besides an experimental evidence, is quite simple. The defining property of the $\mathcal{R}$-matrices is that it should satisfy the Yang-Baxter equation (it is, in fact, nothing but the third Reidemeister move in knot theory), which, for the three-strand braid, looks like $$\mathcal{R}_1\mathcal{R}_2\mathcal{R}_1=\mathcal{R}_2\mathcal{R}_1\mathcal{R}_2.$$ If one also expresses here $\mathcal{R}_2$ through the Racah matrices $$\mathcal{R}_2=U\mathcal{R}_1 U^{\dagger},$$ an equation which relates the Racah coefficients with the $\mathcal{R}$-matrix appears: $$\mathcal{R}_1U\mathcal{R}_1 U^{\dagger}\mathcal{R}_1=U\mathcal{R}_1 U^{\dagger}\mathcal{R}_1 U\mathcal{R}_1 U^{\dagger}.
\label{YBU}$$ Together with the fact that the Racah matrix is, in fact, a rotation matrix (unitary or orthogonal in the real case), this equation can be solved for matrices of the sizes at least up to $6\times 6$ [@IMMMec; @Univ] (see also [@Wentzl]). The solutions are unique for the $\mathcal{R}$-matrix of a generic form [^2] up to inessential choice of signs and are given in their full form in [@IMMMec; @Univ; @Wentzl]. Here we will only provide the answer for the Racah matrix of the size $2\times 2$ as an example.
If the $\mathcal{R}$-matrix is $$\mathcal{R}_1=\left(\begin{array}{rr}\lambda_1 & \\ & \lambda_2\end{array}\right)=\sqrt{|\lambda_1\lambda_2|}\left(\begin{array}{rr}\xi_1 & \\ & \xi_2\end{array}\right),\ \xi_1=\frac{\lambda_1}{\sqrt{|\lambda_1\lambda_2|}},\ \xi_2=\frac{\lambda_2}{\sqrt{|\lambda_1\lambda_2|}},$$ where $\xi_1$ and $\xi_2$ are the normalized eigenvalues of the $\mathcal{R}$-matrix, the corresponding Racah matrix is $$U=\left(\begin{array}{cc}
-\frac{1}{\xi_1-\xi_2} & \frac{\sqrt{\xi_1^2+1+\xi_2^2}}{\xi_1-\xi_2} \\
\frac{\sqrt{\xi_1^2+1+\xi_2^2}}{\xi_1-\xi_2} & \frac{1}{\xi_1-\xi_2}
\end{array}\right).
\label{1cEH}$$
In [@AMcabling] a similar conjecture was made for links, i.e. for the three-strand braids with different representations on different strands. Below we discuss the eigenvalue conjecture in two cases: for two-component and three component links. Let components be colored with representations $R_1$, $R_2$ and (for three-component links) $R_3$. In the case of two components, there are two different sets of Racah coefficients: $U_{112}$ and $U_{121}$, which depend on the placement of different components in the braid, as well as there are two diagonal $\mathcal{R}$-matrices: $\mathcal{R}_{11}$ and $\mathcal{R}_{12}$. In the case of links, it is more useful to solve not the Yang-Baxter equations themselves, but the equations which emerge from the cabling procedure (see [@AMcabling] for details). In the two component case, the Racah matrices satisfy the equations $$\begin{array}{l}
U_{121}\mathcal{R}_{12}U_{112}^{\dagger}\mathcal{R}_{11} U_{112}=\mathcal{R}_{(12)\times 1},
\\
U_{112}\mathcal{R}_{12}U_{121}\mathcal{R}_{12} U_{112}^{\dagger}=\mathcal{R}_{(11)\times 1}.
\end{array}$$ In contrast with (\[YBU\]), at the right hand side here there are new $\mathcal{R}$-matrices acting in higher representations, and they appear due to the cabling procedure application. As such, their form is a kind of unknown variable here, although it is known that they should be diagonal and this property is in fact enough to solve these equations. If one expresses the $\mathcal{R}$-matrices through one eigenvalue each[^3] (this can be done for the normalized $\mathcal{R}$-matrices): $$\mathcal{R}_{11}=\left(\begin{array}{rr}\xi_{11} & \\ & -\xi^{-1}_{11}\end{array}\right),\ \mathcal{R}_{12}=\left(\begin{array}{rr}\xi_{12} & \\ & -\xi^{-1}_{12}\end{array}\right),$$ the Racah matrices are as follows: $$\begin{array}{l}
U_{112}=\left(\begin{array}{cc}
-\sqrt{\cfrac{(\xi_{11}^2-\xi_{12}^4)}{(1-\xi_{12}^4)(1+\xi_{11}^2)}} & \sqrt{\cfrac{(1-\xi_{11}^2\xi_{12}^4)}{(1-\xi_{12}^4)(1+\xi_{11}^2)}}
\\ \sqrt{\cfrac{(1-\xi_{11}^2\xi_{12}^4)}{(1-\xi_{12}^4)(1+\xi_{11}^2)}} & \sqrt{\cfrac{(\xi_{11}^2-\xi_{12}^4)}{(1-\xi_{12}^4)(1+\xi_{11}^2)}}
\end{array}\right),
\\ \\
U_{121}=\left(\begin{array}{cc}
-\cfrac{\xi_{12}^2(1-\xi_{11}^2)}{\xi_{11}(1-\xi_{12}^4)} & \cfrac{\sqrt{(\xi_{11}^2-\xi_{12}^4)(1-\xi_{12}^4\xi_{11}^2)}}{\xi_{11}(1-\xi_{12}^4)}
\\
\cfrac{\sqrt{(\xi_{11}^2-\xi_{12}^4)(1-\xi_{12}^4\xi_{11}^2)}}{\xi_{11}(1-\xi_{12}^4)} & \cfrac{\xi_{12}^2(1-\xi_{11}^2)}{\xi_{11}(1-\xi_{12}^4)}
\end{array}\right).
\end{array}
\label{2cEH}$$
Similarly for three different representations, there are three different Racah matrices: $U_{123}$, $U_{132}$ and $U_{213}$, as well as three $\mathcal{R}$-matrices: $\mathcal{R}_{12}$, $\mathcal{R}_{13}$ and $\mathcal{R}_{23}$. Three equations defining the Racah coefficients are $$\begin{array}{l}
U_{123} \mathcal{R}_{23}U_{132}^{\dagger}\mathcal{R}_{13} U_{213}^{\dagger}=\mathcal{R}_{(12)\times 3},
\\
U_{132} \mathcal{R}_{23}U_{123}^{\dagger}\mathcal{R}_{12} U_{213} =\mathcal{R}_{(13)\times 2},
\\
U_{123}^{\dagger}\mathcal{R}_{12}U_{213} \mathcal{R}_{13} U_{132} =\mathcal{R}_{(23)\times 1}.
\end{array}$$ These equations give the following Racah coefficients: $$U_{123}=\left(\begin{array}{cc}
-\sqrt{\cfrac{(\xi_{12}^2-\xi_{13}^2\xi_{23}^2)(\xi_{23}^2-\xi_{12}^2\xi_{13}^2)}{\xi_{13}^2(1-\xi_{12}^4)(1-\xi_{23}^4)}} &
\sqrt{\cfrac{(\xi_{13}^2-\xi_{12}^2\xi_{23}^2)(1-\xi_{12}^2\xi_{13}^2\xi_{23}^2)}{\xi_{13}^2(1-\xi_{12}^4)(1-\xi_{23}^4)}}
\\
\sqrt{\cfrac{(\xi_{13}^2-\xi_{12}^2\xi_{23}^2)(1-\xi_{12}^2\xi_{13}^2\xi_{23}^2)}{\xi_{13}^2(1-\xi_{12}^4)(1-\xi_{23}^4)}} &
\sqrt{\cfrac{(\xi_{12}^2-\xi_{13}^2\xi_{23}^2)(\xi_{23}^2-\xi_{12}^2\xi_{13}^2)}{\xi_{13}^2(1-\xi_{12}^4)(1-\xi_{23}^4)}}
\end{array}\right),
\label{3cEH}$$ and the remaining two matrices can be easily constructed from this one by simply interchanging the indices. One can definitely easily check that (\[1cEH\]) is a particular example of (\[2cEH\]), when the two representations coincide, as well as (\[2cEH\]) is a particular case of (\[3cEH\]).
### $\underline{[1]\otimes[1]\otimes[2]}$ {#underline1otimes1otimes2 .unnumbered}
Since the matrix in section \[ex112\] is of size $2\times 2$, one can also use the eigenvalue hypothesis in order to calculate it. In this case, the $\mathcal{R}$-matrices in the diagonal form are
$$\mathcal{R}_{[1][1]}=\left(\begin{array}{rr}q & \\ & -q^{-1}\end{array}\right),\ \mathcal{R}_{12}=\sqrt{q}\left(\begin{array}{rr}q^{3/2} & \\ & -q^{-3/2}\end{array}\right),$$
then $$U_{[1][1][2]}=\left(\begin{array}{cc}
-\sqrt{\cfrac{q^2-q^6}{(1-q^6)(1+q^2)}} & \sqrt{\cfrac{(1-q^8)}{(1-q^6)(1+q^2)}}
\\ \sqrt{\cfrac{(1-q^8)}{(1-q^6)(1+q^2)}} & \sqrt{\cfrac{(q^2-q^6)}{(1-q^6)(1+q^2)}}
\end{array}\right)
=\left(\begin{array}{cc}
-\sqrt{\cfrac{1}{[3]}} & \sqrt{\cfrac{[4]}{[2][3]}}
\\ \sqrt{\cfrac{[4]}{[2][3]}} & \sqrt{\cfrac{1}{[3]}}
\end{array}\right).$$ This answer definitely coincides with the one calculated using representation theory in (\[u112ans\]).
HOMFLY polynomials for multicolored links\[s.HOMFLY\]
=====================================================
Using these calculated inclusive multicolored Racah matrices, one can evaluate the multicolored HOMFLY polynomials for links that have three-strand braid presentations. Here we discuss some simple examples: the Whitehead link, the Borromean rings and link L7a3 in the Thistlethwaite Link Table [@twi].
Whitehead link
--------------
The Whitehead link is a two-component link. It has the following braid representation:
(160,80)(-70,-10) (0,48)[(1,0)[20]{}]{} (0,24)[(1,0)[20]{}]{} (0,0)[(1,0)[44]{}]{} (0,51)[$R_2$]{} (0,27)[$R_1$]{} (0,3)[$R_1$]{} (20,48)[(1,-1)[10]{}]{} (44,24)[(-1,1)[10]{}]{} (20,24)[(1,1)[24]{}]{} (24,46)[$\mathcal{R}_{12}$]{} (44,24)[(1,-1)[24]{}]{} (44,0)[(1,1)[10]{}]{} (68,24)[(-1,-1)[10]{}]{} (44,48)[(1,0)[24]{}]{} (48,-3)[$\mathcal{R}_{12}$]{} (68,48)[(1,-1)[10]{}]{} (92,24)[(-1,1)[10]{}]{} (68,24)[(1,1)[24]{}]{} (68,0)[(1,0)[24]{}]{} (72,46)[$\mathcal{R}_{11}$]{} (92,24)[(1,-1)[24]{}]{} (92,0)[(1,1)[10]{}]{} (116,24)[(-1,-1)[10]{}]{} (92,48)[(1,0)[24]{}]{} (96,-3)[$\mathcal{R}_{12}$]{} (116,48)[(1,-1)[10]{}]{} (140,24)[(-1,1)[10]{}]{} (116,24)[(1,1)[24]{}]{} (118,46)[$\mathcal{R}_{12}$]{} (116,0)[(1,0)[44]{}]{} (140,24)[(1,0)[20]{}]{} (140,48)[(1,0)[20]{}]{}
Since it has two components, one needs two $\mathcal{R}$- and Racah matrices. As we explained above, there are two diagonal $\mathcal{R}$-matrices: $\mathcal{R}_{12}$ that stands for crossings of representations $R_1$ and $R_2$, and $\mathcal{R}_{11}$ that stands for crossing between representations $R_1$ and $R_1$. The two Racah matrices $U_{112}$ and $U_{121}$ correspond accordingly to the placements of representations $R_1R_1R_2$ and $R_1R_2R_1$.
The answer for the HOMFLY polynomial is then given by the character expansion [@MMMI]: $$H_{R_1R_2}^{\text{Whitehead}}=\sum\limits_{Q\vdash R_1\otimes R_1\otimes R_2} S_Q^*(A,q) B_Q,$$ where $S_Q^*(A,q)$ is the quantum dimension[^4] of the representation $Q$ of $SU_q(N)$, and $B_Q$ is a trace of the product of $\mathcal{R}$ and Racah matrices for the representation $Q$: $$B=Tr\left(\mathcal{R}_{12} U_{121} \mathcal{R}^{-1}_{12} U_{112}^{\dagger} \mathcal{R}_{11} U_{112} \mathcal{R}^{-1}_{12} U_{121} \mathcal{R}_{12} \right).$$
The eigenvalues of the diagonal $\mathcal{R}$-matrix depend on the representations appearing in the decomposition of the product $R_1\otimes R_1$ for $\mathcal{R}_{11}$ and $R_1\otimes R_2$ for $\mathcal{R}_{12}$. For $\mathcal{R}_{12}$, these eigenvalues are generally equal to $$\lambda_i=q^{\varkappa_{Q_i}-\varkappa_{R_1}-\varkappa_{R_2}}\ \ \text{for } Q_i\vdash R_1\otimes R_2,Q\vdash Q_i\otimes R_1.$$ Unfortunately, signs of the eigenvalues cannot be determined uniquely.
First, let us consider the intrinsic causes and repeat in this concrete case the argument of section 5. Since $$\mathcal{R}_{12}: R_1 \otimes R_2\ \rightarrow\ \ R_2 \otimes R_1,$$ $\mathcal{R}_{12}$ is diagonal, if one chooses some $u$ and $v$ as bases in $R_1\otimes R_2$ and $R_2\otimes R_1$ in such a way that $$\mathcal{R}_{12} \cdot u= \lambda v.$$ As clear from this equation, signs of the eigenvalues depend on the choice of signs of $u$ and $v$, and they remain unchanged only when the two bases change their signs together. In this case, by the definition of the rotation matrix, the Racah matrix $U_{121}$ is the same. Therefore, each choice of signs of the eigenvalues corresponds to only one Racah matrix.
If the representation $Q$ has multiplicity 1 in $R_1 \otimes R_2 \otimes R_1$, there is no problem because the Racah matrix is trivial. But for those representations $Q$ of multiplicity $n\geq 2$, things become more complicated.
Let us denote those eigenvalues to be $\lambda_i(1\leq i \leq n)$. If we fix the sign of $\lambda_1$, there are $2^{n-1}$ ways to combine the signs of different eigenvalues, and it is possible to have several combinations resulting in the same Racah matrix. However, in the highest weight vector method, the Racah matrix is obtained uniquely without any references to $\mathcal{R}$ matrix, so the complexity lies in the determination of the “right” $\mathcal{R}$ matrices among those possibilities with the unique Racah matrix.
Actually, as we explained earlier, one good thing is that the signs of eigenvalues of knots are easier to be determined. If one studies knots in representation $R=R_1=R_2$, each irreducible representation $Q_i\vdash R\otimes R$ lies either in the symmetric or antisymmetric square, which determines the sign of $\lambda_Q$: it is plus if $Q$ belongs to the symmetric square and minus otherwise. See [@MMMS21] for more details.
These formulae together with the Racah matrices calculated in the present paper are enough to evaluate several (multi)colored HOMFLY polynomials for the Whitehead link. Most of them are given in Appendix \[A.w\]. Here we present only one non-trivial example:
$$\begin{array}{r}
\{A\}H_{[2,1][1]}=A^{-2}\big(q^{4}-q^{2}-q^{-2}+q^{-4}\big)+\big(-q^{8}+2q^{6}-3q^{4}+4q^{2}-5+4q^{-2}-3q^{-4}+2q^{-6}-q^{-8}\big)+
\\ \\
A^{2}\big(q^{4}-q^{2}+1-q^{-2}+q^{-4}\big).
\end{array}$$
### Properties of the answers {#properties-of-the-answers .unnumbered}
These polynomials possess several properties which can be used to check the answer.
The main property of the link polynomials is that if one “normalizes” the answer by dividing it by the quantum dimensions of the representations on different components of the link then this normalized answer has no quantum numbers in the denominator, only terms of the form $(Aq^i-A^{-1}q^{-i})$ remain there. For the Whitehead link, the “normalized” polynomial is defined as $$\mathcal{H}_{Q_1,Q_2}=\frac{H_{Q_1,Q_2}}{S_{Q_1}^*S_{Q_2}^*}.$$
The second property of the Whitehead invariant is that if one takes a particular group $SU(N)$ (i.e. puts $A=q^N$) and if representation $Q_1$ is trivial in this group (as it happens for representation $[1,1]$ in $SU(2)$ and for representation $[1,1,1]$ in $SU(3)$), then the answer for the HOMFLY polynomial becomes just the answer for the unknot in representation $Q_2$: $$H_{[r^N],Q_2}|_{A=q^N}=S^*_{Q_2}(A=q^N,q).$$ The same is of course true if $Q_2$ is trivial, then the result is the quantum dimension of representation $Q_1$. This property is only true for the links with all components unknotted by themselves. Otherwise, the result would be the HOMFLY polynomial of the component rather than the polynomial of the unknot (see a detailed discussion of these properties in [@BJLMMMS]).
The third property, which is general for all knots and links is that the transposition of Young diagrams corresponds to the substitution $q\rightarrow -q^{-1}$. For example: $$\begin{array}{l}
H_{[2],[2]}(A,q)=H_{[1,1],[1,1]}(A,-q^{-1});
\\
H_{[3],[2,1]}(A,q)=H_{[1,1,1],[2,1]}(A,-q^{-1});
\\
\text{etc.}
\end{array}$$
The fourth property is peculiar for the Whitehead link. Although it is not obvious from the braid picture, this link is actually symmetric under permutation of its components meaning that $H_{R_1R_2}=H_{R_2R_1}$.
Borromean rings
---------------
The second example is the Borromean rings. The braid representation of this link is
(160,80)(-70,-10) (0,48)[(1,0)[20]{}]{} (0,24)[(1,0)[20]{}]{} (0,0)[(1,0)[44]{}]{} (0,51)[$R_1$]{} (0,27)[$R_2$]{} (0,3)[$R_3$]{} (20,48)[(1,-1)[10]{}]{} (44,24)[(-1,1)[10]{}]{} (20,24)[(1,1)[24]{}]{} (24,46)[$\mathcal{R}_{12}$]{} (44,24)[(1,-1)[24]{}]{} (44,0)[(1,1)[10]{}]{} (68,24)[(-1,-1)[10]{}]{} (44,48)[(1,0)[24]{}]{} (48,-3)[$\mathcal{R}_{13}$]{} (68,48)[(1,-1)[10]{}]{} (92,24)[(-1,1)[10]{}]{} (68,24)[(1,1)[24]{}]{} (68,0)[(1,0)[24]{}]{} (72,46)[$\mathcal{R}_{23}$]{} (92,24)[(1,-1)[24]{}]{} (92,0)[(1,1)[10]{}]{} (116,24)[(-1,-1)[10]{}]{} (92,48)[(1,0)[24]{}]{} (96,-3)[$\mathcal{R}_{12}$]{} (116,48)[(1,-1)[10]{}]{} (140,24)[(-1,1)[10]{}]{} (116,24)[(1,1)[24]{}]{} (116,0)[(1,0)[24]{}]{} (118,46)[$\mathcal{R}_{13}$]{} (140,24)[(1,-1)[24]{}]{} (140,0)[(1,1)[10]{}]{} (164,24)[(-1,-1)[10]{}]{} (144,-3)[$\mathcal{R}_{23}$]{} (164,0)[(1,0)[20]{}]{} (164,24)[(1,0)[20]{}]{} (140,48)[(1,0)[44]{}]{}
It has three components, thus, one needs three $\mathcal{R}$- and Racah matrices. There are three diagonal $\mathcal{R}$-matrices: $\mathcal{R}_{12}$, $\mathcal{R}_{13}$ and $\mathcal{R}_{23}$ (see also section \[incrac\]). Similarly, there are three Racah matrices[^5]: $U_{123}$, $U_{132}$ and $U_{213}$.
The answer for the HOMFLY polynomial is then given by formula [@MMMI]: $$H_{R_1R_2R_3}^{\text{Borromean}}=\sum\limits_{Q\vdash R_1\otimes R_2\otimes R_3} S_Q^*(A,q) B_Q,$$ $B_Q$ here is given by the trace of the product of $\mathcal{R}$- and Racah matrices in representation $Q$: $$B=Tr\left(\mathcal{R}_{12} U_{213} \mathcal{R}^{-1}_{13} U_{132} \mathcal{R}_{23} U_{123}^{\dagger} \mathcal{R}^{-1}_{12} U_{213}^{\dagger} \mathcal{R}_{13} U_{132} \mathcal{R}^{-1}_{23} U_{123}^{\dagger} \right).$$
The eigenvalues of the diagonal $\mathcal{R}$-matrix depend on the representations appearing in the decomposition of the product $R_1\otimes R_1$ for $\mathcal{R}_{11}$ and $R_1\otimes R_2$ for $\mathcal{R}_{12}$. Generally these eigenvalues are equal to $$\lambda_i=q^{\varkappa_{Q_i}-\varkappa_{R_1}-\varkappa_{R_2}}\ \ \text{for}\ Q_i\vdash R_1\otimes R_2.$$ For the same reason as in the case of the Whitehead link, their signs are not uniquely defined.
These formulae together with the Racah matrices calculated in this paper are enough to evaluate several (multi)colored HOMFLY polynomials for the Borromean rings. Most of them are given in Appendix \[A.b\]. Here we present only one non-trivial example: $$\begin{array}{l}
\{A\}^2H_{[2,1],[1],[1]}=\big(q^{10}-4q^{8}+8q^{6}-12q^{4}+15q^{2}-18+15q^{-2}-12q^{-4}+8q^{-6}-4q^{-8}+q^{-10}\big)+
\\ \\
+A^{-2}\big(-q^{6}+3q^{4}-3q^{2}+3-3q^{-2}+3q^{-4}-q^{-6}\big)+A^{2}\big(-q^{6}+3q^{4}-3q^{2}+3-3q^{-2}+3q^{-4}-q^{-6}\big).
\end{array}$$
### Properties of the answers {#properties-of-the-answers-1 .unnumbered}
These polynomials possess several properties which can be used to check the answer.
For Borromean rings, the “normalized” polynomial is defined as $$\mathcal{H}_{Q_1,Q_2,Q_3}=\frac{H_{Q_1,Q_2,Q_3}}{S_{Q_1}^*S_{Q_2}^*S_{Q_3}^*},$$ and its main property is again that it has no quantum numbers in the denominator, only terms of the form $(Aq^i-A^{-1}q^{-i})$ remain there.
The second property is again behaviour under the transposition of Young diagrams. For example: $$\begin{array}{l}
H_{[2],[2],[1,1]}(A,q)=H_{[1,1],[1,1],[2]}(A,-q^{-1});
\\
H_{[3],[2,1],[1,1,1]}(A,q)=H_{[1,1,1],[2,1],[3]}(A,-q^{-1});
\\
\text{etc.}
\end{array}$$
The third property is peculiar for the Borromean rings: it is symmetric under permutating any of its components, e.g. $H_{R_1R_2R_3}=H_{R_2R_1R_3}$ etc.
L7a3
----
The third example is link L7a3 in the Thistlethwaite link table [@twi]. It is a two-component link, which has a three-strand representation
(160,80)(-70,-10) (0,48)[(1,0)[18]{}]{} (0,24)[(1,0)[18]{}]{} (0,0)[(1,0)[34]{}]{} (0,51)[$R_1$]{} (0,27)[$R_2$]{} (0,3)[$R_2$]{} (18,48)[(1,-1)[30]{}]{} (18,24)[(1,1)[10]{}]{} (34,0)[(1,1)[24]{}]{} (22,46)[$\mathcal{R}_{12}$]{} (42,48)[(-1,-1)[10]{}]{} (64,0)[(-1,1)[12]{}]{} (64,24)[(1,0)[8]{}]{} (42,2)[$\mathcal{R}_{12}$]{} (42,48)[(1,0)[30]{}]{} (64,0)[(1,0)[52]{}]{} (57,24)[(1,0)[16]{}]{} (72,48)[(1,-1)[24]{}]{} (72,24)[(1,1)[10]{}]{} (96,48)[(-1,-1)[10]{}]{} (76,46)[$\mathcal{R}_{22}$]{} (42,48)[(1,0)[30]{}]{} (96,48)[(1,0)[10]{}]{} (96,24)[(1,0)[10]{}]{} (106,24)[(1,1)[10]{}]{} (106,48)[(1,-1)[24]{}]{} (110,46)[$\mathcal{R}_{22}$]{} (130,48)[(-1,-1)[10]{}]{} (130,48)[(1,0)[10]{}]{} (130,24)[(1,0)[10]{}]{} (143,46)[$\mathcal{R}_{22}$]{} (140,48)[(1,-1)[28]{}]{} (140,24)[(1,1)[10]{}]{} (164,48)[(-1,-1)[10]{}]{} (164,48)[(1,0)[10]{}]{} (174,48)[(1,-1)[24]{}]{} (198,24)[(1,0)[30]{}]{} (64,0)[(1,0)[90]{}]{} (154,0)[(1,1)[32]{}]{} (188,0)[(-1,1)[15]{}]{} (188,0)[(1,0)[41]{}]{} (201,48)[(-1,-1)[12]{}]{} (201,48)[(1,0)[28]{}]{} (178,46)[$\mathcal{R}_{12}$]{} (163,2)[$\mathcal{R}_{12}$]{}
It has two components, thus one needs two $\mathcal{R}$- and Racah matrices. There are two diagonal $\mathcal{R}$-matrices: $\mathcal{R}_{12}$ that stands for crossings of representations $R_1$ and $R_2$ and $\mathcal{R}_{22}$ that stands for crossings of representations $R_2$ and $R_2$ (see also section \[incrac\]). The two Racah matrices $U_{122}$ and $U_{212}$ correspond accordingly to the placements of representations $R_1R_2R_2$ and $R_2R_1R_2$ .
The answer for the HOMFLY polynomial is then given by the character expansion [@MMMI]: $$H_{R_1R_2}^{\text{L7a3}}=\sum\limits_{Q\vdash R_1\otimes R_2\otimes R_2} S_Q^*(A,q) B_Q,$$ $B_Q$ here is given by the trace of the following product of $\mathcal{R}$ and Racah matrices for representation $Q$: $$B=Tr\left(\mathcal{R}_{12} U_{212} \mathcal{R}^{-1}_{12} U_{122} \mathcal{R}^{3}_{22} U_{122}^{\dagger} \mathcal{R}^{-1}_{12} U_{212} \mathcal{R}_{12} \right).$$
The eigenvalues of the diagonal $\mathcal{R}$-matrix depend on the representations appearing in the decomposition of the product $R_1\otimes R_2$ for $\mathcal{R}_{12}$ and $R_2\otimes R_2$ for $\mathcal{R}_{22}$. Generally this eigenvalues are equal to $$\lambda_i=q^{\varkappa_{Q_i}-\varkappa_{R_1}-\varkappa_{R_2}}\ \ \text{for}\ Q_i\vdash R_1\otimes R_2.$$ These formulae together with the Racah matrices calculated in this paper are enough to calculate several (multi)colored HOMFLY polynomials for link L7a3. Most of them are given in Appendix \[A.l\]. Here we present only one non-trivial example: $$\begin{array}{r}
\{A\}H_{[3],[1]}=A^{-2}\big(q^{4}-q^{2}+2-2q^{-2}+q^{-4}-q^{-6}+q^{-8}\big)+A^{2}\big(q^{8}-q^{6}+q^{4}-q^{2}+1+q^{-4}\big)+
\\ \\
+\big(-q^{10}+q^{8}-q^{6}+2q^{4}-3q^{2}+1-3q^{-2}+2q^{-4}-q^{-6}+q^{-8}-q^{-10}\big).
\end{array}$$
### Properties of the answers {#properties-of-the-answers-2 .unnumbered}
These polynomials possess several properties which can be used to check the answer.
For link L7a3, the “normalized” polynomial is defined as $$\mathcal{H}_{Q_1,Q_2}=\frac{H_{Q_1,Q_2}}{S_{Q_1}^*S_{Q_2}^*},$$ and its main property is again that it has no quantum numbers in the denominator, only terms of the form $(Aq^i-A^{-1}q^{-i})$ remain there.
The second property of the L7a3 polynomial is that if one takes a particular group $SU(N)$ (i.e. puts $A=q^N$) and if representation $Q_1$ is trivial in this group (as it happens for representation $[1,1]$ in the $SU(2)$ and $[1,1,1]$ in representation $SU(3)$), then the answer for the HOMFLY polynomial becomes just the answer for the unknot in representation $Q_2$: $$H_{[r^N],Q_2}|_{A=q^N}=S^*_{Q_2}(A=q^n,q).$$ The same is of course true if $Q_2$ is trivial, then the result is the quantum dimension of representation $Q_1$. This property is only true for the links with all components unknotted by themselves. Otherwise, the result would be HOMFLY polynomial of the component rather than the polynomial of the unknot.
The third property is again the behaviour under the transposition of Young diagrams. For example: $$\begin{array}{l}
H_{[2],[2]}(A,q)=H_{[1,1],[1,1]}(A,-q^{-1});
\\
H_{[2,1],[3]}(A,q)=H_{[2,1],[1,1,1]}(A,-q^{-1});
\\
\text{etc.}
\end{array}$$
Exclusive Racah matrices\[s.exex\]
==================================
Since this paper is devoted mainly to evaluating the inclusive Racah matrices, we will not delve deep into the story of the exclusive ones. However, we will present one example of the calculation of exclusive link Racah matrix using the trick described in s.\[exrac\], namely we will discuss the matrix $S_{\bar{[1]},[1],[2]}$.
To use the trick, one should first calculate the HOMFLY polynomial for the two-parametric series of links $$H_{[2],[1],[1]}^{(2m,-1|2n,\pm 1)}\!\!\!\!\!=\!\!\!\!\!\!\!\!\!\!\sum\limits_{Q\vdash [2]\otimes [1]\otimes [1]}\!\!\!\!\!\!\!\!\!\! S^*_Q \text{Trace}\left(
\mathcal{R}^{2m}_{[1],[2]} U_{[1],[1],[2]}^{\dagger} \mathcal{R}^{-1}_{[1],[1]} U_{[1],[1],[2]} \mathcal{R}^{2n}_{[1],[2]} U_{[1],[1],[2]}^{\dagger} \mathcal{R}^{-1}_{[1],[1]} U_{[1],[1],[2]}
\right)_Q\!\!\!=\!\!\!\!\!\sum\limits_{{Y\vdash [1]\otimes [2]}\atop{Z\vdash [1]\otimes [2]}}\!\!\!\!\! h_{YZ}\lambda_Y^{2m}\lambda_Z^{2n}.$$ The Racah matrices needed for this calculation one can get also from the eigenvalue hypothesis, see s.\[s.eig\]. This leads to a matrix of coefficients $h_{YZ}$: $$h_{YZ}=\begin{array}{c|cc}
Y\backslash Z & [3] & [2,1]
\\
\hline
\\
\ [3] & \cfrac{\{Aq^2\}(A^3q^2-Aq^4+Aq^2-Aq^{-4})}{\{A\}^2[3]_q^2} & \cfrac{\{Aq^2\}\{Aq^{-1}\}[2]_q}{\{A\}^2[3]_q^2}
\\
\ [2,1] & \cfrac{\{Aq^2\}\{Aq^{-1}\}[2]_q}{\{A\}^2[3]_q^2} & \cfrac{\{Aq^{-1}\}[2]_q(A^3+A^3q^{-2}-Aq^4-A+Aq^{-2}-Aq^{-4})}{\{A\}^2[3]_q^2}
\end{array}$$ After dividing its elements by the square roots of dimensions $h_{YZ}/\sqrt{d_Y d_Z}$, the resulting matrix can be diagonalized by the matrix $S_{\bar{[1]},[1],[2]}$: $$S_{\bar{[1]},[1],[2];\bar{X}_1Y}\cfrac{h_{YZ}}{\sqrt{d_Y d_Z}}S_{\bar{[1]},[1],[2];\bar{X}_2Z}^{\dagger}=\text{diagonal matrix}_{\bar{X}_1,\bar{X}_2}.$$ Since the Racah matrix is unitary, this is a set of linear equations for $S_{\bar{[1]},[1],[2]}$, the solution being $$S_{\bar{[1]},[1],[2];\bar{X}Y}=\begin{array}{c|cc}
\bar{X}\backslash Y & [3] & [2,1] \\\hline\\
\varnothing & -\sqrt{\cfrac{\{Aq\}\{A\}\{Aq^{-1}\}}{[3]_q}} & \sqrt{\cfrac{\{Aq^2\}\{Aq\}\{A\}}{[2]_q[3]_q}}
\\
\text{adjoint} & \sqrt{\cfrac{\{Aq^2\}\{Aq\}\{A\}}{[2]_q[3]_q}} & \sqrt{\cfrac{\{Aq\}\{A\}\{Aq^{-1}\}}{[3]_q}}
\end{array}$$ This answer is in perfect agreement with the answer obtained in [@RacahRama].
Conclusion
==========
In this paper, we have calculated the inclusive Racah matrices for all representations up to size $3$ using the highest weight method. Using these matrices, we have evaluated the HOMFLY polynomials for the Whitehead link, Borromean rings and link L7a3. Also calculated inclusive Racah matrices allowed us to check the earlier suggested method of calculation of the exclusive Racah matrices from the inclusive ones: we have evaluated the exclusive Racah matrix $S_{\bar{[1]},[1],[2]}$ and checked that it coincides with the known answer for this matrix.
Acknowledgements {#acknowledgements .unnumbered}
================
The work was partly supported by the grant of the Foundation for the Advancement of Theoretical Physics “BASIS” (A.Mor. and A.S.), by RFBR grants 16-01-00291 (A.Mir.), 16-02-01021 (A.Mor.), 17-01-00585 (An.Mor.) and 16-31-60082-mol-a-dk (A.S.), by joint grants 17-51-50051-YaF, 15-51-52031-NSC-a, 16-51-53034-GFEN, 16-51-45029-IND-a (A.M.’s and A.S.), by grant NSFC (11425104,11611130015) (C.B., J.J. and J.L.) the grant of Ningde Normal University (Grant No. 2017Y07) (J.J.). Chengming Bai, Jianjian Jiang and Jinting Liang thank ITEP for the hospitality and for useful discussions while they visited there in the summer of 2017.
Link Polynomials\[a.pol\]
=========================
In this Appendix, we list the answers for the normalized link polynomials calculated using the Racah coefficients provided in this paper. The normalized polynomials are made from unnormalized ones by dividing them by dimensions: $$\mathcal{H}_{1,2}=\cfrac{H_{1,2}}{d_1 d_2},\ \ \ \ \ \mathcal{H}_{1,2,3}=\cfrac{H_{1,2,3}}{d_1 d_2 d_3}.$$ We use the notation for them in the matrix form suggested in [@Inds8]. The matrix describes the coefficients of a polynomial in $A^2$ and $q^2$ as in the following example:
$$\nonumber
q^{10}A^{16}+2q^{12}A^{16}+3q^{10}A^{18}+4q^{12}A^{18}\longrightarrow
q^{10}A^{16} \left(\begin{array}{rr}
3 & 4 \\
& \\
1 & 2 \\
\end{array}\right).$$
Whitehead link\[A.w\]
---------------------
$$\mathcal{H}_{[1],[1]}=\frac{1}{q^{4}A^{2}\{A\}}
\left(\begin{array}{rrrrr}
0 & 1 & -1 & 1 & 0 \\
&&&& \\
-1 & 2 & -3 & 2 & -1 \\
&&&& \\
0 & 1 & -2 & 1 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[2]}=\frac{1}{q^{6}A^{2}\{A\}}
\left(\begin{array}{rrrrrrr}
0 & 0 & 1 & 0 & -1 & 1 & 0 \\
&&&&&& \\
-1 & 1 & 1 & -3 & 1 & 1 & -1 \\
&&&&&& \\
0 & 1 & -1 & -1 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2]}=\frac{1}{q^{11}A^{4}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 2 & -1 & -1 & 1 & 0 \\
&&&&&&&&&&&& \\
0 & 0 & -1 & 1 & 1 & -4 & 1 & 4 & -4 & -2 & 3 & 0 & -1 \\
&&&&&&&&&&&& \\
1 & -2 & 1 & 4 & -7 & 0 & 10 & -6 & -5 & 6 & 0 & -2 & 1 \\
&&&&&&&&&&&& \\
-1 & 1 & 3 & -5 & -2 & 8 & -2 & -5 & 3 & 1 & -1 & 0 & 0 \\
&&&&&&&&&&&& \\
0 & 1 & -2 & -1 & 4 & -1 & -2 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[1,1]}=\frac{1}{q^{6}A^{1}\{A\}}
\left(\begin{array}{rrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & 0 \\
&&&&&&&& \\
-1 & 0 & 2 & 0 & -3 & 0 & 2 & 0 & -1 \\
&&&&&&&& \\
0 & 0 & 1 & 0 & -2 & 0 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[3]}=\frac{1}{q^{8}A^{2}\{A\}}
\left(\begin{array}{rrrrrrrrr}
0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 & 0 \\
&&&&&&&& \\
-1 & 1 & 0 & 1 & -3 & 1 & 0 & 1 & -1 \\
&&&&&&&& \\
0 & 1 & -1 & 0 & -1 & 1 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[2,1]}=\frac{1}{q^{8}A^{2}\{A\}}
\left(\begin{array}{rrrrrrrrr}
0 & 0 & 1 & -1 & 1 & -1 & 1 & 0 & 0 \\
&&&&&&&& \\
-1 & 2 & -3 & 4 & -5 & 4 & -3 & 2 & -1 \\
&&&&&&&& \\
0 & 0 & 1 & -1 & 0 & -1 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[3]}=\frac{1}{q^{15}A^{4}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 1 & 1 & -1 & -1 & 1 & 0 \\
&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & 0 & 2 & 0 & -3 & -2 & 3 & 3 & -3 & -3 & 1 & 2 & 0 & -1 \\
&&&&&&&&&&&&&&&& \\
1 & -1 & -2 & 3 & 3 & -3 & -6 & 2 & 9 & -1 & -7 & -1 & 4 & 2 & -2 & -1 & 1 \\
&&&&&&&&&&&&&&&& \\
-1 & 0 & 3 & 1 & -5 & -3 & 5 & 5 & -3 & -5 & 1 & 3 & 0 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&& \\
0 & 1 & -1 & -2 & 1 & 2 & 1 & -2 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2,1]}=\frac{1}{q^{13}A^{4}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 1 & 0 & 0 & -1 & 1 & 0 & 0 \\
&&&&&&&&&&&&&& \\
0 & 0 & -1 & 1 & 0 & -1 & 0 & -1 & 2 & -1 & -1 & 0 & 0 & 1 & -1 \\
&&&&&&&&&&&&&& \\
1 & -2 & 2 & -1 & 1 & -1 & -1 & 5 & -4 & 0 & -1 & 2 & 1 & -2 & 1 \\
&&&&&&&&&&&&&& \\
-1 & 2 & -1 & 1 & -3 & 0 & 4 & -1 & 0 & -3 & 2 & 1 & -1 & 0 & 0 \\
&&&&&&&&&&&&&& \\
0 & 0 & 1 & -2 & 0 & 1 & 1 & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$H_{[2],[1,1,1]}=\frac{1}{q^{8}A^{1}\{A\}}
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&& \\
-1 & 0 & 1 & 1 & 0 & -3 & 0 & 1 & 1 & 0 & -1 \\
&&&&&&&&&& \\
0 & 0 & 0 & 1 & 0 & -1 & -1 & 0 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[3],[2,1]}=\frac{1}{q^{13}A^{3}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 2 & -1 & 0 & -1 & 1 & 0 & 0 \\
&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & 0 & 1 & 1 & -1 & -3 & 1 & 0 & 4 & -4 & 1 & -3 & 3 & -1 & 1 & -1 \\
&&&&&&&&&&&&&&&&&& \\
1 & -1 & -1 & 1 & 1 & 3 & -6 & 1 & -2 & 8 & -2 & -2 & -4 & 2 & 3 & 0 & -1 & -1 & 1 \\
&&&&&&&&&&&&&&&&&& \\
-1 & 1 & 0 & 2 & -1 & -3 & -1 & 1 & 6 & -1 & -3 & -3 & 2 & 2 & 0 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&& \\
0 & 0 & 1 & -1 & -1 & -1 & 2 & 2 & -1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[3],[1,1,1]}=\frac{1}{q^{6}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrr}
0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&&&& \\
-1 & 0 & 0 & 2 & 0 & 0 & -3 & 0 & 0 & 2 & 0 & 0 & -1 \\
&&&&&&&&&&&& \\
0 & 0 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 1 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[3],[3]}=\frac{1}{q^{23}A^{6}\{A\}\{Aq\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 1 & 1 & -1 & -2 & 1 & 2 & 0 & -1 & -1 & 1 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & -3 & 1 & 3 & 0 & -5 & -3 & 4 & 5 & -2 & -4 & -2 & 3 & 1 & 0 & -1 \\
&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 1 & -1 & -1 & 2 & 2 & -2 & -6 & 4 & 10 & -1 & -12 & -6 & 12 & 11 & -5 & -10 & -2 & 7 & 3 & -2 & -2 & 0 & 1 \\
&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & -1 & 2 & -1 & -2 & 1 & 5 & -1 & -12 & 0 & 18 & 7 & -18 & -20 & 14 & 22 & -1 & -18 & -7 & 10 & 6 & -3 & -3 & 0 & 2 & -1 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&& \\
1 & -1 & -2 & 1 & 5 & 1 & -11 & -5 & 14 & 13 & -9 & -23 & 2 & 20 & 8 & -11 & -11 & 4 & 6 & 0 & -2 & -1 & 1 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&& \\
-1 & 1 & 2 & 1 & -6 & -4 & 7 & 9 & -2 & -14 & -2 & 9 & 7 & -4 & -6 & 1 & 2 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 1 & -2 & -1 & 2 & 3 & 0 & -6 & 0 & 3 & 2 & -1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2,1],[2,1]}=\frac{1}{q^{20}A^{9}\{A\}\{Aq\}\{A/q\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 1 & -2 & 3 & -4 & 5 & -5 & 5 & -4 & 3 & -2 & 1 & 0 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & -1 & 2 & -4 & 7 & -11 & 14 & -18 & 20 & -21 & 20 & -18 & 14 & -11 & 7 & -4 & 2 & -1 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&& \\
1 & -3 & 8 & -17 & 29 & -43 & 63 & -81 & 94 & -105 & 111 & -105 & 94 & -81 & 63 & -43 & 29 & -17 & 8 & -3 & 1 \\
&&&&&&&&&&&&&&&&&&&& \\
-2 & 7 & -15 & 29 & -52 & 77 & -103 & 133 & -160 & 174 & -177 & 174 & -160 & 133 & -103 & 77 & -52 & 29 & -15 & 7 & -2 \\
&&&&&&&&&&&&&&&&&&&& \\
1 & -4 & 9 & -16 & 28 & -45 & 61 & -77 & 95 & -105 & 106 & -105 & 95 & -77 & 61 & -45 & 28 & -16 & 9 & -4 & 1 \\
&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & -1 & 3 & -3 & 3 & -8 & 10 & -5 & 8 & -14 & 8 & -5 & 10 & -8 & 3 & -3 & 3 & -1 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 1 & -4 & 6 & -6 & 9 & -12 & 9 & -6 & 6 & -4 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
Borromean rings\[A.b\]
----------------------
$$\mathcal{H}_{[1],[1],[1]}=\frac{1}{q^{6}A^{2}\{A\}^2}
\left(\begin{array}{rrrrrrr}
0 & -1 & 4 & -5 & 4 & -1 & 0 \\
&&&&&& \\
1 & -4 & 7 & -10 & 7 & -4 & 1 \\
&&&&&& \\
0 & -1 & 4 & -5 & 4 & -1 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[1],[2]}=\frac{1}{q^{8}A^{2}\{A\}^2}
\left(\begin{array}{rrrrrrrrr}
0 & 0 & -1 & 3 & -1 & -2 & 3 & -1 & 0 \\
&&&&&&&& \\
1 & -3 & 2 & 3 & -8 & 3 & 2 & -3 & 1 \\
&&&&&&&& \\
0 & -1 & 3 & -2 & -1 & 3 & -1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[2],[2]}=\frac{1}{q^{11}A^{3}\{A\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrr}
0 & 0 & 0 & 0 & -1 & 2 & 2 & -4 & 1 & 2 & -1 & 0 \\
&&&&&&&&&&& \\
0 & 0 & 2 & -4 & -2 & 8 & -6 & -5 & 6 & -1 & -2 & 1 \\
&&&&&&&&&&& \\
-1 & 2 & 1 & -6 & 5 & 6 & -8 & 2 & 4 & -2 & 0 & 0 \\
&&&&&&&&&&& \\
0 & 1 & -2 & -1 & 4 & -2 & -2 & 1 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2],[1,1]}=\frac{1}{q^{13}A^{3}\{A\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & -1 & 1 & 3 & -2 & -3 & 3 & 1 & -1 & 0 & 0 \\
&&&&&&&&&&&&& \\
0 & 0 & 2 & -2 & -6 & 6 & 6 & -9 & -5 & 5 & 3 & -3 & -1 & 1 \\
&&&&&&&&&&&&& \\
-1 & 1 & 3 & -3 & -5 & 5 & 9 & -6 & -6 & 6 & 2 & -2 & 0 & 0 \\
&&&&&&&&&&&&& \\
0 & 0 & 1 & -1 & -3 & 3 & 2 & -3 & -1 & 1 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[1],[3]}=\frac{1}{q^{10}A^{2}\{A\}^2}
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 0 & -1 & 3 & -2 & 2 & -3 & 3 & -1 & 0 \\
&&&&&&&&&& \\
1 & -3 & 3 & -2 & 4 & -8 & 4 & -2 & 3 & -3 & 1 \\
&&&&&&&&&& \\
0 & -1 & 3 & -3 & 2 & -2 & 3 & -1 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[3],[3]}=\frac{1}{q^{15}A^{3}\{A\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 & 2 & -4 & 2 & -1 & 2 & -1 & 0 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 2 & -4 & 2 & -4 & 8 & -8 & 3 & -6 & 6 & -2 & 1 & -2 & 1 \\
&&&&&&&&&&&&&&& \\
-1 & 2 & -1 & 2 & -6 & 6 & -3 & 8 & -8 & 4 & -2 & 4 & -2 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&& \\
0 & 1 & -2 & 1 & -2 & 4 & -2 & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[1],[2,1]}=\frac{1}{q^{10}A^{2}\{A\}^2}
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & -1 & 3 & -3 & 3 & -3 & 3 & -1 & 0 & 0 \\
&&&&&&&&&& \\
1 & -4 & 8 & -12 & 15 & -18 & 15 & -12 & 8 & -4 & 1 \\
&&&&&&&&&& \\
0 & 0 & -1 & 3 & -3 & 3 & -3 & 3 & -1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[1,1,1],[3]}=\frac{1}{q^{14}A^{2}\{A\}^2}
\left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 0 & 0 & -1 & 2 & -1 & 2 & -3 & 2 & -1 & 2 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&& \\
1 & -2 & 1 & -2 & 4 & -2 & 2 & -6 & 2 & -2 & 4 & -2 & 1 & -2 & 1 \\
&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & 2 & -1 & 2 & -3 & 2 & -1 & 2 & -1 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1,1],[1,1],[3]}=\frac{1}{q^{15}A^{3}\{A\}^2\{Aq^{-1}\}}
\left(\begin{array}{rrrrrrrrrrrrrrrr}
0 & 0 & 0 & -1 & 1 & 2 & -1 & -1 & -1 & 2 & 1 & -1 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&& \\
1 & -1 & -2 & 1 & 2 & 3 & -4 & -7 & 2 & 4 & 2 & -4 & -2 & 2 & 0 & 0 \\
&&&&&&&&&&&&&&& \\
0 & 0 & -2 & 2 & 4 & -2 & -4 & -2 & 7 & 4 & -3 & -2 & -1 & 2 & 1 & -1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 1 & -1 & -2 & 1 & 1 & 1 & -2 & -1 & 1 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1,1],[3],[3]}=\frac{1}{q^{17}A^{3}\{A\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & 2 & -2 & -2 & 1 & 1 & 1 & -1 & 0 & 0 \\
&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 2 & -2 & -2 & -2 & 4 & 4 & -5 & -3 & -3 & 4 & 2 & -1 & -1 & -1 & 1 \\
&&&&&&&&&&&&&&&&& \\
-1 & 1 & 1 & 1 & -2 & -4 & 3 & 3 & 5 & -4 & -4 & 2 & 2 & 2 & -2 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&& \\
0 & 0 & 1 & -1 & -1 & -1 & 2 & 2 & -2 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2],[2,1]}=\frac{1}{q^{20}A^{4}\{A\}^2\{Aq\}^2}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 1 & -4 & 3 & 6 & -9 & -1 & 6 & 4 & -4 & -8 & 8 & 2 & -4 & 1 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & -1 & 3 & 1 & -11 & 10 & 5 & -9 & -3 & -6 & 19 & -6 & -13 & 6 & -1 & 7 & -5 & -3 & 4 & -1 \\
&&&&&&&&&&&&&&&&&&&& \\
1 & -4 & 4 & 4 & -13 & 9 & 2 & 3 & -8 & -11 & 32 & -11 & -8 & 3 & 2 & 9 & -13 & 4 & 4 & -4 & 1 \\
&&&&&&&&&&&&&&&&&&&& \\
-1 & 4 & -3 & -5 & 7 & -1 & 6 & -13 & -6 & 19 & -6 & -3 & -9 & 5 & 10 & -11 & 1 & 3 & -1 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 1 & -4 & 2 & 8 & -8 & -4 & 4 & 6 & -1 & -9 & 6 & 3 & -4 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[1,1,1],[3]}=\frac{1}{q^{8}A^{1}\{A\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&& \\
-1 & 0 & 1 & 1 & 0 & -3 & 0 & 1 & 1 & 0 & -1 \\
&&&&&&&&&& \\
0 & 0 & 0 & 1 & 0 & -1 & -1 & 0 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1,1,1],[3],[3]}=\frac{1}{q^{19}A^{3}\{A\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 3 & -2 & 0 & -3 & 3 & 0 & 1 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 2 & -2 & 0 & -6 & 6 & 0 & 6 & -9 & 0 & -5 & 5 & 0 & 3 & -3 & 0 & -1 & 1 \\
&&&&&&&&&&&&&&&&&&& \\
-1 & 1 & 0 & 3 & -3 & 0 & -5 & 5 & 0 & 9 & -6 & 0 & -6 & 6 & 0 & 2 & -2 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 1 & -1 & 0 & -3 & 3 & 0 & 2 & -3 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2],[2]}=\frac{1}{q^{18}A^{4}\{A\}^2\{Aq\}^2}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 1 & -4 & 2 & 11 & -16 & -4 & 25 & -11 & -14 & 13 & 1 & -4 & 1 & 0 \\
&&&&&&&&&&&&&&&&&& \\
0 & 0 & -1 & 3 & 2 & -15 & 11 & 24 & -43 & -7 & 56 & -27 & -34 & 33 & 5 & -16 & 3 & 3 & -1 \\
&&&&&&&&&&&&&&&&&& \\
1 & -4 & 3 & 10 & -25 & 7 & 47 & -54 & -25 & 86 & -25 & -54 & 47 & 7 & -25 & 10 & 3 & -4 & 1 \\
&&&&&&&&&&&&&&&&&& \\
-1 & 3 & 3 & -16 & 5 & 33 & -34 & -27 & 56 & -7 & -43 & 24 & 11 & -15 & 2 & 3 & -1 & 0 & 0 \\
&&&&&&&&&&&&&&&&&& \\
0 & 1 & -4 & 1 & 13 & -14 & -11 & 25 & -4 & -16 & 11 & 2 & -4 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2],[3]}=\frac{1}{q^{22}A^{4}\{A\}^2\{Aq\}^2}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -3 & -1 & 9 & -2 & -10 & 2 & 10 & 2 & -11 & -2 & 9 & -1 & -3 & 1 & 0 \\
&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & 2 & 4 & -9 & -7 & 19 & 10 & -27 & -19 & 28 & 24 & -26 & -23 & 17 & 16 & -9 & -8 & 4 & 2 & -1 \\
&&&&&&&&&&&&&&&&&&&&&& \\
1 & -3 & -1 & 11 & -4 & -21 & 9 & 33 & -6 & -48 & 2 & 60 & 2 & -48 & -6 & 33 & 9 & -21 & -4 & 11 & -1 & -3 & 1 \\
&&&&&&&&&&&&&&&&&&&&&& \\
-1 & 2 & 4 & -8 & -9 & 16 & 17 & -23 & -26 & 24 & 28 & -19 & -27 & 10 & 19 & -7 & -9 & 4 & 2 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&& \\
0 & 1 & -3 & -1 & 9 & -2 & -11 & 2 & 10 & 2 & -10 & -2 & 9 & -1 & -3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[3],[3]}=\frac{1}{q^{28}A^{5}\{A\}^2\{Aq\}^2\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & -3 & 5 & 7 & -4 & -12 & 0 & 16 & 3 & -12 & -5 & 5 & 6 & -3 & -2 & 1 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 3 & 8 & -9 & -17 & 7 & 33 & 1 & -49 & -17 & 44 & 31 & -28 & -36 & 12 & 25 & -1 & -12 & -3 & 5 & 1 & -1 \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 1 & 0 & -7 & 0 & 20 & 4 & -39 & -19 & 55 & 52 & -56 & -78 & 40 & 89 & -3 & -75 & -17 & 48 & 20 & -22 & -13 & 7 & 7 & -3 & -2 & 1 \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\
-1 & 2 & 3 & -7 & -7 & 13 & 22 & -20 & -48 & 17 & 75 & 3 & -89 & -40 & 78 & 56 & -52 & -55 & 19 & 39 & -4 & -20 & 0 & 7 & 0 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\
1 & -1 & -5 & 3 & 12 & 1 & -25 & -12 & 36 & 28 & -31 & -44 & 17 & 49 & -1 & -33 & -7 & 17 & 9 & -8 & -3 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\
0 & -1 & 2 & 3 & -6 & -5 & 5 & 12 & -3 & -16 & 0 & 12 & 4 & -7 & -5 & 3 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
L7a3 link\[A.l\]
----------------
Since link L7a3 is asymmetric in two components, it is important to fix the order of the components. Here the first component is the unknot and the second one is the trefoil.
$$\mathcal{H}_{[1],[1]}=\frac{1}{q^{6}A^{2}\{A\}}
\left(\begin{array}{rrrrrrr}
0 & 1 & -1 & 2 & -1 & 1 & 0 \\
&&&&&& \\
-1 & 2 & -4 & 3 & -4 & 2 & -1 \\
&&&&&& \\
0 & 1 & -2 & 3 & -2 & 1 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[2]}=\frac{1}{q^{8}A^{3}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrr}
0 & 0 & 1 & 0 & 0 & 2 & -1 & 0 & 2 & 0 & -1 & 1 & 0 \\
&&&&&&&&&&&& \\
-1 & 1 & -1 & -3 & 3 & -3 & -4 & 3 & 0 & -4 & 1 & 1 & -1 \\
&&&&&&&&&&&& \\
1 & 0 & -2 & 4 & 1 & -4 & 4 & 2 & -2 & 0 & 1 & 0 & 0 \\
&&&&&&&&&&&& \\
0 & -1 & 1 & 1 & -3 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[1]}=\frac{1}{q^{8}A^{2}\{A\}}
\left(\begin{array}{rrrrrrrrr}
0 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 \\
&&&&&&&& \\
-1 & 1 & 0 & -2 & 1 & -2 & 0 & 1 & -1 \\
&&&&&&&& \\
0 & 1 & -1 & 0 & 1 & -1 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[2]}=\frac{1}{q^{19}A^{4}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 1 & 2 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&& \\
-1 & 0 & 2 & -2 & -4 & 2 & 1 & -4 & -2 & 0 & -1 & -1 & 0 & -1 & -1 & 0 \\
&&&&&&&&&&&&&&& \\
0 & 1 & 1 & -1 & -1 & 3 & 3 & -1 & -1 & 4 & 2 & 1 & 0 & 0 & 1 & 1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & -1 & 1 & 1 & -2 & -2 & 1 & 1 & -2 & -2 & 1 & 0 & -1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -2 & 1 & 2 & -2 & 0 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[1,1]}=\frac{1}{q^{18}A^{3}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 1 & 2 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
&&&&&&&&&&&&&& \\
-1 & 0 & 2 & -1 & -4 & 1 & 2 & -2 & -3 & 0 & 0 & -1 & 0 & 0 & -1 \\
&&&&&&&&&&&&&& \\
0 & 0 & 1 & 1 & -2 & -1 & 4 & 1 & -3 & 1 & 2 & 1 & -1 & 0 & 1 \\
&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 2 & -1 & -2 & 2 & 0 & -1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1],[3]}=\frac{1}{q^{10}A^{4}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 2 & -1 & 1 & 1 & 2 & -1 & 0 & 0 & 2 & 0 & 0 & -1 & 1 & 0 \\
&&&&&&&&&&&&&&&&&&&&& \\
0 & -1 & 1 & -2 & 0 & -3 & 2 & -5 & -2 & -3 & 4 & -4 & -3 & -4 & 3 & 0 & 0 & -4 & 1 & 0 & 1 & -1 \\
&&&&&&&&&&&&&&&&&&&&& \\
1 & 0 & 0 & 0 & 4 & 1 & -1 & 0 & 6 & 4 & -1 & -3 & 3 & 3 & 2 & -2 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&& \\
-1 & 0 & 0 & 2 & -4 & -1 & -1 & 4 & -3 & -2 & -2 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&& \\
0 & 1 & -1 & 0 & -1 & 3 & -1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[3],[1]}=\frac{1}{q^{10}A^{2}\{A\}}
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 0 & 1 & 0 & 1 & -1 & 1 & -1 & 1 & 0 \\
&&&&&&&&&& \\
-1 & 1 & -1 & 2 & -3 & 1 & -3 & 2 & -1 & 1 & -1 \\
&&&&&&&&&& \\
0 & 1 & -1 & 1 & -2 & 2 & -1 & 1 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2,1],[1]}=\frac{1}{q^{10}A^{2}\{A\}}
\left(\begin{array}{rrrrrrrrrrr}
0 & 0 & 1 & -1 & 2 & -2 & 2 & -1 & 1 & 0 & 0 \\
&&&&&&&&&& \\
-1 & 2 & -4 & 6 & -8 & 7 & -8 & 6 & -4 & 2 & -1 \\
&&&&&&&&&& \\
0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[3]}=\frac{1}{q^{19}A^{4}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 1 & 2 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&& \\
-1 & 0 & 2 & -2 & -4 & 2 & 1 & -4 & -2 & 0 & -1 & -1 & 0 & -1 & -1 & 0 \\
&&&&&&&&&&&&&&& \\
0 & 1 & 1 & -1 & -1 & 3 & 3 & -1 & -1 & 4 & 2 & 1 & 0 & 0 & 1 & 1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & -1 & 1 & 1 & -2 & -2 & 1 & 1 & -2 & -2 & 1 & 0 & -1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -2 & 1 & 2 & -2 & 0 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[2],[1,1,1]}=\frac{1}{q^{34}A^{4}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 0 & 1 & 2 & 0 & -1 & 0 & 2 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&& \\
-1 & 0 & 1 & 1 & -1 & -4 & -1 & 2 & 2 & -2 & -5 & -3 & 1 & 1 & -2 & -4 & -1 & -1 & 0 & -1 & -1 & 0 & -1 & 0 \\
&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 1 & 1 & 0 & -2 & -1 & 3 & 5 & 1 & -4 & -1 & 4 & 5 & 1 & -1 & 0 & 3 & 1 & 1 & -1 & 1 & 1 \\
&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 2 & 1 & -3 & -3 & 0 & 3 & 0 & -2 & -2 & 0 & 0 & 0 & 0 & -1 \\
&&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & -1 & 1 & 2 & -1 & -1 & 0 & 1 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[3],[2]}=\frac{1}{q^{19}A^{4}\{A\}\{Aq\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & -2 & 1 & 2 & -1 & -1 & 1 & 0 \\
&&&&&&&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & 0 & 1 & -1 & -1 & -1 & -2 & -2 & 0 & 2 & -3 & -5 & 1 & 4 & -1 & -4 & 0 & 2 & 0 & -1 \\
&&&&&&&&&&&&&&&&&&&&&& \\
1 & -1 & -1 & 3 & 0 & -1 & 2 & 0 & -2 & 1 & 9 & 3 & -9 & -2 & 11 & 3 & -6 & -3 & 4 & 3 & -2 & -1 & 1 \\
&&&&&&&&&&&&&&&&&&&&&& \\
-1 & 0 & 2 & -1 & -2 & 2 & -1 & -4 & 2 & 6 & -3 & -10 & 1 & 9 & -1 & -7 & 0 & 3 & 0 & -1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&&&&&&&& \\
0 & 1 & -1 & -1 & 2 & -1 & -1 & 2 & 1 & -1 & -3 & 2 & 3 & -2 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[1,1,1],[2]}=\frac{1}{q^{12}A^{3}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrrrrrr}
0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&&& \\
-1 & 0 & -1 & 0 & 2 & -2 & -1 & -2 & -1 & 1 & 0 & -2 & -1 & 0 & 1 & 0 & -1 \\
&&&&&&&&&&&&&&&& \\
1 & 0 & -1 & 1 & 0 & 2 & 1 & -2 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 0 & 0 \\
&&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & 0 & 1 & -1 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0
\end{array}\right).$$
$$\mathcal{H}_{[3],[1,1,1]}=\frac{1}{q^{19}A^{4}\{A\}}
\left(\begin{array}{rrrrrrrrrrrrrrrr}
0 & 0 & 1 & 0 & -1 & 1 & 2 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
&&&&&&&&&&&&&&& \\
-1 & 0 & 2 & -2 & -4 & 2 & 1 & -4 & -2 & 0 & -1 & -1 & 0 & -1 & -1 & 0 \\
&&&&&&&&&&&&&&& \\
0 & 1 & 1 & -1 & -1 & 3 & 3 & -1 & -1 & 4 & 2 & 1 & 0 & 0 & 1 & 1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & -1 & -1 & 1 & 1 & -2 & -2 & 1 & 1 & -2 & -2 & 1 & 0 & -1 \\
&&&&&&&&&&&&&&& \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -2 & 1 & 2 & -2 & 0 & 1 & 0 & 0
\end{array}\right).$$
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[^1]: There is a simple formula for this eigenvalue $$\varkappa_Y=1/2\sum_i Y_i (Y_i+1-2i)$$ which, in accordance with the Schur-Weyl duality, is associated with the value of character $\chi_Y([2])$ of the symmetric group $S_{|Y|}$ in representation described by the Young diagram $Y$ on the cycle of length 2
[^2]: Of course, there are degenerate solutions which correspond to the Racah matrices of smaller sizes, e.g. the matrix of the size $6\times 6$ can be “block-diagonal” and be divided into two or more matrices of smaller sizes. These matrices would definitely solve the Yang-Baxter equation too, and their smaller compartments would satisfy their own smaller version of the eigenvalue conjecture. Hence, the solution is unique if one considers the Racah matrix which “mixes” the eigenvalues (representations) in $\mathcal{R}$-matrix.
[^3]: The eigenvalues are chosen as being of different signs due to the legacy reasons: for low representations, they usually have different signs. However, the answer actually works for eigenvalues of the same sign as well. One should choose the corresponding normalized eigenvalue as being imaginary, and it can be checked that the formulae still give correct link polynomials (see discussion in the previous section).
[^4]: For these quantities, there is a simple hook formula which allows one to easily calculate them: $$S^*_Q(A,q)=\prod_{(i,j)\in Q}
\frac{Aq^{i-j}-A^{-1}q^{j-i}}{q^{h_{i,j}}-q^{-h_{i,j}}}.
\begin{picture}(105,15)(-35,-15)
\put(0,0){\line(1,0){70}}
\put(0,-10){\line(1,0){70}}
\put(0,-20){\line(1,0){60}}
\put(0,-30){\line(1,0){40}}
\put(0,-40){\line(1,0){20}}
\put(0,-50){\line(1,0){20}}
\put(0,0){\line(0,-1){50}}
\put(10,0){\line(0,-1){50}}
\put(20,0){\line(0,-1){50}}
\put(30,0){\line(0,-1){30}}
\put(40,0){\line(0,-1){30}}
\put(50,0){\line(0,-1){20}}
\put(60,0){\line(0,-1){20}}
\put(70,0){\line(0,-1){10}}
\put(15,-15){\makebox(0,0)[cc]{\textbf{x}}}
\put(15,5){\makebox(0,0)[cc]{$i$}}
\put(-5,-15){\makebox(0,0)[cc]{$j$}}
\qbezier(19,-11)(45,20)(55,-15)
\put(40,10){\makebox(0,0)[cc]{$k$}}
\qbezier(11,-19)(-17,-40)(15,-45)
\put(60,-40){\makebox(0,0)[lc]{$h_{i,j}=k+l+1$}}
\end{picture}$$
$[n]_q$ denotes the quantum number, i.e. $[n]_q\equiv\frac{q^n-q^{-n}}{q-q^{-1}}$.
[^5]: $U_{123}$ here corresponds to the Racah matrix which transforms from the basis $(R_1\otimes R_2)\otimes R_3$ to the basis $R_1\otimes (R_2\otimes R_3)$. Thus $U_{321}=U_{123}^{\dagger}$.
|
---
abstract: 'We develop a theoretical framework that combines measurements of galaxy-galaxy lensing, galaxy clustering, and the galaxy stellar mass function in a self-consistent manner. While considerable effort has been invested in exploring each of these probes individually, attempts to combine them are still in their infancy. These combinations have potential to elucidate the galaxy-dark matter connection and the galaxy formation physics that is responsible for it, as well as to constrain cosmological parameters, and to test the nature of gravity. In this paper, we focus on a theoretical model that describes the galaxy-dark matter connection based on standard halo occupation distribution techniques. Several key modifications enable us to extract additional parameters that determine the stellar-to-halo mass relation and to simultaneously fit data from multiple probes while allowing for independent binning schemes for each probe. We construct mock catalogs from numerical simulations to investigate the effects of sample variance and covariance for each probe. Finally, we analyze how trends in each of the three observables impact the derived parameters of the model. In particular, we investigate various features of the observed galaxy stellar mass function(low-mass slope, “plateau”, knee, and high-mass cut-off) and show how each feature is related to the underlying relationship between stellar and halo mass. We demonstrate that the observed “plateau” feature in the stellar mass function at $M_*\sim 2\times10^{10}$ M$_{\odot}$ is due to the transition that occurs in the stellar-to-halo mass relation at $M_h\sim 10^{12}$ M$_{\odot}$ from a low-mass power-law regime to a sub-exponential function at higher stellar mass.'
author:
- 'Alexie Leauthaud, Jeremy Tinker, Peter S. Behroozi, Michael T. Busha, Risa H. Wechsler'
title: 'A theoretical framework for combining techniques that probe the link between galaxies and dark matter.'
---
Introduction
============
Improved measurements of the link between galaxies and the dark matter distribution will benefit a variety of cosmological applications but will also provide important clues about the role that dark matter plays in galaxy formation process. Although multiple techniques have been developed for this purpose, no single method has yet emerged as the ultimate tool and all suffer from various drawbacks. The goal of this paper is to develop the theoretical foundations required to combine multiple probes into a single tool that will provide more powerful constraints on the galaxy-dark matter connection. This paper extends and complements a growing body of work on this topic [@Seljak:2000; @Guzik:2001; @Guzik:2002; @Berlind:2002; @Tasitsiomi:2004; @Mandelbaum:2005a; @Mandelbaum:2006c; @Yoo:2006; @Cacciato:2009; @Tinker:2011].
At present, there are only two observational techniques capable of [*directly*]{} probing the dark matter halos of galaxies out to large radii (above 50 kpc): galaxy-galaxy lensing [e.g., @Brainerd:1996; @McKay:2001; @Hoekstra:2004; @Sheldon:2004; @Mandelbaum:2006; @Mandelbaum:2006c; @Heymans:2006; @Johnston:2007; @Leauthaud:2010] and the kinematics of satellite galaxies [@McKay:2002; @Prada:2003; @Brainerd:2003; @van-den-Bosch:2004; @Conroy:2007; @Becker:2007; @Norberg:2008; @More:2009; @More:2011]. Galaxy-galaxy lensing (hereafter “g-g lensing”) utilizes subtle distortions induced in the shapes and orientations of distant background galaxies in order to measure foreground mass distributions. The satellite kinematic method uses satellite galaxies as test particles to trace out the dark matter velocity field. Neither method can probe the halos of individual galaxies. Instead, both techniques must stack an ensemble of foreground galaxies in order to extract a signal. Nonetheless, with the advent of data-sets large enough to provide statistically significant samples, improvements in photometric redshift techniques, and spectroscopic follow-up programs, both methods have emerged as powerful probes of the galaxy-dark matter connection and have truly evolved into mature techniques over the last decade.
In addition to these two direct probes, there are also several popular indirect methods to infer the galaxy-dark matter connection from the statistics of galaxy clustering. For example, numerous authors have employed a statistical model to describe the probability distribution $P(N|M_h)$ that a halo of mass $M_h$ is host to N galaxies above some threshold in luminosity or stellar-mass. This statistical model, commonly known as the halo occupation distribution (HOD), has been considerably successful at interpreting the clustering properties of galaxies [e.g., @Seljak:2000; @Peacock:2000; @Scoccimarro:2001; @Berlind:2002; @Bullock:2002; @Zehavi:2002; @Zehavi:2005; @Zheng:2005; @Zheng:2007; @Tinker:2007; @Wake:2011; @Zehavi:2010; @White:2011]. The HOD provides a description of the spatial distribution of galaxies at all scales, but it is usually inferred observationally by modeling measurements of the two-point correlation function of galaxies, $\xi_{gg}(r)$. Since they were introduced a decade ago, HOD models have progressively increased in fidelity and complexity owing to stronger observational constraints but also to the availability of larger, high-resolution cosmological N-body simulations of the dark matter. For example, analytical descriptions of the form and evolution of the halo mass function and the large-scale halo bias, both of which are key ingredients for HOD models, are approaching percent-level precision [e.g., @Tinker:2008; @Tinker:2010]. A variety of extensions to the basic HOD framework have also been proposed. For example, the conditional luminosity function $\Phi (L|M_h)dL$ specifies the average number of galaxies of luminosity $L\pm dL/2$ that reside in a halo of mass $M_h $ [e.g., @Yang:2003; @van-den-Bosch:2003; @Vale:2004; @Cooray:2006; @van-den-Bosch:2007; @Vale:2008] and the conditional stellar mass function $\Phi(M_{*}|M_h)dM_*$ describes the average number of galaxies with stellar masses in the range $M_{*}\pm dM_{*}$ as a function of host halo mass $M_h$ [e.g., @Yang:2009; @Moster:2010; @Behroozi:2010]. Furthermore, a number of studies are also starting to take into account, not only the simple expectation values of the underlying relations, but also the scatter between the observable and halo-mass [e.g., @More:2011; @Behroozi:2010; @Moster:2010], a crucial ingredient for a complete description of the galaxy-dark matter connection.
Finally, halo mass constraints from the galaxy stellar mass function (hereafter “SMF”) have also been derived by assuming that there is there is a monotonic correspondence between halo mass (or circular velocity) and galaxy stellar mass (or luminosity) [e.g., @Kravtsov:2004; @Vale:2004; @Tasitsiomi:2004; @Vale:2006a; @Conroy:2009; @Drory:2009; @Moster:2010; @Behroozi:2010; @Guo:2010]. This particular technique, often referred to as “abundance matching”, is economic in terms of data requirements since it only considers the observed stellar mass (or luminosity) function. However, prior knowledge about the mass distribution of halos (and substructure within those halos) from cosmological N-body simulations is necessary as well as the assumption that field halos and subhalos of the same halo mass contain galaxies of the same stellar mass.
While considerable effort has been invested in exploring each of these probes individually, attempts to [*combine*]{} them in a fully consistent way are still in their infancy. Nevertheless, savvy combinations hold great potential to not only elucidate the evolution of the galaxy-dark matter connection, and consequently the galaxy formation physics responsible for it, but to also constrain fundamental physics, including the cosmological model and the nature of gravity. For example, measurements of small-scale galaxy clustering alone do not yield cosmological constraints unless coupled with probes that are sensitive to the mass scales of dark matter halos (e.g., cluster mass-to-light ratios, satellite kinematics, g-g lensing, etc.) [@van-den-Bosch:2003a; @Tinker:2005; @Seljak:2005]. In particular, @Yoo:2006 and @Cacciato:2009 have shown that the combination of g-g lensing and galaxy clustering is sensitive to $\Omega_m$ and $\sigma_8$. Conceptually, this sensitivity arises from the fact that this particular combination simultaneously probes the shape and amplitude of the halo mass function at small scales and the overall matter density and the bias of the galaxy sample at large scales.
Other combinations can be sensitive to parameters in modified gravity theories. A generic metric theory of gravity has two scalar potentials, $\phi$, which affects the clustering and dynamics of galaxies, and $\psi$, which affects the lensing of light around galaxies. Combining probes of g-g lensing with clustering and/or satellite dynamics allows a test of the general relativity (GR) prediction that $\psi = \phi$ as well as the Poisson equations which relate these potentials to the underlying density distribution. For example, @Reyes:2010 have used a combination of g-g lensing, galaxy clustering, and redshift space distortions to place limits on possible modifications to GR on $\sim$10 Mpc scales.
Tests of gravity on smaller scales, though complicated by the fact that structures have undergone non-linear evolution, are interesting in several respects. First, *all of our direct probes* of the dark matter are either intrinsically limited to small scales (e.g., a few Mpc for satellite kinematics) or have significantly larger signals on small scales (e.g., below 10 Mpc for both cosmic shear and g-g lensing). Second, there are many alternative gravity theories which predict unique and interesting modifications on these scales [@Smith:2009; @Hui:2009; @Schmidt:2010; @Jain:2010].
In this paper we develop the theoretical framework necessary to constrain the galaxy-dark matter connection by combining measurements of galaxy clustering, g-g lensing, and the galaxy SMF. The formalism outlined in this paper could also be applied to model satellite kinematics (see @More:2011). For this work, we adopt the standard HOD framework but with several key modifications. For instance, a procedure often adopted in clustering studies is to fit a set of HOD parameters (typically three to five) independently to the clustering signal and number density for each galaxy sample. However, adopting this strategy would require selecting a common binning scheme for all probes. In practice, we would like to avoid using a single binning scheme because various probes have different signal-to-noise (S/N) requirements. We therefore modify the standard HOD model so that we can simultaneously fit data from multiple probes while allowing for independent binning schemes for each probe. Also, since we are interested in the galaxy-dark matter connection, we modify the HOD model so as to specifically include a parameterization for the stellar-to-halo mass relation (hereafter “SHMR”). In Leauthaud et al. 2011b (hereafter Paper II), we demonstrate that this model provides an excellent fit to g-g lensing, galaxy clustering, and stellar mass function measurements in the COSMOS survey from $z=0.2$ to $z=1.0$.
For 2 deg$^2$ surveys such as COSMOS, the finite sample size of the observational data set is also an important concern. We will present an estimate of the sample variance using mock surveys from numerical simulations. We will also estimate the covariance of the data for each observational measure. We will demonstrate that this is especially important for modeling the SMF, an effect that is usually not incorporated into most analyses. The finite sample size in COSMOS biases clustering measurements through the integral constraint, an effect we will also model through our mock surveys.
The layout of this paper is as follows. To begin with, we introduce the parametric form used to model the SHMR in $\S$\[the\_shmr\]. Next, in $\S$\[hodtheory\], we present the general HOD framework and our extensions to this model. In $\S$\[application\_of\_model\], we show how this model can be used to simultaneously fit g-g lensing, galaxy clustering, and SMF measurements. In $\S$\[parameter\_influance\], we describe the influence of each model parameter on the three observables. We then construct a set of mocks catalogs designed to mimic the COSMOS survey and describe the behaviour of the covariance matrices for the three probes in $\S$\[mocks\]. Finally, we draw up our conclusions in $\S$\[conclusions\].
We assume a WMAP5 $\Lambda$CDM cosmology with $\Omega_{\rm m}=0.258$, $\Omega_\Lambda=0.742$, $\Omega_{\rm b}h^2=0.02273$, $n_{\rm
s}=0.963$, $\sigma_{8}=0.796$, $H_0=72$ km s$^{-1}$ Mpc$^{-1}$ [@Hinshaw:2009]. Unless stated otherwise, all distances are expressed in physical Mpc. The letter $M_h$ denotes halo mass. The halo radius is noted $R_h$. In this paper, halo mass is defined as $M_{200b}\equiv M(<R_{200b})=200\bar{\rho} \frac{4}{3}\pi R_{200b}^3$ where $R_{200b}$ is the radius at which the mean interior density is equal to 200 times the mean matter density ($\bar{\rho}$). We note however that our theoretical framework is valid for any reasonable choice of halo definition. Stellar mass is noted $M_{*}$.
The stellar-to-halo mass relation for central galaxies {#the_shmr}
======================================================
To begin with, we present the mathematical function that we use to model the SHMR and we describe the influence of each of the five parameters that regulate its shape. We will assume that the SHMR is specifically valid for “central” galaxies which are located by definition at the center of their parent halos. Dark matter halos also contain smaller bound density peaks that orbit around the center of the potential well. These substructures are commonly referred to as sub-halos; these sub-halos are the likely sites of “satellite” galaxies that have been accreted onto their parent halos. The abundance matching technique commonly assumes that satellite galaxies follow the same SHMR as centrals provided that halo mass is defined at the epoch when satellites were accreted onto their parent halos (M$_{\rm acc}$), rather than the current sub-halo mass [@Conroy:2006; @Moster:2010; @Behroozi:2010]. However, this pre-supposes that satellite stellar growth occurs at a similar rate as centrals of equivalent halo mass (that is to say with a halo mass equal to M$_{\rm acc}$). Since one might expect that satellites and centrals experience distinct stellar growth rates, we model central and satellite galaxies separately in order to keep our model as general as possible.
In $\S$ \[ncen\_model\], we will show how the SHMR can be used to predict the central occupation function and then we will introduce the model for satellite galaxies in $\S$ \[model\_nsat\].
Functional form for the SHMR
----------------------------
Let us consider the conditional stellar mass function (the analog of the conditional luminosity function) which represents the number of galaxies with $M_*$ in the range $M_*\pm dM_*/2$ at fixed halo mass and is noted $\Phi(M_{*}|M_h)$ [e.g., @Yang:2009; @Moster:2010; @Behroozi:2010]. The conditional stellar mass function can be divided into a central component and a satellite component: $\Phi(M_{*}|M_h)=\Phi_c(M_{*}|M_h)+\Phi_s(M_{*}|M_h)$. $\Phi_c(M_{*}|M_h)$ is the conditional stellar mass function for central galaxies, and it will be our mathematical representation of the SHMR. Note that in our model, the halo mass in the term $\Phi_s(M_{*}|M_h)$ refers to the host halo mass.
In addition to the shape and evolution of the mean SHMR, astrophysical processes are expected to induce an intrinsic scatter in stellar mass at fixed halo mass which it is important to take into consideration when defining a functional form for $\Phi_c(M_{*}|M_h)$. Another non negligible source of scatter can be the measurement error associated with the determination of stellar masses. In the absence of strong observational or theoretical guidance for the form and magnitude of the total scatter (intrinsic plus measurement), we adopt a stochastic model where $\Phi_c(M_{*}|M_h)$ is a log-normal probability distribution function (hereafter “PDF”) with a log-normal scatter noted $\sigma_{\rm log M_{*}}$. Since we have assumed a log-normal functional form, $\Phi_c(M_{*}|M_h)$ can be written as:
$$\Phi_c(M_{*}|M_h) = \frac{1}{\ln(10)\sigma_{\rm log
M_{*}}\sqrt{2\pi}} \times\hspace{0.65\columnwidth}$$
$$\label{lognormal_equation}
\quad \exp\left[-\frac{\left[\log_{10}(M_*)-\log_{10}(f_{\textsc{shmr}}(M_h))\right]^2}{2\sigma_{\rm log
M_{*}}^2}\right]$$
where $f_{\textsc{shmr}}$ represents the logarithmic mean of the stellar mass given the halo mass for the $\Phi_c$ distribution function. Equation \[lognormal\_equation\] is normalized such that the integral of $\Phi_c(M_{*}|M_h)$ over $M_*$ is equal to 1.
To model $\Phi_c$ we must specify a functional form for both $f_{\textsc{shmr}}$ and for $\sigma_{\rm log M_{*}}$. There is increasing evidence to suggest that low and high mass galaxies have different stellar-to-halo mass ratios, probably as a result of multiple feedback mechanisms that operate at distinct mass scales and regulate star formation. We therefore require a SHMR that is flexible enough to capture such variations. We adopt the functional form presented in @Behroozi:2010 (hereafter “B10”) which has been shown to reproduce the local SDSS stellar mass function using the abundance matching technique. In practice, $f_{\textsc{shmr}}(M_h)$ is mathematically defined following via its inverse function:
$$\log_{10}(f_{\textsc{shmr}}^{-1}(M_\ast)) = \log_{10}(M_h) = \hspace{0.65\columnwidth}$$
$$\label{shmr}
\quad \log_{10}(M_1) + \beta\,\log_{10}\left(\frac{M_\ast}{M_{\ast,0}}\right) +
\frac{\left(\frac{M_\ast}{M_{\ast,0}}\right)^\delta}{1 + \left(\frac{M_{\ast}}{M_{\ast,0}}\right)^{-\gamma}} - \frac{1}{2}$$
where $M_{1}$ is a characteristic halo mass, $M_{*,0}$ is a characteristic stellar mass, $\beta$ is the low mass end slope, $\gamma$ controls the transition region, and $\delta$ controls the massive end slope. Details regarding the justification of this functional form can be found in $\S$ 3.4.3 of B10.
Note that a variety of similar functional forms have been proposed by previous authors. For example, the interested reader can look at Equation 2 in @Moster:2010 and Equation 20 in @Yang:2009.
In contrast to B10, we do not parametrize the redshift evolution of this functional form. Instead, in Paper II, we bin the data into three redshift bins and check for redshift evolution in the parameters a posteriori. Another difference with respect to B10 is that we assume that Equation \[shmr\] is only relevant for central galaxies whereas B10 assume that the SHMR also applies to satellite galaxies, provided that the halo mass of a satellite galaxy is defined as M$_{\rm acc}$.
It is important to note that our SHMR traces the location of the mean-log stellar mass: $f_{\textsc{shmr}}(M_h)\equiv \langle
\log_{10}(M_*(M_h))\rangle$. Other authors may report the mean stellar mass, $\langle M_*(M_h)\rangle$, or even the mean halo mass at fixed stellar mass, $\langle M_h(M_*)\rangle$ [e.g., @Conroy:2007]. These averaging systems will yield different results in the presence of scatter. For example, $\langle
M_h(M_*)\rangle$ will be biased low compared to $\langle
\log_{10}(M_*(M_h))\rangle$ if $\sigma_{\log M_*}$ is non zero. This bias will increase with $\sigma_{\log M_*}$ and for larger values of the high-mass slope of $f_{\textsc{shmr}}^{-1}$.
In Figure \[mh\_ms\_vary\_p\] we illustrate the impact of the five parameters that determine $f_{\textsc{shmr}}$ on the shape on the SHMR. A brief description is as follows:
- [**$M_{1}$**]{} controls the characteristic halo mass; increasing $M_1$ will result in larger halos hosting galaxies at a given stellar mass. In Figure \[mh\_ms\_vary\_p\] (which represents $M_*$ on the x-axis and $M_h$ on the y-axis): $M_1$ controls the y-axis amplitude (halo mass) of $f_{\textsc{shmr}}^{-1}$ so that a constant change in $M_{1}$ leads to a constant up/down logarithmic shift in $f_{\textsc{shmr}}^{-1}$. Note that this constant logarithmic shift may not be visually obvious because the slope of the SHMR increases sharply at $M_*>10^{11}$ M$_{\odot}$.
- $M_{*,0}$ controls the characteristic stellar mass; increasing $M_{*,0}$ will result in smaller halos hosting galaxies at a given stellar mass. In Figure \[mh\_ms\_vary\_p\], $M_{*,0}$ controls the x-axis amplitude of $f_{\textsc{shmr}}^{-1}$.
- $\beta$ controls the low mass power-law slope of $f_{\textsc{shmr}}^{-1}$. When $\beta$ increases, the low mass end slope becomes steeper.
- The $\delta$ parameter regulates how rapidly $f_{\textsc{shmr}}^{-1}$ climbs at high $M_*$. Indeed, $f_{\textsc{shmr}}^{-1}$ asymptotes to a sub-exponential function at high $M_*$ which signifies that $f_{\textsc{shmr}}^{-1}$ climbs more rapidly than a power-law function but less rapidly than an exponential function (see discussion in B10).
- $\gamma$ controls the transition regime between the low-mass power-law regime and the high-mass sub-exponential behaviour. A larger value of $\gamma$ corresponds to a more sharp transition between the two regimes.
A quantity that is of particular interest is the mass (we refer here to both $M_*$ and $M_h$) at which the ratio $M_h/M_*$ reaches a minimum. This minimum is of noteworthy importance for galaxy formation models because it marks the mass at which the accumulated stellar growth of the central galaxy has been the most efficient. In this paper, and in subsequent papers, we will refer to the stellar mass, halo mass, and ratio at which this minimum occurs as the “pivot stellar mass”, $M_{*}^{\rm piv}$, the “pivot halo mass”, $M_{h}^{\rm piv}$, and the “pivot ratio”, $(M_h/M_*)^{\rm
piv}$. Note that $M_{*}^{\rm piv}$ and $M_{h}^{\rm piv}$ are not simply equal to $M_{1}$ and to $M_{*,0}$. Indeed, the mathematical formulation of the SHMR is such that the pivot masses depend on all five parameters. The three parameters that have the strongest effect on the pivot masses are $M_{1}$, $M_{*,0}$, and $\gamma$. For example, as can be seen in the right hand panel of Figure \[mh\_ms\_vary\_p\], $M_{*}^{\rm piv}$ and $M_{h}^{\rm piv}$ are inversely proportional to $\gamma$. To lesser extent, the two remaining parameters, $\beta$ and $\delta$ also have a small influence on the pivot masses.
Scatter between stellar and halo mass {#scatter_shmr}
-------------------------------------
We now turn our attention to the second component of $\Phi_c(M_{*}|M_h)$ which is the scatter in stellar mass at fixed halo mass, $\sigma_{\rm log M_{*}}$. The total measured scatter will have two components: an intrinsic component (noted $\sigma_{\rm log
M_{*}}^{\rm i}$) and a measurement error component due to redshift, photometry, and modeling uncertainties in stellar mass measurements (noted $\sigma_{\rm log M_{*}}^{\rm m}$). It is reasonable to assume that the intrinsic scatter component is independent of the measurement error component. Assuming Gaussian error distributions, we can write that:
$$(\sigma_{\rm log M_{*}})^2=(\sigma_{\rm log M_{*}}^{\rm i})^2+(\sigma_{\rm log M_{*}}^{\rm m})^2.$$
While the error distribution for the stellar mass estimate for any single galaxy may be non Gaussian, in this work we are only concerned with stacked ensembles. B10 have tested that the error distribution for a stacked ensemble is Gaussian to good approximation and that small non-Gaussian wings in this distribution are not likely to affect this type of analysis.
In practice, since the data are always binned according to $M_*$, the observables are actually sensitive to the scatter in halo mass at fixed stellar mass which we note $\sigma_{\rm log M_h}$. It will therefore be useful to understand the link between $\sigma_{\rm log
M_h}$ and $\sigma_{\rm log M_{*}}$. If the SHMR is a power law, the relationship between $\sigma_{\rm log M_h}$ and $\sigma_{\rm log
M_{*}}$ is simply:
$$\sigma_{\rm log M_h}=\sigma_{\rm log M_{*}}\frac{{\rm d}(\log_{10} M_h)}{{\rm d}(\log_{10} M_* )}.
\label{sigmas}$$
For example, if there is a power law relation between halo mass and stellar mass such that $M_h=M_*^\eta$ then $\sigma_{\rm log M_h}=\eta
\times \sigma_{\rm log M_{*}}$. In our case, the SHMR behaves like a power-law at low $M_*$. At high $M_*$, however, ${\rm
d}(\log_{10}M_h)/{\rm d}(\log_{10}M_*)$ increases as a function of $M_*$. Therefore if $\sigma_{\rm log M_{*}}$ is constant, $\sigma_{\rm
log M_h}$ will be equal to $\sigma_{\rm log M_h}=\beta \times
\sigma_{\rm log M_{*}}$ at low $M_*$ but then will continuously increase with $M_*$ at a rate set by $\gamma$ and $\delta$.
If we adopt the best fit model to the SHMR from Paper II in the redshift range $0.22<z<0.48$, we find that the power-law index of the SHMR increases steeply at $\log_{10}(M_*)>11$ so that $\sigma_{\rm log
M_h}$ becomes quite large. For example, $\sigma_{\rm log M_h}\sim
0.46$ dex at $\log_{10}(M_*)=11$ and $\sigma_{\rm log M_h}\sim 0.7$ dex at $\log_{10}(M_*)=11.5$. In practical terms, this implies that the most massive galaxies do not necessarily live in the most massive halos. For example, a galaxy with $M_* \sim 2\times 10^{11}$ M$_{\odot}$ could be the central galaxy of of group with $M_h \sim
10^{13}-10^{14}~{\rm M}_{\odot}$, or could also be the central galaxy of a cluster with $M_h>10^{15}~{\rm M}_{\odot}$. The increase of $\sigma_{\rm log M_h}$ with $M_*$ will lead to a noticeable effect in the g-g lensing, clustering, and stellar mass functions at large $M_*$ that is analogous to Eddington bias. This effect will be discussed in further detail in $\S$ \[parameter\_influance\].
HOD Framework {#hodtheory}
=============
In this section, we show how $\Phi_c(M_{*}|M_h)$ can be used to determine the central halo occupation function and we introduce five new parameters to describe the satellite occupation function.
Halo Occupation Functions {#thehodmodel}
-------------------------
In this paper, we assume that stellar mass is used to implement the HOD model since it is expected to be a more faithful tracer of halo mass than galaxy luminosity.
Consider a galaxy sample such that $M_*>M_{*}^{t_1}$ (a “threshold” sample). The [*central occupation function*]{}, noted $\langle N_{\rm
cen}(M_h|M_{*}^{t_1})\rangle$, is the average number of central galaxies in this sample that are hosted by a halo of mass $M_h$. The [*satellite occupation function*]{}, noted $\langle N_{\rm
sat}(M_h|M_{*}^{t_1})\rangle$, is the equivalent function for satellite galaxies.
In what follows, we focus on the appropriate equations for threshold samples. In Paper II however, we will use “binned” samples ($M_{*}^{t_1}<M_*<M_{*}^{t_2}$) to calculate the g-g lensing and the SMF. We therefore note that the occupation functions for binned samples are trivially derived from the occupation function for threshold samples via:
$$\langle N_{\rm cen}(M_h|M_{*}^{t_1},M_{*}^{t_2}) \rangle= \langle
N_{\rm cen}(M_h|M_{*}^{t_1})\rangle-\langle N_{\rm cen}(M_h|M_{*}^{t_2})\rangle,$$
and
$$\langle N_{\rm sat}(M_h|M_{*}^{t_1},M_{*}^{t_2}) \rangle= \langle
N_{\rm sat}(M_h|M_{*}^{t_1})\rangle-\langle N_{\rm sat}(M_h|M_{*}^{t_2})\rangle.$$
Functional form for ${\langle N_{\rm cen} \rangle}$ {#ncen_model}
---------------------------------------------------
For a threshold sample of galaxies, $\langle N_{\rm
cen}(M_h|M_{*}^{t_1})\rangle$ is fully specified given $\Phi_c(M_{*}|M_h)$ according to:
$$\langle N_{\rm cen}(M_h|M_{*}^{t_1})\rangle = \int_{M_{*}^{t_1}}^{\infty} \Phi_c(M_*|M_h) {\rm d}M_* .
\label{ncen_and_phic}$$
Because the integral of $\Phi_c(M_{*}|M_h)$ over $M_*$ is equal to 1, $\langle N_{\rm cen}(M_h|M_{*}^{t_1})\rangle$ will vary between 0 and 1.
To begin with, let us make the simplifying assumption that $\sigma_{\rm log M_{*}}$ is constant. Because $\Phi_c$ is parametrized as a log-normal distribution, the central occupation function can be analytically derived from Equation \[ncen\_and\_phic\] by considering the cumulative distribution function of the Gaussian:
$$\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle = \hspace{0.65\columnwidth}$$
$$\label{ncen}
\frac{1}{2}\left[ 1-\mbox{erf}\left(\frac{\log_{10}(M_{*}^{t_1}) - \log_{10}(f_{\textsc{shmr}}(M_h)) }{\sqrt{2}\sigma_{\rm log M_{*}}} \right)\right],$$
where erf is the error function defined as:
$$\label{erf_funct}
\mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^{x}e^{-t^2} {\rm d}t.$$
It is important to note that Equation \[ncen\] is only valid when $\sigma_{\rm log M_{*}}$ is constant. In the more general case where $\sigma_{\rm log M_{*}}$ varies, ${\langle N_{\rm cen} \rangle}$ can nonetheless be calculated by numerically integrating Equation \[ncen\_and\_phic\]. In Paper II we will consider cases in which $\sigma_{\rm log M_{*}}$ varies due to the effects of stellar mass dependant measurement errors. In this case, we will numerically integrate Equation \[ncen\_and\_phic\] to calculate ${\langle N_{\rm cen} \rangle}$ (see $\S$ 4.2 in Paper II).
We note that most readers may be more familiar with a simplified version of Equation \[ncen\] that assumes that $f_{\textsc{shmr}}(M_h)$ is a power law. We will now describe the assumptions made in order to obtain the more commonly employed equation for ${\langle N_{\rm cen} \rangle}$ from Equation \[ncen\].
If we make the assumption that $f_{\textsc{shmr}}(M_h) \propto M_h^p$ and we define $M_{\rm min}$ such that $M_{\rm min}\equiv
f_{\textsc{shmr}}^{-1}(M_{*}^{t_1})$ (in other terms, $M_{\rm min}$ is the inverse of the SHMR relation for the stellar mass threshold $M_{*}^{t_1}$) then using Equation \[ncen\] we can write that:
$$\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle = \hspace{0.65\columnwidth}$$
$$\frac{1}{2}\left[ 1-\mbox{erf}\left(\frac{\log_{10}(M_{\rm min}^p) - \log_{10}(M_h^p) }{\sqrt{2}\sigma_{\rm log M_{*}}} \right)\right].$$
If we now use the fact that $\rm{erf}(-x)= -\rm{erf}(x)$ and if we [*define*]{} $\widetilde{\sigma}_{\rm{log}M}$ such that $\widetilde{\sigma}_{\rm{log}M}\equiv \sigma_{\rm log M_{*}}/p$ we can write that:
$$\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle = \hspace{0.65\columnwidth}$$
$$\frac{1}{2}\left[ 1+\mbox{erf}\left(\frac{\log_{10}(M_h) - \log_{10}(M_{\rm min}) }{\sqrt{2}\widetilde{\sigma}_{\rm{log}M}} \right)\right],
\label{ncen2}$$
which is a commonly employed formula for ${\langle N_{\rm cen} \rangle}$. Firstly, it is important to note that Equation \[ncen2\] is only an approximation for ${\langle N_{\rm cen} \rangle}$ for the case when the SHMR is a power-law and is certainly not valid over a large range of stellar masses. Secondly, $\widetilde{\sigma}_{\rm{log}M}$ can be interpreted as the scatter in halo mass at fixed stellar mass if and only if the SHMR is a power-law and if $\sigma_{\rm log M_{*}}$ is constant. Since there is accumulating evidence that the SHMR is not a single power law (and the same is in general true for the relationship between halo mass and galaxy luminosity), we recommend using Equation \[ncen\] instead of Equation \[ncen2\].
Figures \[study\_mmin\] and \[study\_mmin2\] illustrate the difference in ${\langle N_{\rm cen} \rangle}$ when Equation \[ncen\] is used to describe clustering instead of Equation \[ncen2\]. To make this figure, we have assumed the parameter set: $\log_{10}(M_{1})= 12.71$, $\log_{10}(M_{*,0})= 11.04$, $\beta= 0.467$, $\delta= 0.62$, $\gamma=
1.89$, and $\sigma_{\rm log M_{*}}$=0.25. For each stellar mass threshold in Figure \[study\_mmin\], we determine the values of ${M_{\rm min}}$ and $\widetilde{\sigma}_{\rm{log}M}$ in Equation \[ncen2\] such that the number density of central galaxies and the bias of those galaxies is the same as that achieved with Equation \[ncen\]. Thus our procedure mimics what one would obtain through analysis of the clustering and space density of such samples, assuming that the satellite occupation would be the same in either analysis.
Figure \[study\_mmin\] reveals that because the SHMR has a sub-exponential behaviour at $\log_{10}(M_*)\gtrsim 10.5$, ${\langle N_{\rm cen} \rangle}$ begins to deviate from a simple erf function for high stellar mass samples and therefore is not well described by Equation \[ncen2\]. Assuming that Equation \[ncen\] correctly represents ${\langle N_{\rm cen} \rangle}$, the error made on $M_{\rm min}$ can be of order 10% to 40% at $M_h=f_{\textsc{shmr}}^{-1}(M_{*}^{t_1})\gtrsim 10^{12}$ $M_{\odot}$ if Equation \[ncen2\] is used to describe ${\langle N_{\rm cen} \rangle}$ instead of Equation \[ncen\].
We note that this does not invalidate Equation \[ncen2\] as a possible parameterization of the central occupation function. However, interpreting $M_{\rm min}$ in Equation \[ncen2\] as $f_{\textsc{shmr}}^{-1}(M_{*}^{t_1})$ can result in a 10-40% error in the true mean halo mass (with larger errors for $\sigma_{\rm log
M_{*}}>0.25$). Also, the “scatter” ($\widetilde{\sigma}_{\rm{log}M}$) constrained by this parametrization is not equal to the scatter in a log-normal distribution of stellar mass at fixed halo mass. Finally, one troublesome aspect of using the erf functional form in Equation \[ncen2\] is that ${\langle N_{\rm cen} \rangle}$ curves for different stellar mass thresholds may actually cross at low halo mass, implying the unphysical condition that halos of mass $M_h$ have a “negative” amount of galaxies between two threshold values. This is seen at $M_h\sim 10^{11.5}$ M$_{\odot}$ in Figure \[study\_mmin\]. For values of $\sigma_{\rm log M_{*}}$ larger than what we have assumed here, this effect will occur at even higher halo mass. Using Equation \[ncen\] with a model for the SHMR prevents this from occurring, and ${\langle N_{\rm cen} \rangle}$ for various galaxy samples can be calculated self-consistently.
Functional form for ${\langle N_{\rm sat} \rangle}$ {#model_nsat}
----------------------------------------------------
In addition to the five parameters introduced to model ${\langle N_{\rm cen} \rangle}$ and $\sigma_{\rm log M_{*}}$, we introduce five new parameters to model ${\langle N_{\rm sat} \rangle}$. In order to simultaneously fit g-g lensing, clustering, and stellar mass function measurements that employ different binning schemes, we require a model for ${\langle N_{\rm sat} \rangle}$ that is independent of any given binning scheme. For this reason, we parameterize the satellite function with threshold samples. The number of satellite galaxies in a bin of stellar mass is determined by a simple difference of two threshold samples. This also eliminates the need to integrate over stellar mass, as required when working explicitly through the conditional stellar mass function.
Numerical simulations demonstrate that the occupation of sub-halos (e.g., @Kravtsov:2004 [@Conroy:2006]) and satellite galaxies in cosmological hydrodynamic simulations (@Zheng:2005) follow a power law at high host halo mass, then fall off rapidly when the mean occupation becomes significantly less than unity. Thus we parametrize the satellite occupation function as a power of host mass with an exponential cutoff and scaled to ${\langle N_{\rm cen} \rangle}$ as follows:
$$\langle N_{\rm sat}(M_h|M_{*}^{t_1}) \rangle = \hspace{0.65\columnwidth}$$
$$\label{e.nsat}
\langle N_{\rm cen}(M_h|M_{*}^{t_1}) \rangle \left(\frac{M_h}{M_{\rm sat}}\right)^{{\alpha_{\rm sat}}} \exp\left(\frac{-M_{\rm cut}}{M_h}\right),$$
where $\alpha_{\rm sat}$ represents the power-law slope of the satellite mean occupation function, $M_{\rm sat}$ defines the amplitude of the power-law, and $M_{\rm cut}$ sets the scale of the exponential cut-off. Here, $M_h$ refers to the host halo mass of satellite galaxies.
Observational analyses have demonstrated that there is a self-similarity in occupation functions such that $M_{\rm sat}/M_{\rm
min}\approx constant$ for luminosity-defined samples (@Zehavi:2005 [@Zheng:2007; @Tinker:2007; @Zheng:2009; @Zehavi:2010; @Abbas:2010]), where $M_{\rm min}$ is taken from Equation \[ncen2\] and is conceptually similar to $f_{\textsc{shmr}}^{-1}(M_{*}^{t_1})$ (modulo a $10-40\%$ difference as shown in Figure \[study\_mmin2\]) and where $M_{*}^{t_1}$ is the stellar mass threshold of the sample. Instead of simply modelling $M_{\rm sat}$ and $M_{\rm cut}$ as constant factors of $\fshmr$, we add flexibility to our model by enabling $M_{\rm sat}$ and $M_{\rm
cut}$ to vary as power law functions of $\fshmr$:
$$\frac{M_{\rm sat}}{10^{12} M_{\odot}}= B_{\rm sat} \left(\frac{\fshmr}{10^{12} M_{\odot}}\right)^{\beta_{\rm sat}},$$
and
$$\label{mcut_eq}
\frac{M_{\rm cut}}{10^{12} M_{\odot}}= B_{\rm cut} \left(\frac{\fshmr}{10^{12} M_{\odot}}\right)^{\beta_{\rm cut}}.$$
@Zheng:2007 find that $M_{\rm sat}/M_{\rm min} \sim 18$ for SDSS and $M_{\rm sat}/M_{\rm min} \sim 16$ using luminosity defined samples in DEEP2. For $B_{\rm cut}$, the expectation is that the cutoff mass scale occurs at $M_{\rm min}<M_{\rm cut}<M_{\rm sat}$, although it can be significantly smaller.
Total stellar mass as a function of halo mass
---------------------------------------------
Using this model, one can also compute the total amount of stellar mass in galaxies (summing the contribution from both centrals and satellites) as a function of halo mass $M_h$. To begin with, let us consider the total stellar mass as a function of halo mass in some stellar mass bin: $M_{*}^{\rm tot}(M_h|M_*^{t1},M_*^{t2})$. The expression for $M_{*}^{\rm tot}(M_h|M_*^{t1},M_*^{t2})$ is given by:
$$M_{*}^{\rm tot}(M_h|M_*^{t1},M_*^{t2}) = \hspace{0.65\columnwidth}$$
$$\label{tot_msat_1}
\int_{M_*^{t1}}^{M_*^{t2}}\left[\Phi_c(M_*|M_h)+\Phi_s(M_*|M_h)\right]M_*{\rm d}M_* .$$
However, in the previous section we have only specified the analytic form for $\Phi_c$ (Equation \[lognormal\_equation\]) but not for $\Phi_s$. Indeed, in $\S$ \[model\_nsat\] we outlined an analytic model for ${\langle N_{\rm sat} \rangle}$ but calculating the analytic derivative of ${\langle N_{\rm sat} \rangle}$ would be tedious. Thankfully, however, we do not specifically need to know the functional form of $\Phi_s$ in order to calculate Equation \[tot\_msat\_1\]. By using the integration by parts rule, we can re-write Equation \[tot\_msat\_1\] in a more convenient form as follows:
$$M_{*}^{\rm tot}(M_h|M_*^{t1},M_*^{t2}) = \hspace{0.65\columnwidth}$$
$$\begin{aligned}
&& \int_{M_*^{t1}}^{M_*^{t2}} \langle N_{\rm cen}(M_h|M_{*}) \rangle {\rm d}M_* - \left[ \langle N_{\rm cen}(M_h|M_{*}) \rangle M_*\right]_{M_*^{t1}}^{M_*^{t2}} \nonumber\\
&+&\int_{M_*^{t1}}^{M_*^{t2}} \langle N_{\rm sat}(M_h|M_{*}) \rangle
{\rm d}M_* - \left[ \langle N_{\rm sat}(M_h|M_{*}) \rangle
M_*\right]_{M_*^{t1}}^{M_*^{t2}}. \nonumber\\
&&
\label{tot_msat_2}\end{aligned}$$
This equation provides us with a convenient way to calculate the total stellar mass locked up in galaxies with $M_*^{t1}<M_*<M_*^{t2}$ as a function of halo mass $M_h$.
Summary of model parameters
---------------------------
In total, we have introduced six parameters to model the central occupation function ($M_{1}$, $M_{*,0}$, $\beta$, $\delta$, $\gamma$, $\sigma_{\rm log M_{*}}$) and five parameters to model the satellite occupation function ($\beta_{\rm sat}$, $B_{\rm sat}$, $\beta_{\rm
cut}$, $B_{\rm cut}$, $\alpha$). In addition, one could introduce a model for $\sigma_{\rm log M_{*}}$ or assume that the scatter is constant in which case there would be a total of eleven parameters for this model. In Figure 8 of Paper II, we show the two dimensional marginalized distributions for this parameter set using data from the COSMOS survey. The model described in this paper provides an excellent fit to COSMOS data. Figure 8 in Paper II demonstrates that this model is reasonably free of parameter degeneracies. A summary and description of these parameters can be found in Table \[free\_params\]. Figure \[hod\_bins\_illustration\] gives an illustration of the central and satellite occupation functions for galaxy samples in bins and thresholds of stellar mass.
[llll]{}\
\[-1.5ex\] Parameter & Unit & Description & ${\langle N_{\rm cen} \rangle}$ or ${\langle N_{\rm sat} \rangle}$\
\[1ex\]\
$M_{1}$ & $M_{\sun}$ & Characteristic halo mass in the SHMR &${\langle N_{\rm cen} \rangle}$\
$M_{*,0}$ & $M_{\sun}$ & Characteristic stellar mass in the SHMR &${\langle N_{\rm cen} \rangle}$\
$\beta$ & none & Faint end slope in the SHMR &${\langle N_{\rm cen} \rangle}$\
$\delta$ & none & Controls massive end slope in the SHMR &${\langle N_{\rm cen} \rangle}$\
$\gamma$ & none & Controls the transition regime in the SHMR &${\langle N_{\rm cen} \rangle}$\
$\sigma_{\rm log M_{*}}$ & dex & Log-normal scatter in stellar mass at fixed halo mass&${\langle N_{\rm cen} \rangle}$\
$\beta_{\rm sat}$ & none & Slope of the scaling of $M_{\rm sat}$ &${\langle N_{\rm sat} \rangle}$\
$B_{\rm sat}$ & none & Normalization of the scaling of $M_{\rm sat}$ &${\langle N_{\rm sat} \rangle}$\
$\beta_{\rm cut}$ & none & Slope of the scaling of $M_{\rm cut}$ &${\langle N_{\rm sat} \rangle}$\
$B_{\rm cut}$ & none & Normalization of the scaling of $M_{\rm cut}$ &${\langle N_{\rm sat} \rangle}$\
$\alpha_{\rm sat}$ & none & Power-law slope of the satellite occupation function&${\langle N_{\rm sat} \rangle}$\
How to derive the SMF, g-g lensing, and clustering from the model {#application_of_model}
=================================================================
We now describe how the model outlined in the previous section yields analytic descriptions for the g-g lensing, clustering, and stellar mass function which can then be fit simultaneously to observations.
Analytical model for the stellar mass function
----------------------------------------------
The stellar mass function is typically calculated in bins in stellar mass. Let us consider the stellar mass bin $M_*^{t_1}<M_*<M_*^{t_2}$. The abundance of galaxies within this stellar mass bin, $\Phi_{SMF}(M_*^{t_1},M_*^{t_2})$, is simply obtained from our model and the halo mass function, ${\rm
d}n/{\rm{d}}M_h$, according to:
$$\Phi_{SMF}(M_*^{t_1},M_*^{t_2}) \hspace{0.65\columnwidth}$$
$$\begin{aligned}
&=&\int_{0}^{\infty} \left[\int_{M_*^{t_1}}^{M_*^{t_2}}
\Phi(M_{*}|M_h) {\rm{d}}M_* \right] \frac{{\rm{d}}n}{{\rm{d}}M_h}{\rm{d}}M_h\nonumber\\
&=&\int_0^{\infty} \ntotb
\frac{{\rm{d}}n}{{\rm{d}}M_h}{\rm{d}}M_h,\end{aligned}$$
where we recall that $\Phi(M_{*}|M_h)$ represents the conditional stellar mass function and $\langle N_{\rm tot}\rangle$ is the total occupation function (including both satellites and centrals).
The Lensing Observable, $\Delta\Sigma$
--------------------------------------
The shear signal induced by a given foreground mass distribution on a background source galaxy will depend on the transverse proper distance between the lens and the source and on the redshift configuration of the lens-source system. A lens with a projected surface mass density, $\Sigma(r)$, will create a shear that is proportional to the [ *surface mass density contrast*]{}, $\Delta\Sigma(r)$:
$$\Delta \Sigma(r)\equiv\overline{\Sigma}(< r)-\overline{\Sigma}(r)=\Sigma_{\rm crit}\times\gamma_t(r).
\label{dsigma}$$
Here, $\overline{\Sigma}(< r)$ is the mean surface density within proper radius $r$, $\overline{\Sigma}(r)$ is the azimuthally averaged surface density at radius $r$ [e.g., @Miralda-Escude:1991; @Wilson:2001], and $\gamma_t$ is the tangentially projected shear. The geometry of the lens-source system intervenes through the [*critical surface mass density*]{}, $\Sigma_\mathrm{crit}$, which depends on the angular diameter distances to the lens ($D_{\rm OL}$), to the source ($D_{\rm OS}$), and between the lens and source ($D_{\rm LS}$):
$$\Sigma_\mathrm{crit} = \frac{c^2}{4\pi G_{{{\rm N}}}}\,
\frac{D_\mathrm{OS}}{D_\mathrm{OL}\,D_\mathrm{LS}}\;,
\label{sigma_crit}$$
where $G_{\rm N}$ represents Newton’s constant.
Relationship between $\Delta\Sigma$, the density field, and correlation functions {#ds_correlation_funct}
---------------------------------------------------------------------------------
Consider two different populations characterised respectively by $\delta_a$ and $\delta_b$. The [*two-point cross-correlation function*]{} of $\delta_a$ and $\delta_b$ at comoving position $\vec
r_{\rm co}$ is given by:
$$\xi_{ab}(\vec r_{\rm co}) \equiv \langle \delta_a(\vec r_{\rm
co})\delta_{b}(\vec x_{\rm co}+\vec r_{\rm co}) \rangle .$$
For example, if $\delta_{\rm g}$ and $\delta_{\rm dm}$ are respectively the over-densities of galaxies and dark matter, then we can characterize their relative distributions via the [*galaxy-mass cross-correlation function*]{} which is noted $\xi_{\rm gm}$ and is equal to:
$$\xi_{\rm gm}(\vec{r_{\rm co}}) = \langle \delta_{\rm g} (\vec{r_{\rm co}})
\delta_{\rm dm}(\vec{x_{\rm co}}+\vec{r_{\rm co}})\rangle .$$
Similarly, $\xi_{\rm gg}$ refers to the [*galaxy auto-correlation function*]{}.
In the following, $\vec r_{\rm co}$ is the three dimensional comoving distance, $\vec r_{||,{\rm co}}$ is the projected comoving line-of sight distance, and $\vec r_{p,{\rm co}}$ is the projected comoving transverse distance:
$$r_{\rm co}=\sqrt{r_{p,{\rm co}}^2+r_{||,{\rm co}}^2}.
\label{r_p_co}$$
In Paper II, we will employ physical coordinates for g-g lensing measurements whereas for clustering we will use comoving coordinates. In previous work, @Mandelbaum:2006c have used comoving coordinates and @Johnston:2007 have used physical coordinates. Therefore, our g-g lensing formulas more closely resemble those of @Johnston:2007. The relationship between comoving and physical distances is simply $r_{\rm co}=r_{\rm ph} (1+z)$. In a similar fashion to Equation \[r\_p\_co\], we can write that:
$$r_{\rm ph}=\sqrt{r_{p,{\rm ph}}^2+r_{||,{\rm ph}}^2}.
\label{r_p_ph}$$
In comoving coordinates, the density field is $\rho(r_{\rm
co},z)=\overline \rho(1+\xi_{\rm gm}(r_{\rm co},z))$ where $\overline \rho=\rho_{c,0} \Omega_{m,0}$ is the average density of matter in the Universe. Since $\xi_{\rm gm}$ is often expressed in comoving coordinates, we first derive $\Sigma$ in comoving units (noted $\Sigma_{\rm co}$) and then we transform $\Sigma$ into physical units (noted $\Sigma_{\rm ph}$) before computing $\Delta\Sigma$. For a lens at redshift $z_L$, the projected surface mass density, $\Sigma$, is obtained by integrating the 3d density over the line-of-sight:
$$\Sigma_{\rm co}(r_{\rm p,co},z_L) \hspace{0.65\columnwidth}$$
$$\begin{aligned}
&=& \int \rho\left(\sqrt{r_{\rm p,co}^2+r_{||,\rm co}^2},z_L\right){\rm d}r_{\rm ||,co} \nonumber\\
&=& \rho_{c,0} \Omega_{m,0} \int \left[1+\xi_{\rm
gm}\left(\sqrt{r_{\rm p,co}^2+r_{||,\rm
co}^2},z_L\right)\right]{\rm d}r_{\rm ||,co}, \nonumber\\
&&
\label{sig_and_rho}\end{aligned}$$
where $r_{\rm p,co}$ and $r_{\rm ||,co}$ refer respectively to the comoving transverse and line-of-sight distance from the lens. In principle, this integral should extend from the redshift of the observer ($z_O$) to the redshift of the source ($z_S$). However, [$ \xi_{\rm gm} $]{} falls off rapidly enough that in practice, the redshift evolution of [$ \xi_{\rm gm} $]{} can be neglected and integrating only out to a distance of $r_{\rm ||,co}=50$ Mpc is sufficient. Furthermore, for the purpose of computing $\Delta\Sigma$, the constant term in Equation \[sig\_and\_rho\] can be dropped (due to the subtraction in $\Delta\Sigma$) and the mean excess projected density $\overline\Sigma(r)$ can be approximated by the radial integral:
$$\overline\Sigma_{\rm co}(r_{\rm p,co},z_L) = \hspace{0.65\columnwidth}$$
$$2 \rho_{c,0} \Omega_{m,0} \int_0^{50 {\rm Mpc}}\xi_{\rm gm}\left(\sqrt{r_{\rm p,co}^2+r_{||,\rm co}^2},z_L\right){\rm d}r_{\rm ||,co}.
\label{dsigma1_bis}$$
The mean excess projected density is physical units is then $\overline\Sigma_{\rm ph} =\overline\Sigma_{\rm co}\times(1+z)^2$.
The average $\overline\Sigma$ within radius $r$ is equal to:
$$\overline\Sigma_{\rm ph}(<r_{\rm p,ph}) = \frac{2}{r_{\rm p,ph}^2}
\int_0^{r_{\rm p,ph}}\overline \Sigma_{\rm ph}(r^\prime)r^\prime {\rm
d}r^\prime .
\label{dsigma2}$$
Finally, $\Delta\Sigma$ is obtained by combining Equations \[dsigma\], \[dsigma1\_bis\] and \[dsigma2\].
Analytical modeling of $\xi_{gg}$ and $w(\theta)$ {#model_for_xi_gg}
-------------------------------------------------
Our model for calculating the autocorrelation function of galaxies is based on the model given in [@Tinker:2005] [also see @Zheng:2004]. As described above, the HOD is broken into central and satellite galaxy occupation functions. Thus, in the HOD context, pairs of galaxies come from two distinct terms; pairs within a single halo and pairs between galaxies in two different halos. The total correlation function is
$$\xi_{gg}(r_{\rm co}) + 1 = \left[\xi^{1h}_{gg}(r_{\rm co})+1\right] + \left[\xi^{2h}_{gg}(r_{\rm co})+1\right],$$
where $1h$ and $2h$ refers to “one-halo” and “two-halo” terms, respectively. The one-halo correlation function is written as
$$1+\xi_{gg}^{1h}(r_{\rm co})= \hspace{0.65\columnwidth}$$
$$\quad \frac{1}{2\pi r_{\rm co}^2\overline{n}_g^2}
\int dM_h \frac{dn}{dM_h}\frac{\langle N(N-1)\rangle_M}{2}
\frac{1}{2R_{h}} F^\prime\left(\frac{r_{\rm co}}{2R_{h}}\right),$$
where $\bar{n}_g$ is the space density of galaxies in the sample being modeled, $\langle N(N-1)\rangle_M$ is the second moment of the distribution of galaxies within halos as a function of halo mass, and $F^\prime(x)$ is the radial distribution of pairs within the halo normalized to unity. Within a halo, pairs of galaxies can be between the central galaxy and a satellite, or between two satellite galaxies. The radial pair profile is different for these two combinations, thus we express their relative contributions to $\xi_{gg}^{1h}(r_{\rm co})$ as
$$\begin{aligned}
\frac{\langle N(N-1)\rangle_M}{2}F^\prime(x) & = & \langle N_{\rm
cen}N_{\rm sat} \rangle_MF^\prime_{\rm cs}(x) \nonumber \\ & &
+ \frac{\langle N_{\rm sat}(N_{\rm sat}-1)\rangle_M}{2}F^\prime_{\rm ss}(x),\nonumber \\
&&\end{aligned}$$
where $F^\prime_{cs}$ is the pair distribution for central-satellite pairs and $F^\prime_{ss}$ is the equivalent for satellite-satellite pairs. The former is related to the density profile of satellite galaxies, and the latter is related to the density profile convolved with itself (analytic expressions for this convolution can be found in @Sheth:2001a). Here we assume that the radial distribution of satellite galaxies is the same as the dark matter, for which we assume the profile form of Navarro-Frenk-White [NFW; @Navarro:1997] using the mass-concentration relation of @Munoz-Cuartas:2010. Since we assume that satellites trace the dark matter, $F^\prime_{cs}$ will be equal to the quantity $F^\prime_{c}$ that we will introduce in Equation \[eq:nf\] and $F^\prime_{ss}$ will be equal to $F^\prime_{s}$.
Central galaxies only exist as one or zero objects in a halo, thus they have no second moment. For the second moment of satellite galaxies, we assume Poisson statistics about $\langle N_{\rm
sat}\rangle$, which is in good agreement with results from numerical simulations (@Kravtsov:2004 [@Zheng:2005]). Possible deviations from Poisson behavior [@Busha:2010; @Boylan-Kolchin:2010] mainly affect clustering for galaxy samples where a majority of the satellites originate in halos with $M_h<M_{\rm sat}$ because in this case, ${\langle N_{\rm sat} \rangle}$ drops to ${\langle N_{\rm sat} \rangle}\lesssim 1$. Luminous Red Galaxies (LRGs) fall into this category for example. Indeed, satellite galaxies in LRG samples mainly originate in $M_h<M_{\rm sat}$ halos because $M_{\rm
sat}$ is close to the exponential cut-off in the halo mass function (for LRGs, $M_{\rm sat} \sim 4\times10^{14}$ M$_{\odot}$). However, for the types of samples that we consider in Paper II, deviations from Poisson statistics should not significantly effect the clustering predictions of the HOD.
A detailed description of the two-halo term can be found in [@Tinker:2005]. Briefly, in the regime where $r>R_{h}$ of massive halos, this term can be expressed as
$$\label{e.2halo_approx}
\xi_{gg}^{2h}(r_{\rm co}) = b_g^2\zeta^2(r_{\rm co})\xi_m(r_{\rm co}),$$
where $\xi_m(r_{\rm co})$ is the non-linear matter correlation function and $b_g$ is the large-scale bias of galaxies in the sample, and $\zeta(r_{\rm co})$ is the scale dependence of dark matter halo bias. For $\xi_m(r_{\rm co})$, we use the fitting function of [@Smith:2003]. For $\zeta(r_{\rm co})$, we use the fitting function of [@Tinker:2005]. The galaxy bias is computed from the HOD by
$$b_g = \bar{n}_g^{-1}\int b(M_h)\langle N\rangle_M\frac{dn}{dM_h}dM_h,$$
where $dn/dM_h$ is the halo mass function, for which we use @Tinker:2008, and $b(M_h)$ is the halo bias function, for which we use [@Tinker:2010].
In the regime where $r< R_{h}$, Equation \[e.2halo\_approx\] breaks down due to the effects of halo exclusion; i.e., the effect that the center of one halo cannot exist within the virial radius of another halo (and still be considered a “two-halo” pair). This is explained in detail in [@Tinker:2005]. Because the mass function and bias relation used in this analysis are taken from numerical results based on spherical-overdensity (SO) halo catalogs (@Tinker:2008 [@Tinker:2010]), the halo exclusion must be modified to match this halo definition. In the SO halo finding algorithm of [@Tinker:2008], halos are allowed to overlap so long as the center of one halo is not contained within the radius of another halo. Thus, the minimum separation of two halos with radii $R_1\ge R_2$ is $R_1$, rather than the sum of the two radii, as done in [@Tinker:2005]. For a projected statistic like $w(\theta)$, this makes only a small difference in clustering at the 1-halo 2-halo transition, but significantly speeds up computation of the 2-halo term.
Once we have calculated $\xi_{gg}(r_{\rm co})$ for a given HOD model, we compute the observable $w(\theta)$ by:
$$\label{e.wth}
w(\theta) = \int dz\,N^2(z)\, \frac{dr_{\rm co}}{dz} \int dx\,
\xi\left(\sqrt{x^2 + r_{\rm co}^2\theta^2}\right),$$
where $N(z)$ is the normalized redshift distribution of the galaxy sample, $r_{\rm co}$ is the comoving radial coordinate at redshift $z$ and $dr_{\rm co}/dz = (c/H_0)/\sqrt{\Omega_{\rm
m}(1+z)^3+\Omega_\Lambda}$.
Figure \[illustration\_wtheta1\] shows the breakdown of the angular correlation function into the one-halo and two-halo terms for low-mass and high-mass galaxy samples.
Analytical modeling of $\xi_{gm}$ and $\Delta\Sigma$ {#model_for_xi_gm}
----------------------------------------------------
We have shown in $\S$ \[ds\_correlation\_funct\] that the lensing observable, $\Delta\Sigma$, can be obtained from $\xi_{gm}$ by performing two integrals (Equations \[dsigma1\_bis\] and \[dsigma2\]). Following the approach of @Yoo:2006, $\xi_{gm}$ is computed from the HOD and $\Delta\Sigma$ is obtained by combining Equations \[dsigma\], \[dsigma1\_bis\] and \[dsigma2\]. Thus $\Delta\Sigma$ is fully specified given our model. Note that the calculation of $\xi_{gm}$ is performed in comoving units and then the projections of Equations \[dsigma1\_bis\] and \[dsigma2\] to obtain $\Delta\Sigma$ are performed in physical units (to match our measured g-g lensing signal which is computed in physical units in Paper II).
Typically, $\xi_{gm}$ is decomposed into a one-halo and a two-halo term
$$1+\xi_{gm}(r_{\rm co}) = [1+\xi_{gm}^{1h}(r_{\rm co})]+[1+\xi_{gm}^{2h}(r_{\rm co})],$$
where the one-halo term represents galaxy-matter pairs from single halos and is dominant on small scales ($\lesssim$Mpc) and the two-halo term corresponds to pairs from distinct halos and is dominant on larger scales ($\gtrsim$Mpc). The one-halo term is obtained according to:
$$1+\xi_{gm}^{1h}(r_{\rm co}) = \hspace{0.65\columnwidth}$$
$$\label{eq:1h}
\quad \frac{1}{4\pi r_{\rm co}^2 \overline{n}_g}\int_{0}^{\infty}\frac{{\rm
d}n}{{\rm
d}M_h}\frac{M_h}{\overline{\rho}}\frac{1}{2R_{h}}\langle
N_{tot}\rangle F'\left(\frac{r_{\rm co}}{2R_{h}}\right){\rm d}M_h .$$
Further details about the origin of Equation \[eq:1h\] are presented in the Appendix.
The product $\overline{n}_g F'$ is split into two terms:
$$\overline{n}_g F'(x)=\overline{n}_c F'_{c}(x)+ \overline{n}_s F'_{s}(x),
\label{eq:nf}$$
where $F'_{c}$ is linked to the density profiles of dark matter halos (see Appendix) and $F'_{s}$ is related to the convolution of the dark matter density profile and the satellite galaxy distribution. To calculate $F'_{c}$ we assume spherical NFW profiles truncated at $R_{h}$ and we adopt the @Munoz-Cuartas:2010 mass-concentration relation for a WMAP5 cosmology. To calculate $F'_{s}$ we assume that the satellite galaxy distribution follows the dark matter distribution. Given this assumption, $F'_{s}$ is simply related to the convolution of the NFW profile with itself.
In this paper, we neglect the contribution to $\Delta\Sigma$ from sub-halos; @Yoo:2006 have shown this component to be negligible at the 10% level.
We calculate the two-halo term as described in [@Yoo:2006], with two major exceptions. First, as stated in the previous section, we are using the halo exclusion from the SO halo definition. Second, because the halo mass function of [@Tinker:2008] and halo bias function of [@Tinker:2010] are normalized such that integrating over all $M_h$ produces the mean matter density and a bias of unity, there is no need to employ the “break mass” from @Yoo:2006 (see their Equation 16). In the limit where $r>R_{h}$ of massive halos,
$$\xi_{gm}^{2h}(r_{\rm co}) = b_g\zeta(r_{\rm co})\xi_m(r_{\rm co}),$$
analogous to Equation \[e.2halo\_approx\].
In addition to the two terms presented above, we add another component to the modeling of $\Delta\Sigma$ which is absent in @Yoo:2006, namely the contribution to $\Delta\Sigma$ from the baryons of the central galaxy which can be non negligible on very small scales ($<$ 50 kpc). Although the baryons typically follow Sérsic profiles [@Sersic:1963], at the scales of interest for this study, well above a few effective radii ($>$ 20 kpc), the lensing contribution of the baryons can be modeled by a simple point-source, scaled to $\langle M_{*}\rangle$, the average stellar mass of the galaxies in the sample:
$$\Delta\Sigma_{\rm stellar}(r)=\frac{\langle M_{*}\rangle }{\pi r^2}.$$
In total, the final g-g lensing signal is modelled as the sum of three terms: $\Delta\Sigma_{\rm tot}=\Delta\Sigma_{\rm
stellar}+\Delta\Sigma_{1h}+\Delta\Sigma_{2h}$. Note that the one halo and two halo terms can also be decomposed into central and satellite contributions but for simplicity, we have grouped these terms together.
In order to illustrate the various terms that contribute to the g-g lensing signal, we have plotted the signal in Figure \[illustration\_gg\_lensing\].
Influence of the model parameters on the observables {#parameter_influance}
====================================================
In the previous section we have outlined how our model can be used to analytically predict the SMF, g-g lensing, and clustering signals. We will now investigate how each parameter in the model affects the three observables. For this exercise, we adopt the best fit model parameters for $0.48<z<0.74$ from Paper II and we vary each parameter in turn by $2\sigma$ around the best fit model. For this section, we assume that $\sigma_{\rm log M_{*}}$ is constant. We also assume that $\alpha_{\rm
sat}$ is constant and we set $\alpha_{\rm sat}=1$ in this section since this is also the assumption that we make in Paper II. In total, we therefore study the effects of ten parameters: $M_{1}$, $M_{*,0}$, $\beta$, $\delta$, $\gamma$, $\sigma_{\rm log M_{*}}$, $\beta_{\rm
sat}$, $B_{\rm sat}$, $\beta_{\rm cut}$, and $B_{\rm cut}$. The results are shown in Figures \[p\_variation\_massive\] and \[p\_variation\_low\] and are described in further detail below.
Effect of parameters on the SMF
-------------------------------
The influence of each model parameter on the observed SMF is shown in the right hand column in Figure \[p\_variation\_massive\] (the same column is reproduced in Figure \[p\_variation\_low\]). The data point with an error bar represents the typical error bar for a COSMOS-like survey where the error bar includes sample variance computed from a series of mock catalogs (described in the following section). The first point worth mentioning here is that the errors on the SMF are relatively small compared to the clustering and the lensing. It is always the case that a measurement of a one-point statistic from a given set of data is more precise than a measurement of a two-point (or higher) statistic. The implies that the SMF will in general play an important role in constraining the parameters of the SHMR. However, in follow-up work we will investigate the sensitivity of this probe combination to cosmological parameters and models of modified gravity for example. In these studies, the clustering and the g-g lensing will play a critical role despite their typically larger errors bars.
A second noteworthy point in Figure \[p\_variation\_massive\] is that the SMF appears to be sensitive to all ten parameters whereas certain parameters such as $M_{*,0}$ and $\beta$ have very little effect on the clustering and lensing signals. Coupled with the fact that the SMF has fairly small error bars, this implies that the SMF will have quite a lot of constraining power on the overall model compared to the g-g lensing for example.
The effects of $M_{1}$ and $M_{*,0}$ on the SMF are fairly intuitive. $M_{1}$ roughly induces an up/down shift in the amplitude of the SMF and $M_{*,0}$ corresponds to a left/right shift in the SMF. A larger value of $B_{\rm sat}$ implies that there are fewer satellite galaxies in high mass halos for a given stellar mass threshold. Thus we see that an increase in $B_{\rm sat}$ corresponds to a decrease in the amplitude of the SMF due to the fact that the contribution from satellite galaxies has decreased. Similar arguments apply to $B_{\rm cut}$. From Figure \[p\_variation\_massive\] we can anticipate that if the SMF were only used to constrain this model, degeneracies would occur between $B_{\rm sat}$, $B_{\rm cut}$, and $M_{1}$. Fortunately, the satellite parameters have a significant effect on both the clustering and the g-g lensing so this degeneracy should be broken when all three probes are used in conjunction.
In Figure \[smf\_dip\_feature\] we highlight the effects of four particular parameters on the SMF: $\beta$, $\gamma$, $\delta$, and $\sigma_{\rm log M_{*}}$. The $\beta$ parameter affects the low-mass slope of the SMF so that a larger value of $\beta$ corresponds to a steeper low-mass slope in the SMF. The $\gamma$ parameter (we recall that this controls the transition region of the SHMR as can be seen from Figure \[mh\_ms\_vary\_p\]) has an interesting effect since it regulates a “plateau” feature in the SMF at $\log_{10}(M_*)\sim
10.5$. In fact, this feature in the SMF has been noticed already and discussed in detail for example in @Drory:2009. In @Drory:2009, this feature was described as a “dip” because at these scales, the SMF is below the best fit Schecter function. However, we note that “dip” is a somewhat misleading name for this feature since it could also be taken to mean that ${\rm
d}N/{\rm d}\log_{10}(M_*)$ does not decrease monotonically with $M_*$. A close inspection of our SMFs in Paper II shows that there is no evidence in the data for an actual “dip” in ${\rm d}N/{\rm
d}\log_{10}(M_*)$. Instead, the data are more consistent with a flattening of ${\rm d}N/{\rm d}\log_{10}(M_*)$ around $\log_{10}(M_*)\sim 10.5$. Therefore, we would like to suggest that this feature should be described as a “plateau” in the SMF rather than a “dip”.
In Figure \[smf\_dip\_feature2\] we show the link between the dark matter halo mass function, the SHMR, and the SMF. In this Figure, we have used the fact that ${\rm d}N/{\rm d}\log_{10}M_* = {\rm d}N/{\rm
d}\log_{10}M_h\times({\rm d}\log_{10}M_h/{\rm d}\log_{10}M_*)$ so that the various functions can be linked “by eye” by drawing a box between the four different panels. We have illustrated how to link the various functions with the dash-dash lines in Figure \[smf\_dip\_feature2\] at the scale of the pivot stellar mass. Figure \[smf\_dip\_feature2\] shows that the “plateau” feature is caused by the transition that occurs in the SHMR at $M_h\sim 10^{12}$ M$_{\odot}$ from a low-mass power-law regime to a sub-exponential function at higher stellar mass.
Finally, the scatter in stellar mass at fixed halo mass has a noticeable effect on the SMF at the high mass end which is also commonly referred to as Eddington bias. A larger value of $\sigma_{\rm
log M_{*}}$ will lead to an inflated observed SMF at large stellar masses.
Effect of parameters on g-g lensing
-----------------------------------
The g-g lensing signal is dominated by the central one-halo term roughly on scales below $0.3$ Mpc and by the satellite one-halo term roughly on scales above $0.3$ Mpc and below a few Mpc (see Figure \[illustration\_gg\_lensing\]). Thus, the effects of the four parameters that regulate the satellite occupation function ($B_{\rm
sat}$, $B_{\rm cut}$, $\beta_{\rm sat}$, $\beta_{\rm cut}$) have a strong scale-dependant effect on the g-g lensing signal. For example, $B_{\rm sat}$ controls the power-law amplitude of ${\langle N_{\rm sat} \rangle}$. A smaller value of $B_{\rm sat}$ will reduce the ratio $M_{\rm sat}/\fshmr$ and will therefore increase the number of satellites in a sample. This will lead to an increase in the g-g lensing signal from 0.3-1 Mpc due to the increased amplitude of the one-halo satellite term. Similarly, a smaller value of $\beta_{\rm sat}$ will also reduce the ratio $M_{\rm sat}/\fshmr$ and by consequence, will increase the one-halo satellite term. This effect is more pronounced in Figure \[p\_variation\_low\] which illustrates a low-stellar mass sample compared to Figure \[p\_variation\_massive\] which illustrates a high stellar mass sample.
Another parameter worth discussing here is $\sigma_{\rm log
M_{*}}$. Figure \[p\_variation\_massive\] demonstrates that $\sigma_{\rm log M_{*}}$ has a stronger effect on the lensing signal for high stellar mass samples compared to low stellar mass samples. As discussed in $\S$ \[scatter\_shmr\], this is simply due to the fact that the data are binned according to $M_*$. The observables are therefore sensitive to the scatter in halo mass at fixed stellar mass, $\sigma_{\rm log M_h}$. At fixed $\sigma_{\rm log M_{*}}$, $\sigma_{\rm log M_h}$ will increase with $M_*$. As a result, the effects of scatter are more prominent in g-g lensing measurements for high stellar mass samples.
Effect of parameters on clustering
----------------------------------
To first order, the stellar mass function and the clustering of galaxies are tethered; more massive halos are both more clustered and less abundant. This is also true of galaxies because rare, massive galaxies live in such halos. If the amplitude of the stellar mass function increases, the clustering as a function of stellar mass decreases. This is especially true at masses above the knee in the stellar mass function. From Figure \[smf\_dip\_feature\], increasing $\delta$ or $\sigma_{\rm log M_{*}}$ increases the abundance of high-mass galaxies. Given that the number of halos is fixed, this can only mean that massive galaxies are occupying less massive, less clustered halos.
There are several parameters that have a direct influence on the clustering of galaxies without changing the stellar mass function appreciably. The parameters $B_{\rm sat}$ and $\beta_{\rm sat}$ are the most important in this regard. They control the “shoulder” in the HOD, defined conceptually as the increase in halo mass, relative to $\fshmr$, before satellites begin to enter the sample. Quantitatively, this is expressed as the ratio $M_{\rm sat}
/\fshmr$, as shown in Equation \[e.nsat\]. Reducing this ratio increases the number of satellite galaxies in a sample, which in turn increases the large-scale bias of a sample and significantly enhances the clustering within the one-halo term. The parameters $B_{\rm cut}$ and $\beta_{\rm cut}$ have a more subtle effect on clustering. If the cutoff mass, defined by Equation \[mcut\_eq\], is below $\fshmr$, then $M_{\rm cut}$ has no effect on clustering. But as $M_{\rm cut}$ increases, satellite galaxies are removed from low-mass halos. If the density of satellites is held fixed, increasing $M_{\rm cut}$ redistributes satellites into more massive halos. This will increase the large-scale bias and change the shape of the one-halo term such that the correlation function deviates from a pure power law form (see the Appendix in @Zheng:2009).
Mock catalogs, sample variance, and covariance {#mocks}
==============================================
In this section we construct mock catalogs in order to investigate the effects of sample variance and covariance associated with measurements of g-g lensing, clustering, and the SMF. Sample variance occurs due to the finite nature of the volume encompassed by any given survey. Because of limited volume, any given survey may yield a biased measurement of the number density of galaxies and halos compared to the full universe. The error bars on all three observables must therefore reflect this additional source of error. Also, the data-points in all three observables will be correlated to some degree. Consider the SMF for example. A region of space with high matter density will have an increased abundance of galaxies of nearly all masses. For $w(\theta)$, because it is a projection of $\xi_{gg}(r)$—which is itself a correlated quantity—multiple physical scales will contribute to each bin in $\theta$. For $\Delta\Sigma$, we will show that the data points are correlated on scales where satellite galaxies contribute to the lensing signal.
To investigate both the sample variance and the covariance associated with all three observables, we use numerical simulations to construct a series of mock catalogs for a COSMOS-like survey. Since the volume of COSMOS is relatively small, the effects of variance and covariance will be quite apparent (whereas the effects would decrease if we simulated a larger fiducial survey) and so COSMOS is well suited for our purpose. In addition, we will also use these mock catalogs in Paper II to analyze the actual COSMOS data.
COSMOS-like mocks are created from a single simulation (named “Consuelo”) 420 $h^{-1}$ Mpc on a side, resolved with 1400$^3$ particles, and a particle mass of 1.87$\times 10^{9}$ $h^{-1}$ M$_{\odot}$. This simulation can robustly resolve halos with masses above $\sim 10^{11}$ $h^{-1}$ M$_{\odot}$ and is part of the Las Damas suite (McBride et al. in prep). We create mocks for three redshift intervals: $z_1=[0.22,0.48]$, $z_2=[0.48,0.74]$, and $z_3=[0.74,1]$. For each redshift interval, we construct a series of mocks created from random lines of sight through the simulation volume that have the same area as COSMOS and the same comoving length for the given redshift slice. This yields 405 independent mocks for the $z_1$ bin, 172 mocks for the $z_2$ bin, and 109 mocks for the $z_3$ bin. For each redshift bin, mocks are created from the simulation output at the median redshift of the bin.
Halos within the simulation are identified with the friends-of-friends halo finder (@Davis:1985) with a linking length of b=0.2. For each redshift interval, halos are populated with our best-fit model from Paper II. We use the mock-to-mock variance and covariance to estimate a covariance matrix for the stellar mass function, for $w(\theta)$ (using a series of stellar mass thresholds), and for $\Delta\Sigma$ (using a series of stellar mass bins). Although the data points [*between*]{} the different quantities will be correlated to some degree (as well as the bins in $w(\theta)$ and $\Delta\Sigma$) we ignore that covariance as we do not have enough simulation volume to estimate the [*uber*]{}-covariance matrix of all \[N\] data points in each redshift bin.
Figure \[smf\_covar\_matrix\] shows the correlation coefficient matrix for the SMF in three redshifts bins for a COSMOS-like survey. The first-order effect of sample variance on the SMF is to correlate all of the data points so that globally, the SMF will shift up and down for different realizations of a COSMOS-like survey.
Figure \[clustering\_covar\_matrix\] shows the correlation coefficient matrix for the galaxy clustering for $0.22<z<0.48$ and for three stellar mass thresholds: $\log_{10}(M_*)>9.3$, $\log_{10}(M_*)>10.3$, and $\log_{10}(M_*)>11.1$. The data are more correlated at larger scales where galaxy pairs come from the two-halo term. As shown earlier, clustering at these scales is proportional to the matter clustering $\xi_m(r)$. Patches of the universe that exist in an over- or under-density tend to have higher or lower clustering in their matter. This will be reflected in the clustering of the halos and thus the two-halo term for the galaxies. In the one-halo term, Poisson fluctuations of the number of satellites become more important and the data are less correlated at these scales. Overall, as the density of the galaxy sample becomes smaller, shot noise will dominate on all scales. This can be seen in the progression from left to right in the examples in Figure \[clustering\_covar\_matrix\].
Figure \[gg\_sample\_variance\] illustrates the effect of sample variance on g-g lensing signals for various stellar mass bins and for $0.22<z<0.48$. Figure \[z1\_lensing\_covar\_matrix\] shows the associated correlation coefficient matrices. The key point to note here is that the sample variance for g-g lensing is dominated by the one-halo satellite term on scales of about 100 kpc to 1 Mpc. The impact of this term becomes more apparent in galaxy samples with lower stellar masses as the contribution from the one-halo central term decreases. The fact that the one-halo satellite term has a large sample variance compared to the one-halo central term can be understood as follows. Consider a sample of galaxies in a given stellar mass bin. The galaxies that are satellites in this sample will tend to live in more massive halos than the galaxies that are centrals (this can be seen in Figure \[hod\_bins\_illustration\] for example). Since more massive halos are more rare than less massive halos at fixed survey volume, this explains the large one-halo satellite sample variance.
Finally, we also use mock catalogs to estimate the effects of the integral constraint (IC) [@Groth:1977] on clustering measurements for a small area survey. Due to spatial fluctuations in the number density of galaxies, the mean correlation function measured from an ensemble of samples will be smaller than the correlation function measured from a single contiguous sample of the same volume as the sum of the ensemble sample. This attenuation of $w(\theta)$ becomes relevant on angular scales significant with respect to the sample size. For large surveys like the SDSS, the IC is not an issue on scales of interest. For a pencil-beam survey like COSMOS, however, the IC must be taken into account when modeling the clustering. We estimate the IC correction to our $w(\theta)$ measurements through the use of the mock galaxy distributions described previously. The results are shown in Figure \[ic\_correction\]. For COSMOS, our fitting functions for the IC correction are:
$$f_{IC} = \exp(\log_{10}(\theta)/3.4)^{2.5},$$
$$f_{IC} = \exp(\log_{10}(\theta)/3.2)^{4.1},$$
for $0.22<z<0.48$ and $0.48<z<0.74$ respectively and where $\theta$ is expressed in arc-seconds. For $0.74<z<1$ there is sufficient volume such that $f_{IC}=1$. We note that these fitting functions are only valid for $\theta<10^3$ arcseconds.
Summary and conclusions {#conclusions}
=======================
The goal of this paper is to develop the theoretical framework necessary to combine measurements of galaxy-galaxy lensing, galaxy clustering, and the galaxy stellar mass function, into a single and more robust probe of the galaxy-dark matter connection. We have achieved this goal by introducing several key modifications to the standard HOD framework. To begin with, we have modified the standard HOD model so as to fit all three probes simultaneously and independantly of the selected binning scheme. Next, since we are interested in the galaxy-dark matter connection, we have also modified the HOD model so as to specifically include the stellar-to-halo mass relation (SHMR). In a companion paper, Leauthaud et al. 2011b, we demonstrate that the model presented here provides an excellent fit to galaxy-galaxy lensing, galaxy clustering, and stellar mass functions measured in the COSMOS survey from $z=0.2$ to $z=1.0$.
Nonetheless, while the promise of combined dark matter probes in studying galaxy formation, gravity and cosmology is clear, we must ensure that our parametric description of the SHMR is sophisticated enough to capture its possible behavior. There are a number of questions that remain to be answered in order to achieve this goal. For example, is $P(M_*|M_h)$ well described by a log-normal distribution and is the scatter in $P(M_*|M_h)$ constant or does it vary with halo mass? Do the parameters that describe $P(M_*|M_h)$ vary with redshift and galaxy type? Can we marginalize over uncertainties related to the shapes and concentrations of dark matter halos? What exactly do we learn from various probe combinations? The challenges are steep but with increasing large data-sets such as the Dark Energy Survey, the Large Synoptic Survey Telescope, the HyperSuprime Cam survey, and EUCLID, refined and sophisticated models can be built and constrained by the data. Although the model presented in this paper is sophisticated enough to describe COSMOS data, it is clear that further refinements will be necessary given the statistical precision of up-coming surveys. Improving models such as the one presented in this paper by using insights provided, for example, by semi-analytic models of galaxy formation and dark matter N-body simulations, is clearly a worthy pursuit.
[**Acknowledgments**]{}
We thank Jaiyul Yoo for help with the Appendix and Kevin Bundy for useful discussions and for reading the manuscript. AL acknowledges support from the Chamberlain Fellowship at LBNL and from the Berkeley Center for Cosmological Physics. This research received partial support from the U.S. Department of Energy under contract number DE-AC02-76SF00515. RHW and PSB received additional support from NASA Program HST-AR-12159.A, provided through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. MTB and RHW also thank their collaborators on the LasDamas project for critical input on the Consuelo simulation, which was performed on the Orange cluster at SLAC.
Further details on the origin of Equation 34
============================================
We consider it useful to provide some more details on the origin of Equation \[eq:1h\] and in particular, on the link between the NFW profile and $F'_{c}$ and $F'_{s}$. This might be useful for those who are not familiar with the notations of galaxy clustering studies. To begin with, consider a central galaxy that is associated with a NFW dark matter halo at redshift $z_L$ and with halo mass $M$. The NFW profile, $\rho_{\textsc{nfw}}$, is given by:
$$\frac{\rho_{\textsc{nfw}}(r, z_L)}{\rho_{\rm crit}} = \frac{\delta}{(r/R_s)\left(1+r/R_s\right)^2},$$
where $\delta$ is a characteristic (dimensionless) density and $R_s$ is the NFW scale radius. The relation between $\delta$ and the NFW concentration parameter $c$ is:
$$\delta=\frac{\Delta}{3}\times\frac{c^3}{\ln(1+c)-c/(1+c)},
\label{delta_nfw}$$
where $\Delta$ is a chosen over-density (for example, $\Delta$ is often set to 200). The NFW radius is noted $R_h$ and is equal to $R_h=c\times R_s$. The projected surface mass density of this lens, $\Sigma$, is computed by taking the integral of $\rho_{\textsc{nfw}}$ over the line-of-sight:
$$\Sigma_{\rm co}(r_{\rm p,co},z_L|M) = \int
\rho_{\textsc{nfw}}\left(\sqrt{r_{\rm p,co}^2+r_{||,\rm co}^2},z_L\right){\rm
d}r_{\rm ||,co}.
\label{nfw_projection}$$
Analytical expressions for the projection of $\rho_{\textsc{nfw}}$ to $\Sigma$ can be found in @Wright:2000 for example.
Instead of a single galaxy, now consider an ensemble of central galaxies characterized by the central occupation function ${\langle N_{\rm cen} \rangle}$. We note $\overline{n}_c$ such that:
$$\overline{n}_{c} \equiv \int {\langle N_{\rm cen} \rangle}\frac{{\rm d}n}{{\rm d}M} {\rm d}M.$$
The probability that a galaxy in this selection lives in a halo of mass $M$ is:
$$P(M) = {\langle N_{\rm cen} \rangle}\frac{1}{\overline{n}_{c}} \frac{{\rm d}n}{{\rm d}M}.
{\rm d}M.$$
The average surface mass density of the galaxy ensemble is:
$$\begin{aligned}
\Sigma_{\rm co}(r_{\rm p,co},z_L) &=& \int \int \Sigma_{\rm
co}(r_{\rm p,co},z_L|M) P(M) {\rm d}M{\rm d}r_{\rm ||,co} \nonumber\\
&=& \int \int \rho_{\textsc{nfw}}\left(\sqrt{r_{\rm
p,co}^2+r_{||,\rm co}^2},z_L\right) {\langle N_{\rm cen} \rangle}\frac{1}{\overline{n}_{c}} \frac{{\rm d}n}{{\rm d}M} {\rm d}M{\rm d}r_{\rm ||,co}.
\label{app_eq1}\end{aligned}$$
Let us now define $F'_{c}$ such that:
$$\rho_{\textsc{nfw}}(r, z_L) = \frac{1}{4\pi r^2}\times M \times \frac{1}{2R_{h}}\times F'_{c}\left(\frac{r}{2R_{h}}\right).
\label{app_eq2}$$
Combining Equation \[app\_eq1\] and Equation \[app\_eq2\] we obtain:
$$\Sigma_{\rm co}(r_{\rm p,co},z_L) = \frac{1}{4\pi r^2
\overline{n}_c}\int \int \frac{{\rm d}n}{{\rm
d}M}\frac{M}{\overline{\rho}}\frac{1}{2R_{h}}n_{\rm tot}F'_{c}\left(\frac{\sqrt{r_{\rm p,co}^2+r_{||,\rm
co}^2}}{2R_{h}}\right) {\rm d}M{\rm d}r_{\rm ||,co}.$$
Finally, Equation \[eq:1h\] is obtained by considering a galaxy sample that contains both central and satellite galaxies. In this case $\overline{n}_g$ is defined as:
$$\overline{n}_{g} \equiv \int \left({\langle N_{\rm cen} \rangle}+{\langle N_{\rm sat} \rangle}\right) \frac{{\rm d}n}{{\rm d}M} {\rm d}M.$$
$F'_{s}$ is defined in a similar fashion to Equation \[app\_eq2\] but $\rho_{\textsc{nfw}}$ is replaced with the convolution of the NFW profile with itself. Analytic expressions for the convolution of the truncated NFW profile with itself can be found in the Appendix of @Sheth:2001a.
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